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Monolix 2024 User Guide

  1. Monolix Documentation
  2. Data
    1. Defining a data set
    2. Data Format
    3. Data formatting
      1. Data formatting presets
    4. Filtering a data set
    5. Handling censored (BLQ) data
    6. Mapping between the data and the model
  3. Structural model
    1. Libraries of models
    2. Writing your own model
    3. Models for continuous outcomes
      1. Typical PK models
        1. Single route of administration
        2. Multiple routes of administration
        3. Multiple doses to steady-state
      2. Mixture of structural models
      3. Joint models for continuous outcomes
    4. Models for non continuous outcomes
      1. Time-to-event data models
      2. Count data model
      3. Categorical data model
      4. Joint models for non continuous outcomes
    5. Extensions
      1. Using a scale factor
      2. Using regression variables
      3. Delayed differential equations
      4. Outputs and Tables
      5. How to compute AUC using in Monolix and Mlxtran
  4. Statistical Model
    1. Observation (error) model
    2. Individual model
      1. Introduction
      2. Probability distribution of the individual parameters
      3. Bounded parameter distribution
      4. Model for individual covariates
      5. Complex parameter-covariate relationships
      6. Inter occasion variability (IOV)
      7. Inter-occasion variability and effect of guar gum on alcohol concentration in blood
      8. Mixture of distributions
  5. Tasks and results
    1. Estimation tasks
      1. Initialization
        1. Check initial estimates and auto init
      2. Population parameter estimation using SAEM
        1. The convergence indicator
        2. Simulated annealing
        3. Bayesian estimation
      3. EBEs
      4. Conditional distribution
        1. Understanding shrinkage and how to circumvent it
        2. Statistical tests
        3. Proposal
      5. Standard error using the Fisher Information Matrix
      6. Log Likelihood estimation
    2. Algorithms convergence assessment
    3. Model building
      1. Statistical tests for model assessment
      2. Automatic statistical model building (covariate model, correlation model, error model)
      3. Automatic complete statistical model building
      4. Automatic covariate model building
      5. Automatic variability model building
      6. Statistical model building with SAMBA
    4. Bootstrap
    5. Output files
    6. Reproducibility of MonolixSuite results
    7. Comments
  6. Plots
    1. Overview
      1. Generating and exporting plots
      2. Interacting with the plots
      3. Transferring plot formatting
      4. Export charts
    2. Data
      1. Observed data
      2. Covariate Viewer
      3. Bivariate Data Viewer
    3. Model for the observations
      1. Individual fits
      2. Observation versus Prediction
      3. Scatter plot of the residuals
      4. Distribution of the residuals
    4. Model for the individual parameters
      1. Distribution of the individual parameters
      2. Distribution of the random effects
      3. Correlation between the random effects
      4. Individual parameters versus covariates
    5. Predictive checks and predictions
      1. Visual predictive checks
        1. VPC for Time-To-Event data
        2. Correcting the VPC for missing observations
      2. Numerical predictive checks
      3. BLQ predictive checks
      4. Prediction distribution
    6. Convergence diagnosis
      1. SAEM
      2. MCMC
      3. Importance sampling
    7. Tasks results
      1. Likelihood contribution
      2. Standard errors of the estimates
  7. Reporting
    1. Reporting examples
    2. List of reporting keywords
    3. List of plot settings
    4. List of table placeholder settings
  8. Export a Monolix project to PKanalix and Simulx
  9. R functions to run Monolix
    1. R-package installation and initialization
    2. Rsmlx
    3. Installation
    4. Examples using R functions
    5. Additional connectors for Monolix
    6. Creating forest plots
  10. FAQ
    1. List of known bugs
    2. Submission of Monolix analysis to regulatory agencies
    3. Running Monolix using a command line

1.Monolix Documentation

Version 2024

This documentation is for Monolix starting from 2018 version.
©Lixoft

Monolix

Monolix (Non-linear mixed-effects models or “MOdèles NOn LInéaires à effets miXtes” in French) is a platform of reference for model based drug development. It combines the most advanced algorithms with unique ease of use. Pharmacometricians of preclinical and clinical groups can rely on Monolix for population analysis and to model PK/PD and other complex biochemical and physiological processes. Monolix is an easy, fast and powerful tool for parameter estimation in non-linear mixed effect models, model diagnosis and assessment, and advanced graphical representation. Monolix is the result of a ten years research program in statistics and modeling, led by Inria (Institut National de la Recherche en Informatique et Automatique) on non-linear mixed effect models for advanced population analysis, PK/PD, pre-clinical and clinical trial modeling & simulation.

 









Objectives

The objectives of Monolix are to perform:

  1. Parameter estimation for nonlinear mixed effects models
  2. Model selection and diagnosis
  3. Easy description of pharmacometric models (PK, PK-PD, discrete data) with the Mlxtran language
  4. Goodness of fit plots

An interface for ease of use

Monolix can be used either via a graphical user interface (GUI) or a command-line interface (CLI) for powerful scripting. This means less programming and more focus on exploring models and pharmacology to deliver in time. The interface is depicted as follows:

The GUI consists of 7 tabs.

Each of these tabs refer to a specific section on this website. An advanced description of available plots is also provided.

 

2.Data

2.1.Defining a data set

To start a new Monolix project, you need to define a dataset by loading a file in the Data tab, and load a model in the structural model tab. The project can be saved only after defining the data and the model.

Supported file types: Supported file types include .txt, .csv, and .tsv files. Starting with version 2024, additional Excel and SAS file types are supported: .xls, .xlsx, .sas2bdat, and .xpt files in addition to .txt, .csv, and .tsv files.

The data set format expected in the Data tab is the same as for the entire MonolixSuite, to allow smooth transitions between applications. The columns available in this format and example datasets are detailed on this page. Briefly:

  • Each line corresponds to one individual and one time point
  • Each line can include a single measurement (also called observation), or a dose amount (or both a measurement and a dose amount)
  • Dosing information should be indicated for each individual, even if it is identical for all.

Your dataset may not be originally in this format, and you may want to add information on dose amounts, limits of quantification, units, or filter part of the dataset. To do so, you should proceed in this order:

  • Formatting: If needed, format your data first by loading the dataset in the Data Formatting tab. Briefly, it allows to:
    • to deal with several header lines
    • merge several observation columns into one
    • add censoring information based on tags in the observation column
    • add treatment information manually or from external sources
    • add more columns based on another file
  • Loading a new data set: If the data is already in the right format, load it directly in the Data tab (otherwise use the formatted dataset created by data formatting).
  • Observation types: Specify if the observation is of type continuous, count/categorical or event.
  • Labeling: label the columns not recognized automatically to indicate their type and click on ACCEPT.
  • Filtering: If needed, filter your dataset to use only part of it in the Filters tab
  • Explore: The interpreted dataset is displayed in Data, and Plots and covariate statistics are generated.

If you have already defined a dataset in Datxplore or in PKanalix, you can skip all those steps in Monolix and create a new project by importing a project from Datxplore or PKanalix.

Loading a new data set

To load a new data set, you have to go to “Browse” your data set (green frame), tag all the columns (purple frame), define the observation types in Data Information (blue frame), and click on the blue button ACCEPT  as on the following. If the dataset does not follow a formatting rule, the dataset will not be accepted, but errors will guide you to find what is missing and could be added by data formatting.

Observation types

There are three types of observations:

  • continuous: The observation is continuous and can take any value within a range. For example, a concentration is a continuous observation.
  • count/categorical: The observation values can take values only in a finite categorical space. For example, the observation can be a categorical observation (an effect can be observed as “low”, “medium”, or “high”) or a count observation over a defined time (the number of epileptic crisis in a week).
  • event: The observation is an event, for example the time of death.

For each OBSERVATION ID, the type of observations must be specified by the user in the interface. Depending on the choice, the data will be displayed in the Observed Data plot in different ways (e.g spaghetti plot for continuous data and Kaplan-Meier plot for event data). The mapping of model outputs and observations from the dataset will also take into account the data type (e.g a model outptu of type “event” can only be mapped to an observation that is also an “event”). Once a model has been selected, the choice of the data types are locked (because they are enforced by the model output type).

Regressor settings

If columns have been tagged as REGRESSORS, the interpolation method for the regressors can be chosen. In the dataset, regressors are defined only for a finite number of time points. In between those time points, the regressor values can be interpolated in two different ways:

  • last carried forward: if we have defined in the dataset two times for each individual with \(reg_A\) at time \(t_A\) and \(reg_B\) at time \(t_B\)
    • for \(t\le t_A\), \(reg(t)=reg_A\)  [first defined value is used]
    • for \(t_A\le t<t_B\), \(reg(t)=reg_A\) [previous value is used]
    • for \(t>t_B\), \(reg(t)=reg_B\) [previous value is used]
  • linear interpolation: the interpolation is:
    • for \(t\le t_A\), \(reg(t)=reg_A\)  [first defined value is used]
    • for \(t_A\le t<t_B\), \(reg(t)=reg_A+(t-t_A)\frac{(reg_B-reg_A)}{(t_B-t_A)}\) [linear interpolation is used]
    • for \(t>t_B\), \(reg(t)=reg_B\) [previous value is used]

The interpolation is used to obtain the regressor value at times not defined in the dataset. This is necessary to integrate ODE-based models (which are using an internal adaptative time step), or obtain prediction on a fine grid for the plots (e.g in the Individuals fits) for instance.

When some dataset lines have a missing regressor value (dot “.”), the same interpolation method is used.

Labeling (column tagging)

The column type suggested automatically by Monolix based on the headers in the data can be customized in the preferences. By clicking on Settings>Preferences, the following windows pops up.

In the DATA frame, you can add or remove preferences for each column.
To remove a preference, double-click on the preference you would like to remove. A confirmation window will be proposed.
To add a preference, click on the header type you consider, add a name in the header name and click on “ADD HEADER” as on the following figure.


Notice that all the preferences are shared between Monolix, Datxplore, and PKanalix.

Starting from the version 2024, it is also possible to update the preferences with the columns tagged in the opened project, by clicking on the icon in the top left corner of the table:


Clicking on the icon will open a modal with the option to choose which of the tagged headers a user wants to add to preferences:

Dataset load times

Starting with the 2024 version, it is possible to improve the project load times, especially for projects with large datasets, but saving the data as a binary file. This option is available in Settings>Preferences and will save a copy of the data file in binary format in the results folder. When reloading a project, the dataset will be read from the binary file, which will be faster. If the original dataset file has been modified (compared to the binary), a warning message will appear, the binary dataset will not be used and the original dataset fiel will be loaded instead.

Resulting plots and tables to explore the data

Once the dataset is accepted:

  • Plots are automatically generated based on the interpreted dataset to help you proceed with a first data exploration before running any task.

 

  • The interpreted dataset appears in Data tab, which incorporates all changes after formatting and filtering.

 

  • Covariate Statistics appear in a section of the data tab.

 

 

All the covariates (if any) are displayed and a summary of the statistics is proposed. For continuous covariates, minimum, median and maximum values are proposed along with the first and third quartile, and the standard deviation. For categorical covariates, all the modalities are displayed along with the number of each. Note the “Copy table” button that allows to copy the table in Word and Excel. The format and the display of the table will be preserved.

Importing a project from Datxplore or PKanalix

It is possible to import a project from Datxplore or PKanalix. For that, go to Project>New project for Datxplore/PKanalix (as in the green box of the following figure). In that case, a new project will be created and all the DATA frame will already be filled by the information from the Datxplore or PKanalix project.

2.2.Data Format

In Monolix, a dataset should be loaded in the Data tab to create a project. To be accepted in the Data tab, the dataset should be in a specific format described below. If your dataset is not in the right format, in most cases, it is possible to format it in a few steps in the data formatting tab, to incorporate the missing information. Once the dataset is accepted and once a model is loaded, it is possible to filter the dataset to remove outliers or focus on a particular group. The data set format used in Monolix is the same as for the entire MonolixSuite, to allow smooth transitions between applications. In this format:
  • Each line corresponds to one individual and one time point.
  • Each line can include a single measurement (also called observation), or a dose amount (or both a measurement and a dose amount).
  • Dosing information should be indicated for each individual in a specific column, even if it is the same treatment for all individuals.
  • Headers are free but there can be only one header line.
  • Different types of information (dose, observation, covariate, etc) are recorded in different columns, which must be tagged with a column type (see below).
The column types are very similar and compatible with the structure used by the Nonmem software (the differences are listed here). This is specified when the user defines each column type in the data set as in the following picture. Notice that Monolix often provides an initial guess of the type of the column depending on the name.

Description of column-types

The first line of the data set must be a header line, defining the names of the columns. The columns names are completely free. In the MonolixSuite applications, when defining the data, the user will be asked to assign each column to a column-type (see here for an example of this step). The column type will indicate to the application how to interpret the information in that column. The available column types are given below. Column-types used for all types of lines: Column-types used for response-lines: Column-types used for dose-lines:

Order of events

There are prioritization rules in place in case of various event types occurring at the same time. The order of row numbers in the data set is not important, and same is true for the order of administration and empty/reset macros in model files. The sequence of events will always be the following:
  1. regressors are updated,
  2. reset done by EVID=3 or EVID=4 is performed,
  3. dose is administered,
  4. empty/reset done by macros is performed,
  5. observation is made.

2.3.Data formatting

Monolix versions prior to 2023R1 do not include the data formatting module. Instead we provide an Excel macro to adapt the format of your dataset for Monolix.

The dataset format that is used in Monolix is the same as for the entire MonolixSuite, to allow smooth transitions between applications. In this format, some rules have to be fullfilled, for example:

  • Each line corresponds to one individual and one time point.
  • Each line can include a single measurement (also called observation), or a dose amount (or both a measurement and a dose amount).
  • Dose amount should be indicated for each individual dose in a column AMOUNT, even if it is identical for all.
  • Headers are free but there can be only one header line.

If your dataset is not in this format, in most cases, it is possible to format it in a few steps in the data formatting tab, to incorporate the missing information.

In this case, the original dataset should be loaded in the “Format data” box, or directly in the “Data Formatting” tab, instead of the “Data” tab. In the data formatting module, you will be guided to build a dataset in the MonolixSuite format, starting from the loaded csv file. The resulting formatted dataset is then loaded in the Data tab as if you loaded an already-formatted dataset in “Data” directly. Then as for defining any dataset, you can tag columns, accept the dataset, and once accepted,  the Filters tab can be used to select only parts of this dataset for analysis. Note that units and filters are neither information to be included in the data file, nor part of the data formatting process.

The original datafile is NOT modified by Monolix. Formatting operations are saved in the Monolix project and applied to the data when the project is loaded. The formatted dataset is not saved by default, but it can be exported by the user as a CSV file.

Jump to:

  1. Data formatting workflow
  2. Dataset initialization (mandatory step)
  3. Creating occasions from a SORT column to distinguish different sets of measurements within each subject, (eg formulations).
  4. Selecting an observation type (required to add a treatment)
  5. Merging observations from several columns
  6. Specifying censoring from censoring tags eg “BLQ” instead of a number in an observation column. Demo project DoseAndLOQ_manual.mlxtran
  7. Adding doses in the dataset Demo project DoseAndLOQ_manual.mlxtran
  8. Reading censoring limits or dosing information from the dataset “by category” or “from data”. Demo projects DoseAndLOQ_byCategory.mlxtran and DoseAndLOQ_fromData.mlxtran
  9. Creating occasions from dosing intervals to analyze separately the measurements following different doses.Demo project doseIntervals_as_Occ.mlxtran
  10. Handling urine data to merge start and end times in a single column.
  11. Adding new columns from an external file, eg new covariates, or individual parameters estimated in a previous analysis. Demo warfarin_PKPDseq_project.mlxtran
  12. Exporting the formatted dataset

 

 

1. Data formatting workflow

When opening a new project, two Browse buttons appear. The first one, under “Data file”, can be used to load a dataset already in a MonolixSuite-standard format, while the second one, under “Format data”, allows to load a dataset to format in the Data formatting module.

After loading a dataset to format, data formatting operations can be specified in several subtabs: Initialization, Observations, Treatments and Additional columns.

  • Initialization is mandatory and must be filled before using the other subtabs.
  • Observations is required to enable the Treatments tab.

After Initialization has been validated by clicking on “Next”, a button “Preview” is available from any subtab to view in the Data tab the formatted dataset based on the formatting operations currently specified.

 

2. Dataset initialization

The first tab in Data formatting is named Initialization. This is where the user can select header lines or lines to exclude (in the blue area on the screenshot below) or tag columns (in the yellow area).

Selecting header lines or lines to exclude

These settings should contain line numbers for lines that should be either handled as column headers or that should be excluded.

  • Header lines: one or several lines containing column header information. By default, the first line of the dataset is selected as header. If several lines are selected, they are merged by data formatting into a single line, concatenating the cells in each column.
  • Lines to exclude (optional): lines that should be excluded from the formatted dataset by data formatting.

 

 

Tagging mandatory columns

Only the columns corresponding to the following tabs must be tagged in Initialization, while all the other columns should keep the default UNDEFINED tag:

  • ID (mandatory): subject identifiers
  • TIME (mandatory): the single time column
  • SORT (optional): one or several columns containing SORT variables can be tagged as SORT. Occasions based on these columns will be created in the formatted dataset as described in Section 3.
  • START, END and VOLUME (mandatory in case of urine data): these column tags replace the TIME tag in case of urine data, if the urine collection time intervals are encoded in the dataset with two time columns for the start and end times of the intervals. In that case there should also be a column with the urine volume in each interval. See Section 10 for more details.

Initialization example

  • demo CreateOcc_AdmIdbyCategory.pkx (Monolix demo in the folder 0.data_formatting, here imported into Monolix. The screenshot below focuses on the formatting initialization and excludes other elements present in the demo):

In this demo the first line of the dataset is excluded because it contains a description of the study. The second line contains column headers while the third line contains column units. Since the MonolixSuite-standard format allows only a single header line, lines 2 and 3 are merged together in the formatted dataset.

 

3. Creating occasions from a SORT column

A SORT variable can be used to distinguish different sets of measurements (usually concentrations) within each subject, that should be analyzed separately by Monolix (for example: different formulations given to each individual at different periods of time, or multiple doses where concentration profiles are available to be analyzed following several doses).

In Monolix, these different sets of measurements must be distinguished as OCCASIONS (or periods of time), via the OCCASION column-type. However, a column tagged as OCCASION can only contain integers with occasion indexes. Thus, if a column with a SORT variable contains strings, its format must be adapted by Data formatting, in the following way:

  • the user must tag the column as SORT in the Initialization subtab of Data formatting,
  • the user validates the Initialization with “Next”, then clicks on “Preview” (after optionally defining other data formatting operations),
  • the formatted data is shown in Data: the column tagged as SORT is automatically duplicated. The original column is automatically tagged as CATEGORICAL COVARIATE in Data, while the duplicated column, which has the same name appended with “_OCC”, is tagged as OCCASION. This column contains occasion indexes instead of strings.

Example:

  • demo CreateOcc_AdmIdbyCategory.pkx (PKanalix demo in the folder 0.data_formatting, here imported into Monolix. The screenshot below focuses on the formatting of occasions and excludes other elements present in the demo):

The image below shows lines 25 to 29 from the dataset from the CreateOcc_AdmIdbyCategory.pkx demo, where covariate columns have been removed to simplify the example. This dataset contains two sets of concentration measurements for each individual, corresponding to two different drug formulations administered on different periods. The sets of concentrations are distinguished with the FORM column, which contains “ref” and “test” categories (reference/test formulations). The column is tagged as SORT in Data formatting Initialization. After clicking on “Preview”, we can see in the Data tab that a new column named FORM_OCC has been created with occasion indexes for each individual: for subject 1, FORM_OCC=1 corresponds to the reference formulation because it appears first in the dataset, and FORM_OCC=2 corresponds to the test formulation because it appears in second in the dataset.

 

4. Selecting an observation type

The second subtab in Data formatting allows to select one or several observation types. An observation type corresponds to a column of the dataset, that contains a type of measurements (usually drug concentrations, but it can also be PD measurements for example). Only columns that have not been tagged as ID, TIME or SORT are available as observation type.

This action is optional and can have several purposes:

  • If doses must be added by Data formatting (see Section 7), specifying the column containing observations is mandatory, to avoid duplicating observations on new dose lines.
  • If several observation types exist in different columns (for example: concentrations for different analytes, or measurements for PK and PD), they must be specified in Data formatting to be merged into a single observation column (see Section 5).
  • In the MonolixSuite-standard format, the column containing observations can only contain numbers, and no string except “.” for a missing observation. Thus if this column contains strings in the original dataset, it must be adapted by Data formatting, with two different cases:
    • if the strings are tags for censored observations (usually BLQ: below the limit of quantification), they can be specified in Data formatting to adapt the encoding of the censored observations (see Section 6),
    • any other string in the column is automatically replaced by “.” by Data formatting.

5. Merging observations from several columns

The MonolixSuite-standard format allows a single column containing all observations (such as concentrations or PD measurements). Thus if a dataset contains several observation types in different columns (for example: concentrations for different analytes, or measurements for PK and PD), they must be specified in Data formatting to be merged into a single observation column.

In that case, different settings can be chosen in the area marked in orange in the screenshot below:

  • The user must choose between distinguishing observation types with observation ids or occasions.
  • The user can unselect the option “Duplicate information from undefined columns”.

 

As observation ids

After selecting the “Distinguish observation types with: observation ids” option and clicking “Preview,” the columns for different observation types are combined into a single column called “OBS.” Each row of the dataset is duplicated for each observation type, with one value per observation type. Additionally, an “OBSID” column is created, with the name of the observation type corresponding to the measurement on each row.

This option is recommended for joint modeling of observation types, such as CA in Monolix or population modeling in Monolix. It is important to note that NCA cannot be performed on two different observation ids simultaneously, so it is necessary to choose one observation id for the analysis.

Example:

  • demo merge_obsID_ParentMetabolite.pkx PKanalix demo in the folder 0.data_formatting, here imported into Monolix. The screenshot below focuses on the formatting of observations and excludes other elements present in the demo):

This demo involves two columns that contain drug parent and metabolite concentrations. When merging both observation types with observation ids, a new column called OBSID is generated with categories labeled as “PARENT” and “METABOLITE.”

 

As occasions

After selecting the “Distinguish observation types with: occasions” option and clicking “Preview,” the columns for different observation types are combined into a single column called “OBS.” Each row of the dataset is duplicated for each observation type, with one value per observation type. Additionally, two columns are created: an “OBSID_OCC” column with the index of the observation type corresponding to the measurement on each row, and an “OBSID_COV” with the name of the observation type.

This option is recommended for NCA, which can be run on different occasions for each individual. However, joint modeling of the observation types with CA or population modeling with Monolix cannot be performed with this option.

Example:

 

  • demo merge_occ_ParentMetabolite.pkx (PKanalix demo in the folder 0.data_formatting, here imported into Monolix. The screenshot below focuses on the formatting of observations and excludes other elements present in the demo):

This demo involves two columns that contain drug parent and metabolite concentrations. When merging both observation types with occasions, two new columns called OBSID_OCC and OBSID_COV are generated with OBSID_OCC=1 corresponding to OBSID_COV=”PARENT” catand OBSID_OCC=2 corresponding to OBSID_COV=”METABOLITE.”

Duplicate information from undefined columns

When merging two observation columns into a single column, all other columns will see their lines duplicated. The data formatting will know how to treat columns which have been tagged in the Initialization tab, but not the other columns (header “UNDEFINED”) which are not used for data formatting. A checkbox enables to decide if the information from these columns should be duplicated on the new lines, or if “.” should be used instead. The default option is to duplicate information, because in general, the undefined columns correspond to covariates with one value per individual, so this value is the same for the two lines that correspond to the same id.

 

 

It is rare that you need to uncheck this box. An example where you should not duplicate the information is if you already have a column Amount in the MonolixSuite format, so with a dose amount only at the dosing time, and “.” everywhere else. If you do not want to specify amount again in data formatting, and simply want to merge observation columns as observation ids, you should not duplicate the lines of the Amount column which is undefined. Indeed, the dose amounts have been administered only once.

 

 

 

 

 

6. Specifying censoring from censoring tags

In the MonolixSuite-standard format, censored observations are encoded with a 1or -1 flag in a column tagged as CENSORING in the Data tab, while exact observations have a 0 flag in that column. In addition, on rows for censored observations, the LOQ is indicated in the observation column: it is the LLOQ (lower limit of quantification) if CENSORING=1 or the ULOQ (upper limit of quantification) if CENSORING=-1. Finally, to specify a censoring interval, an additional column tagged as LIMIT in the Data tab must exist in the dataset, with the other censoring bound.

The Data Formatting module can take as input a dataset with censoring tags directly in the observation column, and adapt the dataset format as described above. After selecting one or several observation types in the Observations subtab (see Section 4), all strings found in the corresponding columns are displayed in the “Censoring tags” on the right of the observation types. If at least one string is found, the user can then define some censoring associated with an observation type and with one or several censoring tags with the button “Add censoring”. 3 types of censoring can be defined:

  • LLOQ: this corresponds to left-censoring, where the censored observation is below a lower limit of quantification (LLOQ), that must specified by the user. In that case Data Formatting replaces the censoring tags in the observation column by the LLOQ, and creates a new CENS column tagged as CENSORING in the Data tab, with 1 on rows that had censoring tags before formatting, and 0 on other rows.
  • ULOQ: this corresponds to right-censoring, where the censored observation is above an upper limit of quantification (ULOQ), that must specified by the user. Here Data Formatting replaces the censoring tags in the observation column by the ULOQ, and creates a new CENS column tagged as CENSORING in the Data tab, with -1 on rows that had censoring tags before formatting, and 0 on other rows.
  • Interval: this is for interval-censoring, where the user must specify two bound of a censoring interval, to which the censored observation belong. Data Formatting replaces the censoring tags in the observation column by the upper bound of the interval, and creates two new columns: a CENS column tagged as CENSORING in the Data tab, with 1 on rows that had censoring tags before formatting, and 0 on other rows, and a LIMIT column with the lower bound of the censoring interval on rows that had censoring tags before formatting, and “.” on other rows.

 

For each type of censoring, available options to define the limits are:

  • Manual“: limits are defined manually, by entering the limit values for all censored observations.
  • By category“: limits are defined manually for different categories read from the dataset.
  • From data“: limits are directly read from the dataset.

The options “by category” and “from data” are described in detail in Section 8.

Example:

  • demo DoseAndLOQ_manual.mlxtran (the screenshot below focuses on the formatting of censored observations and excludes other elements present in the demo):

In this demo there are two censoring tags in the CONC column: BLQ1 (from Study 1) and BLQ2 (from Study 2), that correspond to different LLOQs. An interval censoring is defined for each censoring tag, with manual limits, where LLOQ=0.06 for BLQ1 and LLOQ=0.1 for BLQ2, and the lower limit of the censoring interval being 0 in both cases.

 

7. Adding doses in the dataset

Datasets in MonolixSuite-standard format should contain all information on doses, as dose lines. An AMOUNT column records the amount of the administrated doses on dose-lines, with “.” on response-lines. In case of infusion, an INFUSION DURATION or INFUSION RATE column records the infusion duration or rate. If there are several types of administration, an ADMINISTRATION ID column can distinguish the different types of doses with integers.

If doses are missing from a dataset, the Data Formatting module can be used to add dose lines and dose-related columns: after initializing the dataset, the user can specify one or several treatments in the Treatments subtab. The following operations are then performed by Data Formatting:

  • a new dose line is inserted in the dataset for each defined dose, with the dataset sorted by subject and times. On such a dose line, the values from the next line are duplicated for all columns, except for the observation column in which “.” is used for the dose line.
  • A new column AMT is created with “.” on all lines except on dose lines, on which dose amounts are used. The AMT column is automatically tagged as AMOUNT in the Data tab.
  • If administration ids have been defined in the treatment, an ADMID column is created, with “.” on all lines except on dose lines, on which administration ids are used. The ADMID column is automatically tagged as ADMINISTRATION ID in the Data tab.
  • If an infusion duration or rate has been defined, a new INFDUR (for infusion duration) or INFRATE (for infusion rate) is created, with “.” on all lines except on dose lines. The INFDUR column is automatically tagged as INFUSION DURATION in the Data tab, and the INFRATE column is automatically tagged as INFUSION RATE.

 

Starting from the 2024R1 version, different dosing schedules can be defined for different individuals, based on information from other columns, which can be useful in cases when different cohorts received different dosing regimens, or when working with data pooled from multiple studies. To define a treatment just for a specific cohort or study, the dropdown on the top of the treatment section can be used:

For each treatment, the dosing schedule can defined as:

  • regular: for regularly spaced dosing times, defined with the start time, inter-dose internal, and number of doses. A “repeat cycle” option allows to repeat the regular dosing schedule to generate a more complex regimen.
  • manual: a vector of one or several dosing times, each defined manually. A “repeat cycle” option allows to repeat the manual dosing schedule to generate a more complex regimen.
  • external: an external text file with columns id (optional), occasions (optional), time (mandatory), amount (mandatory), admid (administration id, optional), tinf or rate (optional), that allows to define individual doses.

While dose amounts, administration ids and infusion durations or rates are defined in the external file for external treatments, available options to define them for treatments of type “manual” or “regular” are:

  • Manual“: this applies the same amount (or administration id or infusion duration or rate) to all doses.
  • By category“: dose amounts (or administration id or infusion duration or rate) are defined manually for different categories read from the dataset.
  • From data“: dose amounts (or administration id or infusion duration or rate) are directly read from the dataset.

The options “by category” and “from data” are described in detail in Section 8.

There is a “common settings” panel on the right:

  • dose intervals as occasions: this creates a column to distinguish the dose intervals as different occasions (see Section 9).
  • infusion type: If several treatments correspond to infusion administration, they need to share the same type of encoding for infusion information: as infusion duration or as infusion rate.

 

Example:

  • demo DoseAndLOQ_manual.mlxtran (the screenshot below focuses on the formatting of doses and excludes other elements present in the demo):

In this demo, doses are initially not included in the dataset to format. A single dose at time 0 with an amount of 600 is added for each individual by Data Formatting. This creates a new AMT column in the formatted dataset, tagged as AMOUNT.

 

 

8. Reading censoring limits or dosing information from the dataset

When defining censoring limits for observations (see Section 6) or dose amounts, administration ids, infusion duration or rate for treatments (see Section 7), two options allow to define different values for different rows, based on information already present in the dataset: “by category” and “from data”.

By category

It is possible to define manually different censoring limits, dose amounts, administration ids, infusion durations, or rates for different categories within a dataset’s column. After selecting this column in the “By category” drop-down menu, the different modalities in the column are displayed and a value must be manually assigned each modality.

  • For censoring limits, the censoring limit used to replace each censoring tag depends on the modality on the same row.
  • For doses, the value chosen for the newly created column (AMT for amount, ADMID for administration id, INFDUR for infusion duration, INFRATE for infusion rate) on each new dose line depends on the modality on the first row found in the dataset for the same individual and the same time as the dose, or the next time if there is no line in the initial dataset at that time, or the previous time if no time is found after the dose.

Example:

  • demo DoseAndLOQ_byCategory.mlxtran (the screenshot below focuses on the formatting of doses and excludes other elements present in the demo):

In this demo there are three studies distinguished in the STUDY column with the categories “SD_400mg”, “SD_500mg” and “SD_600mg”. In Data Formatting, a single dose is manually defined at time 0 for all individuals, with different amounts depending the STUDY category. In addition, censoring interval is defined for the censoring tags BLQ, with an upper limit of the censoring interval (lower limit of quantification) that also depends on the STUDY category. Three new columns – AMT for dose amounts, CENS for censoring tags (0 or 1), and LIMIT for the lower limit of the censoring intervals – are created by Data Formatting. A new dose line is then inserted at time 0 for each individual.

 

From data

The option “From data” is used to directly read censoring limits, dose amounts, administration ids, infusion durations, or rates from a dataset’s column. The column must contain either numbers or numbers inside strings. In that case, the first number found in the string is extracted (including decimals with .).

  • For censoring limits, the censoring limit used to replace each censoring tag is read from the selected column on the same row.
  • For doses, the value chosen for the newly created column (AMT for amount, ADMID for administration id, INFDUR for infusion duration, INFRATE for infusion rate) on each new dose line is read from the selected column on the first row found in the dataset for the same individual and the same time as the dose, or the next time if there is no line in the initial dataset at that time, or the previous time if no time is found after the dose.

Example:

  • demo DoseAndLOQ_fromData.mlxtran (the screenshot below focuses on the formatting of doses and censoring and excludes other elements present in the demo):

In this demo there are three studies distinguished in the STUDY column with the categories “SD_400mg”, “SD_500mg” and “SD_600mg”. In Data Formatting, a single dose is manually defined at time 0 for all individuals, with the amount read the STUDY column. In addition, censoring interval is defined for the censoring tags BLQ, with an upper limit of the censoring interval (lower limit of quantification) read from the LLOQ_mg_L column. Three new columns – AMT for dose amounts, CENS for censoring tags (0 or 1), and LIMIT for the lower limit of the censoring intervals – are created by Data Formatting. A new dose line is then inserted at time 0 for each individual, with amount 400, 500 or 600 for studies SD_400mg, SD_500mg and SD_600mg respectively.

 

 

9. Creating occasions from dosing intervals

The option “Dose intervals as occasions” in the Treatments subtab of Data Formatting allows to create an occasion column to distinguish dose intervals. This is useful if the sets of measurements following different doses should be analyzed independently for a same individual. From the 2024 version on, an additional option “Duplicate observations at dose times into each occasion” is available. This option allows to duplicate the observations which are at exactly the same time as dose, such that they appear both as last point of the previous occasion and first point of the next occasion. This setting is recommended for NCA, but not recommended for population PK modeling.

 

Example:

  • demo doseIntervals_as_Occ.mlxtran (Monolix demo in the folder 0.data_formatting, here imported into Monolix):

This demo imported from a Monolix demo has an initial dataset in Monolix-standard format, with multiple doses encoded as dose lines with dose amounts in the AMT column. When using this dataset directly into Monolix or Monolix, a single analysis is done on each individual concentration profile considering all doses, which means that NCA would be done on the concentrations after the last dose only, and modeling (CA in Monolix or population modeling in Monolix) would be estimated with a single set of parameter values for each individual. If instead we want to run separate analyses on the sets of concentrations following each dose, we need to distinguish them as occasions with a new column added with the Data Formatting module. To this end, we define the same treatment as in the initial dataset with Data Formatting (here as regular multiple doses) with the option “Dose intervals as occasions” selected. After clicking Preview, Data Formatting adds two new columns: an AMT1 column with the new doses, to be tagged as AMOUNT instead of the AMT column that will now be ignored, and a DOSE_OCC column to be tagged as OCCASION.

 

Example:

  • demo CreateOcc_duplicateObs.pkx imported to Monolix:

 

10. Handling urine data

In Monolix-standard format, the start and end times of urine collection intervals must be recorded in a single column, tagged as TIME column-type, where the end time of an interval automatically acts as start time for the next interval (see here for more details). If a dataset contains start and end times in two different columns, they can be merged into a single column by Data Formatting. This is done automatically by tagging these two columns as START and END in the Initialization subtab of Data Formatting (see Section 2). In addition the column containing urine collection volume must be tagged as VOLUME.

Example:

  • demo Urine_LOQinObs.pkx (Monolix demo here imported into Monolix):

11. Adding new columns from an external file

The last subtab is used to insert additional columns in the dataset from a separate file. The external file must contain a table with a column named ID or id with the same subject identifiers as in the dataset to format, and other columns with a header name and individual values (numbers or strings). There can be only one value per individual, which means that the additional columns inserted in the formatted dataset can contain only a constant value within each individual, and not time-varying values.

Examples of additional columns that can be added with this option are:

  • individual parameters estimated in a previous analysis, to be read as regressors to avoid estimating them. Time-varying regressors are not handled.
  • new covariates.

If occasions are defined in the formatted dataset, it is possible to have an occasion column in the external file and values defined per subject-occasion.

Example:

  • demo warfarin_PKPDseq_project.mlxtran (Monolix demo in the folder 0.data_formatting, here imported into Monolix):

This demo imported from a Monolix demo has an initial PKPD dataset in Monolix-standard format. The option “Additional columns” is used to add the PK parameters estimated on the PK part of the data in another Monolix project.

12. Exporting the formatted dataset

Once data formatting is done and the new dataset is accepted, the project can be saved and it is possible to export the formatted dataset as a csv file from the main menu Export > Export formatted data.

Note that if you did some data formatting directly in Monolix (instead of Monolix), the possibility to save the project and export the formatted data is enabled only after loading a model in the structural model tab.

 

 

2.3.1.Data formatting presets

Starting from the 2024R1 version, if users need to perform the same or similar data formatting steps with each new project, a data formatting preset can be created and applied for new projects. A typical workflow for using data formatting presets is as follows:
  1. Data formatting steps are performed on a specific project.
  2. Steps are saved in a data formatting preset.
  3. Preset is reapplied on multiple new projects (manually or automatically).

Creating a data formatting preset

After data formatting steps are done on a project, a preset can be created by clicking on the New button in the bottom left corner of the interface, while in the Data formatting tab:
Clicking on this button will open a pop-up window with a form that contains four parts:
  1. Name: contains a name of the preset, this information will be used to distinguish between presets when applying them.
  2. Configuration: contains three checkboxes (initialization, observations, treatments). Users can choose which steps of data formatting will be saved in the preset. If option “External” was used in creating treatments, the external file paths will not be saved in the preset.
  3. Description: a custom description can be added.
  4. Use this preset as default to format data in all MonolixSuite apps: if ticked, the preset will be automatically applied every time a data set is loaded for data formatting in PKanalix and Monolix.
After saving the preset by clicking on Create, the description will be updated with the summary of all the steps performed in data formatting. Important to note is that if a user wants to save censoring information in the data formatting preset, and the censored tags change between projects (e.g., censoring tag <LOQ=0.1> with varying numbers is used in different projects), the option “Use all automatically detected tags” should be used in the Observations tab. This way, no specific censoring tags will be saved in the preset, but they will be automatically detected and applied each time a preset is applied.

Applying a preset

A preset can be applied manually using the Apply button in the bottom left corner of the data formatting tab. Clicking on the Apply button will open a dropdown selection of all saved presets and the desired preset can be chosen. The Apply button is enabled only after the initialization step is performed (ID and TIME columns are selected and the button Next was clicked). This allows PKanalix to fall back to the initialized state, if the initialization step from a preset fails (e.g., if ID and TIME column header saved in the presets do not exist in the new data set). After applying a preset, all data formatting steps saved in the preset will try to be applied. If it is not possible to apply certain steps (e.g., some censoring tags or column headers saved in a preset do not exist), the error message will appear.

Managing presets

Presets can be created, deleted, edited, imported and exported by clicking on Settings > Manage Presets > Data Formatting.
The pop-window allows users to:
  1. Create presets: clicking on the button New has the same behavior as clicking on the button New in the Data formatting tab (described in the section “Creating a data formatting preset”).
  2. Delete presets: clicking on the button Delete will permanently delete a preset. Clicking on this button will not ask for confirmation.
  3. Edit presets: by clicking on a preset in the left panel, the preset information will appear in the right panel. Name and preset description of the presets can then be updated, and a preset can be selected to automatically format data in PKanalix and Monolix.
  4. Export presets: a selected preset can be exported as a lixpst file which can be shared between users and imported using the Import button.
  5. Import presets: a user can import a preset from a lixpst file exported from another computer.
Additionally, an option to pin presets (always show them on top) can be used by clicking on the icon  to facilitate the usage of presets when a user has a lot of them.

2.4.Filtering a data set

Starting on the 2020 version, it is possible to filter your data set to only take a subpart into account in your modelization. It allows to make filters on some specific IDs, times, measurement values,… It is also possible to define complementary filters and also filters of filters. It is accessible through the filters item on the data tab.

Creation of a filter

To create a filter, you need to click on the data set name. You can then create a “child”. It corresponds to a subpart of the data set where you will define your filtering actions.

You can see on the top (in the green rectangle) the action that you will complete and you can CANCEL, ACCEPT, or ACCEPT & APPLY with the bottoms on the bottom.

Filtering actions

In all the filtering actions, you need to define

  • An action: it corresponds to one of the following possibilities: select ids, remove ids, select lines, remove lines.
  • A header: it corresponds to the column of the data set you wish to have an action on. Notice that it corresponds to a column of the data set that was tagged with a header.
  • An operator: it corresponds to the operator of choice (=, ≠, < ≤, >, or ≥).
  • A value. When the header contains numerical values, the user can define it. When the header contains strings, a list is proposed.

For example, you can

  • Remove the ID 1 from your study:

    In that case, all the IDs except ID = 1 will be used for the study.
  • Select all the lines where the time is less or equal 24:

    In that case, all lines with time strictly greater that 24 will be removed. If a subject has no measurement anymore, it will be removed from the study.
  • Select all the ids where SEX equals F:

    In that case, all the male will be removed of the study.
  • Remove all Ids where WEIGHT less or equal 65:

    In that case, only the subjects with a weight over 65 will be kept for the study.

In any case, the interpreted filter data set will be displayed in the data tab.

Filters with several actions

In the previous examples, we only did one action. It is also possible to do several actions to define a filter. We have the possibility to define UNION and/or INTERSECTION of actions.

INTERSECTION

By clicking by the + and – button on the right, you can define an intersection of actions. For example, by clicking on the +, you can define a filter corresponding to intersection of

  • The IDs that are different to 1.
  • The lines with the time values less than 24.

Thus in that case, all the lines with a time less than 24 and corresponding to an ID different than 1 will be used in the study. If we look at the following data set as an example


Initial data set
Resulting data set after action: select IDs ≠ 1
Considered data set for the study
as the intersection of the two actions
Resulting data set after action: select lines with time ≤ 24
UNION

By clicking by the + and – button on the bottom, you can define an union of actions. For example, in a  data set with a multi dose, I can focus on the first and the last dose. Thus, by clicking on the +, you can define a filter corresponding to union of

  • The lines where the time is strictly less than 12.
  • The lines where the time is greater than 72.


Initial data set

Resulting data set after action:
select lines where the time is strictly less than 12

Considered data set for the study
as the union of the three actions

Resulting data set after action:
select lines where the time is greater than 72

Resulting data set after action:
select lines where amt equals 40

Notice that, if just define the first two actions, all the dose lines at a time in ]12, 72[ will also be removed. Thus, to keep having all the doses, we need to add the condition of selecting the lines where the dose is defined.

In addition, it is possible to do any combination of INTERSECTION and UNION.

Other filers: filter of filter and complementary filters

Based on the definition of a filter, it is possible to define two other actions. By clicking on the filter, it is possible to create

  • A child: it corresponds to a new filter with the initial filter as the source data set.
  • A complement: corresponds to the complement of the filter. For example, if you defined a filter with only the IDs where the SEX is F, then the complement corresponds to the IDs where the SEX is not F.

2.5.Handling censored (BLQ) data











Objectives: learn how to handle easily and properly censored data, i.e. data below (resp. above) a lower (resp.upper) limit of quantification (LOQ) or below a limit of detection (LOD).


Projects: censoring1log_project, censoring1_project, censoring2_project, censoring3_project, censoring4_project


Introduction

Censoring occurs when the value of a measurement or observation is only partially known. For continuous data measurements in the longitudinal context, censoring refers to the values of the measurements, not the times at which they were taken. For example, the lower limit of detection (LLOD) is the lowest quantity of a substance that can be distinguished from its absence. Therefore, any time the quantity is below the LLOD, the “observation” is not a measurement but the information that the measured quantity is less than the LLOD. Similarly, in longitudinal studies of viral kinetics, measurements of the viral load below a certain limit, referred to as the lower limit of quantification (LLOQ), are so low that their reliability is considered suspect. A measuring device can also have an upper limit of quantification (ULOQ) such that any value above this limit cannot be measured and reported.
As hinted above, censored values are not typically reported as a number, but their existence is known, as well as the type of censoring. Thus, the observation y^{(r)}_{ij} (i.e., what is reported) is the measurement y_{ij} if not censored, and the type of censoring otherwise.
We usually distinguish between three types of censoring: left, right and interval. In each case, the SAEM algorithm implemented in Monolix properly computes the maximum likelihood estimate of the population parameters, combining all the information provided by censored and non censored data.

Theory

In the presence of censored data, the conditional density function needs to be computed carefully. To cover all three types of censoring (left, right, interval), let I_{ij} be the (finite or infinite) censoring interval existing for individual i at time t_{ij}. Then,

$$\displaystyle p(y^{(r)}|\psi)=\prod_{i=1}^{N}\prod_{j=1}^{n_i}p(y_{ij}|\psi_i)^{1_{y_{ij}\notin I_{ij}}}\mathbb{P}(y_{ij}\in I_{ij}|\psi_i)^{1_{y_{ij}\in I_{ij}}}$$

where

$$\displaystyle \mathbb{P}(y_{ij}\in I_{ij}|\psi_i)=\int_{I_{ij}} p_{y_{ij}|\psi_i} (u|\psi_i)du$$

We see that if y_{ij} is not censored (i.e. 1_{y_{ij}\notin I_{ij}}=1), its contribution to the likelihood is the usual p(y_{ij}|\psi_i), whereas if it is censored, the contribution is \mathbb{P}(y_{ij}\in I_{ij}|\psi_i).
For the calculation of the likelihood, this is equivalent to the M3 method in NONMEM when only the CENSORING column is given, and to the M4 method when both a CENSORING column and a LIMIT column are given.

Censoring definition in a data set

In the dataset format used by Monolix and PKanalix, censored information is included in this way:

  • The censored measurement should be in the OBSERVATION column.
  • In an additional CENSORING column, put 0 if the observation is not censored, and 1 or – 1 depending if the measurement given in the observation column is a lower or an upper limit.
  • Optionally, include a LIMIT column to set the other limit.

To quickly include censoring information to your dataset by using BLQ tags in the observation column, you can use data formatting.

Examples are provided below and here.

 

PK data below a lower limit of quantification

Left censored data

  • censoring1log_project (data = ‘censored1log_data.txt’, model = ‘pklog_model.txt’)

PK data are log-concentration in this example. The limit of quantification of 1.8 mg/l for concentrations becomes log(1.8)=0.588 for log-concentrations. The column of observations (Y) contains either the LLOQ for data below the limit of quantification (BLQ data) or the measured log-concentrations for non BLQ data. Furthermore, Monolix uses an additional column CENSORING to indicate if an observation is left censored (CENS=1) or not (CENS=0). In this example, subject 1 has two BLQ data at times 24h and 30h (the measured log-concentrations were below 0.588 at these times):

The plot of individual fits displays BLQ (red band) and non BLQ data (blue dots) together with the predicted log-concentrations (purple line) on the whole time interval:

Notice that the band goes from .8 to -Infinity as no bound has been specified (no LIMIT column was proposed).
For diagnosis plots such as VPC, residuals of observations versus predictions, Monolix samples the BLQ data from the conditional distribution

$$p(y^{BLQ} | y^{non BLQ}, \hat{\psi}, \hat{\theta})$$

where \hat{\theta} and \hat{\psi} are the estimated population and individual parameters. This is done by adding a residual error on top of the prediction, using a truncated normal distribution to make sure that the simulated BLQ remains within the censored interval. This is the most efficient way to take into account the complete information provided by the data and the model for diagnosis plots such as VPCs:


A strong bias appears if LLOQ is used instead for the BLQ data (if you choose LOQ instead of simulated in the display frame of the settings) :

Notice that ignoring the BLQ data entails a loss of information as can be seen below (if you choose no in the “Use BLQ” toggle):

As can be seen below, imputed BLQ data is also used for residuals (IWRES on the left) and for observations versus predictions (on the right)

More on these diagnosis plots

Impact of the BLQ in residuals and observations versus predictions plots

A strong bias appears if LLOQ is used instead for the BLQ data for these two diagnosis plots:

while ignoring the BLQ data entails a loss of information:

BLQ predictive checks

The BLQ predictive check is a diagnosis plot that displays the fraction of cumulative BLQ data (blue line) with a 90% prediction interval (blue area).

Interval censored data

  • censoring1_project (data = ‘censored1_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)

We use the original concentrations in this project. Then, BLQ data should be treated as interval censored data since a concentration is know to be positive. In other word, a data reported as BLQ data means that the (non reported) measured concentration is between 0 and 1.8mg/l. The value in the observation column 1.8 indicates the value, the value in the CENSORING column indicates that the value in the observation column is the upper bound. An additional column LIMIT reports the lower limit of the censored interval (0 in this example):


Remarks

  • if this column is missing, then BLQ data is assumed to be left-censored data that can take any positive and negative value below LLOQ.
  • the value of the limit can vary between observations of the same subject.

Monolix will use this additional information to estimate the model parameters properly and to impute the BLQ data for the diagnosis plots.
Plot of individual fits now displays LLOD at 1.8 with a red band when a PK data is censored. We see that the band lower limit is at 0 as defined in the limit column.

PK data below a lower limit of quantification or below a limit of detection

  • censoring2_project (data = ‘censored2_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)

Plot of individual fits now displays LLOQ or LLOD with a red band when a PK data is censored. We see that the band lower limits depend on the observation.

PK data below a lower limit of quantification and PD data above an upper limit of quantification

  • censoring3_project (data = ‘censored3_data.txt’, model = ‘pkpd_model.txt’)

We work with PK and PD data in this project and assume that the PD data may be right censored and that the upper limit of quantification is ULOQ=90. We use CENS=-1 to indicate that an observation is right censored. In such case, the PD data can take any value above the upper limit reported in column Y (here the YTYPE column of type OBSERVED ID defines the type of observation, YTYPE=1 and YTYPE=2 are used respectively for PK and PD data):

Plot of individual fits for the PD data now displays ULOQ and the predicted PD profile:

We can display the cumulative fraction of censored data both for the PK and the PD data (on the left and right respectively):

Combination of interval censored PK and PD data

  • censoring4_project (data = ‘censored4_data.txt’, model = ‘pkpd_model.txt’)

We assume in this example

  • 2 different censoring intervals(0,1) and (1.2, 1.8) for the PK,
  • a censoring interval (80,90) and right censoring (>90) for the PD.

Combining columns CENS, LIMIT and Y allow us to combine efficiently these different censoring processes:

This coding of the data means that, for subject 1,

  • PK data is between 0 and 1 at time 30h (second blue frame),
  • PK data is between 1.2 and 1.8 at times 0.5h and 24h (first blue frame for time .5h),
  • PD data is between 80 and 90 at times 12h and 16h (second green frame for time 12h),
  • PD data is above 90 at times 4h and 8h (first green frame for time 4h).

Plot of individual fits for the PK and the PD data displays the different limits of these censoring intervals (PK on the left and PD on the right):

Other diagnosis plots, such as the plot of observations versus predictions, adequately use imputed censored PK and PD data:

Case studies

  • 8.case_studies/hiv_project (data = ‘hiv_data.txt’, model = ‘hivLatent_model.txt’)
  • 8.case_studies/hcv_project (data = ‘hcv_data.txt’, model = ‘hcvNeumann98_model_latent.txt’)

2.6.Mapping between the data and the model





Starting from the 2019 version, it is possible to change the mapping between the data set observations ids and the structural model output. By default and in previous versions, the mapping is done by order, i.e. the first output listed in the output= statement of the model is mapped to the first OBSERVATION ID (ordered alphabetically). It is possible with the interface to set exactly which model output is mapped to which data output. Model output or data outputs can be left unused.

Changing the mapping

If you have more output in the data set (i.e more OBSERVATION IDs) than in the structural model, you can set which data output you will use in the project. In the example below there are two outputs in the data set (managed by the OBSERVATION ID column) and only one output in the structural model, Cc. By default the following mapping is proposed: the data with observation id ‘1’ is mapped to the modle prediction ‘Cc’. The model observation (with error model) is called ‘CONC’ (the name of the OBSERVATION column, can be edited):

To set the data output to use to observation id ‘2’, you can either:

  • Unlink by clicking on either the dot representing the output ‘1’ of the Data or ‘Cc’ of the structural model, and then draw the line between ‘2’ and ‘Cc’ (as can be seen on the figure below on the left)
  • Directly draw a line from ‘2’ to ‘Cc’ (as can be seen on the figure below on the right). This will automatically undo the link between ‘1’ and ‘Cc’.

And click on the button ACCEPT on the bottom on the window to apply the changes.

The same possibility is proposed if you have more outputs in the structural model, compared to the number of observation ids. If you have a TMDD model with both the free and the total ligand concentration listed as model output and one type of measurement, you can map either the free or the total ligand as can be seen on the following figure with the same actions as described above.

 

Several types of outputs

The mapping is only possible between outputs of same nature (continuous / count-categorical / event), i.e. it is only possible to map a continuous output with a continuous output of the structural model. Thus, mapping a continuous output with a discrete or a time-to-event is not possible.  If you try to link a forbidden combination, the line connecting line will be displayed in red as in the following figure

The type of output is indicated via the shapes:

  • continuous outputs are displayed as circles
  • categorical/count outputs are displayed as squares
  • event outputs are displayed as triangles

Changing the observation name

In the example below, ‘1’ is the observation id used in the data set to identify the data to use, ‘Cc’ is the model output (a prediction, without residual error) and ‘y1’ the observation (with error). ‘y1’ represents the data with observation id ‘1’ and it appears in the labels/legends of the plots. These elements are related by observation model, which formula can be displayed.

For count/categorical and event model outputs, the model observation is defined in the model file directly. The name used in the model file is reused in the mapping interface and cannot be changed.

For continuous outputs, the model file defines the name of the prediction (e.g ‘Cc’), while the model observation (e.g ‘y1’, with error) definition is done in the “Statistical model and tasks” tab of the interface. If there is only one model output, the default observation name is the header of the data set column tagged as OBSERVATION. In case of several model outputs, the observation names are y1, y2, y3, etc. The observation names for continuous outputs can be changed by clicking on the node and “edit observation name”:

3.Structural model

3.1.Libraries of models


Objectives: learn how to use the Monolix libraries of models and use your own models.


Projects: theophylline_project, PDsim_project, warfarinPK_project, TMDD_project, LungCancer_project, hcv_project


For the definition of the structural model, the user can either select a model from the available model libraries or write a model itself using the Mlxtran language.
Discover how to easily choose a model from the libraries via step-by-step selection of its characteristics. An enriched PK, a PD, a joint PKPD, a target-mediated drug disposition (TMDD), and a time to-event (TTE) library are now available.







Model libraries

Five different model libraries are available in Monolix, which we will detail below. To use a model from the libraries, in the Structural model tab, click on Load from library and select the desired library. A list of model files appear, as well as a menu to filter them.  Use the filters and indications in the file name (parameters names) to select the model file you need.

The model files are simply text files that contain pre-written models in Mlxtran language. Once selected, the model appears in the Monolix GUI. Below we show the content of the (ka,V,Cl) model:

The PK library

  • theophylline_project (data = ‘theophylline_data.txt’ , model=’lib:oral1_1cpt_kaVCl.txt’)

The PK library includes model with different administration routes (bolus, infusion, first-order absorption, zero-order absorption, with or without Tlag), different number of compartments (1, 2 or 3 compartments), and different types of eliminations (linear or Michaelis-Menten). More details, including the full equations of each model, can be found on the  PK model library wepage. The PK library models can be used with single or multiple doses data, and with two different types of administration in the same data set (oral and bolus for instance).

The PD and PKPD libraries

  • PDsim_project (data = ‘PDsim_data.txt’ , model=’lib:immed_Emax_const.txt’)

The PD model library contains direct response models such as Emax and Imax with various baseline models, and turnover response models. These models are PD models only and the drug concentration over time must be defined in the data set and passed as a regressor.

  • warfarinPKPD_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_IndirectModelInhibitionKin_TlagkaVClR0koutImaxIC50.txt’)

The PKPD library contains joint PKPD models, which correspond to the combination of the models from the PK and from the PD library. These models contain two outputs, and thus require the definition of two observation identifiers (i.e two different values in the OBSERVATION ID column).

Complete description of the PD and PK/PD model libraries.

The PK double absorption library

The library of double absorption models contains all the combinations for two mixed absorptions, with different types and delays. The absorptions can be specified as simultaneous or sequential, and with a pre-defined or independent order. This library simplifies the selection and testing of different types of absorptions and delays. More details about the library and examples can be found on the dedicated PK double absorption documentation page.

 

The TMDD library

  • TMDD_project (data = ‘TMDD_dataset.csv’ , model=’lib:bolus_2cpt_MM_VVmKmClQV2_outputL.txt’)

The TMDD library contains various models for molecules displaying target-mediated drug disposition (TMDD). It includes models with different administration routes (bolus, infusion, first-order absorption, zero-order absorption, bolus + first-order absorption, with or without Tlag), different number of compartments (1, or 2 compartments), different types of TMDD models (full model, MM approximation, QE/QSS approximation, etc), and different types of output (free ligand or total free+bound ligand). More details about the library and guidelines to choose model can be found on the dedicated TMDD documentation page.

 

The TTE library

  • LungCancer_project (data = ‘lung_cancer_survival.csv’ , model=’lib:gompertz_model_singleEvent.txt’)

The TTE library contains typical parametric models for time-to-event (TTE) data. TTE models are defined via the hazard function, in the library we provide exponential, Weibull, log-logistic, uniform, Gompertz, gamma and generalized gamma models, for data with single (e.g death) and multiple events (e.g seizure) per individual. More details and modeling guidelines can be found on the TTE dedicated webpage, along with case studies.

 

The Count library

The Count library contains the typical parametric distributions to describe count data. More details can be found on the Count dedicated webpage, with a short introduction on count data, the different ways to model this kind of data, and typical models.

The tumor growth inhibition (TGI) library

A wide range of models for tumour growth (TG) and tumour growth inhibition (TGI) is available in the literature and correspond to different hypotheses on the tumor or treatment dynamics. In MonolixSuite2020, we provide a modular TG/TGI model library that combines sets of frequently used basic models and possible additional features. This library permits to easily test and combine different hypotheses for the tumor growth kinetics and effect of a treatment, allowing to fit a large variety of tumor size data.

Complete description of the TGI model library.

Step-by-step example with the PK library

  • theophylline_project (data = ‘theophylline_data.txt’ , model=’lib:oral1_1cpt_kaVCl.txt’)

We would like to set up a one compartment PK model with first order absorption and linear elimination for the theophylline data set. We start by creating a new Monolix project. Next, the Data tab, click browse, and select the theophylline data set (which can be downloaded from the data set documentation webpage). In this example, all columns are already automatically tagged, based on the header names. We click ACCEPT and NEXT and arrive on the Structural model tab, click on LOAD FROM LIBRARY to choose a model from the Monolix libraries. The menu at the top allow to filter the list of models: after selecting an oral/extravascular administration, no delay, first-order absorption, one compartment and a linear elimination, two models remain in the list (ka,V,Cl) and (ka,V,k). Click on the oral1_1cpt_kaVCl.txt file to select it.

After this step, the GUI moves to the Initial Estimates tab, but it is possible to go back to the Structural model tab to see the content of the file:

[LONGITUDINAL]
input = {ka, V, Cl}

EQUATION:
Cc = pkmodel(ka, V, Cl)

OUTPUT:
output = Cc

Back to the Initial Estimates tab, the initial values of the population parameters can be adjusted by comparing the model prediction using the chosen population parameters and the individual data. Click on SET AS INITIAL VALUES when you are done.

In the next tab, the Statistical model & Tasks tab, we propose by default:

At this stage, the monolix project should be saved. This creates a human readable text file with extension .mlxtran, which contains all the information defined via the GUI. In particular, the name of the model appears in the section [LONGITUDINAL] of the saved project file:

<MODEL>
[INDIVIDUAL]
input = {ka_pop, omega_ka, V_pop, omega_V, Cl_pop, omega_Cl}

DEFINITION:
ka = {distribution=lognormal, typical=ka_pop, sd=omega_ka}
V = {distribution=lognormal, typical=V_pop, sd=omega_V}
Cl = {distribution=lognormal, typical=Cl_pop, sd=omega_Cl}

[LONGITUDINAL]
input = {a, b}
file = 'lib:oral1_1cpt_kaVCl.txt'

DEFINITION:
CONC = {distribution=normal, prediction=Cc, errorModel=combined1(a,b)}

3.2.Writing your own model

If the models present in the libraries do not suit your needs, you can write a model yourself. You can either start completely from scratch or adapt a model existing in the libraries. In both cases, the syntax of the Mlxtran language is detailed on the mlxtran webpage. You can also copy-paste models from the mlxtran model example page.

Videos on this page use the application mlxEditor included in previous versions of MonolixSuite. From the 2021R1 version on, the editor is integrated within the interface of Monolix, as shown on the screenshots on this page, and it can also be used as a separate application.





 

Writing a model from scratch

  • 8.case_studies/hcv_project (data = ‘hcv_data.txt’ , model=’hcvNeumann98_model.txt’)

In the Structural model tab, you can click on New model to open the editor integrated within Monolix, and start writing your own model. The new model contains a convenient template defining the main blocks, input parameters and output variables. When you are done, click on Create model button of Monolix to save your new model file. After saving, the model is automatically loaded in the project.

 

You can even create your own library of models. An example of a basic library which includes several viral kinetics model is available in the demos 8.case_studies/model.

 

Note: A button “Check syntax” is available to check that there is no syntax error in the model. In case of an error, informative messages are displayed to help correct the error. The syntax check is also automatically applied before saving the model so that only a model with a valid syntax can be saved.

 

 

Understanding the error messages

The error messages generated when syntax errors are present in the model are very informative and quickly help to get the model right. The most common error messages are explained in detail in this video.





 

Modifying a model from the libraries

Browse existing models from the libraries using the Load from library button, then click the “file” icon next to a model name. This opens a pop-up window where the content of the model file is displayed. Click on Open in editor to open the model file in the MlxEditor. There you can adapt the model, for instance to add a PD model. Be careful to save the new model under a new name, to avoid overwriting the library files.

The video below shows an example of how a scale factor can be added:





See the dedicated webpage for more details on model libraries.

3.3.Models for continuous outcomes

3.3.1.Typical PK models

3.3.1.1.Single route of administration


Objectives: learn how to define and use a PK model for single route of administration.


Projects: bolusLinear_project, bolusMM_project, bolusMixed_project, infusion_project, oral1_project, oral0_project, sequentialOral0Oral1_project, simultaneousOral0Oral1_project, oralAlpha_project, oralTransitComp_project

Introduction

Once a drug is administered, we usually describe subsequent processes within the organism by the pharmacokinetics (PK) process known as ADME: absorption, distribution, metabolism, excretion. A PK model is a dynamical system mathematically represented by a system of ordinary differential equations (ODEs) which describes transfers between compartments and elimination from the central compartment.
See this web animation for more details.
Mlxtran is remarkably efficient for implementing simple and complex PK models:

  • The function pkmodel can be used for standard PK models. The model is defined according to the provided set of named arguments. The pkmodel function enables different parametrizations, different models of absorption, distribution and elimination, defined here and summarized in the following..
  • PK macros define the different components of a compartmental model. Combining such PK components provide a high degree of flexibility for complex PK models. They can also extend a custom ODE system.
  • A system of ordinary differential equations (ODEs) can be implemented very easily.

It is also important to highlight the fact that the data file used by Monolix for PK modelling only contains information about dosing, i.e. how and when the drug is administrated. There is no need to integrate in the data file any information related to the PK model. This is an important remark since it means that any (complex) PK model can be used with the same data file. In particular, we make a clear distinction between administration (related to the data) and absorption (related to the model).

 The pkmodel function

The PK model is defined by the names of the input parameters of the pkmodel function. These names are reserved keywords.

Absorption

  • p: Fraction of dose which is absorbed
  • ka: absorption constant rate (first order absorption)
  • or, Tk0: absorption duration (zero order absorption)
  • Tlag: lag time before absorption
  • or, Mtt, Ktr: mean transit time & transit rate constant

Distribution

  • V: Volume of distribution of the central compartment
  • k12, k21: Transfer rate constants between compartments 1 (central) & 2 (peripheral)
  • or V2, Q2: Volume of compartment 2 (peripheral) & inter compartment clearance, between compartments 1 and 2,
  • k13, k31: Transfer rate constants between compartments 1 (central) & 3 (peripheral)
  • or V3, Q3: Volume of compartment 3 (peripheral) & inter compartment clearance, between compartments 1 and 3.

Elimination

  • k: Elimination rate constant
  • or Cl: Clearance
  • Vm, Km: Michaelis Menten elimination parameters

Effect compartment

  • ke0: Effect compartment transfer rate constant

Intravenous bolus injection

Linear elimination

  • bolusLinear_project

A single iv bolus is administered at time 0 to each patient. The data file bolus1_data.txt contains 4 columns: id, time, amt (the amount of drug in mg) and y (the measured concentration). The names of these columns are recognized as keywords by Monolix:
It is important to note that, in this data file, a row contains either some information about the dose (in which case y = ".") or a measurement (in which case amt = "."). We could equivalently use the data file bolus2_data.txt which contains 2 additional columns: EVID (in the green frame) and IGNORED OBSERVATION (in the blue frame):

Here, the EVENT ID column  allows the identification of an event. It is an integer between 0 and 4. It helps to define the type of line. EVID=1 means that this record describes a dose while EVID=0 means that this record contains an observed value.
On the other hand, the IGNORED OBSERVATION column enables to tag lines for which the information in the OBSERVATION column-type is missing. MDV=1 means that the observed value of this record should be ignored while MDV=0 means that this record contains an observed value. The two data files bolus1_data.txt and bolus2_data.txt contain exactly the same information and provide exactly the same results. A one compartment model with linear elimination is used with this project:

$$\begin{array}{ccl} \frac{dA_c}{dt} &=& – k A_c(t) \\ A_c(t) &= &0  ~~\text{for}~~ t<0 \end{array} $$

Here, \(A_c(t)\) and \(C_c(t)=A_c(t)/V\) are, respectively, the amount and the concentration of drug in the central compartment at time t. When a dose D arrives in the central compartment at time \(\tau\), an iv bolus administration assumes that

$$A_c(\tau^+) = A_c(\tau^-) + D$$

where \(A_c(\tau^-)\) (resp. \(A_c(\tau^+)\)) is the amount of drug in the central compartment just before (resp. after) \(\tau\) Parameters of this model are V and k. We therefore use the model bolus_1cpt_Vk from the Monolix PK library:

[LONGITUDINAL]
input = {V, k}

EQUATION:
Cc = pkmodel(V, k)

OUTPUT:
output = Cc

We could equivalently use the model bolusLinearMacro.txt (click on the button Model and select the new PK model in the library 6.PK_models/model)

[LONGITUDINAL]
input = {V, k}

PK:
compartment(cmt=1, amount=Ac)
iv(cmt=1)
elimination(cmt=1, k)
Cc = Ac/V

OUTPUT:
output = Cc

These two implementations generate exactly the same C++ code and then provide exactly the same results. Here, the ODE system is linear and Monolix uses its analytical solution. Of course, it is also possible (but not recommended with this model) to use the ODE based PK model bolusLinearODE.txt :

[LONGITUDINAL]
input = {V, k}

PK:
depot(target = Ac)

EQUATION:
ddt_Ac = - k*Ac
Cc = Ac/V

OUTPUT:
output = Cc

Results obtained with this model are slightly different from the ones obtained with the previous implementations since a numeric scheme is used here for solving the ODE. Moreover, the computation time is longer (between 3 and 4 time longer in that case) when using the ODE compared to the analytical solution.
Individual fits obtained with this model look nice

but the VPC show some misspecification in the elimination process:

Michaelis Menten elimination

  • bolusMM_project

A non linear elimination is used with this project:

$$\frac{dA_c}{dt} = – \frac{ V_m \, A_c(t)}{V\, K_m + A_c(t) }$$

This model is available in the Monolix PK library as bolus_1cpt_VVmKm:

[LONGITUDINAL]
input = {V, Vm, Km}

PK:
Cc = pkmodel(V, Vm, Km)

OUTPUT:
output = Cc

Instead of this model, we could equivalently use PK macros with bolusNonLinearMacro.txt from the library 6.PK_models/model:

[LONGITUDINAL]
input = {V, Vm, Km}

PK:
compartment(cmt=1, amount=Ac, volume=V)
iv(cmt=1)
elimination(cmt=1, Vm, Km)
Cc = Ac/V

OUTPUT:
output = Cc

or an ODE with bolusNonLinearODE:

[LONGITUDINAL]
input = {V, Vm, Km}

PK:
depot(target = Ac)

EQUATION:
ddt_Ac = -Vm*Ac/(V*Km+Ac) 
Cc=Ac/V

OUTPUT:
output = Cc

Results obtained with these three implementations are identical since no analytical solution is available for this non linear ODE. We can then check that this PK model seems to describe much better the elimination process of the data:

Mixed elimination

  • bolusMixed_project

THe Monolix PK library contains “standard” PK models. More complex models should be implemented by the user in a model file. For instance, we assume in this project that the elimination process is a combination of linear and nonlinear elimination processes:

$$ \frac{dA_c}{dt} = -\frac{ V_m A_c(t)}{V K_m + A_c(t) } – k  A_c(t) $$

This model is not available in the Monolix PK library. It is implemented in bolusMixed.txt:

[LONGITUDINAL]
input = {V, k, Vm, Km}

PK:
depot(target = Ac)

EQUATION:
ddt_Ac = -Vm*Ac/(V*Km+Ac) - k*Ac
Cc=Ac/V

OUTPUT:
output = Cc

This model,  with a combined error model, seems to describe very well the data:

Intravenous infusion

  • infusion_project

Intravenous infusion assumes that the drug is administrated intravenously with a constant rate (infusion rate), during a given time (infusion time). Since the amount is the product of infusion rate and infusion time, an additional column INFUSION RATE or INFUSION DURATION is required in the data file: Monolix can use both indifferently. Data file infusion_rate_data.txt has an additional column rate:

It can be replaced by infusion_tinf_data.txt which contains exactly the same information:

We use with this project a 2 compartment model with non linear elimination and parameters V_1, Q, V_2, V_m, K_m:

$$\begin{aligned} k_{12} &= Q/V_1 \\ k_{21} &= Q/V_2 \\\frac{dA_c}{dt} & = k_{21} \, Ap(t) – k_{12} \, Ac(t)- \frac{ V_m \, A_c(t)}{V_1\, K_m + A_c(t) } \\ \frac{dA_p}{dt} & = – k_{21} \, Ap(t) + k_{12} \, Ac(t) \\ Cc(t) &= \frac{Ac(t)}{V_1} \end{aligned}$$

This model is available in the Monolix PK library as infusion_2cpt_V1QV2VmKm:

[LONGITUDINAL]
input = {V1, Q, V2, Vm, Km}

PK:
V = V1
k12 = Q/V1
k21 = Q/V2
Cc = pkmodel(V, k12, k21, Vm, Km)

OUTPUT:
output = Cc

Oral administration

first-order absorption

  • oral1_project

This project uses the data file oral_data.txt. For each patient, information about dosing is the time of administration and the amount. A one compartment model with first order absorption and linear elimination is used with this project. Parameters of the model are ka, V and Cl. we will then use model oral1_kaVCl.txt from the Monolix PK library

[LONGITUDINAL]
input = {ka, V, Cl}

EQUATION:
Cc = pkmodel(ka, V, Cl)

OUTPUT:
output = Cc

Both the individual fits and the VPCs show that this model doesn’t describe the absorption process properly.

Many options for implementing this PK model with Mlxtran exists:
– using PK macros: oralMacro.txt:

[LONGITUDINAL]
input = {ka, V, Cl}

PK:
compartment(cmt=1, amount=Ac)
oral(cmt=1, ka)
elimination(cmt=1, k=Cl/V) 
Cc=Ac/V

OUTPUT:
output = Cc

– using a system of two ODEs as in oralODEb.txt:

[LONGITUDINAL]
input = {ka, V, Cl}

PK:
depot(target=Ad)

EQUATION:
k = Cl/V
ddt_Ad = -ka*Ad
ddt_Ac =  ka*Ad - k*Ac
Cc = Ac/V

OUTPUT: 
output = Cc

– combining PK macros and ODE as in oralMacroODE.txt (macros are used for the absorption and ODE for the elimination):

[LONGITUDINAL]
input = {ka, V, Cl}

PK:
compartment(cmt=1, amount=Ac)
oral(cmt=1, ka)

EQUATION:
k = Cl/V
ddt_Ac = - k*Ac
Cc = Ac/V

OUTPUT: 
output = Cc

– or equivalently, as in oralODEa.txt:

[LONGITUDINAL]
input = {ka, V, Cl}

PK:
depot(target=Ac, ka)

EQUATION:
k = Cl/V
ddt_Ac = - k*Ac
Cc = Ac/V<

OUTPUT: 
output = Cc

Remark: Models using the pkmodel function or PK macros only use an analytical solution of the ODE system.

zero-order absorption

  • oral0_project

A one compartment model with zero order absorption and linear elimination is used to fit the same PK data with this project. Parameters of the model are Tk0, V and Cl. We will then use model oral0_1cpt_Tk0Vk.txt from the Monolix PK library

[LONGITUDINAL]
input = {Tk0, V, Cl}

EQUATION:
Cc = pkmodel(Tk0, V, Cl)

OUTPUT:
output = Cc


Remark 1: implementing a zero-order absorption process using ODEs is not easy… on the other hand, it becomes extremely easy to implement using either the pkmodel function or the PK macro oral(Tk0).
Remark 2: The duration of a zero-order absorption has nothing to do with an infusion time: it is a parameter of the PK model (exactly as the absorption rate constant ka for instance), it is not part of the data.

sequential zero-order first-order absorption

  • sequentialOral0Oral1_project

More complex PK models can be implemented using Mlxtran. A sequential zero-order first-order absorption process assumes that a fraction Fr of the dose is first absorbed during a time Tk0 with a zero-order process, then, the remaining fraction is absorbed with a first-order process. This model is implemented in sequentialOral0Oral1.txt using PK macros:

[LONGITUDINAL]
input = {Fr, Tk0, ka, V, Cl}

PK:
compartment(amount=Ac)
absorption(Tk0, p=Fr)
absorption(ka, Tlag=Tk0, p=1-Fr)
elimination(k=Cl/V)
Cc=Ac/V

OUTPUT:
output = Cc

Both the individual fits and the VPCs show that this PK model describes very well the whole ADME process for the same PK data:

simultaneous zero-order first-order absorption

  • simultaneousOral0Oral1_project

A simultaneous zero-order first-order absorption process assumes that a fraction Fr of the dose is absorbed with a zero-order process while the remaining fraction is absorbed simultaneously with a first-order process. This model is implemented in simultaneousOral0Oral1.txt using PK macros:

[LONGITUDINAL]
input = {Fr, Tk0, ka, V, Cl}

PK:
compartment(amount=Ac)
absorption(Tk0, p=Fr)
absorption(ka, p=1-Fr)
elimination(k=Cl/V)
Cc=Ac/V

OUTPUT:
output = Cc

alpha-order absorption

  • oralAlpha_project

An \alpha-order absorption process assumes that the rate of absorption is proportional to some power of the amount of drug in the depot compartment:

\frac{dA_d}{dt} = - r \left(A_d(t)\right)^\alpha

This model is implemented in oralAlpha.txt using ODEs:

[LONGITUDINAL]
input = {r, alpha, V, Cl}

PK:
depot(target = Ad)

EQUATION:
dAd = Ad^alpha
ddt_Ad = -r*dAd
ddt_Ac = r*Ad - (Cl/V)*Ac
Cc = Ac/V

OUTPUT:
output = Cc

transit compartment model

  • oralTransitComp_project

A PK model with a transit compartment of transit rate Ktr and mean transit time Mtt can be implemented using the PK macro oral(ka, Mtt, Ktr), or using the pkmodel function, as in oralTransitComp.txt:

[LONGITUDINAL]
input = {Mtt, Ktr, ka, V, Cl}

EQUATION:
Cc = pkmodel(Mtt, Ktr, ka, V, Cl)

OUTPUT:
output = Cc

Using different parametrizations

The PK macros and the function pkmodel use some preferred parametrizations and some reserved names as input arguments: Tlag, ka, Tk0, V, Cl, k12, k21. It is however possible to use another parametrization and/or other parameter names. As an example, consider a 2-compartment model for oral administration with a lag, a first order absorption and a linear elimination. We can use the pkmodel function with, for instance, parameters ka, V, k, k12 and k21:

[LONGITUDINAL]
input = {ka, V, k, k12, k21}

PK:
Cc = pkmodel(ka, V, k, k12, k21)

OUTPUT:
output = Cc

Imagine now that we want i) to use the clearance Cl instead of the elimination rate constant k, ii) to use capital letters for the parameter names. We can still use the pkmodel function as follows:

[LONGITUDINAL]
input = {KA, V, CL, K12, K21}

PK:
Cc = pkmodel(ka=KA, V, k=CL/V, k12=K12, k21=K21)

OUTPUT:
output = Cc

3.3.1.2.Multiple routes of administration


Objectives: learn how to define and use a PK model for multiple routes of administration..


Projects: ivOral1_project, ivOral2_project





Some drugs can display complex absorption kinetics. Common examples are mixed first-order and zero-order absorptions, either sequentially or simultaneously, and fast and slow parallel first-order absorptions. A few examples of those kinds of absorption kinetics are proposed below. Various absorption models are proposed here as examples.

Combining iv and oral administrations – Example 1

  • ivOral1_project (data = ‘ivOral1_data.txt’ , model = ‘ivOral1Macro_model.txt’)

In this example, we combine oral and iv administrations of the same drug. The data file ivOral1_data.txt contains an additional column ADMINISTRATION ID which indicates the route of administration (1=iv, 2=oral)

We assume here a one compartment model with first-order absorption process from the depot compartment (oral administration) and a linear elimination process from the central compartment. We further assume that only a fraction F (bioavailability) of the drug orally administered is absorbed. This model is implemented in ivOral1Macro_model.txt using PK macros:

[LONGITUDINAL]
input = {F, ka, V, k}

PK:
compartment(cmt=1, amount=Ac)
iv(adm=1, cmt=1)
oral(adm=2, cmt=1, ka, p=F)
elimination(cmt=1, k)
Cc = Ac/V

OUTPUT:
output = Cc

A logit-normal distribution is used for bioavability F that takes it values in (0,1). The model properly fits the data as can be seen on the individual fits of the 6 first individuals
Remark: the same PK model could be implemented using ODEs instead of PK macros.
Let \(A_d\) and \(A_c\) be, respectively, the amounts in the depot compartment (gut) and the central compartment (bloodtsream). Kinetics of \(A_d\) and \(A_c\) are described by the following system of ODEs

$$\dot{A}_d(t)  = – k_a A_d(t)~~\text{and}~~ \dot{A}_c(t) = k_a A_d(t) – k A_c(t)$$

The target compartment is the depot compartment (\(A_d\)) for oral administrations and the central compartment (\(A_c\)) for iv administrations. This model is implemented in ivOral1ODE_model.txt using a system of ODEs:

[LONGITUDINAL]
input = {F, ka, V, k}

PK:
depot(type=1, target=Ad, p=F)
depot(type=2, target=Ac)

EQUATION:
ddt_Ad = -ka*Ad
ddt_Ac =  ka*Ad - k*Ac
Cc = Ac/V

OUTPUT:
output = Cc

Solving this ODEs system is less efficient than using the PK macros which uses the analytical solution of the linear system.

Combining iv and oral administrations – Example 2

  • ivOral2_project (data = ‘ivOral2_data.txt’ , model = ‘ivOral2Macro_model.txt’)

In this example (based on simulated PK data), we combine intraveinous injection with 3 different types of oral administrations of the same drug. The datafile ivOral2_data.txt contains column ADM which indicates the route of administration (1,2,3=oral, 4=iv). We assume that one type of oral dose (adm=1) is absorbed into a latent compartment following a zero-order absorption process. The 2 oral doses (adm=2,3) are absorbed into the central compartment following first-order absorption processes with different rates. Bioavailabilities are supposed to be different for the 3 oral doses. There is linear transfer from the latent to the central compartment. A peripheral compartment is linked to the central compartment. The drug is eliminated by a linear process from the central compartment:

This model is implemented in ivOral2Macro_model.txt using PK macros:

[LONGITUDINAL]
input = {F1, F2, F3, Tk01, ka2, ka3, kl, k23, k32, V, Cl}

PK:
compartment(cmt=1, amount=Al)
compartment(cmt=2, amount=Ac)
peripheral(k23,k32)
oral(type=1, cmt=1, Tk0=Tk01, p=F1)
oral(type=2, cmt=2, ka=ka2,   p=F2)
oral(type=3, cmt=2, ka=ka3,   p=F3)
iv(type=4, cmt=2)
transfer(from=1, to=2, kt=kl)
elimination(cmt=2, k=Cl/V)
Cc = Ac/V

OUTPUT:
output = Cc

Here, logit-normal distributions are used for bioavabilities \(F_1\), \(F_2\) and \(F_3\). The model fits the data properly :

Remark: the number and type of doses vary from one patient to another in this example.

3.3.1.3.Multiple doses to steady-state


Objectives: learn how to define and use a PK model with multiple doses or assuming steady-state.


Projects: multidose_project, addl_project, ss1_project, ss2_project, ss3_project


Multiple doses

  • multidose_project (data = ‘multidose_data.txt’ , model = ‘lib:bolus_1cpt_Vk.txt’)

In this project, each patient receives several iv bolus injections. Each dose is represented by a row in the data file multidose_data.txt:

The PK model and the statistical model used in this project properly fit the observed data of each individual. Even if there is no observation between 12h and 72h, predicted concentrations computed on this time interval exhibit the multiple doses received by each patient:

VPCs, which is a diagnosis tool, are based on the design of the observations and therefore “ignore” what may happen between 12h and 72h:

On the other hand, the prediction distribution, which is not a diagnosis tool, computes the distribution of the predicted concentration at any time point:

Additional doses (ADDL)






 

  • addl_project (data = ‘addl_data.txt’ , model = ‘lib:bolus_1cpt_Vk.txt’)

We can note in the previous project, that, for each patient, the interval time between two successive doses is the same (12 hours for each patient) and the amount of drug which is administrated is always the same as well (40mg for each patient). We can take advantage of this design in order to simplify the data file by defining, for each patient, a unique amount (AMT), the number of additional doses which are administrated after the first one (ADDITIONAL DOSES) and the time interval between successive doses (INTERDOSE INTERVAL):

The keywords ADDL and II are automatically recognized by Monolix.

Remarks:

  • Results obtained with this project, i.e. with this data file, are identical to the ones obtained with the previous project.
  • It is possible to combine single doses (using ADDL=0) and repeated doses in a same data file.

Steady-state

  • ss1_project (data = ‘ss1_data.txt’ , model = ‘lib:oral0_1cpt_Tk0VCl.txt’)

The dose, orally administrated at time 0 to each patient, is assumed to be a “steady-state dose” which means that a  “large” number of doses before time 0 have been administrated, with a constant amount and a constant interval dosing, such that steady-state, i.e. equilibrium, is reached at time 0. The data file ss1_data contains a column STEADY STATE which indicates if the dose is a steady-state dose or not and a column INTERDOSE INTERVAL for the inter-dose interval:

Click on Check the initial fixed effects to display the predicted concentration between the last dose administrated at time 0. One can see that the initial concentration is not 0 but the result of the steady state calculation.

We see on this plot that Monolix adds 5 doses before the last dose to reach steady-state. Individual fits display the predicted concentrations computed with these additional doses:

If the dynamics is slow, adding 5 doses before the last dose might not be sufficient. You can adapt the number of doses in the frame data and thus define it for all individuals as on the following figure.

leading to the following check initial fixed effects

  • ss2_project (data = ‘ss2_data.txt’ , model = ‘lib:oral0_1cpt_Tk0VCl.txt’)

Steady-state and non steady-sates doses are combined in this project:

Individual fits display the predicted concentrations computed with this combination of doses:

3.3.2.Mixture of structural models






Objectives: learn how to implement between subject mixture models (BSMM) and within subject mixture models (WSMM).


Projects: bsmm1_project, bsmm2_project, wsmm_project


Introduction

There are two approaches to define a mixture of models:

Several types of mixture models exist, they are useful in the context of mixed effects models. It may be necessary in some situations to introduce diversity into the structural models themselves:

  • Between-subject model mixtures (BSMM) assume that there exists subpopulations of individuals. Different structural models describe the response of each subpopulation, and each subject belongs to one of these subpopulations. One can imagine for example different structural models for responders, nonresponders and partial responders to a given treatment.

The easiest way to model a finite mixture model is to introduce a label sequence (z_i ; 1 \leq i \leq N) that takes its values in {1, 2, \ldots, M} such that z_i = m if subject i belongs to subpopulation m\mathbb{P}(z_i = m) is the probability for subject i to belong to subpopulation m. A BSMM assumes that the structural model is a mixture of M different structural models:

$$f\left(t_{ij}; \psi_i, z_i \right) = \sum_{m=1}^M 1_{z_i = m} f_m\left( t_{ij}; \psi_i \right) $$

In other word, each subpopulation has its own structural model: f_m is the structural model for subpopulation m.

  • Within-subject model mixtures (WSMM) assume that there exist subpopulations (of cells, viruses, etc.) within each patient. In this case, different structural models can be used to describe the response of different subpopulations, but the proportion of each subpopulation depends on the patient.

Then, it makes sense to consider that the mixture of models happens within each individual. Such within-subject model mixtures require additional vectors of individual parameters \pi_i=(\pi_{1,i}, \ldots, \pi_{M,i}) representing the proportions of the M models within each individual i:

$$f\left( t_{ij}; \psi_i, z_i \right) = \sum_{m=1}^M \pi_{m,i} f_m\left( t_{ij}; \psi_i \right)$$

The proportions (\pi_{m,i}) are now individual parameters in the model and the problem is transformed into a standard mixed effects model. These proportions are assumed to be positive and to sum to 1 for each patient.

Between subject mixture models

Supervised learning

  • bsmm1_project (data = ‘pdmixt1_data.txt’, model = ‘bsmm1_model.txt’)

We consider a very simple example here with two subpopulations of individuals who receive a given treatment. The outcome of interest is the measured effect of the treatment (a viral load for instance). The two populations are non responders and responders. We assume here that the status of the patient is known. Then, the data file contains an additional column GROUP. This column is duplicated because Monolix uses it

  • i) as a regression variable (REGRESSOR): it is used in the model to distinguish responders and non responders,
  • ii) as a categorical covariate (CATEGORICAL COVARIATE): it is used to stratify the diagnosis plots.

We can then display the data

and use the categorical covariate GROUP_CAT to split the plot into responders and non responders:
We use different structural models for non responders and responders. The predicted effect for non responders is constant f(t) = A1 while the predicted effect for responders decreases exponentially f(t) = A2 exp(-kt).

The model is implemented in the model file bsmm1_model.txt (note that the names of the regression variable in the data file and in the model script do not need to match):

[LONGITUDINAL]
input = {A1, A2, k, g}
g = {use=regressor}

EQUATION:
if g==1
   f = A1
else
   f = A2*exp(-k*max(t,0))
end

OUTPUT:
output = f

The plot of individual fits exhibit the two different structural models:

VPCs should then be splitted according to the GROUP_CAT

as well as the prediction distribution for non responders and responders:

Unsupervised learning

  • bsmm2_project (data = ‘pdmixt2_data.txt’, model = ‘bsmm2_model.txt’)

The status of the patient is unknown in this project (which means that the column GROUP is not available anymore). Let p be the proportion of non responders in the population. Then, the structural model for a given subject is f1 with probability p and f2 with probability 1-p. The structural model is therefore a BSMM:

[LONGITUDINAL]
input = {A1, A2, k, p}

EQUATION:
f1 = A1
f2 = A2*exp(-k*max(t,0))
f  = bsmm(f1, p, f2, 1-p)

OUTPUT:
output = f

Important:

  • The bsmm function must be used on the last line of the structural model, just before “OUTPUT:”. It is not possible to reuse the variable returned by the bsmm function (here f) in another equation.
  • p is a population parameter of the model to estimate. There is no inter-patient variability on p: all the subjects have the same probability of being a non responder in this example. We use a logit-normal distribution for p  in order to constrain it to be between 0 and 1, but without variability:

p is estimated with the other population parameters:
Then, the group to which a patient belongs is also estimated as the group of highest conditional probability:

$$\begin{aligned}\hat{z}_i &= 1~~~~\textrm{if}~~~~ \mathbb{P}(z_i=1 | (y_{ij}), \hat{\psi}_i, \hat{\theta})> \mathbb{P}(z_i=2 | (y_{ij}),\hat{\psi}_i, \hat{\theta}),\\ &=0~~~~\textrm{otherwise}\end{aligned}$$

The estimated groups can be used as a stratifying variable to split some plots such as VPCs

Bsmm function with ODEs

The bsmm function can also be used with models defined via ODE systems. The syntax in that case follows this example, with model M defined as a mixture of M1 and M2:

M1_0 = ... ; initial condition for M1
ddt_M1 = ... ; ODE for M1
M2_0 = ... ; initial condition for M2
ddt_M2 = ... ; ODE for M2

M = bsmm(M1,p1,M2,1-p1)

 

Unsupervised learning with latent covariates

If the models composing the mixture have a similar structure, it is sometimes possible and easier to implement the mixture with a latent categorical covariate instead of the bsmm function. It also has the advantage of allowing more than two mixture groups, while the bsmm function can only define two mixture groups.

 

Within subject mixture models

  • wsmm_project (data = ‘pdmixt2_data.txt’, model = ‘wsmm_model.txt’)

It may be too simplistic to assume that each individual is represented by only one well-defined model from the mixture. We consider here that the mixture of models happens within each individual and use a WSMM: f = p*f1 + (1-p)*f2

[LONGITUDINAL]
input = {A1, A2, k, p}

EQUATION:
f1 = A1
f2 = A2*exp(-k*max(t,0))
f = wsmm(f1, p, f2, 1-p)

OUTPUT:
output = f

Remark: Here, writing f = wsmm(f1, p, f2, 1-p) is equivalent to writing f = p*f1 + (1-p)*f2
Important: Here, p is an individual parameter: the subjects have different proportions of non responder cells. We use a probit-normal distribution for p in order to constrain it to be between 0 and 1, with variability:

There is no latent covariate when using WSMM: mixtures are continuous mixtures. We therefore cannot split anymore the VPC and the prediction distribution anymore.

3.3.3.Joint models for continuous outcomes


Objectives: learn how to implement a joint model for continuous PKPD data.


Projects: warfarinPK_project, warfarin_PKPDimmediate_project, warfarin_PKPDeffect_project, warfarin_PKPDturnover_project, warfarin_PKPDseq1_project, warfarin_PKPDseq2_project, warfarinPD_project





Introduction

A “joint model” describes two or more types of observation that typically depend on each other. A PKPD model is a “joint model” because the PD depends on the PK. Here we demonstrate how several observations can be modeled simultaneously. We also discuss the special case of sequential PK and PD modelling, using either the population PK parameters or the individual PK parameters as an input for the PD model.

Fitting first a PK model to the PK data

  • warfarinPK_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

The column DV of the data file contains both the PK and the PD measurements: in Monolix this column is tagged as  an OBSERVATION column. The column DVID is a flag defining the type of observation: DVID=1 for PK data and DVID=2 for PD data: the keyword OBSERVATION ID is then used for this column.

We will use the model oral1_1cpt_TlagkaVCl from the Monolix PK library

[LONGITUDINAL]
input = {Tlag, ka, V, Cl}

EQUATION:
Cc = pkmodel(Tlag, ka, V, Cl)

OUTPUT:
output = {Cc}

Only the predicted concentration Cc is defined as an output of this model. Then, this prediction will be automatically associated to the outcome of type 1 (DVID=1) while the other observations (DVID=2) will be ignored.
Remark: any other ordered values could be used for OBSERVATION ID column: the smallest one will always be associated to the first prediction defined in the model.

Simultaneous PKPD modeling

  • warfarin_PKPDimmediate_project (data = ‘warfarin_data.txt’, model = ‘immediateResponse_model.txt’)

It is also possible for the user to write his own PKPD model. The same PK model used previously and an immediate response model are defined in the model file immediateResponse_model.txt

[LONGITUDINAL]
input = {Tlag, ka, V, Cl, Imax, IC50, S0}

EQUATION:
Cc = pkmodel(Tlag, ka, V, Cl)
E = S0 * (1 - Imax*Cc/(Cc+IC50))

OUTPUT:
output = {Cc, E}

Two predictions are now defined in the model: Cc for the PK (DVID=1) and E for the PD (DVID=2).

  • warfarin_PKPDeffect_project (data = ‘warfarin_data.txt’, model = ‘effectCompartment_model.txt’)

An effect compartment is defined in the model file effectCompartment_model.txt

[LONGITUDINAL]
input = {Tlag, ka, V, Cl, ke0, Imax, IC50, S0}

EQUATION:
{Cc, Ce} = pkmodel(Tlag, ka, V, Cl, ke0)
E = S0 * (1 - Imax*Ce/(Ce+IC50))

OUTPUT:
output = {Cc, E}

Ce is the concentration in the effect compartment

  • warfarin_PKPDturnover_project (data = ‘warfarin_data.txt’, model = ‘turnover1_model.txt’)

An indirect response (turnover) model is defined in the model file turnover1_model.txt

[LONGITUDINAL]
input =  {Tlag, ka, V, Cl, Imax, IC50, Rin, kout}

EQUATION:
Cc = pkmodel(Tlag, ka, V, Cl)
E_0 = Rin/kout
ddt_E = Rin*(1-Imax*Cc/(Cc+IC50)) - kout*E

OUTPUT:
output = {Cc, E}

Sequential PKPD modelling

In the sequential approach, a PK model is developed and parameters are estimated in the first step. For a given PD model, different strategies are then possible for the second step, i.e., for estimating the population PD parameters:

Using estimated population PK parameters

  • warfarin_PKPDseq1_project (data = ‘warfarin_data.txt’, model = ‘turnover1_model.txt’)

Population PK parameters are set to their estimated values but individual PK parameters are not assumed to be known and sampled from their conditional distributions at each SAEM iteration. In Monolix, this simply means changing the status of the population PK parameter values so that they are no longer used as initial estimates for SAEM but considered fixed as on the figure below.

To fix parameters, click on the green option button (framed in green) and choose the Fixed method as on the figure below

The joint PKPD model defined in turnover1_model.txt is again used with this project.

Using estimated individual PK parameters

  • warfarin_PKPDseq2_project (data = ‘warfarinSeq_data.txt’, model = ‘turnoverSeq_model.txt’)

In htis case, individual PK parameters are set to their estimated values and used as constants in the PKPD model to fit the PD data. To do so, the individual PK parameters need to be added to the PD dataset (or PK/PD dataset) and tagged as regressors.

The PK project (that was executed before and through which the estimated PK parameters were obtained) contains in the result folder the individual parameter values “..\warfarin_PKPDseq1_project\IndividualParameters\estimatedIndividualParameters.txt”. These estimated PK parameters can be added to the PD dataset by using the data formatting tool integrated in Monolix version 2023R1. Depending which tasks have been run in the PK project, the individual parameters corresponding to the conditional mode (EBEs, with “_mode”), the conditional mean (mean of the samples from the conditional distributions, with “_mean”) and an approximation of the conditional mean obtained at the end of the SAEM step (with “_SAEM”) are available in the file. All columns are added to the PD dataset.

At the data tagging step, the user can choose which individual parameters to use. The most common is to use the EBEs so to tag the columns with “_mode” as regressor and leave the others as IGNORE (purple frame below). By activating the toggle button (green frame), the ignored columns flagged with the keyword IGNORE can be hidden.

We use the same turnover model for the PD data. Here, the PK parameters are defined as regression variables (i.e. regressors).

[LONGITUDINAL]
input =  {Imax, IC50, Rin, kout, Tlag, ka, V, Cl}
Tlag  = {use = regressor}
ka    = {use = regressor}
V     = {use = regressor}
Cl    = {use = regressor}

EQUATION:
Cc = pkmodel(Tlag,ka,V,Cl)
E_0 = Rin/kout
ddt_E= Rin*(1-Imax*Cc/(Cc+IC50)) - kout*E

OUTPUT:
output = {E}

As you can see, the names of the regressors do not match the parameter names. The regressors are matched by order (not by name) between the data set and the model input statement.

If there are multiple observation types in the data as well as different response vectors in the output statement of the structural model, then these must be mapped accordingly in the mapping panel.

Fitting a PKPD model to the PD data only

  • warfarinPD_project (data = ‘warfarinPD_data.txt’, model = ‘turnoverPD_model.txt’)

In this example, only PD data is available. Nevertheless, a PKPD model – where only the effect is defined as a prediction – can be used for fitting this data and thus defined in the OUTPUT section.

[LONGITUDINAL]
input =  {Tlag, ka, V, Cl, Imax, IC50, Rin, kout}

EQUATION:
Cc = pkmodel(Tlag, ka, V, Cl)
E_0 = Rin/kout
ddt_E = Rin*(1-Imax*Cc/(Cc+IC50)) - kout*E

OUTPUT:
output = {E}

Case studies

  • 8.case_studies/PKVK_project (data = ‘PKVK_data.txt’, model = ‘PKVK_model.txt’)
  • 8.case_studies/hiv_project (data = ‘hiv_data.txt’, model = ‘hivLatent_model.txt’)

3.4.Models for non continuous outcomes

3.4.1.Time-to-event data models






Objectives: learn how to implement a model for (repeated) time-to-event data with different censoring processes.


Projects: tte1_project, tte2_project, tte3_project, tte4_project, rtteWeibull_project, rtteWeibullCount_project


Introduction

Here, observations are the “times at which events occur”. An event may be one-off (e.g., death, hardware failure) or repeated (e.g., epileptic seizures, mechanical incidents, strikes). Several functions play key roles in time-to-event analysis: the survival, hazard and cumulative hazard functions. We are still working under a population approach here so these functions, detailed below, are thus individual functions, i.e., each subject has its own. As we are using parametric models, this means that these functions depend on individual parameters \((\psi_i)\).

  • The survival function \(S(t, \psi_i)\) gives the probability that the event happens to individual i after time \(t>t_{\text{start}}\):

    $$S(t,\psi_i) = \mathbb{P}(T_i>t; \psi_i) $$

  • The hazard function \(h(t,psi_i)\) is defined for individual i as the instantaneous rate of the event at time t, given that the event has not already occurred:

    $$h(t, \psi_i) = \lim_{dt \to 0} \frac{S(t, \psi_i) – S(t + dt, \psi_i)}{ S(t, \psi_i)  dt} $$

    This is equivalent to

    $$h(t, \psi_i) = -\frac{d}{dt} \left(\log{S(t, \psi_i)}\right)$$

  • Another useful quantity is the cumulative hazard function \(H(a,b; \psi_i)\), defined for individual i as

$$H(a,b; \psi_i) = \int_a^b h(t,\psi_i) dt $$

Note that \(S(t, \psi_i) = e^{-H(t_{\text{start}},t; \psi_i)}\). Then, the hazard function \(h(t,\psi_i)\) characterizes the problem, because knowing it is the same as knowing the survival function \(S(t, \psi_i)\). The probability distribution of survival data is therefore completely defined by the hazard function.

Time-to-event (TTE) models are thus defined in Monolix via the hazard function. Monolix also holds a TTE library that contains typical hazard functions for time-to-event data. More details and modeling guidelines can be found on the TTE dedicated webpage, along with case studies.

 

Formatting of time-to-event data in the MonolixSuite



In the data set, exactly observed events, interval censored events and right censoring are recorded for each individual. Contrary to other softwares for survival analysis, the MonolixSuite requires to specify the time at which the observation period starts. This allows to define the data set using absolute times, in addition to durations (if the start time is zero, the records represent durations between the start time and the event).

The column TIME also contains the end of the observation period or the time intervals for interval-censoring. The column OBSERVATION contains an integer that indicates how to interpret the associated time. The different values for each type of event and observation are summarized in the table below:

The figure below summarizes the different situations with examples:

For instance for single events, exactly observed (with or without right censoring), one must indicate the start time of the observation period (Y=0), and the time of event (Y=1) or the time of the end of the observation period if no event has occurred (Y=0). In the following example:

ID TIME Y
1   0   0
1  34   1
2   0   0
2  80   0

the observation period lasts from starting time t=0 to the final time t=80. For individual 1, the event is observed at t=34, and for individual 2, no event is observed during the period. Thus it is noticed that at the final time (t=80), no event had occurred. Using absolute times instead of duration, we could equivalently write:

ID TIME Y
1  20   0
1  54   1
2  33   0
2  113  0

The duration between start time and event (or end of the observation period) are the same as before, but this time we record the day at which the patients enter the study and the days at which they have events or leave the study. Different patients may enter the study at different times.

Examples for repeated events, and interval censored events are available on the data set documentation page.

 

Single event

To begin with, we will consider a one-off event. Depending on the application, the length of time to this event may be called the survival time (until death, for instance), failure time (until hardware fails), and so on. In general, we simply say “time-to-event”. The random variable representing the time-to-event for subject i is typically written Ti.

 

Single event exactly observed or right censored

  • tte1_project (data = tte1_data.txt , model=lib:exponential_model_singleEvent.txt)

The event time may be exactly observed at time \(t_i\), but if we assume that the trial ends at time \(t_{\text{stop}}\), the event may happen after the end. This is “right censoring”. Here, Y=0 at time t means that the event happened after t and Y=1 means that the event happened at time t. The rows with t=0 are included to show the trial start time \(t_{\text{start}}=0\):
By clicking on the button Observed data, it is possible to display the Kaplan Meier plot (i.e. the empirical survival function) before fitting any model:


A very basic model with constant hazard is used for this data:

[LONGITUDINAL]
input = Te
  
EQUATION:
h = 1/Te

DEFINITION:
Event = {type=event, maxEventNumber=1, hazard=h}

OUTPUT:
output = {Event}

Here, Te is the expected time to event. Specification of the maximum number of events is required both for the estimation procedure and for the diagnosis plots based on simulation, such as the predicted interval for the Kaplan Meier plot which is obtained by Monte Carlo simulation:

 

Single event interval censored or right censored

  • tte2_project (data = tte2_data.txt , model=exponentialIntervalCensored_model.txt)

We may know the event has happened in an interval \(I_i\) but not know the exact time \(t_i\). This is interval censoring. Here, Y=0 at time t means that the event happened after t and Y=1 means that the event happened before time t.
Event for individual 1 happened between t=10 and t=15. No event was observed until the end of the experiment (t=100) for individual 5. We use the same basic model, but we now need to specify that the events are interval censored:

[LONGITUDINAL]
input = Te  

EQUATION:
h = 1/Te

DEFINITION:
Event = {type=event, maxEventNumber=1, eventType=intervalCensored, hazard = h
         intervalLength=5     ; used for the plots (not mandatory)
}

OUTPUT:
output = Event

Repeated events

Sometimes, an event can potentially happen again and again, e.g., epileptic seizures, heart attacks. For any given hazard function h, the survival function S for individual i now represents the survival since the previous event at \(t_{i,j-1}\), given here in terms of the cumulative hazard from \(t_{i,j-1}\) to \(t_{i,j}\):

$$S(t_{i,j} | t_{i,j-1}; \psi_i) = \mathbb{P}(T_{i,j} > t_{i,j} | T_{i,j-1} = t_{i,j-1}; \psi_i) = \exp(-\int_{t_{i,j-1}}^{t_{i,j}}h(t,\psi_i) dt)$$

 

Repeated events exactly observed or right censored

  • tte3_project (data = tte3_data.txt , model=lib:exponential_model_repeatedEvents.txt)

A sequence of \(n_i\) event times is precisely observed before \(t_{\text{stop}} = 200\): We can then display the Kaplan Meier plot for the first event and the mean number of events per individual:

After fitting the model, prediction intervals for these two curves can also be displayed on the same graph as on the following

 

Repeated events interval censored or right censored

  • tte4_project (data = tte4_data.txt , model=exponentialIntervalCensored_repeated_model.txt)

We do not know the exact event times, but the number of events that occurred for each individual in each interval of time.

 

User defined likelihood function for time-to-event data

  • weibullRTTE (data = weibull_data.txt , model=weibullRTTE_model.txt)

A Weibull model is used in this example:

[LONGITUDINAL]
input = {lambda, beta}  

EQUATION: 
h = (beta/lambda)*(t/lambda)^(beta-1)

DEFINITION:
Event = {type=event, hazard=h, eventType=intervalCensored,
         intervalLength=5}

OUTPUT:
output = Event
  • weibullCount (data = weibull_data.txt , model=weibullCount_model.txt)

Instead of defining the data as events, it is possible to consider the data as count data: indeed, we count the number of events per interval. An additional column with the start of the interval is added in the data file and defined as a regression variable. We then use a model for count data (see rtteWeibullCount_model.txt).

3.4.2.Count data model


Objectives: learn how to implement a model for count data.


Projects: count1a_project, count1a_project, count1a_project, count2_project


Introduction

Longitudinal count data is a special type of longitudinal data that can take only nonnegative integer values {0, 1, 2, …} that come from counting something, e.g., the number of seizures, hemorrhages or lesions in each given time period . In this context, data from individual j is the sequence \(y_i=(y_{ij},1\leq j \leq n_i)\) where \(y_{ij}\) is the number of events observed in the jth time interval \(I_{ij}\).
Count data models can also be used for modeling other types of data such as the number of trials required for completing a given task or the number of successes (or failures) during some exercise. Here, \(y_{ij}\) is either the number of trials or successes (or failures) for subject i at time \(t_{ij}\). For any of these data types we will then model \(y_i=(y_{ij},1 \leq j \leq n_i)\) as a sequence of random variables that take their values in {0, 1, 2, …}.  If we assume that they are independent, then the model is completely defined by the probability mass functions \(\mathbb{P}(y_{ij}=k)\) for \(k \geq 0\) and \(1 \leq j \leq n_i\). Here, we will only consider  parametric distributions for count data.

Formatting of count data in the MonolixSuite

Count data can only take non-negative integer values that come from counting something, e.g., the number of trials required for completing a given task. The task can for instance be repeated several times and the individual’s performance followed. In the following data set:

ID TIME Y
1 0 10
1 24 6
1 48 5
1 72 2

10 trials are necessary the first day (t=0), 6 the second day (t=24), etc. Count data can also represent the number of events happening in regularly spaced intervals, e.g the number of seizures every week. If the time intervals are not regular, the data may be considered as repeated time-to-event interval censored, or the interval length can be given as regressor to be used to define the probability distribution in the model.
One can see the epilepsy attacks data set for a more practical example.

Modling count data in the MonolixSuite

Link to the detailed description of the library of count models integrated within Monolix.

Count data with constant distribution over time

  • count1a_project (data = ‘count1_data.txt’, model = ‘count_library/poisson_mlxt.txt’)

A Poisson model is used for fitting the data:

[LONGITUDINAL]
input = lambda

DEFINITION:
Y = {type = count,  log(P(Y=k)) = -lambda + k*log(lambda) - factln(k) }

OUTPUT:
output = Y

Residuals for noncontinuous data reduce to NPDEs. We can compare the empirical distribution of the NPDEs with the distribution of a standardized normal distribution either with the pdf (top) or the cdf (bottom):

VPCs for count data compare the observed and predicted frequencies of the categorized data over time:

  • count1b_project (data = ‘count1_data.txt’, model = ‘count_library/poissonMixture_mlxt.txt’)

A mixture of two Poisson distributions is used to fit the same data. For that, we define the probability of k occurrences as the weigthed sum of two Poisson distributions with two expected numbers of occurrences lambda1 and lambda2. The structural model file writes

[LONGITUDINAL]
input = {lambda1, alpha, mp}

EQUATION:
lambda2 = (1+alpha)*lambda1

DEFINITION:
Y = { type = count,
       P(Y=k) = mp*exp(-lambda1 + k*log(lambda1) - factln(k)) + (1-mp)*exp(-lambda2 + k*log(lambda2) - factln(k)) 
}

OUTPUT:
output = Y

Thus, the parameter alpha has to be strictly positive to ensure different expected number of occurrences in the two poisson distributions and mp has to be in [0, 1] to ensure the probability is correctly defined. Thus those parameters should be defined with lognormal and probitnormal distribution respectively as shown on the following figure.

We see on the VPC below that the data set is well modeled using this mixture of Poisson distributions.

In addition, we can compute the prediction distribution of the modalities as on the following figure

Count data with time varying distribution

  • count2_project (data = ‘count2_data.txt’, model = ‘count_library/poissonTimeVarying_mlxt.txt’)

The distribution of the data changes with time in this example:

We then use a Poisson distribution with a time varying intensity:

[LONGITUDINAL]
input =  {a,b}
                           
EQUATION:
lambda= a*exp(-b*t)
                           
DEFINITION:
y = {type=count, P(y=k)=exp(-lambda)*(lambda^k)/factorial(k)}

OUTPUT:
output = y

This model seems to fit the data very well:

3.4.3.Categorical data model


Objectives: learn how to implement a model for categorical data, assuming either independence or a Markovian dependence between observations.


Projects: categorical1_project, categorical2_project, markov0_project, markov1a_project, markov1b_project, markov1c_project, markov2_project, markov3a_project, markov3b_project


Introduction

Assume now that the observed data takes its values in a fixed and finite set of nominal categories \(\{c_1, c_2,\ldots , c_K\}\). Considering the observations \((y_{ij},\, 1 \leq j \leq n_i)\) for any individual \(i\) as a sequence of conditionally independent random variables, the model is completely defined by the probability mass functions \(\mathbb{P}(y_{ij}=c_k | \psi_i)\) for \(k=1,\ldots, K\) and \(1 \leq j \leq n_i\). For a given (i,j), the sum of the K probabilities is 1, so in fact only K-1 of them need to be defined. In the most general way possible, any model can be considered so long as it defines a probability distribution, i.e., for each k, \(\mathbb{P}(y_{ij}=c_k | \psi_i) \in [0,1]\), and \(\sum_{k=1}^{K} \mathbb{P}(y_{ij}=c_k | \psi_i) =1\). Ordinal data further assumed that the categories are ordered, i.e., there exists an order \(\prec\) such that

$$c_1 \prec c_2,\prec \ldots \prec c_K $$

We can think, for instance, of levels of pain (low \(\prec\) moderate \(\prec\) severe) or scores on a discrete scale, e.g., from 1 to 10. Instead of defining the probabilities of each category, it may be convenient to define the cumulative probabilities \(\mathbb{P}(y_{ij} \preceq c_k | \psi_i)\) for \(k=1,\ldots ,K-1\), or in the other direction: \(\mathbb{P}(y_{ij} \succeq c_k | \psi_i)\) for \(k=2,\ldots, K\). Any model is possible as long as it defines a probability distribution, i.e., it satisfies

$$0 \leq \mathbb{P}(y_{ij} \preceq c_1 | \psi_i) \leq \mathbb{P}(y_{ij} \preceq c_2 | \psi_i)\leq \ldots \leq \mathbb{P}(y_{ij} \preceq c_K | \psi_i) =1 .$$

It is possible to introduce dependence between observations from the same individual by assuming that \((y_{ij},\,j=1,2,\ldots,n_i)\) forms a Markov chain. For instance, a Markov chain with memory 1 assumes that all that is required from the past to determine the distribution of \(y_{ij}\) is the value of the previous observation \(y_{i,j-1}\)., i.e., for all \(k=1,2,\ldots ,K\),

$$\mathbb{P}(y_{ij} = c_k\,|\,y_{i,j-1}, y_{i,j-2}, y_{i,j-3},\ldots,\psi_i) = \mathbb{P}(y_{ij} = c_k | y_{i,j-1},\psi_i)$$

Formatting of categorical data in the MonolixSuite





In case of categorical data, the observations at each time point can only take values in a fixed and finite set of nominal categories. In the data set, the output categories must be coded as integers, as in the following example:

ID TIME Y
1 0.5 3
1 1 0
1 1.5 2
1 2 2
1 2.5 3

One can see the respiratory status data set and the warfarin data set for example for more practical examples on a categorical and a joint continuous and categorical data set respectively.

 

Ordered categorical data

  • categorical1_project (data = ‘categorical1_data.txt’, model = ‘categorical1_model.txt’)

In this example, observations are ordinal data that take their values in {0, 1, 2, 3}:

  • Cumulative odds ratio are used in this example to define the model

$$\textrm{logit}(\mathbb{P}(y_{ij} \leq k))= \log \left( \frac{\mathbb{P}(y_{ij} \leq k)}{1 – \mathbb{P}(y_{ij} \leq k )} \right)$$

where

$$\begin{array}{ccl} \text{logit}(\mathbb{P}(y_{ij} \leq 0)) &=& \theta_{i,1}\\ \text{logit}(\mathbb{P}(y_{ij} \leq 1)) &=& \theta_{i,1}+\theta_{i,2}\\ \text{logit}(\mathbb{P}(y_{ij} \leq 2)) &=& \theta_{i,1}+\theta_{i,2}+\theta_{i,3}\end{array}$$

This model is implemented in categorical1_model.txt:

[LONGITUDINAL]
input = {th1, th2, th3}

DEFINITION:
level = { type = categorical,  categories = {0, 1, 2, 3},
  logit(P(level<=0)) = th1
  logit(P(level<=1)) = th1 + th2
  logit(P(level<=2)) = th1 + th2 + th3
}

A normal distribution is used for \(\theta_{1}\), while log-normal distributions for \(\theta_{2}\) and \(\theta_{3}\) ensure that these parameters are positive (even without variability). Residuals for noncontinuous data reduce to NPDE’s. We can compare the empirical distribution of the NPDE’s with the distribution of a standardized normal distribution:

VPC’s for categorical data compare the observed and predicted frequencies of each category over time:

The prediction distribution can also be computed by Monte-Carlo:

 

Ordered categorical data with regression variables

  • categorical2_project (data = ‘categorical2_data.txt’, model = ‘categorical2_model.txt’)

A proportional odds model is used in this example, where PERIOD and DOSE are used as regression variables (i.e. time-varying covariates)

 

Discrete-time Markov chain

If observation times are regularly spaced (constant length of time between successive observations), we can consider the observations \((y_{ij},j=1,2,\ldots,n_i)\) to be a discrete-time Markov chain.

  • markov0_project (data = ‘markov1a_data.txt’, model = ‘markov0_model.txt’)

In this project, states are assumed to be independent and identically distributed:

\( \mathbb{P}(y_{ij} = 1) = 1 – \mathbb{P}(y_{ij} = 2) = p_{i,1} \)

Observations in markov1a_data.txt take their values in {1, 2}.

  • markov1a_project (data = ‘markov1a_data.txt’, model = ‘markov1a_model.txt’)

Here,

\(\begin{aligned}\mathbb{P}(y_{i,j} = 1 | y_{i,j-1} = 1) = 1 – \mathbb{P}(y_{i,j} = 2 | y_{i,j-1} = 1) = p_{i,11}\\ \mathbb{P}(y_{i,j} = 1 | y_{i,j-1} = 2) = 1 – \mathbb{P}(y_{i,j} = 2 | y_{i,j-1} = 2) = p_{i,12} \end{aligned}\)

[LONGITUDINAL]
input = {p11, p21}
DEFINITION:
State = {type = categorical,  categories = {1,2},  dependence = Markov
  P(State=1|State_p=1) = p11
  P(State=1|State_p=2) = p21
}

The distribution of the initial state is not defined in the model, which means that, by default,

\( \mathbb{P}(y_{i,1} = 1) = \mathbb{P}(y_{i,1} = 2) = 0.5 \)

  • markov1b_project (data = ‘markov1b_data.txt’, model = ‘markov1b_model.txt’)

The distribution of the initial state, \(p = \mathbb{P}(y_{i,1} = 1)\), is estimated in this example

DEFINITION:
State = {type = categorical,  categories = {1,2},  dependence = Markov
  P(State_1=1)= p
  P(State=1|State_p=1) = p11
  P(State=1|State_p=2) = p21
}
  • markov3a_project (data = ‘markov3a_data.txt’, model = ‘markov3a_model.txt’)

Transition probabilities change with time in this example. We then define time varying transition probabilities in the model:

[LONGITUDINAL]
input = {a1, b1, a2, b2}
EQUATION:
lp11 = a1 + b1*t/100
lp21 = a2 + b2*t/100
DEFINITION:
State = {type = categorical, categories = {1,2}, dependence = Markov
  logit(P(State=1|State_p=1)) = lp11
  logit(P(State=1|State_p=2)) = lp21
}
  • markov2_project (data = ‘markov2_data.txt’, model = ‘markov2_model.txt’)

Observations in markov2_data.txt take their values in {1, 2, 3}. Then, 6 transition probabilities need to be defined in the model.

 

Continuous-time Markov chain

The previous situation can be extended to the case where time intervals between observations are irregular by modeling the sequence of states as a continuous-time Markov process. The difference is that rather than transitioning to a new (possibly the same) state at each time step, the system remains in the current state for some random amount of time before transitioning. This process is now characterized by transition rates instead of transition probabilities:

\( \mathbb{P}(y_{i}(t+h) = k,|,y_{i}(t)=\ell , \psi_i) = h \rho_{\ell k}(t,\psi_i) + o(h),\qquad k \neq \ell .\)

The probability that no transition happens between \(t\) and \(t+h\) is

\( \mathbb{P}(y_{i}(s) = \ell, \forall s\in(t, t+h) | y_{i}(t)=\ell , \psi_i) = e^{h , \rho_{\ell \ell}(t,\psi_i)} .\)

Furthermore, for any individual i and time t, the transition rates \((\rho_{\ell,k}(t, \psi_i))\) satisfy for any \(1\leq \ell \leq K\),

\( \sum_{k=1}^K \rho_{\ell k}(t, \psi_i) = 0\)

Constructing a model therefore means defining parametric functions of time \((\rho_{\ell,k})\) that satisfy this condition.

  • markov1c_project (data = ‘markov1c_data.txt’, model = ‘markov1c_model.txt’)

Observation times are irregular in this example. Then, a continuous time Markov chain should be used in order to take into account the Markovian dependence of the data:

DEFINITION:
State = { type = categorical,  categories = {1,2}, dependence = Markov
  transitionRate(1,2) = q12
  transitionRate(2,1) = q21
}
  • markov3b_project (data = ‘markov3b_data.txt’, model = ‘markov3b_model.txt’)

Time varying transition rates are used in this example.

3.4.4.Joint models for non continuous outcomes


Objectives: learn how to implement a joint model for continuous and non continuous data.


Projects: warfarin_cat_project, PKcount_project, PKrtte_project


Joint model for continuous PK and categorical PD data

  • warfarin_cat_project (data = ‘warfarin_cat_data.txt’, model = ‘PKcategorical1_model.txt’)

In this example, the original PD data has been recorded as 1 (Low), 2 (Medium) and 3 (High).

More details about the data

International Normalized Ratio (INR) values are commonly used in clinical practice to target optimal warfarin therapy. Low INR values (<2) are associated with high blood clot risk and high ones (>3) with high risk of bleeding, so the targeted value of INR, corresponding to optimal therapy, is between 2 and 3.

Prothrombin complex activity is inversely proportional to the INR. We can therefore associate the three ordered categories for the INR to three ordered categories for PCA: Low PCA values if PCA is less than 33% (corresponding to INR>3), medium if PCA is between 33% and 50% (INR between 2 and 3) and high if PCA is more than 50% (INR<2).

The column dv contains both the PK and the new categorized PD measurements. Instead of modeling the original PD data, we can model the probabilities of each of these categories, which have direct clinical interpretations. The model is still a joint PKPD model since this probability distribution is expected to depend on exposure, i.e., the plasmatic concentration predicted by the PK model. We introduce an effect compartment to mimic the effect delay. Let \(y_{ij}^{(2)}\) be the PCA level for patient i at time \(t_{ij}^{(2)}\). We can then use a proportional odds model for modeling this categorical data:

$$\begin{array}{ccl}\text{logit} \left(\mathbb{P}(y_{ij}^{(2)} \leq 1 | \psi_i)\right) &= &\alpha_{i} + \beta_{i} Ce(t_{ij}^{(2)},\phi_i^{(1)}) \\ \text{logit} \left(\mathbb{P}(y_{ij}^{(2)} \leq 2 | \psi_i)\right) &=& \alpha_{i} + \gamma_{i} + \beta_{i}Ce(t_{ij}^{(2)},\phi_i^{(1)}) \\ \text{logit} \left(\mathbb{P}(y_{ij}^{(2)} \leq 3 | \psi_i)\right) &= & 1,\end{array}$$

where \(C_e(t,\phi_i^{(1)})\) is the predicted concentration of warfarin in the effect compartment at time t for patient i with PK parameters \(\phi_i^{(1)}\). This model defines a probability distribution for \(y_{ij}\) if \(\gamma_i\geq 0\).
If \(\beta_i>0\), the probability of low PCA at time \(t_{ij}^{(2)}\) (\(y_{ij}^{(2)}=1\)) increases along with the predicted concentration \(Ce(t_{ij}^{(2)},\phi_i^{(1)})\). The joint model is implemented in the model file PKcategorical1_model.txt

[LONGITUDINAL]
input = {Tlag, ka, V, Cl, ke0, alpha, beta, gamma}

EQUATION:
{Cc,Ce} = pkmodel(Tlag,ka,V,Cl,ke0)
lp1 = alpha + beta*Ce
lp2 = lp1+ gamma         ; gamma >= 0

DEFINITION:
Level = {type=categorical, categories={1,2,3}
    logit(P(Level<=1)) = lp1
    logit(P(Level<=2)) = lp2
}
OUTPUT:
output = {Cc, Level}

See Categorical data model for more details about categorical data models.

Joint model for continuous PK and count PD data

  • PKcount_project (data = ‘PKcount_data.txt’, model = ‘PKcount1_model.txt’)

The data file used for this project is PKcount_data.txt where the PK and the count PD measurements are simulated. We use a Poisson distribution for the count data, assuming that the Poisson parameter is function of the predicted concentration. For any individual i, we have

$$\lambda_i(t) = \lambda_{0,i} \left( 1 – \frac{Cc_i(t)}{Cc_i(t) + IC50_i} \right)$$

where \(Cc_i(t)\) is the predicted concentration for individual i at time t and

$$ \log\left(P(y_{ij}^{(2)} = k)\right) = -\lambda_i(t_{ij}) + k\,\log(\lambda_i(t_{ij})) – \log(k!)$$

The joint model is implemented in the model file PKcount1_model.txt

[LONGITUDINAL]
input = {ka, V, Cl, lambda0, IC50} 

EQUATION:
Cc = pkmodel(ka,V,Cl)
lambda=lambda0*(1 - Cc/(IC50+Cc)) 

DEFINITION:
Seizure = {type = count, 
           log(P(Seizure=k)) = -lambda + k*log(lambda) - factln(k)
}

OUTPUT:
output = {Cc,Seizure}

See Count data model for more details about count data models.

Joint model for continuous PK and time-to-event data

  • PKrtte_project (data = ‘PKrtte_data.txt’, model = ‘PKrtteWeibull1_model.txt’)

The data file used for this project is PKrtte_data.txt where the PK and the time-to-event data are simulated. We use a Weibull model for the events count data, assuming that the baseline is function of the predicted concentration. For any individual i, we define the hazard function as

$$h_i(t) = \gamma_{i}  Cc_i(t)  t^{\beta-1}$$

where \(Cc_i(t)\) is the predicted concentration for individual i at time t. The joint model is implemented in the model file PKrtteWeibull1_model.txt

[LONGITUDINAL]
input  = {ka, V, Cl, gamma, beta}  

EQUATION:
Cc = pkmodel(ka, V, Cl)
if t<0.1
  haz = 0
else
  haz = gamma*Cc*(t^(beta-1))
end

DEFINITION:
Hemorrhaging  = {type=event, hazard=haz}

OUTPUT:
output = {Cc, Hemorrhaging}

See Time-to-event data model for more details about time-to-event data models.

3.5.Extensions

3.5.1.Using a scale factor






By default the models from the library do not include a scale factor. Units of the estimated parameters will depend on the units of the data set. For instance:

  • Concentration in mg/L, amount in mg, time in hours => volumes are in L and clearances in L/h
  • Concentration in ng/mL, amount in mg, time in minutes => volume are in 10^3 L and clearances in 10^3 L/min

The files from the library can easily be modified to include a scale factor:

Step 1: load a model from the library.

Step 2: in the “Structural model” tab, click “Open in editor”.

Step 3: add a scaling of the concentration. If the dose is in mg and I want the volume in L, then the concentration Cc will be in mg/L. If my observations in the data set are in ng/mL (i.e μg/L), I need to multiply Cc by 1000 (green highlight). Do not forget to output the scaled concentration instead of the original one (pink highlight).

Step 4: save the file under a new name (to avoid overwriting the library model files).

Step 5: load the saved model file.

 

3.5.2.Using regression variables







Objectives: learn how to define and use regression variables (time varying covariates).


Projects: reg1_project, reg2_project


Introduction

A regression variable is a variable x which is a given function of time, which is not defined in the model but which is used in the model. x is only defined at some time points \(t_1, t_2, \ldots, t_m\) (possibly different from the observation time points), but x is a function of time that should be defined for any t (if is used in an ODE for instance, or if a prediction is computed on a fine grid). Then, Mlxtran defines the function x by interpolating the given values \((x_1, x_2, \ldots, x_m)\). In the current version of Mlxtran, interpolation is performed by using the last given value:

\( x(t) = x_j \quad~~\text{for}~~t_j \leq t < t_{j+1} \)

The way to introduce it in the Mlxtran longitudinal model is defined here.

Regressor definition in a data set

It is possible to have in a data set one or several columns with column-type REGRESSOR. Within a given subject-occasion, string “.” will be interpolated (last value carried forward interpolation is used) for observation and dose-lines. Lines with no observation and no dose but with regressor values are also taken into account by Monolix for regressor interpolation.

Several points have to be noticed:

  • The name of the regressor in the data set and the name of the regressor used in the longitudinal model do not need to be identical.
  • If there are several regressors, the mapping will be done by order of definition.
  • Regressors can only be used in the longitudinal model.

Continuous regression variables

  • reg1_project (data = reg1_data.txt , model=reg1_model.txt)

We consider a basic PD model in this example, where some concentration values are used as a regression variable. The data set is defined as follows

[LONGITUDINAL]
input = {Emax, EC50, Cc}
Cc    = {use=regressor}

EQUATION:
E = Emax*Cc/(EC50 + Cc)

OUTPUT:
output = E

As explained in the previous subsection, there is no name correspondance between the regressor in the data set and the regressor in the model file. Thus, in that case, the values of Cc with respect to time will be taken from the y1 column.
In addition, in that case, the predicted effect is therefore piece wise constant because

  • the regressor interpolation is performed by using the last given value, and then Cc is piece wise constant.
  • The effect model is direct with respect to the concentration.

Thus, it changes at the time points where concentration values are provided:

Categorical regression variables

  • reg2_project (data = reg2_data.txt , model=reg2_model.txt)

The variable \(z_{ij}\) takes its values in {1, 2} in this example and represents the state of individual i at time \(t_{ij}\). We then assume that the observed data \(y_{ij}\) has a Poisson distribution with parameter lambda1 if \(z_{ij}=1\) and parameter lambda2 if \(z_{ij}=2\). z is known in this example: it is then defined as a regression variable in the model:

[LONGITUDINAL]
input = {lambda1, lambda2, z}
z = {use=regressor}
                           
EQUATION:
if z==0
   lambda=lambda1
else
   lambda=lambda2
end

DEFINITION:
y = {type=count, 
     log(P(y=k)) = -lambda + k*log(lambda) - factln(k)
}

OUTPUT:
output = y

3.5.3.Delayed differential equations


Objectives: learn how to implement a model with ordinary differential equations (ODE) and delayed differential equations (DDE).


Projects: tgi_project, seir_project





Ordinary differential equations based model

  • tgi_project (data = tgi_data.txt , model = tgi_model.txt)

Here, we consider the tumor growth inhibition (TGI) model proposed by Ribba et al. (Ribba, B., Kaloshi, G., Peyre, M., Ricard, D., Calvez, V., Tod, M., . & Ducray, F., *A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy*. Clinical Cancer Research, 18(18), 5071-5080, 2012.). This model is defined by a set of ordinary differential equations

 \begin{aligned}\frac{dC}{dt} &= - k_{de} C(t) \\\frac{dP_T}{dt} &= \lambda P_T(t)(1- P^{\star}(t)/K) + k_{QPP}Q_P(t) -k_{PQ} P_T(t) -\gamma \, k_{de} P_T(t)C(t) \\ \frac{dQ_T}{dt} &= k_{PK} P_T(t) -\gamma k_{de} Q_T(t)C(t) \\\frac{dQ_P}{dt} &= \gamma k_{de} Q_T(t)C(t) - k_{QPP} Q_P(t) -\delta_{QP} Q_P(t)\end{aligned}

where P^\star(t) = P_T(t) + Q_T(t) + Q_P(t) is the total tumor size. This set of ODEs is valid for t greater than 0, while

 \begin{aligned} C(0) &= 0 \\ P_T(0) &= P_{T0} \\ Q_T(0) &= Q_0 \\ Q_P(0) &= 0 \end{aligned} 

This model (derivatives and initial conditions) can easily be implemented with Mlxtran:

DESCRIPTION: Tumor Growth Inhibition (TGI) model proposed by Ribba et al
A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy. 
Clinical Cancer Research, 18(18), 5071-5080, 2012.

Variables 
- PT: proliferative equiescent tissue
- QT: nonproliferative equiescent tissue
- QP: damaged quiescent cells 
- C:  concentration of a virtual drug encompassing the 3 chemotherapeutic components of the PCV regimen

Parameters
- K      : maximal tumor size (should be fixed a priori)
- KDE    : the rate constant for the decay of the PCV concentration in plasma
- kPQ    : the rate constant for transition from proliferation to quiescence
- kQpP   : the rate constant for transfer from damaged quiescent tissue to proliferative tissue 
- lambdaP: the rate constant of growth for the proliferative tissue
- gamma  : the rate of damages in proliferative and quiescent tissue
- deltaQP: the rate constant for elimination of the damaged quiescent tissue
- PT0    : initial proliferative equiescent tissue
- QT0    : initial nonproliferative equiescent tissue

[LONGITUDINAL]
input = {K, KDE, kPQ, kQpP, lambdaP, gamma, deltaQP, PT0, QT0}

PK:
depot(target=C)

EQUATION:
; Initial conditions
t0    = 0
C_0   = 0
PT_0  = PT0
QT_0  = QT0
QP_0  = 0

; Dynamical model
PSTAR   = PT + QT + QP
ddt_C   = -KDE*C
ddt_PT  = lambdaP*PT*(1-PSTAR/K) + kQpP*QP - kPQ*PT - gamma*KDE*PT*C
ddt_QT  = kPQ*PT - gamma*KDE*QT*C
ddt_QP  = gamma*KDE*QT*C - kQpP*QP - deltaQP*QP

OUTPUT:
output = PSTAR

Remark: t0, PT_0 and QT_0 are reserved keywords that define the initial conditions.

Then, the graphic of individual fits clearly shows that the tumor size is constant until t=0 and starts changing according to the model at t=0.

Don’t forget the initial conditions!

  • tgiNoT0_project (data = tgi_data.txt , model = tgiNoT0_model.txt)

The initial time t0 is not specified in this example. Since t0 is missing, Monolix uses the first time value encountered for each individual. If, for instance, the tumor size has not been computed before 5 for the individual fits, then t0=5 will be used for defining the initial conditions for this individual, which introduces a shift in the plot:

As defined here, the following rule applies

  • When no starting time t0 is defined in the Mlxtran model for Monolix then by default t0 is selected to be equal to the first dose or the first observation, whatever comes first.
  • If t0 is defined, a differential equation needs to be defined.

Conclusion: don’t forget to properly specify the initial conditions of a system of ODEs!

Delayed differential equations based model

A system of delay differential equations (DDEs) can be implemented in a block EQUATION of the section [LONGITUDINAL] of a script Mlxtran. Mlxtran provides the command delay(x,T) where x is a one-dimensional component and T is the explicit delay. Therefore, DDEs with a nonconstant past of the form

$$ \begin{array}{ccl} \frac{dx}{dt} &=& f(x(t),x(t-T_1), x(t-T_2), …), ~~\text{for}~~t \geq 0\ x(t) &=& x_0(t) ~~~~\text{for}~~\text{min}(T_k) \leq t \leq 0 \end{array} $$

can be solved. The syntax and rules are explained here.

  • seir_project (data = seir_data.txt , model = seir_model.txt)

The model is a system of 4 DDEs and defined with the following mode:

DESCRIPTION: SEIR model, using delayed differential equations.
"An Epidemic Model with Recruitment-Death Demographics and Discrete Delays", Genik & van den Driessche (1999)

Decomposition of the total  population into four epidemiological classes 
S (succeptibles), E (exposed), I (infectious), and  R (recovered)

The parameters corresponds to 
- birthRate: the birth rate,
- deathRate: the natural death rate,
- infectionRate: the contact rate of infective individuals,
- recoveryRate: the rate of recovery,
- excessDeathRate: the excess death rate for infective individuals

There is a time delay in the model:
- tauImmunity: a temporary immunity delay

[LONGITUDINAL]
input = {birthRate, deathRate, infectionRate, recoveryRate, excessDeathRate, tauImmunity, tauLatency}

EQUATION:
; Initial conditions
t0    = 0
S_0 = 15
E_0 = 0
I_0  = 2
R_0 = 3

; Dynamical model
N = S + E + I + R

ddt_S = birthRate - deathRate*S - infectionRate*S*I/N + recoveryRate*delay(I,tauImmunity)*exp(-deathRate*tauImmunity)
ddt_E = infectionRate*S*I/N - deathRate*E - infectionRate*delay(S,tauLatency)*delay(I,tauLatency)*exp(-deathRate*tauLatency)/(delay(I,tauLatency)+delay(S,tauLatency)+delay(E,tauLatency)+delay(R,tauLatency))
ddt_I = -(recoveryRate+excessDeathRate+deathRate)*I + infectionRate*delay(S,tauLatency)*delay(I,tauLatency)*exp(-deathRate*tauLatency)/(delay(I,tauLatency)+delay(S,tauLatency)+delay(E,tauLatency)+delay(R,tauLatency))
ddt_R = recoveryRate*I - deathRate*R - recoveryRate*delay(I,tauImmunity)*exp(-deathRate*tauImmunity)

OUTPUT:
output = {S, E, I, R}

Introducing these delays allows to obtain nice fits for the 4 outcomes, including (R_{ij}) (corresponding to the output y4):

Case studies

  • 8.case_studies/arthritis_project

3.5.4.Outputs and Tables


Objectives: learn how to define outputs and create tables from the outputs of the model.


Projects: tgi_project, tgiWithTable_project


About the OUTPUT block

  • tgi_project (data = ‘tgi_data.txt’ , model=’tgi_model.txt’)

We use the Tumor Growth Inhibition (TGI) model proposed by Ribba et al. in this example (Ribba, B., Kaloshi, G., Peyre, M., Ricard, D., Calvez, V., Tod, M., . & Ducray, F., A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy. Clinical Cancer Research, 18(18), 5071-5080, 2012.)

DESCRIPTION: Tumor Growth Inhibition (TGI) model proposed by Ribba et al
A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy. 
Clinical Cancer Research, 18(18), 5071-5080, 2012.

Variables 
- PT: proliferative equiescent tissue
- QT: nonproliferative equiescent tissue
- QP: damaged quiescent cells 
- C:  concentration of a virtual drug encompassing the 3 chemotherapeutic components of the PCV regimen

Parameters
- K      : maximal tumor size (should be fixed a priori)
- KDE    : the rate constant for the decay of the PCV concentration in plasma
- kPQ    : the rate constant for transition from proliferation to quiescence
- kQpP   : the rate constant for transfer from damaged quiescent tissue to proliferative tissue 
- lambdaP: the rate constant of growth for the proliferative tissue
- gamma  : the rate of damages in proliferative and quiescent tissue
- deltaQP: the rate constant for elimination of the damaged quiescent tissue
- PT0    : initial proliferative equiescent tissue
- QT0    : initial nonproliferative equiescent tissue

[LONGITUDINAL]
input = {K, KDE, kPQ, kQpP, lambdaP, gamma, deltaQP, PT0, QT0}

PK:
depot(target=C)

EQUATION:
; Initial conditions
t0    = 0
C_0   = 0
PT_0  = PT0
QT_0  = QT0
QP_0  = 0

; Dynamical model
PSTAR   = PT + QT + QP
ddt_C   = -KDE*C
ddt_PT  = lambdaP*PT*(1-PSTAR/K) + kQpP*QP - kPQ*PT - gamma*KDE*PT*C
ddt_QT  = kPQ*PT - gamma*KDE*QT*C
ddt_QP  = gamma*KDE*QT*C - kQpP*QP - deltaQP*QP

OUTPUT:
output = PSTAR

PSTAR is the tumor size predicted by the model. It is therefore used as a prediction for the observations in the project.
At the end of the scenario or of SAEM, individual predictions of the tumor size PSTAR are computed using the individual parameters available. Thus, individual predictions of the tumor size PSTAR are computed using both the conditional modes (indPred_mode), the conditional mean (indPred_mean), and the conditional means estimated during the last iterations of SAEM (indPred_SAEM) and saved in the table predictions.txt.
Notice that the population prediction is also proposed.

Remark: the same model file tgi_model.txt can be used with different tools, including Mlxplore or Simulx (see this Shiny application for instance).

 

 

 

Add additional outputs in tables







  • tgiWithTable_project (data = ‘tgi_data.txt’ , model=’tgiWithTable_model.txt’)

We can save in the tables additional variables defined in the model, such as PT, Q and QP for instance, by adding a block OUTPUT: in the model file:

OUTPUT:
output = PSTAR
table  = {PT, QT, QP}

An additional file tables.txt now includes the predicted values of these variables for each individual (columns PT_mean, QT_mean, QP_mean, PT_mode, QT_mode, QP_mode, PT_popPred, QT_popPred, QP_popPred, PT_popPred_medianCOV, QT_popPred_medianCOV, QP_popPred_medianCOV, PT_SAEM, QT_SAEM, and QP_SAEM.

Notice that only continuous variable are possible for variable in table.

Good to know: it is not allowed to do calculations directly in the output or table statement. The following example is not possible:

; not allowed:
OUTPUT:
output = {Cser+Ccsf}

It has to be replaced by:

EQUATION:
Ctot = Cser+Ccsf
OUTPUT:
output = {Ctot}

 

3.5.5.How to compute AUC using in Monolix and Mlxtran

Computing additional outputs such as AUC and Cmax in the structural model is possible but requires a particular syntax, because it is not possible to redefine a variable in Mlxtran. For example, it would not be possible to directly update the variable Cmax = Cc when Cc>Cmax. But it is possible to compute Cmax via an ODE, where Cmax increases like Cc at each time where Cc>Cmax. This page gives detailed examples for the AUC on full time interval or specific interval, Cmax, nadir, and other derived variables such as duration above a threshold.

Often the Area under the PK curve (AUC) is needed as an important PK metric to link with the pharmacodynamic effect. We show here how to:

  • compute the AUC within the mlxtran model file
  • output the AUC calculations for later analysis.

Calculation of the AUC can be done in the EQUATION section of the Mlxtran model.  If a dataset contains the AUC observations, then the calculation in the EQUATION section can be used as an output in the output={} definition (matched to observations of the data set). Or it can be saved for post-treatment using table={}, as described here.

AUClast from t=0 to t=tlast

The following code is the basic implementation of the AUC for a 1-compartmental model with the absorption rate ka. It integrates the concentration profile from the start to the end of the observation period.

[LONGITUDINAL]
input = {ka, V, Cl}

PK:
Cc=pkmodel(ka,V,Cl)

EQUATION:
odeType=stiff
ddt_AUC = Cc

OUTPUT:
output = {Cc}
table  = {AUC}

 

AUC in a time interval

The following code computes the AUC between two time points t1 and t2. The idea is to compute the AUC_0_t1 and AUC_0_t2 using if/else statements and then do the difference between the two.

[LONGITUDINAL]
input = {ka, V, Cl}
PK:
depot(ka, target=Ac)

EQUATION:
odeType = stiff
useAnalyticalSolution=no

Cc = pkmodel(ka,V,Cl)

t1=24
t2=48
AUCt1_0 = 0
if(t < t1)
   dAUCt1 = Cc
else
   dAUCt1 = 0
end
ddt_AUCt1 = dAUCt1

AUCt2_0 = 0
if(t < t2)
   dAUCt2 = Cc
else
   dAUCt2 = 0
end
ddt_AUCt2 = dAUCt2

AUC_t1_t2 = AUCt2 - AUCt1

OUTPUT:
output = {Cc}
table  = {AUC_t1_t2}

Note that the t==tDose would not work because the integrator does not necessarily evaluate the time exactly at the times of doses. Thus the test t==tDose might not be tested at all during the computation.
Also note that, although partial AUC can be calculated by comparing time with the both boundaries in the if statement (e.g. if (t>20 and t<40)), this might not be the best practice when concentration is calculated using an analytical solution, because ODE solver used for calculating AUC could be deceived by AUC remaining 0 for a period of time and might increase the integration time step to a value larger than the difference between time boundaries.

 

Dose-interval AUC (AUCtau)

The following code compute the AUCtau for each dose interval. At each dose the AUC is reset to zero and the concentration is integrated until the next administration.

[LONGITUDINAL]
input = {ka, V, Cl}

PK:
Cc = pkmodel(ka,V,Cl)

; Reset the ODE variable AUCtau to zero each time there is a dose
empty(target=AUCtau)
EQUATION: 
odeType=stiff
ddt_AUCtau = Cc 

OUTPUT: 
output = {Cc} 
table = {AUCtau}
Since AUCtau is equal to 0 at each dosing time as it has just been reset, the AUC in the last interdose interval should actually be read just before that time.
Here we provide a Simulx project with an example of a PK model where AUCtau is calculated as additional lines in the model and defined as an output of the simulations.

Computing the Cmax or nadir in the structural model

Cmax

Cmax can be calculated directly in the structural model by integrating the increase of the concentration. Alternatively, the Cmax can also be calculated in Simulx using the Outcomes&Endpoints tab.

Cmax for models with first-order absorption

The following example shows how to do it in case of a one-compartment model with first-order absorption and linear elimination. Absorption processes defined via the depot macro must be replaced by explciit ODEs, while the depot macro is used to add the dose as a bolus into the depot compartment.

[LONGITUDINAL]
input = {Cl, ka, V}

PK:
depot(target=Ad)

EQUATION:

; initial conditions
t_0  = 0
Ad_0 = 0
Ac_0 = 0

ddt_Ad = -ka*Ad
ddt_Ac = ka*Ad - k*Ac
Cc = Ac/V

; Calculation of Cmax
slope  = ka*Ad -k*Ac
Cmax_0 = 0
if slope > 0 && Cc > Cmax
   x = slope/V
else
   x = 0
end

ddt_Cmax = x

Cmax for model with bolus administration or infusion

If the dose is administered as bolus, it is necessary to add in the model a very short infusion. This prevents from the instantaneous increase of the concentration. The following example shows this situations in case of a three-compartments model with linear elimination. The duration of the “short infusion” can be adapted with respect to the time scale by modifying dT=0.1.

If the doses are administered via iv infusion, then dT=0.1 can be replaced by dT = inftDose, which reads the infusion duration from the data.

[LONGITUDINAL]
input = {Cl, V1, Q2, V2, Q3, V3}

EQUATION:

; Parameter transformations 
V   = V1 
k   = Cl/V1 
k12 = Q2/V1 
k21 = Q2/V2 
k13 = Q3/V1 
k31 = Q3/V3

; initial conditions
t_0  = 0
Ac_0 = 0
A2_0 = 0
A3_0 = 0

;short pseudo-infusion duration
dT = 0.1 ; use dT=inftDose if the administration is infusion: inftDose is the infusion duration from the last dose read from the data

; infusion input to Ac
if t < tDose+dT
   input = amtDose/dT ;amtDose is the last dose amount read from the data
else
   input = 0
end

dAc = input -k*Ac - k12*Ac - k13*Ac + k21*A2 + k31*A3
ddt_Ac = dAc
ddt_A2 = k12*Ac - k21*A2
ddt_A3 = k13*Ac - k31*A3
Cc = Ac/V

; Calculation of AUC
AUC_0 = 0
ddt_AUC = Cc

; Calculation of Cmax
Cmax_0 = 0
if dAc > 0 && Cc > Cmax
   x = dAc/V
else
   x = 0
end

ddt_Cmax = x

OUTPUT:
output = {Ac, Cc}
table  = {Cmax, AUC}

Cmax for models with transit compartments

The transit compartments defined via the depot() macro must be replaced by explicit ODEs. The amount in the last (nth) transit compartment is described via an analytical solution using the keyword amtDose to get the dose amount and (t-tDose) to get the time elapsed since the last dose. The calculation of Cmax then follows the same logic as for a first-order absorption.

[LONGITUDINAL]
input={F, Ktr, Mtt, ka, V, Cl}

EQUATION:
k = Cl/V
odeType=stiff

; transit model with explicit ODEs
n = max(0, Mtt*Ktr - 1)
An = F * amtDose * exp( n * log(Ktr*(t-tDose)) - factln(n) - Ktr * (t-tDose))

t_0 = 0
Aa_0 = 0
Ac_0 = 0

ddt_Aa = Ktr*An - ka*Aa
ddt_Ac = ka*Aa - k*Ac
Cc = Ac/V
; Calculation of Cmax
slope  = ka*Aa - k*Ac
Cmax_0 = 0
if slope > 0 && Cc > Cmax
   x = slope/V
else
   x = 0
end
ddt_Cmax = x

OUTPUT:
output = {Cc}
table = {Cmax}

Nadir

This example shows how to compute tumor size (variable TS) at time of nadir:

[LONGITUDINAL]
input = {ka, Vl, Cl, TS0, kge, kkill, lambda}
EQUATION:

odeType=stiff
Cc = pkmodel(ka, V, Cl)

; ODE system for tumor growth dynamics
t_0 = 0
TS_0 = TS0
TSDynamics = (kge*TS)-kkill*TS*Cc*exp(-lambda*t)
ddt_TS = TSDynamics

; ===== computing TS at nadir
if TSDynamics < 0 && TS < TS_atNadir
  x = TSDynamics
else
  x = 0
end
TS_atNadir_0 = TS0
ddt_TS_atNadir = x

OUTPUT:
output = {TS}
table = {TS_atNadir}

Here is an example of simulation for TS and TS_atNadir with a multiple dose schedule:

 

 

Time above a threshold

Here we compute the time that a variable C spends above some threshold, which could be a toxicity threshold for example. This piece of code comes in the structural model, after the dynamics of the variable C has already been defined.

TimeAboveThreshold_0 = 0
if C > Cthreshold
  xTime = 1
else
  xTime = 0
end
ddt_TimeAboveThreshold = xTime


Time since disease progression

In this full structural model example, tumor size TS is described with an exponential growth, and a constant treatment effect since time=0 with a log-kill killing hypothesis and a Claret exponential resistance term. Several additional variables are computed:

  • TS_atNadir: the tumor size at the time of nadir,
  • PC_from_Nadir: the percent change of TS between the time of nadir and the current time,
  • DP: predicted disease progression status (1 if TS has increased more than >20% and >5-mm from last nadir, 0 otherwise),
  • TDP: time since disease progression (duration since last time when DP is switched from 0 to 1)
[LONGITUDINAL]
input = {TS0, kge, kkill, lambda}

EQUATION:
odeType=stiff

;initial conditions:
;t_0 = 0 ;this should be uncommented if the initial time is 0 for all subjects 
TS_0 = TS0

K = kkill*exp(-lambda*t)

;Saturation for TS at 1e12 to avoid infinite values
if TS>1e12
  TSDynamics = 0
  else
  if t<0 ; before treatment
    TSDynamics = (kge*TS)
  else ; after treatment
    TSDynamics = (kge*TS)-K*TS
  end
end

ddt_TS = TSDynamics

; Computing time to nadir (TS decreases after treatment start at time=0, then increases again because of resistance)
if TSDynamics<0
  x1=1
else
  x1=0
end
TimeToNadir_0=t0
ddt_TimeToNadir = x1 ; x1 is used as intermediate variable because it is not possible to define an ODE inside an if/else statement

; Computing TS at time of nadir
if TSDynamics < 0 & TS < TS_atNadir
  x2 = TSDynamics
else
  x2 = 0
end
TS_atNadir_0 = TS0
ddt_TS_atNadir = x2 ;

; Computing DP: predicted disease progression status at the previous visit (1 if TS has increased more than >20% and >5-mm from last nadir, 0 otherwise)
PC_from_Nadir = (TS/TS_atNadir-1)*100
if t>TimeToNadir & PC_from_Nadir > 20 & (TS-TS_atNadir) > 5
  DP = 1
else
  DP = 0
end

; Computing TDP = time since disease progression: duration since last time DP was switched from 0 to 1
if DP==1
  x3 = 1
else
  x3 = 0
end
TDP_0 = 0
ddt_TDP = x3

OUTPUT: 
output = {TS}
table = {TS_atNadir, PC_from_Nadir, DP, TDP}

Simulx can be conveniently used to simulate each intermediate variable with typical parameters (the example model and Simulx project can be downloaded here):

4.Statistical Model

The statistical model tab in Monolix enables to define the statistical model and run estimation tasks.

The statistical model includes

  • the observation model, which combines the error model and the distribution of the observations.
  • the individual model, combining
    • distributions for the individual parameters
    • which parameters have inter-individual variability (random effects)
    • correlations between the individual parameters
    • covariate effects on the individual parameters

 

4.1.Observation (error) model


Objectives: learn how to use the predefined residual error models.


Projects: warfarinPK_project, bandModel_project, autocorrelation_project, errorGroup_project







Introduction

For continuous data, we are going to consider scalar outcomes (\(y_{ij} \in \mathbb{R}\)) and assume the following general model:

$$y_{ij}=f(t_{ij},\psi_i)+ g(t_{ij},\psi_i,\xi)\varepsilon_{ij}$$

for i from 1 to N, and j from 1 to \(\text{n}_{i}\), where \(\psi_i\) is the parameter vector of the structural model f for individual i. The residual error model is defined by the function g which depends on some additional vector of parameters \(\xi\). The residual errors \((\varepsilon_{ij})\) are standardized Gaussian random variables (mean 0 and standard deviation 1). In this case, it is clear that \(f(t_{ij}, \psi_i)\) and \(g(t_{ij}, \psi_i, \xi)\) are the conditional mean and standard deviation of \(y_{ij}\), i.e.,

$$\mathbb{E}(y_{ij} | \psi_i) = f(t_{ij}, \psi_i)~~\textrm{and}~~\textrm{sd}(y_{ij} | \psi_i)= g(t_{ij}, \psi_i, \xi)$$

Available error models

In Monolix, we only consider the function g to be a function of the structural model f, i.e. \(g(t_{ij}, \psi_i, \xi)= g(f(t_{ij}, \psi_i), \xi)\)  leading to an expression of the observation model of the form

$$y_{ij}=f(t_{ij},\psi_i)+ g(f(t_{ij}, \psi_i), \xi)\varepsilon_{ij}$$

The following error models are available:

  • constant : \(y = f + a \varepsilon\). The function g is constant, and the additional parameter is \(\xi=a\)
  • proportional : \(y = f + bf^c \varepsilon\). The function g is proportional to the structural model f, and the additional parameters are \(\xi = (b,c)\). By default, the parameter c is fixed at 1 and  the additional parameter is \xi = b.
  • combined1 : \(y = f + (a+ bf^c) \varepsilon\). The function g is a linear combination of a constant term and a term proportional to the structural model f, and the additional parameters are \(\xi = (a, b)\) (by default, the parameter c is fixed at 1).
  • combined2 : \(y = f + \sqrt{a^2+ b^2(f^c)^2} \varepsilon\). The function g is a combination of a constant term and a term proportional to the structural model f (g = bf^c), and the additional parameters are \(\xi = (a, b)\) (by default, the parameter c is fixed at 1).

Notice that the parameter c is fixed to 1 by default. However, it can be unfixed and estimated.
The assumption that the distribution of any observation \(y_{ij}\) is symmetrical around its predicted value is a very strong one. If this assumption does not hold, we may want to transform the data to make it more symmetric around its (transformed) predicted value. In other cases, constraints on the values that observations can take may also lead us to transform the data.

Available transformations

The model can be extended to include a transformation of the data:

$$u(y_{ij})=u(f(t_{ij},\psi_i)) + g(u(f(t_{ij},\psi_i)) ,\xi) $$

As we can see, both the data \(y_{ij}\) and the structural model are transformed by the function u so that \(f(t_{ij}, \psi_i)\) remains the prediction of \(y_{ij}\). Classical distributions are proposed as transformation:

  • normal: u(y) = y. This is equivalent to no transformation.
  • lognormalu(y) = log(y). Thus, for a combined error model for example, the corresponding observation model writes \(\log(y) = \log(f) + (a + b\log(f)) \varepsilon\). It assumes that all observations are strictly positive. Otherwise, an error message is thrown. In case of censored data with a limit, the limit has to be strictly positive too.
  • logitnormalu(y) = log(y/(1-y)). Thus, for a combined error model for example, the corresponding observation model writes \(\log(y/(1-y)) = \log(f/(1-f)) + (a + b\log(f/(1-f)))\varepsilon\). It assumes that all observations are strictly between 0 and 1. It is also possible to modify these bounds and not “impose” them to be 0 and 1, i.e. to define the logit function between a minimum and a maximum: the function u becomes u(y) = log((y-y_min)/(y_max-y)). Again, in case of censored data with a limit, the limits too must belong strictly to the defined interval.

Any interrogation on what is the formula behind your observation model? There is a button FORMULA on the interface as on the figure below where the observation model is described linking the observation (named CONC in that case) and the prediction (named Cc in that case). Note that \(\epsilon\) is noted e here.

Remarks: In previous Monolix version, only the error was available. Thus, what happens to the errors that are not proposed anymore? Is it possible to have “exponential”, “logit”, “band(0,10)”, and “band(0,100)”? Yes, in this version, we choose to split the observation model between its error model and its distribution. The purpose is to have a more unified vision of models and increase the number of possibilities. Thus, here is how to configure new projects with the previous error model definition.

  • “exponential” is an observation model with a constant error model and a lognormal distribution.
  • “logit” is an observation model with a constant error model and a logitnormal distribution.
  • “band(0,10)” is an observation model with a constant error model and a logitnormal distribution with min and max at 0 and 10 respectively.
  • “band(0,100)” is an observation model with a constant error model and a logitnormal distribution with min and max at 0 and 100 respectively.

 Defining the residual error model from the Monolix GUI

A menu in the frame Statistical model|Tasks of the main GUI allows one to select both the error model and the distribution as on the following figure (in green and blue respectively)

A summary of the statistical model which includes the residual error model can be displayed by clicking on the button formula.

Some basic residual error models

  • warfarinPKlibrary_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

The residual error model used with this project for fitting the PK of warfarin is a combined error model, i.e. \(y_{ij} = f(t_{ij}, \psi_i))+ (a+bf(t_{ij}, \psi_i)))\varepsilon_{ij}\)Several diagnosis plots can then be used for evaluating the error model. The observation versus prediction figure below seems ok.

Remarks:

  • Figures showing the shape of the prediction interval for each observation model available in Monolix are displayed here.
  • When the residual error model is defined in the GUI, a bloc DEFINITION: is then automatically added to the project file in the section [LONGITUDINAL] of <MODEL> when the project is saved:
DEFINITION:
y1 = {distribution=normal, prediction=Cc, errorModel=combined1(a,b)}

Residual error models for bounded data

  • bandModel_project (data = ‘bandModel_data.txt’, model = ‘lib:immed_Emax_null.txt’)

In this example, data are known to take their values between 0 and 100. We can use a constant error model and a logitnormal for the transformation with bounds (0,100) if we want to take this constraint into account.

In the Observation versus prediction plot, one can see that the error is smaller when the observations are close to 0 and 100 which is normal. To see the relevance of the predictions, one can look at the 90% prediction interval. Using a logitnormal distribution, we have a very different shape of this prediction interval to take that specificity into account.

VPCs obtained with this error model do not show any mispecification

This residual error model is implemented in Mlxtran as follows:

DEFINITION:
effect = {distribution=logitnormal, min=0, max=100, prediction=E, errorModel=constant(a)}

Autocorrelated residuals

For any subject i, the residual errors \((\varepsilon_{ij},1 \leq j \leq n_i)\) are usually assumed to be independent random variables. The extension to autocorrelated errors is possible by assuming, that \((\varepsilon_{ij})\) is a stationary autoregressive process of order 1, AR(1), which autocorrelation decreases exponentially:

$$ \textrm{corr}(\varepsilon_{ij},\varepsilon_{i,{j+1}}) = r_i^{(t_{i,j+1}-t_{ij})}$$

where \(0 \leq r_i \leq 1\) for each individual i. If \(t_{ij}=j\) for any (i,j), then \(t_{i,j+1}-t_{i,j}=1\) and the autocorrelation function \(\gamma_i\)for individual i is given by

$$\gamma_i(\tau) = \textrm{corr}(\varepsilon_{ij}, \varepsilon_{i,j+\tau}) = r_i^{\tau}$$

The residual errors are uncorrelated when \(r_i=0\).

  • autocorrelation_project (data = ‘autocorrelation_data.txt’, model = ‘lib:infusion_1cpt_Vk.txt’)

Autocorrelation is estimated since the checkbox r is ticked in this project:Estimated population parameters now include the autocorrelation r:
Important remarks:

  • Monolix accepts both regular and irregular time grids.
  • For a proper estimationg of the autocorrelation structure of the residual errors, rich data is required (i.e. a large number of time points per individual) .
  • To add autocorrelation, the user should either use the connectors, or write it directly in the Mlxtran
    • add “autoCorrCoef=r” in definition “DV = {distribution=normal, prediction=Cc, errorModel=proportional(b), autoCorrCoef=r}”  for example
    • add “r” as an input parameter.

Using different error models per group/study






  • errorGroup_project (data = ‘errorGroup_data.txt’, model = ‘errorGroup_model.txt’)

Data comes from 3 different studies in this example. We want to have the same structural model but use different error models for the 3 studies. A solution consists in defining the column STUDY with the reserved keyword OBSERVATION ID. It will then be possible to define one error model per outcome:
Here, we use the same PK model for the 3 studies:

[LONGITUDINAL]
input = {V, k}

PK:
Cc1 = pkmodel(V, k)
Cc2 = Cc1
Cc3 = Cc1

OUTPUT:
output = {Cc1, Cc2, Cc3}

Since 3 outputs are defined in the structural model, one can now define 3 error models in the GUI:
Different residual error parameters are estimated for the 3 studies. One can remark than, even if 2 proportional error models are used for the 2 first studies, different parameters b1 and b2 are estimated:

4.2.Individual model

4.2.1.Model for the individual parameters: introduction

What is the individual model and where is it defined in Monolix?

The population approach considers that parameters of the structural model can have a different value for each individual, and the way these values are distributed over individuals and impacted by covariate values is defined in the individual model. The individual model is defined in the lower part of the statistical model tab. This model includes

 

Theory for the individual model

A model for observations depends on a vector of individual parameters \(\psi_i\). As we want to work with a population approach, we now suppose that \(\psi_i\) comes from some probability distribution \(p_{{\psi_i}}\).

In this section, we are interested in the implementation of individual parameter distributions \((p_{{\psi_i}}, 1\leq i \leq N)\). Generally speaking, we assume that individuals are independent. This means that in the following analysis, it is sufficient to take a closer look at the distribution \(p_{{\psi_i}}\) of a unique individual i. The distribution \(p_{{\psi_i}}\) plays a fundamental role since it describes the inter-individual variability of the individual parameter \(\psi_i\). In Monolix, we consider that some transformation of the individual parameters is normally distributed and is a linear function of the covariates:

\(h(\psi_i) = h(\psi_{\rm pop})+ \beta \cdot ({c}_i – {c}_{\rm pop}) + \eta_i \,, \quad \eta_i \sim {\cal N}(0,\Omega).\)

This model gives a clear and easily interpreted decomposition of the variability of \(h(\psi_i)\) around \(h(\psi_{\rm pop})\), i.e., of \(\psi_i\) around \(\psi_{\rm pop}\):

The component \(\beta \cdot ({c}_i – {c}_{\rm pop})\) describes part of this variability by way of covariates \({c}_i\) that fluctuate around a typical value \({c}_{\rm pop}\).
The random component \(\eta_i\) describes the remaining variability, i.e., variability between subjects that have the same covariate values. By definition, a mixed effects model combines these two components: fixed and random effects. In linear covariate models, these two effects combine. In the present context, the vector of population parameters to estimate is \(\theta = (\psi_{\rm pop},\beta,\Omega)\). Several extensions of this basic model are possible:

We can suppose for instance that the individual parameters of a given individual can fluctuate over time. Assuming that the parameter values remain constant over some periods of time called occasions, the model needs to be able to describe the inter-occasion variability (IOV) of the individual parameters.
If we assume that the population consists of several homogeneous subpopulations, a straightforward extension of mixed effects models is a finite mixture of mixed effects models, assuming for instance that the distribution \(p_{{\psi_i}}\) is a mixture of distributions.

 

 

4.2.2.Probability distribution of the individual parameters







Objectives: learn how to define the probability distribution and the correlation structure of the individual parameters.


Projects: warfarin_distribution1_project, warfarin_distribution2_project, warfarin_distribution3_project, warfarin_distribution4_project


Introduction

One way to extend the use of Gaussian distributions is to consider that some transformation of the parameters in which we are interested is Gaussian, i.e., assume the existence of a monotonic function \(h\) such that \(h(\psi)\) is normally distributed. Then, there exists some \(\omega\) such that, for each individual i:

\(h(\psi_i) \sim {\cal N}(h(\bar{\psi}_i), \omega^2)\)

where \(\bar{\psi}_i\) is the predicted value of \(\psi_i\). In this section, we consider models for the individual parameters without any covariate. Then, the predicted value of \(\psi_i\) is the \(\bar{\psi}_i = \psi_{\rm pop}\) and

\(h(\psi_i) \sim {\cal N}(h(\psi_{pop}), \omega^2)\)

The transformation \(h\) defines the distribution of \(\psi_i\). Some predefined distributions/transformations are available in Monolix:

  • Normal distribution in ]-inf,+inf[:

In that case, \(h(\psi_i) = \psi_i\).
Note: the two mathematical representations for normal distributions are equivalent:

\( \psi_i \sim {\cal N}(\bar{\psi}_{i}, \omega^2) ~~\Leftrightarrow~~ \psi_i = \bar{\psi}_i + \eta_i, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2).\)

  • Log-normal distribution in ]0,+inf[:

In that case, \(h(\psi_i) = log(\psi_i)\). A log-normally random variable takes positive values only. A log-normal distribution looks like a normal distribution for a small variance \(\omega^2\). On the other hand, the asymmetry of the distribution increases when \(\omega^2\) increases.
Note: the two mathematical representations for log-normal distributions are equivalent:

\(\log(\psi_i) \sim {\cal N}(\log(\bar{\psi}_{i}), \omega^2) ~~\Leftrightarrow~~ \log(\psi_i)=\log(\bar{\psi}_{i})+\eta_i~~\Leftrightarrow~~ \psi_i = \bar{\psi}_i e^{\eta_i}, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2).\)

\(\bar{\psi}_i\) represents the typical value (fixed effect) and \(\omega\) the standard deviation of the random effects, which is interpreted as the inter-individual variability. Note that \(\bar{\psi}_i\) is the median of the distribution (neither the mean, nor the mode).

  • Logit-normal distribution in ]0,1[:

In that case, \(h(\psi_i) = log\left(\frac{\psi_i}{1-\psi_i}\right)\). A random variable \(\psi_i\) with a logit-normal distribution takes its values in ]0,1[. The logit of \(\psi_i\) is normally distributed, i.e.,

\(\text{logit}(\psi_i) = \log \left(\frac{\psi_i}{1-\psi_i}\right) \ \sim \ \ {\cal N}( \text{logit}(\bar{\psi}_i), \omega^2) ~~\Leftrightarrow~~ \text{logit}(\psi_i) = \text{logit}(\bar{\psi}_i) + \eta_i, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2)\)

Note that:

\( m = \text{logit}(\psi_i) = \log \left(\frac{\psi_i}{1-\psi_i}\right) ~~\Leftrightarrow~~ \psi_i = \frac{\exp(m)}{1+\exp(m)} \)

  • Generalized logit-normal distribution in ]a,b[:

In that case, \(h(\psi_i) = log\left(\frac{\psi_i – a}{b-\psi_i}\right)\). A random variable \(\psi_i\) with a logit-normal distribution takes its values in ]a,b[. The logit of \(\psi_i\) is normally distributed, i.e.,

\(\text{logit}_{(a,b)}(\psi_i) = \log \left(\frac{\psi_i – a}{b-\psi_i}\right) \ \sim \ \ {\cal N}( \text{logit}_{(a,b)}(\bar{\psi}_i), \omega^2) ~~\Leftrightarrow~~ \text{logit}_{(a,b)}(\psi_i) = \text{logit}_{(a,b)}(\bar{\psi}_i) + \eta_i, ~~\text{where}~~\eta_i \sim {\cal N}(0,\omega^2)\)

Note that:

\( m = \text{logit}_{(a,b)}(\psi_i) = \log \left(\frac{\psi_i – a}{b-\psi_i}\right) ~~\Leftrightarrow~~ \psi_i = \frac{b \exp(m)+a}{1+\exp(m)} \)

  • Probit-normal distribution:

The probit function is the inverse cumulative distribution function (quantile function) \(\Phi^{-1}\) associated with the standard normal distribution \({\cal N}(0,1)\). A random variable \(\psi\) with a probit-normal distribution also takes its values in ]0,1[.

\(\text{probit}(\psi_i) = \Phi^{-1}(\psi_i) \ \sim \ {\cal N}( \Phi^{-1}(\bar{\psi}_i), \omega^2) .\)

To chose one of these distribution in the GUI, click on the distribution corresponding to the parameter you want to change in the individual model part and choose the corresponding distribution.

Remarks:

  1. If you change your distribution and your population parameter is not valid, then an error message is thrown. Typically, when you want to change your distribution to a log normal distribution, make sure the associated population parameter is strictly positive.
  2. When creating a project, the default proposed distribution is lognormal.
  3. Logit transformations can be generalized to any interval (a,b) by setting \( \psi_{(a,b)} = a + (b-a)\psi_{(0,1)}\) where \(\psi_{(0,1)}\) is a random variable that takes values in (0,1) with a logit-normal distribution. Thus, if you need to have bounds between a and b, you need to modify your structural model to reshape a parameter between 0 and 1 and use a logit or a probit distribution. Examples are shown on this page.
  • “Adapted” Logit-normal distribution: 




Another interesting possibility is to “extend” the logit distribution to be bounded in [a, b] rather than in [0, 1]. It is possible starting from the 2019 version. For that, set your parameter in a logit normal distribution. The setting button appear next to the distribution.

Clicking on it will allow to define your bounds as in the following figure.

Notice that if your parameter initial value is not in [0, 1], the bounds are automatically adapted and the following warning message is proposed  “The initial value of XX is greater than 1: the logit limit is adjusted”

Marginal distributions of the individual parameters

  • warfarin_distribution1_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

We use the warfarin PK example here. The four PK parameters Tlag, ka, V and Cl are log-normally distributed. LOGNORMAL distribution is then used for these four log-normal distributions in the main Monolix graphical user interface:

The distribution of the 4 PK parameters defined in the MonolixGUI is automatically translated into Mlxtran in the project file:

[INDIVIDUAL]
input = {Tlag_pop, omega_Tlag, ka_pop, omega_ka, V_pop, omega_V, Cl_pop, omega_Cl}
DEFINITION:
Tlag = {distribution=lognormal, typical=Tlag_pop, sd=omega_Tlag}
ka = {distribution=lognormal, typical=ka_pop, sd=omega_ka}
V = {distribution=lognormal, typical=V_pop, sd=omega_V}
Cl = {distribution=lognormal, typical=Cl_pop, sd=omega_Cl}

Estimated parameters are the parameters of the 4 log-normal distributions and the parameters of the residual error model:


Here, \(V_{\rm pop} = 7.94\) and \(\omega_V=0.326\) means that the estimated population distribution for the volume is: \(\log(V_i) \sim {\cal N}(\log(7.94) , 0.326^2)\) or, equivalently, \(V_i = 7.94 e^{\eta_i}\) where \(\eta_i \sim {\cal N}(0,0.326^2)\).

Remarks:

  • \(V_{\rm pop} = 7.94\) is not the population mean of the distribution of \(V_i\), but the median of this distribution (in that case, the mean value is 7.985). The four probability distribution functions are displayed figure Parameter distributions:
  • \(V_{\rm pop}\) is not the population mean of the distribution of \(V_i\), but the median of this distribution. The same property holds for the 3 other distributions which are not Gaussian.
  • Here, standard deviations \(\omega_{Tlag}\), \(\omega_{ka}\), \(\omega_V\) and \(\omega_{Cl}\) are approximately the coefficients of variation (CV) of Tlag, ka, V and Cl since these 4 parameters are log-normally distributed with variances < 1.
  • warfarin_distribution2_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

Other distributions for the PK parameters are used in this project:

  • NORMAL for Tlag, we fix the population value \(Tlag_{\text{pop}}\) to 1.5 and the standard deviation \(\omega_{\rm Tlag}\) to 1:
  • NORMAL for ka,
  • NORMAL for V,
  • and LOGNORMAL for Cl

Estimated parameters are the parameters of the 4 transformed normal distributions and the parameters of the residual error model:

Here, \( Tlag_{\rm pop} = 1.5\) and \(\omega_{Tlag}=1\) means that \(Tlag_i \sim {\cal N}(1.5, 1^2)\) while \(Cl_{\rm pop} = .133\) and \(\omega_{Cl}=..29\) means that \(log(Cl_i) \sim {\cal N}(log(.133), .29^2)\). The four probability distribution functions are displayed Figure Parameter distributions:

 

 

Correlation structure of the random effects






Dependency can be introduced between individual parameters by supposing that the random effects \(\eta_i\) are not independent. This means considering them to be linearly correlated.

  • warfarin_distribution3_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

Defining correlation between random effects in the interface

To introduce correlations between random effects in Monolix, one can define correlation groups. For example, two correlation groups are defined on the interface below, between \(\eta_{V,i}\) and \(\eta_{Cl,i}\) (#1 in that case) and between \(\eta_{Tlag,i}\) and \(\eta_{ka,i}\) in an other group (#2 in that case):

To define a correlation between the random effects of V and Cl, you just have to click on the check boxes of the correlation for those two parameters. If you want to define a correlation between the random effects ka and Tlag independently of the first correlation group, click on the + next to CORRELATION to define a second group and click on the check boxes corresponding to the parameters ka and Tlag under the correlation group #2. Notice, that as the random effects of Cl and V are already in the correlation group #1, these random effects can not be used in another correlation group. When three of more parameters are included in a correlation groups, all pairwise correlations will be estimated. It is not instance not possible to estimate the correlation between \(\eta_{ka,i}\) and \(\eta_{V,i}\) and between \(\eta_{Cl,i}\) and \(\eta_{V,i}\) but not between \(\eta_{Cl,i}\) and \(\eta_{ka,i}\).

It is important to mention that the estimated correlations are not the correlation between the individual parameters (between \(Tlag_i\) and \(ka_i\), and between \(V_i\) and \(Cl_i\)) but the (linear) correlation between the random effects (between \(\eta_{Tlag,i}\) and \(\eta_{ka,i}\), and between \(\eta_{V,i}\) and \(\eta_{Cl,i}\)  respectively).

Remarks

  • If the box is greyed, it means that the associated random effects can not be used in a correlation group, as in the following cases
    • when the parameter has no random effects
    • when the random effect of the parameter is already used in another correlation group
  • There are no limitation in terms of number of parameters in a correlation group
  • You can have a look in the FORMULA to have a recap of all correlations
  • In case of inter-occasion variability, you can define the correlation group for each level of variability independently.
  • The initial value for the correlations is zero and cannot be changed.
  • The correlation value cannot be fixed.

Estimated population parameters now include these 2 correlations:

Notice that the high uncertainty on \(\text{corr_ka_Tlag}\) suggests that the correlation between \(\eta_{Tlag,i}\) and \(\eta_{ka,i}\) is not reliable.

How to decide to include correlations between random effects?

The scatterplots of the random effects can hint at correlations to include in the model. This plot represents the joint empirical distributions of each pair of random effects. The regression line (in pink below) and the correlation coefficient (“information” toggle in the settings) permits to visually detect tendencies. If “conditional distribution” (default) is chosen in the display settings, the displayed random effects are calculated using individual parameters sampled from the conditional distribution, which permits to avoid spurious correlations (see the page on shrinkage for more details). If a large correlation is present between a pair of random effects, this correlation can be added to the model in order to be estimated as a population parameter.

Depending on a number of random effects values used to calculate the correlation coefficient, a same correlation value can be more or less significant. To help the user identify significant correlations, Pearson’s correlation tests are performed in the “Result” tab, “Tests” section. If no significant correlation is found, like for the pair \(\eta_{Tlag}\) and \(\eta_{Cl}\) below, the distributions can be assumed to be independent. However, if a significant correlation appears, like for the pair \(\eta_V\) and \(\eta_{Cl}\) below, it can be hypothesized that the distributions are not independent and that the correlation must be included in the model and estimated. Once the correlation is included in the model, the random effects for \(V\) and \(Cl\) are drawn from the joint distribution rather than from two independent distributions.

 

How do the correlations between random effects affect the individual model?

In this example the model has four parameters Tlag, ka, V and Cl. Without correlation, the individual model is:
\(log(Tlag) = log(Tlag_{pop}) + \eta_{Tlag}\)
\(log(ka) = log(ka_{pop}) + \eta_{ka}\)
\(log(V) = log(V_{pop}) + \eta_V\)
\(log(Cl) = log(Cl_{pop}) + \eta_{Cl}\)

The random effects follow normal distributions: \((\eta_{Tlag,i},\eta_{ka,i},\eta_{V,i},\eta_{Cl,i}) \sim \mathcal{N}(0,\Omega)\)
\(\Omega\) is the variance-covariance matrix defining the distributions of the vectors of random effects, here:

\(\Omega = \begin{pmatrix} \omega_{Tlag}^2 & 0 & 0 & 0 \\ 0 & \omega_{ka}^2 & 0 & 0 \\ 0 & 0 & \omega_V^2 & 0 \\ 0 & 0 & 0 & \omega_{Cl}^2 \end{pmatrix}\)

In this example, two correlations between \(\eta_{Tlag}\) and \(\eta_{ka}\) and between \(\eta_{V}\) and \(\eta_{Cl}\) are added to the model. They are defined with two population parameters called \(\text{corr_Tlag_ka}\) and \(\text{corr_V_Cl}\) that appear in the variance-covariance matrix. So the only difference in the individual model is in \(\Omega\), that is now:

\(\Omega = \begin{pmatrix} \omega_{Tlag}^2 & \omega_{Tlag} \omega_{ka} \text{corr_Tlag_ka} & 0 & 0 \\ \omega_{Tlag} \omega_{ka} \text{corr_Tlag_ka} & \omega_{ka}^2 & 0 & 0 \\ 0 & 0 & \omega_V^2 & \omega_{V} \omega_{Cl} \text{corr_V_Cl} \\ 0 & 0 & \omega_{V} \omega_{Cl} \text{corr_V_Cl} & \omega_{Cl}^2 \end{pmatrix}\)

So the correlation matrix is related to the variance-covariance matrix \(\Omega\) as:

$$\text{corr}(\theta_i,\theta_j)=\frac{\text{covar}(\theta_i,\theta_j)}{\sqrt{\text{var}(\theta_i)}\sqrt{\text{var}(\theta_j)}}$$

Why should the correlation be estimated as part of the population parameters?

The effect of correlations is especially important when simulating parameters from the model. This is the case in the VPC or when simulating new individuals in Simulx to assess the outcome of a different dosing scenario for instance. If in reality individuals with a large distribution volume also have a large clearance (i.e there is a positive correlation between the random effects of the volume and the clearance), but this correlation has not been included in the model, then the concentrations predicted by the model for a new cohort of individuals will display a larger variability than they would in reality.

How do the EBEs change after having included correlation in the model?

Before adding correlation in the model, the EBEs or the individual parameters sampled from the conditional distribution may already be correlated, as can be seen in the “correlation between random effects” plot. This is because the individual parameters (EBEs or sampled) are based on the individual conditional distributions, which takes into account the information given by the data. Especially when the data is rich, the data can indicate that individuals with a large volume of distribution also have a large clearance, even if this correlation is not yet included in the model.

Including the correlation in the model as a population parameter to estimate allows to precisely estimate its value. Usually, one can see a stronger correlation for the corresponding pair of random effects when the correlation is included in the model compared to when it is not. In this example, after including the correlations in the individual model, the joint distribution of \(\eta_{V}\) and \(\eta_{Cl}\) displays a higher correlation coefficient (0.439 compared to 0.375 previously):

  • warfarin_distribution4_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

In this example, \(Tlag_i\) does not vary in the population, which means that \(\eta_{Tlag,i}=0\) for all subjects i, while the three other random effects are correlated:

Estimated population parameters now include the 3 correlations between \(\eta_{ka,i}\), \(\eta_{V,i}\) and \(\eta_{Cl,i}\) :

Parameters without random effects

Adding/removing inter-individual variability

By default, all parameters have inter-individual variability. To remove it, click on the checkbox of the random effect column:

How the parameters with no variability are estimated is explained here.

Custom parameter distributions

Some datasets may require more complex parameter distributions that those pre-implemented in the Monolix GUI. This video shows how to implement a lognormal distribution with Box-Cox transformation and how to bound a parameter between two values using a transformed logit distribution (this latter case can be handled automatically from Monolix2019R1).





4.2.3.Bounded parameter distribution

Bounded parameters in the interface

Starting from the 2019 version, it is possible to “extend” the logit distribution to be bounded in [a, b] rather than in [0, 1].  For that, set your parameter in a logit normal distribution. The setting button appear next to the distribution.

Clicking on it will allow to define your bounds as in the following figure.

Notice that if your parameter initial value is not in [0, 1], the bounds are automatically adapted and the following warning message is proposed  “The initial value of XX is greater than 1: the logit limit is adjusted”

 

 

Bounded parameters in the structural model

For versions of Monolix below 2019R1, a bounded parameter distribution, for example between a and b, can not be set directly through the interface, but have to be defined in two steps: (1) an auxiliary parameter and its distribution choice in the GUI, and (2) a transformation of the auxiliary parameter into the parameter of interest in the structural model file.

The same approach should be used in Monolix2019R1 to define a single bound (upper or lower).

Let’s take a simple PK example where a volume V is constrained. The structural model for this example is:

[LONGITUDINAL]
input = {V, k}
EQUATION:
; PK model definition
Cc = pkmodel(V, k)

 

  • Thus, to have a parameter V between two bounds a=1 and b=10, you have to define the structural model as below
    [LONGITUDINAL]
    input = {V_logit, k}
    EQUATION:
    ; PK model definition
    a = 1
    b = 10
    V_bound = a+V_logit*(b-a)
    Cc = pkmodel(V=V_bound, k)
    

    In the “Statistical model & Tasks” tab of the GUI, the distribution for V_logit should be set to LOGIT.

 

  • To have a parameter V larger than a=1 (with ‘a’ different from 0), you have to define the structural model as below
    [LONGITUDINAL]
    input = {V_log, k}
    EQUATION:
    ; PK model definition
    a = 1
    V_bound = a+V_log
    Cc = pkmodel(V=V_bound, k)
    

    In the “Statistical model & Tasks” tab of the GUI, the distribution for V_log should be set to LOGNORMAL.

 

  • To have a parameter V smaller than b=10, you have to define the structural model as below
    [LONGITUDINAL]
    input = {V_log, k}
    EQUATION:
    ; PK model definition
    b = 10
    V_bound = b-V_log
    Cc = pkmodel(V=V_bound, k)
    

    In the “Statistical model & Tasks” tab of the GUI, the distribution for V_log should be set to LOGNORMAL.

Notice that, using that transformation, you have to multiply the standard error of V_logit by (b-a) in the first case to have the standard error of the initial V_bound parameter. It is not necessary for the two other cases as it is an offset. In addition, you can output V_bound for each individual using the table statement.

4.2.4.Model for individual covariates


Objectives: learn how to implement a model for continuous and/or categorical covariates.


Projects: warfarin_covariate1_project, warfarin_covariate2_project, warfarin_covariate3_project, phenobarbital_project







Model with continuous covariates

  • warfarin_covariate1_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

The warfarin data contains 2 individual covariates: weight which is a continuous covariate and sex which is a categorical covariate with 2 categories (1=Male, 0=Female). We can ignore these columns if are sure not to use them, or declare them using respectively the reserved keywords CONTINUOUS COVARIATE  and CATEGORICAL COVARIATE to define continuous and categorical covariate.

Even if these 2 covariates are now available, we can choose to define a model without any covariate by not clicking on any check box in the covariate model.

Here, an unchecked box in the line of the parameter V and the column of the covariate wt means that there is no relationship between weight and volume in the model. A diagnosis plot Individual parameters vs covariates is generated which displays possible relationships between covariates and individual parameters (even if these covariates are not used in the model):

On the figure, we can see a strong correlation between the volume V and both the weight wt and the sex. One can also see a correlation between the clearance and the weight wt. Therefore, the next step is to add some covariate to our model.

  • warfarin_covariate2_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

We decide to use the weight in this project in order to explain part of the variability of \(V_i\) and \(Cl_i\). We will implement the following model for these two parameters:

$$\log(V_i) = \log(V_{\rm pop}) + \beta_V \log(w_i/70) + \eta_{V,i} ~~\text{and}~~\log(Cl_i) = \log(Cl_{\rm pop}) + \beta_{Cl} \log(w_i/70) + \eta_{Cl,i}$$

which means that population parameters of the PK parameters are defined for a typical individual of the population with weight = 70kg.

More details about the model
The model for \(V_{i}\) and \(Cl_{i}\) can be equivalently written as follows:

$$ V_i = V_{\rm pop} ( w_i/70 )^{\beta_V} e^{ \eta_{V,i} } ~~\text{and}~~ Cl_i = Cl_{\rm pop} ( w_i/70 )^{\beta_{Cl}} e^{ \eta_{Cl,i} }$$

The individual predicted values for \(V_i\) and \(Cl_i\) are therefore

$$\bar{V}_i = V_{\rm pop} \left( w_i/70 \right)^{\beta_V} ~~\text{and}~~ \bar{Cl}_i = Cl_{\rm pop} \left( w_i/70 \right)^{\beta_{Cl}} $$

and the statistical model describes how \(V_i\) and \(Cl_i\) are distributed around these predicted values:

$$ \log(V_i)  \sim {\cal N}( \log(\bar{V}_i) , \omega^2_V) ~~\text{and}~~\log(Cl_i) \sim {\cal N}( \log(\bar{Cl}_i) , \omega^2_{Cl}) $$

Here, \(\log(V_i)\) and \(\log(Cl_i)\) are linear functions of \(\log(w_i/70)\): we then need to transform first the original covariate \(w_i\) into \(\log(w_i/70)\) by clicking on the button CONTINUOUS next to ADD COVARIATE (blue button). Then, the following pop up arises

You have to

  • define the name of the covariate you want to add (the blue frame).
  • define the associated equation (the green frame).
  • click on the ACCEPT button

Remarks

  • You can define any formula for your covariate as long as you use mathematical functions available in the Mlxtran language.
  • You can use any covariate available in the list of covariates proposed in the window. Thus, if you have a Height and Weight as covariates, you can directly compute the Body Mass Index.
  • If you go over a covariate with your mouse, all the information (min, mean, median, max and weighted mean) are displayed as a tooltip. The weighted mean is defined as \[ \text{weighted mean} (cov) = \exp \Big( \sum_i \frac{\text{nbObs}_i }{\text{nbObs}} \log(cov_i) \Big)\] where \(\text{nbObs}_i\) is the number of observation for the \(i^{th}\) individual and \(\text{nbObs}\) is the total number of observations.
  • If you click on the covariate name, it will be written in the formula.
  • You can use this formula box to replace missing continuous covariate values by an imputed value. This is explained in the feature of the week #141 below. For example, if your continuous covariate takes only positive values, you can use a negative value for the missing values in your dataset, for example -99, and enter the following formula: max(COV,0) + min(COV,0)/COV*ImputedValue with the desired ImputedValue (COV is thAse name of your covariate).

 


We then define a new covariate model, where \(\log(V_i)\) and \(\log(Cl_i)\) are linear functions of the transformed weight \(lw70_i\) as shown on the following figure:

Notice that by clicking on the button FORMULA, you have the display of all the individual model equations. Coefficients \(\beta_{V}\) and \(\beta_{Cl}\) are now estimated with their s.e. and the p-values of the Wald tests are derived to test if these coefficients are different from 0.
Again, a diagnosis plot Individual parameters vs covariates is generated which displays possible relationships between covariates and individual parameters (even if these covariates are not used in the model) as one can see on the figure below on the left. However, as there are covariates on the model, what is interesting is to see if there still are correlation between the random effects and the covariates as one can see on the figure below on the right.

 

Note: To make it automatically, starting from the 2019 version, there is an arrow next (in purple in the next figure) to the continuous covariate from the data set and propose to add a log transformed covariate centered by the weighted mean.





 

Model with categorical covariates

  • warfarin_covariate3_project (data = ‘warfarin_data.txt’, model = ‘lib:oral1_1cpt_TlagkaVCl.txt’)

We use sex instead of weight in this project, assuming different population values of volume and clearance for males and females. More precisely, we consider the following model for \(V_i\) and \(Cl_i\):

$$\log(V_i) = \log(V_{\rm pop}) + \beta_V 1_{sex_i=F} + \eta_{V,i}~~\text{and}~~\log(Cl_i) = \log(Cl_{\rm pop}) + \beta_{Cl} 1_{sex_i=F} + \eta_{Cl,i}$$

where \(1_{sex_i=F} =1\) if individual i is a female and 0 otherwise. Then, \(V_{\rm pop}\) and \(Cl_{\rm pop}\) are the population volume and clearance for males while \(V_{\rm pop}, e^{\beta_V}\) and \(Cl_{\rm pop} e^{\beta_{Cl}}\) are the population volume and clearance for females. By clicking on the purple button DISCRETE, the following window pops up

You have to

  • define the name of the covariate you want to add (the blue frame).
  • define the associated categories (the green frame).
  • click on the ALLOCATE button to define all the categories.

Then, you can

  • define the name of the categories (the blue frame).
  • define the reference category (the green frame).
  • click on ACCEPT

Then, define the covariate model in the main GUI:

Estimated population parameters, including the coefficients \(\beta_V\) and \(\beta_{Cl}\) are displayed with the results:

We can display the probability distribution functions of the 4 PK parameters using the Individual parameter graphic:

Notice that for the volume and the clearance, the theoretical curve is is not the PDF of a lognormal distribution, due to the impact of the covariate sex.

 

Calculating the typical value for each category

Cl_pop represents the typical value for the reference category (in the example above SEX=0). The typical value for the other categories can be calculated based on the estimated beta parameters:

  • normal distribution: \( Cl_{SEX=1} = Cl_{pop} + beta\_Cl\_SEX\_1 \)
  • lognormal distribution: \( Cl_{SEX=1} = Cl_{pop} \times  e^{beta\_Cl\_SEX\_1 } \)
  • logit distribution: \( F_{SEX=1} = \frac{1}{1+ e^{-\left(  \log \left(\frac{F_{pop}}{1-F_{pop}} \right) + beta\_F\_SEX\_1 \right) }} \)




 

Transforming categorical covariates

  • phenobarbital_project (data = ‘phenobarbital_data.txt’, model = ‘lib:bolus_1cpt_Vk.txt’)

The phenobarbital data contains 2 covariates: the weight and the APGAR score which is considered as a categorical covariate. Instead of using the 10 original levels of the APGAR score, we will transform this categorical covariate and create 3 categories: Low = {1,2,3}, Medium = {4, 5, 6, 7} and High={8,9,10}.

If we assume, for instance that the volume is related to the APGAR score, then \(\beta_{V,Low}\) and \(\beta_{V,High}\) are estimated (assuming that Medium is the reference level).

In that case, one can see that both p-values concerning the transformed APGAR covariate are over .05.

 

The same covariate effect for several parameters

Adding a covariate effect on a parameter in the Individual Model section of the Statistical Model and Tasks tab, creates automatically a corresponding beta parameter (population parameter) that describes the strength of the effect. For example, adding weight WT on a parameter volume V1 creates beta_V1_WT. If you add the same covariate effect on two different parameters, eg. V1 and V2, then Monolix will create two corresponding parameters: beta_V1_WT and beta_V2_WT.

If you need to add the same covariate effect for several parameters, eg. have only one parameter betaWT for both V1 and V2 in the example above, then this covariate effect has to be implemented in the structural model directly. It requires:

  • Reading weight WT from the dataset, so the columnt WT in the dataset has to be tagged as a regressor (not covariate)
  • Defining in the model new volumes with the covariate effect (sample code below, based on a two-compartment model from the PK library)[LONGITUDINAL]
    input = {WT, betaWT, ka, Cl, V1, Q2, V2}
    WT = {use=regressor}
    PK:
    ; Effect of WT on V1, V2
    V1_withWT = V1*exp(betaWT*WT)
    V2_withWT = V2*exp(betaWT*WT)
    ; Parameter transformations
    V = V1_withWT
    k12 = Q2/V1_withWT
    k21 = Q2/V2_withWT
    ; PK model definition
    Cc = pkmodel(ka, V, Cl, k12, k21)
    OUTPUT:
    output = {Cc}
  • In the Individual model in the GUI, set normal distribution for “betaWT” – to allow positive and negative values, and remove random effects from it – they are already included in the parameters V1, V2.
This method gives more flexibility for more complex paramter-covariate relationships,  also to include time-varying covariates in the model. However, remember that when you add covariate effects in the structural model – the statistical tests are not available for these relationships. Also, Monolix is more efficient when covariate effects are added in the GUI, as the algorithm “see” that V and beta_V_WT are related, and it improves the stability.

4.2.5.Complex parameter-covariate relationships

Note that starting from version 2024 on, analytical solutions are used even if a time-varying regressor is used in the model.

Complex parameter-covariate relationships and time-dependent continuous covariates









Covariate-parameter relationships are usually defined via the Monolix GUI, leading for instance to exponential and power law relationships. However more complex parameter-covariate relationships such as Michaelis-Menten or Hill dependencies cannot the defined via the GUI because they cannot be put into the format where the (possibly transformed) covariate is added linearly on the transformed parameter. Similarly, when the covariate value is changing over time and thus not constant for each subject (or each occasion in each subject in case of occasions), the covariate cannot be added to the model via the GUI. In both cases, the effect of the covariate must be tagged as a regressor and the relationship must be defined directly in the model file.

In the following, we will use as an example a Hill relationship between the clearance parameter Cl and the time-varying post-conception age (PCA) covariate, which is a typical way to scale clearance in paediatric pharmacokinetics:

$$Cl_i = Cl_{pop} \frac{PCA^n}{PCA^n+A50^n} e^{\eta_i}$$

where \(Cl_i\) is the parameter value for individual i, \(Cl_{pop}\) the typical clearance for an adult, \(A50\) the PCA for the clearance to reach 50% mature, \(n\) the shape parameter and \(\eta_i\) the random effect for individual i.

Step 1: To make the PCA covariate available as a variable is the model file, the first step is to tag it as a regressor column-type REGRESSOR when loading the data set (instead of using the CONTINUOUS COVARIATE column-type).

Step 2: In the model file, the PCA covariate is passed as input argument and designated as being a regressor. The clearance Cl, the hill shape parameter n, and the A50 are passed as usual input parameters:

[LONGITUDINAL]
input = {..., Cl, n, A50, PCA, ...}
PCA = {use=regressor}

If several regressors are used, be careful that the regressors are matched by order with the data set columns tagged as REGRESSOR (not by name).
The relationship between the clearance Cl and the post-conception age PCA is defined in the EQUATION: block, before ClwithPCA is used (for instance in a simple (V,Cl) model):

EQUATION:
ClwithPCA = Cl * PCA^n / (PCA^n + A50^n)

Cc = pkmodel(Cl=ClwithPCA, V)

Note that the input parameter Cl includes the random effect ( \(Cl = Cl_{pop} e^{\eta_i} \) ), such that only the covariate term must be added. Because the parameter including the covariate effect CLwithPCA is not a standard keyword for macros, one must write Cl=ClwithPCA.

Step 3: The definition of the parameters in the GUI deserves special attention. Indeed the parameters n and A50 characterize the covariate effect and are the same for all individuals: their inter-individual variability must be removed by unselecting the random effects. On the opposite, the parameter Cl keeps its inter-individual variability, corresponding to the \(e^{\eta_i} \) term.

Step 4: When covariates relationships are not defined via the GUI, the p-value corresponding to the Wald test is not automatically outputted. It is however possible to calculate it externally. Assuming that we would like to test if the shape parameter n is significantly different from 1:

$$H_0: \quad \textrm{”}n=1\textrm{”} \quad \textrm{versus} \quad H_1:\quad \textrm{”}n \neq 1\textrm{”}$$

Using the parameter estimate and the s.e outputted by Monolix, we can calculate the Wald statistic:

$$W = \frac{\hat{n}-n_{ref}}{\textrm{s.e}(\hat{n})}$$

with \(\hat{n}\) the estimated value for parameter n, \(n_{ref}\) the reference value for n (here 1) and \(\textrm{s.e}(\hat{n}) \) the standard error for the n estimate.
The test statistic W can then be compared to a standard normal distribution. Below we propose a simple R script to calculate the p-value:

 n_estimated = 1.32
 n_ref = 1
 se_n = 0.12
 
 W = abs(n_estimated - n_ref)/se_n
 
 pvalue = 2 * pnorm(W, mean = 0, sd = 1, lower.tail = FALSE)

Note that the factor 2 is added to do a two-sided test.

 

 

Categorical time-varying covariates

Categorical covariates may also be time-varying, for instance when the covariate represents concomitant medications over the course of the clinical trial or a fed/fasting state at the time of the dose.

In the following, we will use as an example a concomitant medication categorical covariate with 3 categories: no concomitant drug, concomitant drug 1, and concomitant drug 2. We would like to investigate the effect of the concomitant drug covariate on the clearance Cl.

$$Cl_i =  \begin{cases}Cl_{pop} e^{\eta_i} & \textrm{if no concomitant drug} \\ Cl_{pop} (1+\beta_1)e^{\eta_i} & \textrm{if concomitant drug 1} \\ Cl_{pop} (1+\beta_2)e^{\eta_i} & \textrm{if concomitant drug 2} \end{cases}$$

where \(Cl_i\) is the parameter value for individual i, \(Cl_{pop}\) the typical clearance if no concomitant drug, \(\beta_1\) the fractional change in case of concomitant drug 1, \(\beta_2\) the fractional change in case of concomitant drug 2 and \(\eta_i\) the random effect for individual i.

Step 1: Encode the categorical covariate as integers. Indeed, while strings are accepted for the CATEGORICAL COVARIATE column-type, only numbers are accepted for the REGRESSOR column-type. Here we will use 0 = no concomitant medication, 1 = concomitant drug 1 and 2 = concomitant drug 2.

Step 2: To make the COMED covariate available as a variable is the model file, tag it as a column-type REGRESSOR when loading the data set (instead of using the CATEGORICAL COVARIATE column-type).

Step 3: In the model file, the COMED covariate is passed as input argument and designated as being a regressor. The clearance Cl, and the two beta parameters are passed as usual input parameters:

[LONGITUDINAL]
input = {..., Cl, beta1, beta2, COMED, ...}
COMED = {use=regressor}

If several regressors are used, be careful that the regressors are matched by order with the data set columns tagged as REGRESSOR (not by name).

To define the COMED covariate impact, we use a if/else statement in the EQUATION: block. The parameter value taking into account the COMED effect (called ClwithCOMED in this example) can then be used in an ODE system or within macros.

EQUATION:
if COMED==0
ClwithCOMED = Cl 
elseif COMED==1
ClwithCOMED = Cl * (1+beta1)
else
ClwithCOMED = Cl * (1+beta2)
end

Cc = pkmodel(Cl=ClwithCOMED, V)

Note that the input parameter Cl includes the random effect ( \(Cl = Cl_{pop} e^{\eta_i} \) ), such that only the covariate term must be added. Because the parameter including the covariate effect CLwithCOMED is not a standard keyword for macros, one must write Cl=ClwithCOMED.

If the categorical covariate has only two categories encoded as 0 and 1 (for instance COMED=0 for no concomitant medication and COMED=1 for concomitant medication), it is also possible to write the model in a more compact form:

ClwithCOMED = Cl * (1 + beta * COMED)

Step 4: The definition of the parameters in the GUI deserves special attention. Indeed the parameters beta1 and beta2 characterize the covariate effect and are the same for all individuals: their inter-individual variability must be removed by unselecting the random effects. On the opposite, the parameter Cl keeps its inter-individual variability, corresponding to the \(e^{\eta_i} \) term. In addition, we choose a normal distribution for beta1 and beta2 (with a standard deviation of zero as we have removed the random effects) in order to allow both positive and negative values.

 

 

Covariate-dependent parameter

When adding a categorical covariate on a parameter via the GUI, different typical values will be estimated for each group. However, all groups will have the same standard deviation. It can sometimes be useful to consider that the standard deviations also differ between groups. For example, healthy volunteers may have a smaller inter-individual variability than patients.

From the 2018R1 version on, categorical covariates affecting both the typical value and the standard deviation have to be defined directly in the structural model, by using the covariate as a regressor and different parameters depending on the value of the regressor. Using a different parameter for each group permits to estimate a typical value and a standard deviation per group. Note that a regressor can contain only numbers, so the categorical covariate should be encoded with integers rather than strings.

We show below an example, where the fixed effect and standard deviation of the volume V both depend on the covariate SEX. This require the definition of two different parameters VM (for male) and VF (for female).

Step 1: To make the covariate SEX available as a variable is the model file, it has to be tagged as a regressor with column-type REGRESSOR when loading the data set (instead of using the CONTINUOUS COVARIATE and CATEGORICAL COVARIATE column-types).

Step 2: In the model file shown below, the covariate SEX is passed as input argument and designated as being a regressor. Two parameters VM (for male) and VF (for female) are given as input, to be used as the volume V depending on the SEX. The use of VM or VF depending on the SEX value is defined in the EQUATION: block, before V is used (for instance in a simple (V,Cl) model):

[LONGITUDINAL]
input = {Cl, VM, VF, SEX}
SEX = {use=regressor}
EQUATION: 
if SEX==0
V = VM
else
V = VF
end 

Cc = pkmodel(Cl, V)

OUTPUT:
output = {Cc}

Step 3: The distribution of the parameters VM and VF is set as usual in the GUI. A different typical population value and a different standard deviation of the random effects will be estimated for males and females.

Note: As SEX has been tagged as a regressor, it is not available as a covariate in the GUI. If a covariate effect of SEX on another parameter is needed, the column SEX can be duplicated in the dataset so that the duplicate can be tagged as a covariate.

 

 

Covariate-dependent standard deviation

This video shows a variation of the previous solution, where only the standard deviation of the random effect of a parameter is covariate-dependent, while the fixed effect is not affected.





 

Transforming a continuous covariate into a categorical covariate

The Monolix GUI allows to discretize a continuous covariate in order to handle it as a categorical covariate in the model, using binary 0/1 values.





4.2.6.Inter occasion variability (IOV)





 


Objectives: learn how to take into account inter occasion variability (IOV).


Projects: iov1_project, iov1_Evid_project, iov2_project, iov3_project, iov4_project


Introduction

A simple model consists of splitting the study into K time periods or occasions and assuming that individual parameters can vary from occasion to occasion but remain constant within occasions. Then, we can try to explain part of the intra-individual variability of the individual parameters by piecewise-constant covariates, i.e., occasion-dependent or occasion-varying (varying from occasion to occasion and constant within an occasion) ones. The remaining part must then be described by random effects. We will need some additional notation to describe this new statistical model. Let

  • \(\psi_{ik}\) be the vector of individual parameters of individual i for occasion k, where \(1\leq i \leq N\) and \(1\leq k \leq K\).
  • \({c}_{ik}\) be the vector of covariates of individual i for occasion k. Some of these covariates remain constant (gender, group treatment, ethnicity, etc.) and others can vary (weight, treatment, etc.).

Let \(\psi_i = (\psi_{i1}, \psi_{i2}, \ldots , \psi_{iK})\) be the sequence of K individual parameters for individual i. We also need to define:

  • \(\eta_i^{(0)}\), the vector of random effects which describes the random inter-individual variability of the individual parameters,
  • \(\eta_{ik}^{(1)}\), the vector of random effects which describes the random intra-individual variability of the individual parameters in occasion k, for each \(1\leq k \leq K\).

Here and in the following, the superscript (0) is used to represent inter-individual variability, i.e., variability at the individual level, while superscript (1) represents inter-occasion variability, i.e., variability at the occasion level for each individual. The model now combines these two sequences of random effects:

\(h(\psi_{ik}) = h(\psi_{\rm pop})+ \beta(c_{ik} – c_{\rm pop}) + \eta_i^{(0)} + \eta_{ik}^{(1)} \)

Remark: Individuals do not need to share the same sequence of occasions: the number of occasions and the times defining the occasions can differ from one individual to another.

Occasion definition in a data set

There are two ways to define occasions in a data set:

  • Explicitly using an OCCASION column. It is possible to have, in a data set, one or several columns with the column-type OCCASION. It corresponds to the same subject (ID should remain the same) but under different circumstances, occasions. For example, if the same subject has two successive different treatments, it should be considered as the same subject with two occasions. The OCC columns can contain only integers.
  • Implicitly using EVID column. If there is an EVID column with a value 4 then Monolix defines a washout and creates an occasion. Thus, if there are several times where EVID equals 4 for a subject, it will create the same number of occasions. Notice that if EVID equals 4 happens only once at the beginning, only one occasion will be defined and no inter occasion variability would be possible.

There are three kinds of occasions

  • Cross over study: In that case, data is collected for each patient during two independent treatment periods of time, there is an overlap on the time definition of the periods. A column OCCASION can be used to identify the period. An alternative way is to define an EVID column starting for all occasions with EVID equals 4. Both types of definition will be presented in the iov1 example.
  • Occasions with washout: In that case, data is collected for each patient during one period and there is no overlap between the periods. The time is increasing but the dynamical system (i.e. the compartments) is reset when the second period starts. In particular, EVID=4 indicates that the system is reset (washout) for example, when a new dose is administrated.
  • Occasions without washout: In that case, data is collected for each patient during one period and there is no overlap between the periods. The time is increasing and we want to differentiate periods in terms of occasions without any reset of the dynamical system. Multiple doses are administrated to each patient. each period of time between successive doses is defined as a statistical occasion. A column OCCASION is therefore necessary in the data file to define it.

Cross over study

  • iov1_project (data = ‘iov1_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)

In this example, PK data is collected for each patient during two independent treatment periods of time (each one starting at time 0). A column OCCASION is used to identify the study:
This column is defined using the reserved keyword OCCASION. Then, the model associated to the individual parameter is as presented below

First, to define the variability of each parameter on each level, you just have to go on the good level, and you’ll see the associated random effects on each level. On the figure above, we see that all parameters have variability on the ID level, which means that all parameters have inter-individual variability. On the figure below,  we see the OCC level. In the presented case, only the volume V has inter-study variability and thus inter occasion variability. Thus, this is the only one having variability on the occasion level.

In terms of covariates, we then see two parts as displayed below. We see the covariates

  • associated to the level ID (in green). It corresponds to all the covariates that are constant for each subject.
  • associated to the level OCC (in blue). It corresponds to all the covariates that are constant for each occasion but not on each subject.

In the presented case, the treatment TRT varies for each individual. It contains inter-occasion information and is thus displayed with the occasion level. On the other hand, the SEX is constant for each subject. It contains then inter-individual information but no inter-occasion information. It is then displayed with the ID level.

What is the  impact?
Covariates can be associated to the parameter if and only if their level of variability is coherent with the level of variability of the parameter.
In the presented case,

  • TRT has inter-occasion variability. It can only be used with the parameter V that has inter-occasion variability. The two other parameters have only inter-individual variability and can therefore not use this TRT information. The interface is greyed and the user can not add this covariate to the parameters ka and Cl.
  • SEX has only inter-individual variability. It can therefore be associated to any parameter that has inter-individual variability.

The population parameters now include the standard deviations of the random effects for the 2 levels of variability (omega is used fo IIV and gamma for IOV):
Two important features are proposed in the plots. Firstly, in the individual fits, you can split or merge the occasions. When split is done, the name of the subject-occasion is the name of the subject, #, and the name of the occasion.

Secondly, you can use the occasion to split the plots

  • iov1_Evid_project (data = ‘iov1_Evid_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)

Another way to describe this cross over study is to use EVID=4 as explained in the data set definition. In that example, the EVID creates a washout and another occasion.

Occasions with washout

  • iov2_project (data = ‘iov2_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)

The time is increasing in this example, but the dynamical system (i.e. the compartments) is reset when the second period starts. Column EVID provides some information about events concerning dose administration. In particular, EVID=4 indicates that the system is reset (washout) when a new dose is administrated

Monolix automatically proposes to define the treatment periods (between successive resetting) as statistical occasions and introduce IOV, as we did in the previous example. We can display the individual fit by splitting each occasion for each individual


Or by merging the different occasions in a unique plot for each individual:

Remark: If you are modeling a PK as in this example, the washout implies that the occasions are independent. Thus, the cpu time is much faster as we do not have to compute predictions between occasions.

Occasions without washout

  • iov3_project (data = ‘iov3_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)

Multiple doses are administrated to each patient. We consider each period of time between successive doses as a statistical occasion. A column OCCASION is therefore necessary in the data file.

We can color the observed data by their occasion to have a better representation

The model for IIV and IOV can then be defined as usual. The plot of individual fits allows us to check that the predicted concentration is now continuous over the different occasions for each individual:

Multiple levels of occasions

  • iov4_project (data = ‘iov4_data.txt’, model = ‘lib:oral1_1cpt_kaVk.txt’)

We can easily extend such an approach to multiple levels of variability. In this example, columns P1 and P2 define embedded occasions. They are both defined as occasions:

We then define a statistical model for each level of variability.

4.2.7.Inter-occasion variability and effect of guar gum on alcohol concentration in blood

Download data set

This case study uses the MonolixSuite to analyze and model the absorption and elimination of alcohol with or without a dietary additive of guar gum. It focuses in particular on the modeling of inter-occasion variability.

Guar gum, also called guaran, is a polysaccharide extracted from guar beans. As a natural polymer, it has been used for many years as an emulsifier, thickener, and stabilizer in the food industry. In the pharmaceutical sector, guar gum and guar-gum based systems are frequently studied for the development of controlled-released formulations and colon targeted drug delivery systems, as guar gum can protect active molecules from the enzymes and pH in the stomach and small intestine and it can be degraded by intestinal bacteria in the colon. [Aminabhavi, T. M., Nadagouda, M. N., Joshi, S. D., & More, U. A. (2014). Guar gum as platform for the oral controlled release of therapeutics. Expert Opinion on Drug Delivery, 11(5), 753–766.]

Moreover, guar gum may affect the bioavailability of concomitantly administered substances due to its effect on the rate of gastrointestinal transit and gastric emptying.

The goal of this case study is then to assess the effect of guar gum on the absorption and bioavailability of alcohol.

Outline:

 





Data set

 

The data has been published in:

Practical Longitudinal Data Analysis, David J. Hand, Martin J. Crowder, Chapman and Hall/CRC, Published March 1, 1996

It is composed of measurements of blood alcohol concentrations in 7 healthy individuals. All subjects took alcohol at time 0 and gave a blood sample at 14 times over a period of 5 hours. The whole procedure was repeated at a later date but with a dietary additive of guar gum. The two different periods of time are encoded in the data with overlapping times, both starting at time 0.

Although the precise amount of ingested alcohol is unknown, for this case study we assume each amount to be 10g (standard drink).

 

Data exploration in Datxplore

 

The data is loaded in Datxplore to explore it graphically. It appears in the Data tab as above. The time is in hours, the alcohol concentration in the column Y is in mg/L, and the amount in AMT is in mg. Censored observations are indicated with the column CENS tagged as CENSORING.

The two periods of measurements, during which the subjects have received or not guar gum in addition to alcohol are distinguished with the column OCC, which is automatically recognized as OCCASION. In addition, I use the column DIET that I tag as CATCOV to indicate which occasion corresponds to the addition of guargum. This column contains 2 strings: noGuar for occasion 1 and withGuar for occasion 2.

In the Plots tab, the plot of alcohol concentration vs time in log-scale seen below suggests to use a one-compartment model with a first-order absorption. A non-linear elimination appears for some individuals, but the data might not be sufficient to capture it. Thus a linear elimination should be tried as a first model.

The different occasions can be visualized in several ways. First, hovering on a curve highlights the curve in solid yellow and the curve corresponding to the other occasion from the same subject in dashed yellow, as on the figure above.

Second, OCC is available for stratification along with the covariate DIET in the “Stratify” panel , and can thus be used for splitting, coloring or filterting. Below are for example the subplots of the data split by OCC and colored by ID. Each subject-occasion is assigned a color, with matched color shades for subject-occasions corresponding to the same subject. This is convenient to compare at a glance the two occasions for all subjects. The inter-individual variability seems mostly reproduced from one occasion to the other, and concentration levels seem slightly higher for OCC=2.

This global trend can be confirmed in linear scale below. Here the plot is colored by DIET, whose categories are matched with OCC: DIET=noGuar corresponds to OCC=1 and DIET=withGear to OCC=2. Thus, the main inter-occasion variability seen in the data seems to be explained by the covariate DIET.

 

First analysis in PKanalix

 

Non-compartmental analysis

 

As a first analysis, we can check the difference in PK parameters for the two occasions with non-compartmental analysis (NCA) in PKanalix. After loading the previously saved Datxplore project in PKanalix, the following settings are chosen:

  • extravascular administration,
  • “linear up log down” integral method,
  • “missing” for blq after Tmax (censored observations after Tmax are not used in the analysis).

The “Check lambda_z” panel, seen below, allows to check the regressions estimating the elimination slope. The default “adjusted R2” rule selects for each individual the optimal number of points used in the regression to get the best regression. While the plots allow to adjust the selection of points for some individuals, it is not necessary here. The plots already show some variability between individuals in the estimated lambza_z.

Running the NCA gives the lambda_z and other PK parameters for each individual. In the “Plot” tab, the plot of individual parameters vs. covariates is convenient to visualize the variability in the parameters and compare the distributions without and with guar gum. Here the following parameters have been selected: AUCINF_pred, Cl_F_pred, Cmax, HL_Lambda_z, Tmax:

Some difference can be seen between the two conditions for the half-life, the apparent clearance and Cmax, however the parameters also show large variability within each dietary condition.

 

Compartmental analysis in PKanalix

 

Next, a compartmental analysis (CA) can be run to estimate a compartmental model and compare the estimated parameters between the two conditions. PKanalix considers that the subject-occasions are independent, thus the parameters are optimized independently on each individual and on each occasion. This allows to check easily if different values are estimated for the two occasions.

Choosing a one-compartment model with a first-order absorption and a linear elimination gives the following individual fits (after choosing initial values with the “aut-init” button):

The absorption phase is not really well captured. Zooming on this phase can help confirm that the absorption should be delayed. Choosing the same model as before but with a lag time before the absorption now gives good individual fits:

In the plot Individual parameters vs. covariates, the estimated individual parameters show different distributions across the two conditions of DIET, in particular for ka, V and k:

Since the data size is small, it is not clear whether these differences are significant. The effect of DIET on the alcohol kinetics can be assessed more precisely with a detailed population analysis in Monolix.

Moreover, the bioavailability is not explicitly taken into account by this model, because it is not identifiable with only extravascular administrations, so it is included in the apparent volume V. In Monolix, it is possible to use more complex models than the simple PK models from the library, and in particular to add the bioavailability explicitly, allowing to assess in a more meaningful way whether guar gum could have an effect on the relative bioavailability depending on the value of DIET.

Therefore, we export the compartmental model from PKanalix to Monolix.

 

Population modelling in Monolix

 

This opens a Monolix project in which the data and the structural model are set up like in PKanalix.

In “Statistical model & Tasks”, the “Individual model” part is now split in two. The part on the left at the ID level describes the inter-individual variability (IIV), and includes by default a random effect for each parameter. The part on the right (highlighted in the figure below) is dedicated to the OCC variability-level, where it is possible to add random effects at the inter-occasion level: this would create inter-occasion variability (IOV). Since DIET varies from one occasion to another, it appears in this panel to explain part of the inter-occasion variability with a covariate effect. The boxes for adding covariate effects are greyed out, because at this step there is no inter-occasion variability. It is only possible to add covariate effects at the occasion level on parameters that either have inter-occasion variability with random effects at the occasion level, or parameters that have no random effects at the id and occasion levels.

 

Model without inter-occasion variability

 

The first step in this worklow aims at validating the structural model without taking into account differences between the occasions. Thus we keep the statistical model to default, select all tasks in the scenario and save the project as run01.mlxtran.

Estimating this model does not show misspecifications on the plot of Observations vs Predictions in log-log scale:

All parameters are estimated with a good confidence, except omega_ka which is small and with a high rse:

On the individual fits seen below, disabling the option “Split occasions” in “Display” allows to visualize the two occasions on the same plot for each individual. The observed data can be colored by occasion or equivalently by DIET in Stratify.

In this case, the predictions are identical for both occasions and overlap, since no inter-occasion variability is taken into account in the model. The prediction curves are displayed in purple for the first occasion and orange for the second, by default the curve for the first occasion on top, except for the last individual for which the second occasion is on top because it corresponds to a smaller observation period.

The individual fits shows that capturing both occasions with the same prediction is not possible, because there are small non-random variations from one occasion to another, as seen during the data exploration. This could corresponds to variability in the parameters between the occasions, that can be taken into account in the “Statistical model & Tasks” tab, by adding some random effects at the inter-occasion level or defining a covariate effect of DIET. We will first focus on the random effects.

 

Model with relative bioavailability and unexplained inter-occasion variability

 

Each parameter can be modelled with IIV, IOV or both. Physiological considerations can help deciding if a parameter should have variability at each level or not. But in the absence of clear physiological knowledge, a possible approach is also to add a random effect at the occasion level on each parameter for which variability may be relevant, and check if the estimated standard deviation of the random effect is small.

In this case, all parameters may show some inter-occasion variability. Indeed, the elimination can easily show some variations between different periods, and the dietary additive of guar gum might change the values of the parameters Tlag and ka characterizing the absorption. The volume V is unlikely to vary much for one occasion to another, however in this case V corresponds to the apparent volume, that includes the bioavailability of alcohol, which may vary with guar gum. Thus, it would be possible to set a random effect for IOV on V, or alternatively to modify the structural model to include explicitly the bioavailability, and add the random effect at the occasion level on the bioavailability instead of the volume. This is what we are going to do.

Before modifying the structural model, we use the last estimates as new initial values to facilitate the estimation for the next run.

Then, we open the structural model in the editor, and add an argument p=F in the pkmodel macro. This means that the proportion of absorbed drug will be defined by the parameter F, that should also be added in the input list:

[LONGITUDINAL]
input = {Tlag, ka, V, k, F}

EQUATION:
; PK model definition
Cc = pkmodel(Tlag, ka, V, k, p=F)

OUTPUT:
output = Cc

This modified model is then saved under a new name. The compile button is convenient to check that there is no syntax error. The new model can then be loaded in Monolix instead of the previous one.

After loading the model, Monolix brings us to the “Check initial estimates” tab to choose a good initial value for F_pop. Here F is not the absolute bioavailability, but it corresponds to a relative bioavailability between the individuals. Thus F_pop is the reference value for the bioavailability, and it should be fixed to 1. This can be done in the list of the initial estimates, by changing the estimation method for F_pop to “Fixed”:

Now that the model includes the relative bioavailability explicitly, we can consider IOV for F instead of for V. Since V and F are not identifiable together, we should not include IIV for F while there is already IIV for V.

Clicking on Formula displays the model for the individual parameters. For instance, the model for Tlag now includes a random effect eta_OCC_Tlag in addition to the random effect eta_ID_Tlag:

This project is saved as run02.mlxtran and all tasks are run.

The table of population parameters now include the standard deviations of the new random effects at the OCC level, which are called gamma:

There a few high rses for the standard deviations of the random effects, because it is not possible to identify well all the random effects with such a small dataset. The random effects with the smallest standard deviations could probably be removed, such as omega_ka or gamma_F. In a later step, we will check more precisely which random effects should be removed. For now, we will first check the relationships between the random effects and the covariate DIET.

On the individual fits, there are now different individual predictions for each occasion. The colors associated to each value of DIET for the observations can be changed in “Stratify” to match the colors of the predictions. The predictions from occasion 1 are in purple, they correspond to the first category noGuar. The second category withGuar corresponds to the second occasion with orange predictions.

After clicking on Information, the individual parameter values appear on the plots for each occasion (for example here for the two first individuals):

For V, which has only IIV, a single value is estimated for each individual across both occasions. For F, which has only IOV, it is important to note that estimated individual random effects from the distribution defined by gamma_F are independent across ids and occasions, and take into account the fact that F is slightly different for all subject-occasions. So a different value is estimated for each subject-occasion. Thus, the inter-occasion variability represents also an inter-individual variability. For parameters that have both IIV and IOV (Tlag, ka and k), the variability at the id level represents the additional variability between individuals that is common across both occasions.

The individual fits show that the IOV allows to properly capture the observations for each subject-occasion, and the predicted alcohol concentration seems usually higher when individuals have taken guar gum, except for ids 5 and 6. Let’s check this with the other diagnostic plots.

 

Assessing the effect of guar gum on alcohol’s PK

 

First, the plot of individual parameters vs covariates can be used to compare the distributions of each parameters across the two occasions.

Notable differences appear for ka, k and F. The kinetics with guar gum exhibit higher absorption rates and bioavailability, and smaller elimination rates. We can try to implement one or several of these differences in the model with a covariate effect, starting with the hypothesis that guar gum could affect the bioavailability of alcohol.

The statistical tests in Results show that these differences do not correspond to a significant correlation between eta_ka and DIET, but there is a significant correlation with eta_k, and a slightly significant correlation with eta_F. The lack of significance for ka and F is explained by the small size of the data which affects the p-values.

Second, we can have a look at the VPC split by DIET. Here, the prediction intervals are based on simulations that use the IOV included in the model, which is independent from DIET. Thus the prediction intervals are almost identical on each plot, while the empirical curves differ with DIET.

With the 4 bins computed by default, empirical curves are well captured by the prediction intervals, so with this size of data, the small differences caused by guar gum do not cause a visible discrepancy of the model, but when setting the number of bins to 6 (see below), a small discrepancy appears in the absorption phase. Although we should keep in mind that the empirical percentiles represent only a small number of individuals, this is a hint that it could be relevant to take into account an effect of guar gum on the absorption or the bioavailability.

 

Model with inter-occasion variability and occasion-varying covariate effect

 

As a result of this diagnosis, we can now adjust the model, after using the last estimates as new initial values. Based on the diagnostic plots and the biological knowledge on possible mechanisms for the effect of guar gum, we will try to explain part of the IOV by adding a covariate effect of DIET on F. We save this modified project as run03.mlxtran and run all tasks.

The new parameter beta_F_DIET_withGuar is estimated to a small value (0.08) but with a good standard error, and it results in a small decrease of gamma_F (from 0.07 to 0.034):

In the statistical tests, the p-value for the Wald test, which checks whether the parameter is close to 0, is small but the test is not quite significant. In addition, the correlation between F and DIET is significant:

The diagnostic plot also show that this correlation is strong:

Therefore the covariate effect is relevant and should not be removed from the model.

Moreover, the -2*LL and BICc for run03.mlxtran are slightly smaller than run02.mlxtran (2 points of difference), showing that the modified model still captures the data as well as the previous run. This can be seen easily by comparing the runs in Sycomore:

 

 

Estimation without simulated annealing

 

Finally, in the next step we are going to check more precisely whether some random effects are not well estimated and should be removed.

For the next run, we are going to modify the settings of SAEM to disable the simulated annealing:

This option is explained in details in this video. Briefly, it constrains the variance of the random effects to decrease slowly during the estimation, in order to explore a large parameter space to avoid getting stuck in a local maximum. A side-effect of the simulated annealing is that it may keep the omega values artificially high, and prevent the identification of parameters for which the variability is in fact zero. This leads to large values in the standard errors. So when large standard errors are estimated for random effects, like it is the case here for omega_ka and gamma_F, it is recommended to disable the simulated annealing once the estimated parameters are close to the solution.

Before changing the settings, the last estimates should be used as new initial values to start really close to the solution. The modified project is saved as run04.mlxtran and SAEM is run. In the graphical report, omega_ka and gamma_F now decrease to a very small value:

This confirms that there is not enough information in the data to identify the distributions for ka and F. Therefore, we can use the last estimates and then remove the IIV on ka and the IOV on F. The IOV on F can be removed while keeping the covariate effect of DIET, because F has also no IIV.

After doing this, we run the whole scenario again for the new project run05.mlxtran. All the parameters are now estimated with quite good standard errors, considering the small size of the data:

With the covariate effect on DIET on F, the discrepancy in the VPC for the occasions with guar gum is slightly smaller but still present, and is likely due to the variability in the data:

This run is the final model. Despite the small size of the data, it is able to take into account IIV and IOV, and to explain a modest part of the inter-occasion variability in bioavailability of alcohol by the effect of guar gum.

4.2.8.Mixture of distributions





 


Objectives: learn how to implement a mixture of distributions for the individual parameters.


Projects: PKgroup_project, PKmixt_project


Introduction

Mixed effects models allow us to take into account between-subject variability.

One complicating factor arises when data is obtained from a population with some underlying heterogeneity. If we assume that the population consists of several homogeneous subpopulations, a straightforward extension of mixed effects models is a finite mixture of mixed effects models.

There are two approaches to define a mixture of models:

The second approach assumes that the probability distribution of some individual parameters vary from one subpopulation to another one. The introduction of a categorical covariate (e.g., sex, phenotype, treatment, status, etc.) into such a model already supposes that the whole population can be decomposed into subpopulations. The covariate then serves as a label for assigning each individual to a subpopulation.

In practice, the covariate can either be known or not. If it is unknown, the covariate is called a latent covariate and is defined as a random variable with a user-defined number of modalities in the statistical model. Differences in estimation and diagnosis methods appear to deal with this additional random variable: this difference represents a task of unsupervised classification.
Mixture models usually refer to models for which the categorical covariate is unknown and unsupervised classification is needed.
For the sake of simplicity, we will consider a basic model that involves individual parameters \((\psi_i,1\leq i \leq N)\) and observations \((y_{ij}, i \leq N, 1\leq j \leq n_i)\). Then, the easiest way to model a finite mixture model is to introduce a label sequence \((z_i , 1\leq i \leq N)\) that takes its values in \(\{1,2,\ldots,M\}\) such that \(z_i=m\) if subject i belongs to subpopulation m.
In some situations, the label sequence \((z_i , 1\leq i \leq N)\) is known and can be used as a categorical covariate in the model. If \((z_i)\) is unknown, it can be modeled as a set of independent random variables taking their values in \(\{1,2,\ldots,M\}\) where for \(i=1,2,\ldots, N\), \(P(z_i = m)\) is the probability that individual belongs to group m. We will assume furthermore that the \((z_i)\) are identically distributed, i.e., \(P(z_i = m)\) does not depend on i for \(m=1, \ldots, M\).

Mixture of distributions based on a categorical covariate

  • PKgroup_project (data = ‘PKmixt_data.txt’, model = ‘lib:oral1_1cpt_kaVCl.txt’)

The sequence of labels is known as GROUP in this project and comes from the dataset. It is therefore defined as a categorical covariate that classifies  We can then assume, for instance different population values for the volume in the two groups and estimate the population parameters using this covariate model.

Then, this covariate GROUP can be used as a stratification variable and is very important in the modeling.

 

Mixture of distributions based on unsupervised classification with a latent covariate

A latent covariate is defined as a random variable, and the probability of each modality is part of the statistical model and is estimated as well. Methods for estimation and diagnosis are different. After the estimation, for each individual the categorical covariate is not perfectly known, only the probabilities of each modality are estimated.

Note also that latent covariates can be useful to model statistical mixtures of populations, but they provide no biological interpretation for the cause of the heterogeneity in the population since they do not come from the dataset.

Latent covariates can not be handled with IOV.

  • PKmixt_project (data = ‘PKmixt_data.txt’, model = ‘lib:oral1_1cpt_kaVCl.txt’)

We will use the same data with this project but ignoring the column GROUP (which is equivalent to assuming that the label is unknown). If we suspect some heterogeneity in the population, we can introduce a “latent covariate” by clicking on the grey button MIXTURE.

It is possible to change the name and the number of modalities of this latent covariate.
Remark: several latent covariates can be introduced in the model, with different number of categories.

We can then use this latent covariate lcat as any observed categorical covariate. Again, we can assume again different population values for the volume in the two groups by applying it on the volume random effect and estimating the population parameters using this covariate model. Proportions of each group are also estimated, plcat_1 which is the probability to have modality 1:

Once the population parameters are estimated, the sequence of latent covariates, i.e. the group to which belongs each subject, can be estimated together with the individual parameters, as the modes of the conditional distributions.

The sequence of estimated latent covariates lcat can be used as a stratification variable. We can for example display the VPC in the 2 groups:

By plotting the distribution of the individual parameters, we see that V has a bimodal distribution

5.Tasks and results






Monolix tasks

Monolix allows a workflow with several tasks.

On the interface, one can see six different tasks

  • POP. PARAM.: it corresponds to the estimation of the population parameters,
  • EBEs: it corresponds to the estimation of the individual parameters using the conditional mode, i.e. the most probable individual parameters.
  • CONDITIONAL DISTRIBUTION: It corresponds to the draws individual parameters based on the conditional distribution. It allows to compute the mean value of the conditional distribution.
  • STD. ERRORS.: it corresponds to the calculation of the Fisher information matrix and standard errors. Two methods are proposed for it. Either using the linearization method or using the stochastic approximation. The choice between those methods is done with the “Use linearization method” toggle under the tasks.
  • LIKELIHOOD: it corresponds to the explicit calculation of the log-likelihood. A specificity of the SAEM algorithm is that it does not explicitly compute the objective function. Thus, a dedicated task is proposed. Two methods are proposed for it. Either using the linearization method or using the importance sampling. The choice between those methods is done with the “Use linearization method” toggle under the tasks. This toogle is for both STD ERRORS and LIKELIHOOD tasks to be more relevant.
  • PLOTS: it corresponds to the generation of the plots.

Also, different types of results are available in the form of plots and tables. The tasks can be run individually by clicking on the associated button, or you can define a workflow by clicking on the tasks to run (on the small light blue checks) and click on the play button (in green) as proposed on the figure below.
Notice that you can initialize all the parameters and the associated methods in the “Initial Estimates” frame as described here.
Moreover, Monolix includes a convergence assessment tool. It is possible to execute a workflow as defined above but for different, randomly generated, initial values of fixed effects.

Monolix results

All the output files are detailed here.

Monolix-R functions

Monolix is now proposed with an API leading to the possibility to have access to the project exactly by the same way as you would do with the interface. All the functions are described here.

5.1.Estimation tasks

5.1.1.Initialization

Parameter initial estimates and associated methods






Initial values are specified for the fixed effects, for the standard deviations of the random effects and for the residual error parameters. These initial values are available through the frame “Initial estimates” of the interface as can be seen on the following figure. It is recommended to initialize the estimation to have faster convergence.

Initialization of the estimates

Initialization of the “Fixed effects”

The user can modify all the initial values of the fixed effects. When initializing the project, the values are set by default to 1. To change it, the user can click on the parameter and change the value

Notice that when you click on the parameter, an information is provided to tell what value is possible. The constraint depends on the distribution chosen for the parameter. For exemple, if the volume parameter V is defined as lognormal, its initial value should be strictly greater than 0. In that case, if you set a negative value, an error will be thrown and the previous parameter will be displayed.

When a parameter depends on a covariate, initial values for the dependency (named with \(\beta\) prefix, for instance beta_V_SEX_M to add the dependency of SEX, on parameter V) are displayed. The default initial value is 0. In case of a continuous covariate, the covariate is added linearly to the transformed parameter, with a coefficient \(\beta\). For categorical covariates, the initial value for the reference category will be the one of the fixed effect, while for all other categories it will be the initial value for the fixed effect plus the initial value of the \(\beta\), in the transformed parameter space. It is possible to define different initial values for the non-reference categories. The equations for the parameters can be visualized by clicking on button formula in the “Statistical model & Tasks” frame

Initialization of the “Standard deviation of the random effects”

The user can modify all the initial values of the standard deviations of the random effects. The default value is set to 1. We recommend to keep these values high in order for SAEM to have the possibility to explore the domain.

Initialization of the “Residual error parameters”

The user can modify all the initial values of the residual error parameters. There are as many lines as continuous outputs of the model. The default value depends on the parameter (1 for “a”, 0.3 for “b” and 1 for “c”).

What method can I use for the parameters estimations?

For all the parameters, there are several methods for the estimation

  • “Fixed”: the parameter is kept to its initial value and so, it will not be estimated. In that case, the parameter name is set to orange.
  • “Maximum Likelihood Estimation”: The parameter is estimated using maximum likelihood. In that case, the parameter name remains grey. This is the default option
  • “Maximum A Posteriori Estimation”: The parameter is estimated using maximum a posteriori estimation. In that case, the user has to define both a typical value and a standard deviation. For more about this, see here. In that case, the parameter name is colored in purple.

To change the method, click on the right of the parameter as on the following.

A window pops up to choose the method as on the following figure


Notice that you have buttons to fix all the parameters or estimate all on the top right of the window as can be seen on the following figure

How to initialize your parameters?

On the use of last estimates





If you have already estimated the population parameters for this project, then you can use the “Use the last estimates” buttons to use the previous estimates as initial values. The user has the possibility to use all the last estimates or only the fixed effects. The interest of using only the fixed effects is not to have too low initial standard effects and thus let SAEM explore a larger domain for the next run.

Check initial fixed effects

When clicking on the “Check the initial fixed effects”, the simulations obtained with the initial population fixed effects values are displayed for each individual together with the data points, in case of continuous observations. It allows also an automatic initialization in case of a model of the PK library as described here.

5.1.1.1.Check initial estimates and auto init

Check initial fixed effects

The subtab “Check initial estimates” is part of the tab “Initial estimates“.

When clicking on the “Check the initial estimates”, the simulations obtained with the initial population fixed effects values and the individual designs (doses and regressors) are displayed for each individual together with the data points, in case of continuous observations. This feature is very useful to find some “good” initial values. Although Monolix is quite robust with respect to initial parameter values, good initial estimates speed up the estimation. You can change the values of the parameters on the bottom of the screen and see how the agreement with the data change. In addition, you can change the axis to log-scale and choose the same limit on all axes to have a better comparison of the individuals. When you are confident with the initial values, you should click on the “SET AS INITIAL VALUES” button on the top of the frame to validate the selection.

In addition, if you think that there are not enough points for the prediction (if there are a lot of doses for example), you can change the discretization and increase the number of points as displayed in the blue box of the figure.

If several observation ids have been mapped to model outputs (for example a parent and a metabolite, or a PK and a PD observation), you can select which output to look at under the output section on the right:

 

Using the reference in the “check initial estimates”

Starting from the 2019 version, it is possible to add a reference and thus change a parameter to see the impact of the variation of this parameter. In this example, we click on reference to use the current fit as reference and change k12 from 1 to 2 as can be seen on the following figure.

 

 

The solid red curve corresponds to the current curve and the dashed one corresponds to the reference. At any time, you can change the reference to use the current fit, and restore and delete the reference or delete all references by clicking on the icons at the top right of the frame.

 

Automatic initialization of the parameters

Starting from the 2021 version, an auto-init section appears on the right side of the frame:

 

By clicking on RUN, Monolix will compute initial population parameters that best fit the data points, starting from parameter values currently used in the initial estimates panel, and using the data from the 12 first individuals by default, and from all observations mapped to a model output. It is possible to change the set of individuals used in the Ids selection panel just below the RUN button.

The computation is done with a custom optimization method, on the pooled data, without inter-individual variability. The purpose is not to find a perfect match but rather to have all the parameters in the good range for starting the population modeling approach. While auto-init is running, we show the evolution of the cost of the optimization algorithm over the iterations. It is possible to stop the algorithm at any time if you find the cost has decreased sufficiently and you want to have a look at the parameter values. Note that the more individuals are selected, the longer the run will take. Moreover, it may be easier for the auto-init algorithm to find a point in the parameter space that is sensitive to specific model features (eg a third compartment, a complex absorption) if you select only a few individuals (1-3 for instance) for which this feature can be observed.

 

 

After clicking on the button, the population parameters are updated and the corresponding fit is displayed. To use these parameters as initial estimates, you need to click on the button “SET AS INITIAL VALUES”. A new reference appears with previous parameter values so that you can come back to previous values if you are not satisfied with the fit.

Note that the auto-init procedure takes into account the current initial values. Therefore, in the few cases where the auto-init might give poor results, it is possible to improve the results by changing manually the parameter values before running the auto-init again.

5.1.2.Population parameter estimation using SAEM

Purpose

The estimation of the population parameters is the key task in non-linear mixed effect modeling. In Monolix, it is performed using the Stochastic Approximation Expectation-Maximization (SAEM) algorithm [1]. SAEM has been shown to be extremely efficient for both simple and a wide variety of complex models: categorical data [2], count data [3], time-to-event data [4], mixture models [56], differential equation based models, censored data [7], … The convergence of SAEM has been rigorously proven [1] and its implementation in Monolix is particularly efficient. No other algorithms are available in Monolix.

Calculations: the SAEM algorithm





 

 

Running the population parameter estimation task

Overview

The pop-up window which permits to follow the progress of the task is shown below. The algorithm starts with a small number (5 by default) of burn-in iterations for initialization which are displayed in the following way: (note that this step can be so fast that it is not visible by the user)

Afterwards, the evolution of the value for each population parameter over the iterations of the algorithm is displayed. The red line marks the switch from the exploratory phase to the smoothing phase. The exact value at each iterations can be followed by hovering over the curve (as for Cl_pop below). The convergence indicator (in purple) helps to detect that convergence has been reached (see below for more details).

 

Dependencies between tasks

The “Population parameter” estimation task must be launched before running any other task. To skip this task, the user can fix all population parameters. If all population parameters have been set to “fixed”, the estimation will stop after a single iteration and allow the user to continue with the other tasks.

The convergence indicator





The convergence indicator (also sometimes called complete likelihood) is defined as the joint probability distribution of the data and the individual parameters and can be decomposed using Bayes law:

\(\sum_{i=1}^{N_{\text{ind}}}\log\left(p(y_i, \phi_i; \theta)\right)=p(y_i| \psi_i; \theta)p(\psi_i; \theta)\)

Those two terms have an analytical expression and are easy to calculate, using as \(\phi_i\) the individual parameters drawn by MCMC for the current iteration of SAEM. This quantity is calculated at each SAEM step and is useful to assess the convergence of the SAEM algorithm.
The convergence indicator aggregates the information from all parameters and can serve to detect if the SAEM algorithm has already converged or not. When the indicator is stable, that is it oscillates around the same value without drifting, then we can be pretty confident that the maximum likelihood has been achieved. The convergence indicator is used, among other measures, in the auto-stop criteria to switch from the exploratory phase to the smoothing phase.
Note that the likelihood (i.e the objective function) \(\sum_{i=1}^{N_{\text{ind}}}\log\left(p(y_i; \theta)\right)\) cannot be computed in closed form because the individual parameters \(\phi_i\) are unobserved. It requires to integrate over all possible values of the individual parameters. Thus, to estimate the log-likelihood an importance sampling Monte Carlo method is used in a separate task (or an approximation is calculated via linearization of the model).

The simulated annealing





The simulated annealing option (setting enabled by default) permits to keep the explored parameter space large for a longer time (compared to without simulated annealing). This allows to escape local maximums and improve the convergence towards the global maximum.
In practice, the simulated annealing option constrains the variance of the random effects and the residual error parameters to decrease by maximum 5% (by default – the setting “Decreasing rate” can be changed) from one iteration to the next one. As a consequence, the variances decrease more slowly:

The size of the parameter space explored by the SAEM algorithm depends on individual parameters sampled from their conditional distribution via Markov Chain Monte Carlo. If the standard deviation of the conditional distributions is large, the individual parameters sampled at iteration k can be quite far away from those at iteration (k-1), meaning a large exploration of the parameter space. The standard deviation of the conditional distribution depends on the standard deviation of the random effects (population parameters ‘omega’). Indeed, the conditional distribution is \(p(\psi_i|y_i;\hat{\theta})\) with \(\psi_i\) the individual parameters for individual \(i\), \(\hat{\theta}\) the estimated population parameters, and \(y_i\) the data (observations) for individual \(i\). The conditional distribution thus depends on the population parameters, and the larger the population parameters ‘omega’, the larger the standard deviation of the conditional distribution. That’s why we want to keep large ‘omega’ values during the first iterations.

Methods for the parameters without variability





Parameters without variability are not estimated in the same way as parameters with variability. Indeed, the SAEM algorithm requires to draw parameter values from their marginal distribution, which exists only for parameters with variability.
Several methods can be used to estimate the parameters without variability. By default, these parameters are optimized using the Nelder-Mead simplex algorithm (Matlab’s fminsearch method). Other options are also available in the SAEM settings:

  • No variability (default): optimization via Nelder-Mead simplex algorithm
  • Add decreasing variability: an artificial variability (i.e random effects) is added for these parameters, allowing estimation via SAEM. The variability starts at omega=1 and is progressively decreased such that at the end of the estimation process, the parameter has a variability of 1e-5. The decrease in variability is exponential with a rate based on the maximum number of iterations for both the exploratory and smoothing phases. Note that if the autostop is triggered, the resulting variability might me higher.
  • Variability at the first stage: during the exploratory phase of SAEM, an artificial variability is added and progressively forced to 1e-5 (same as above). In the smoothing phase, the Nelder-Mead simplex algorithm is used.

Depending on the specific project, one or the other method may lead to a better convergence. If the default method does not provide satisfying results, it is worth trying the other methods. In terms of computing time, if all parameters are without variability, the first option will be faster because only the Nelder-Mead simplex algorithm will be used to estimate all the fixed effects. If some parameters have random effects, the first option will be slower because the Nelder-Mead and the SAEM algorithm are computed at each step. In that case the second or third option will be faster because only the SAEM algorithm will be required when the artificial variability is added.

Alternatively, the standard deviation of the random effects can be fixed to a small value, for instance 5% for log-normally distributed parameters. (See next section on how to enforce a fixed value). With this method, the SAEM algorithm can be used, and the variability is kept small.

 

Other estimation methods: fixing population parameters or Bayesian estimation

Instead of the default estimation method with SAEM, it is possible to fix a population parameter, or set a prior on the estimate and use Bayesian estimation. This page gives details on these methods and how to use them.

 

 

Outputs

In the graphical user interface

The estimated population parameters are displayed in the POP.PARAM section of the RESULTS tab. Fixed effects names are “*_pop”, the standard deviation of the random effects “omega_*”, parameters of the error model “a”, “b”, “c”, the correlation between random effects “corr_*_*” and parameters associated to covariates “beta_*_*”. The standard deviation of the random effects is also expressed as coefficient of variation (CV) – a feature present in versions Monolix 2023 or above. The CV calculation depends on the parameter distribution:

  • lognormal: \(CV=100*\sqrt{exp(\omega_p^2) -1}\)
  • normal: \(CV=100*\frac{\omega_p}{p_{pop}}\)
  • logitnormal and probitnormal: the CV is computed by Monte-Carlo. 100000 samples X are drawn from the distribution (defined by \(\omega_p\) and \(p_{pop}\)) and the CV is calculated as the ratio of the sample standard deviation over the sample mean: \(CV=100*\frac{sd(X)}{mean(X)}\)

When you run the “Standard errors” task, then the population parameter table contains also the standard error (s.e) and relative standard error (r.s.e). Note that the CV% represents the inter-individual variability, while the RSE% represents the uncertainty on the estimated parameters.

Starting from version 2024, alongside the standard errors, the upper and lower percentiles, here P2.5 and P97.5 of the confidence interval with level 1−\( \alpha\) are computed for population parameters. This calculation depends on the execution of the “Standard errors” task, as it relies on the standard error of the respective parameter. In order to take into account the boundaries of the parameters (e.g omega or V_pop cannot be negative), the SE is transformed into the gaussian domain, the CI in gaussian domain is calculated assuming a normal distribution and finally the CI is backtransformed. Its transformation is determined by the type of parameter and its distribution assumption:

  • Normal distributed parameters: \(CI_{1-\alpha} (\theta _{pop})=[\theta_{pop} + q_{\alpha /2}\times s.e.(\theta_{pop}), ~\theta_{pop} + q_{1 – \alpha /2}\times s.e.(\theta_{pop})]\). This applies for typical values (fixed effects) of normally distributed parameters and covariate effects (betas).
  • Lognormal distributed parameters: \(CI_{1-\alpha} (\theta _{pop})=[\exp(\mu_{pop} + q_{\alpha /2}\times s.e.(\mu_{pop})), ~\exp(\mu_{pop} + q_{1 – \alpha /2}\times s.e.(\mu_{pop}))]\),  where \( \mu_{pop} = \ln(\theta_{pop})\). This applies to typical values (fixed effects) of lognormally distributed parameters, standard deviations of the random effects (omegas), and error model parameters.
  • Logitnormal distributed parameters: \(CI_{1-\alpha} (\theta _{pop})=[logit^{-1}(\mu_{pop} + q_{\alpha /2}\times s.e.(\mu_{pop})), ~logit^{-1}(\mu_{pop} + q_{1 – \alpha /2}\times s.e.(\mu_{pop}))]\),  where \( \mu_{pop} = logit(\theta_{pop})\). This applies to typical values (fixed effects) of logit-normally distributed parameters and correlation parameters.

where \(q_{\alpha /2}\) is the quantile of order \( \alpha /2\) for a standard normal distribution. The confidence interval level \( \alpha\) can be modified in the settings of the “Standard errors” task within the Statistical Model & Tasks tab by clicking on the wrench icon.

Information about the SAEM task performance are below the table (orange frame):

  • The total elapsed time for this task
  • The number of iterations in the exploratory and smoothing phases, along with a message that indicates whether the convergence has been reached (“auto-stop”), or if the algorithm arrived at the maximum number of iterations (“stopped at the maximum number of iterations/auto-stop criteria have not been reached”) or if it was stopped by the user (“manual stop”). (for Monolix versions 2021 or above)

A “Copy table” icon on the top of the table allows to copy it in Excel, Word, etc. The table format and display is kept.

Starting from version 2024R1, if categorical covariate effects are included in the statistical model, a “Display fixed effects by category” toggle allows to add a section named “Fixed effects by category” in the table, with the values of the typical population parameters in each category of the categorical covariates and their SEs, RSEs and confidence intervals.

In the output folder

After having run the estimation of the population parameters, the following files are available:

  • summary.txt: contains the estimated population parameters (and the number of iterations in Monolix2021R1), in a format easily readable by a human (but not easy to parse for a computer)
  • populationParameters.txt: contains the estimated population parameters (by default in csv format), including the CV.
  • populationParametersByGroups.txt: contains the estimated population parameters by categories of categorical covariates included in the statistical model. This file is not generated if there are no categorical covariate effects.
  • predictions.txt: contains for each individual and each observation time, the observed data (y), the prediction using the population parameters and population median covariates value from the data set (popPred_medianCOV), the prediction using the population parameters and individual covariates value (popPred), the prediction using the individual approximate conditional mean calculated from the last iterations of SAEM (indivPred_SAEM) and the corresponding weighted residual (indWRes_SAEM).
  • IndividualParameters/estimatedIndividualParameters.txt: individual parameters corresponding to the approximate conditional mean, calculated as the average of the individual parameters sampled by MCMC during all iterations of the smoothing phase. When several chains are used (see project settings), the average is also done over all chains. Values are indicated as *_SAEM in the file.
    Parameters without variability: 

    • method “no variability” or “variability at the first stage”: *_SAEM represents the value at the last SAEM iteration, so the estimated population parameter plus the covariate effects. In absence of covariate, all individuals have the same value.
    • method “add decreasing variability”: *_SAEM represents the average of all iterations of the smoothing phase. This value can be slightly different from individual to individual, even in te absence of covariates.
  • IndividualParameters/estimatedRandomEffects.txt: individual random effects corresponding to the approximate conditional mean, calculated using the last estimations of SAEM (*_SAEM).
    For parameters without variability, see above.

More details about the content of the output files can be found here.

Settings





The settings are accessible through the interface via the button next to the parameter estimation task.

Burn-in phase:

The burn-in phase corresponds to an initialization of SAEM: individual parameters are sampled from their conditional distribution using MCMC using the initial values for the population parameters (no update of the population parameter estimates).
Note: the meaning of the burn-in phase in Monolix is different to what is called burn-in in Nonmem algorithms.

  • Number of iterations (default: 5): number of iterations of the burn-in phase

Exploratory phase

  • Auto-stop criteria (default: yes): if ticked, auto-stop criteria are used to automatically detect convergence during the exploratory phase. If convergence is detected, the algorithm switches to the smoothing phase before the maximum number of iterations. The criteria take into account the stability of the convergence indicator, omega parameters and error model parameters.
  • Maximum number of iterations (default: 500, if auto-stop ticked): maximum number of iterations for the exploratory phase. Even if the auto-stop criteria are not fulfilled, the algorithm switches to the smoothing phase after this maximum number of iterations. A warning message will be displayed in the GUI if the maximum number of iterations is reached while the auto-stop criteria are not fulfilled.
  • Minimum number of iterations (default: 150, if auto-stop ticked): minimum number of iterations for the exploratory phase. This value also corresponds to the interval length over which the auto-stop criteria are tested. A larger minimum number of iterations means that the auto-stop criteria are harder to reach.
  • Number of iterations (default: 500, if auto-stop unticked): fixed number of iterations for the exploratory phase.
  • Step-size exponent (default: 0): The value, comprised between 0 and 1, represents memory of the stochastic process, i.e how much weight is given at iteration k to the value of the previous iteration compared to the new information collected. A value 0 means no memory, i.e the parameter value at iteration k is built based on the information collected at that iteration only, and does not take into account the value of the parameter at the previous iteration.
  • Simulated annealing (default: enabled): the Simulated Annealing version of SAEM permits to better explore the parameter space by constraining the standard deviation of the random effects to decrease slowly.
  • Decreasing rate for the variance of the residual errors (default: 0.95, if simulated annealing enabled): the residual error variance (parameter “a” for a constant error model for instance) at iteration k is constrained to be larger than the decreasing rate times the variance at the previous iteration.
  • Decreasing rate for the variance of the individual parameters (default: 0.95, if simulated annealing enabled): the variance of the random effects at iteration k is constrained to be larger than the decreasing rate times the variance at the previous iteration.

Smoothing phase

  • Auto-stop criteria (default: yes): if ticked, auto-stop criteria are used to automatically detect convergence during the smoothing phase. If convergence is detected, the algorithm stops before the maximum number of iterations.
  • Maximum number of iterations (default: 200, if auto-stop ticked): maximum number of iterations for the smoothing phase. Even if the auto-stop criteria are not fulfilled, the algorithm stops after this maximum number of iterations.
  • Minimum number of iterations (default: 50, if auto-stop ticked): minimum number of iterations for the smoothing phase. This value also corresponds to the interval length over which the auto-stop criteria is tested. A larger minimum number of iterations means that the auto-stop criteria is harder to reach.
  • Number of iterations (default: 200, if auto-stop unticked): fixed number of iterations for the smoothing phase.
  • Step-size exponent (default: 0.7): The value, comprised between 0 and 1, represents memory of the stochastic process, i.e how much weight is given at iteration k to the value of the previous iteration compared to the new information collected.  The value must be strictly larger than 0.5 for the smoothing phase to converge. Large values (close to 1) will result in a smoother parameter trajectory during the smoothing phase, but may take longer to converge to the maximum likelihood estimate.

 

Methodology for parameters without variability (if parameters without variability are present in the model):

The SAEM algorithm requires to draw parameter values from their marginal distribution, which does not exist for parameters without variability. These parameters are thus estimated via another method, which can be chosen among:

  • No variability (default choice): After each SAEM iteration, the parameter without variability are optimized using the Nelder-Mead simplex algorithm. The absolute tolerance (stopping criteria) is 1e-4 and the maximum number of iterations 20 times the number of parameters to calculate via this algorithm.
  • Add decreasing variability: an artificial variability is added for these parameters, allowing estimation via SAEM. The variability is progressively decreased such that at the end of the estimation process, the parameter has a variability of 1e-5.
  • Variability in the first stage: during the exploratory phase, an artificial variability is added and progressively forced to 1e-5 (same as above). In the smoothing phase, the Nelder-Mead simplex optimization algorithm is used.

Handling parameters without variability is also discussed here.

Set all SAEM iterations to zero in one click

The icon on the bottom left provides a shortcut to set all SAEM iterations to zero (in all three phases). This is convenient if the user wish to skip the estimation of the population parameters and keep the initial estimates as population estimates to run the other tasks, since running SAEM first is mandatory. It is not equivalent to fixing all population parameters, since standard errors are not estimated for fixed parameters, while they will be estimated for the initial estimates in this case. The action can be easily reversed with the second shortcut to reset default SAEM iterations.

 

Good practice, troubleshooting and tips

Choosing to enable or disable the simulated annealing

As the simulated annealing option permits to more surely find the global maximum, it should be used during the first runs, when the initial values may be quite far from the final estimates.
On the other side, the simulated annealing option may keep the omega values artificially high, even after a large number of SAEM iterations. This may prevent the identification of parameters for which the variability is in fact zero and lead to NaN in the standard errors. So once good initial values have been found and there is no risk to fall in a local maximum, the simulated annealing option can be disabled. Below we show an example where removing the simulated annealing permits to identify parameters for which the inter-individual variability can be removed.

Example: The dataset used in the tobramycin case study is quite sparse. In these conditions, we expect that estimating the inter-individual variability for all parameters will be difficult. In this case, the estimation can be done in two steps, as shown below for a two-compartments model on this dataset:

  • First, we run SAEM with the simulated annealing option (default setting), which facilitates the convergence towards the global maximum. All four parameters V, k, k12 and k21 have random effects. The estimated parameters are shown below:

The parameters omega_k12 and omega_k21 have high standard errors, suggesting that the variability is difficult to estimate. The omega_k12 and omega_k21 values themselves are also high (100% inter-individual variability), suggesting that they may have been kept too high due to the simulated annealing.

  • As a second step, we use the last estimates as new initial values (as shown here), and run SAEM again after disabling the simulated annealing option. On the plot showing the convergence of SAEM, we can see omega_V, omega_k12 and omega_k21 decreasing to very low values. The data is too sparse to correctly identify the inter-individual variability for V, k12 and k21. Thus, their random effects can be removed, but the random effect of k can be kept.

Note that because the omega_V, omega_k12 and omega_k21 parameters decrease without stabilizing, the convergence indicator does the same.

 

5.1.2.1.The convergence indicator

When you launch the estimation of the population parameters, you can see the evolution of the population parameter estimates over the iterations of the SAEM algorithm but also the convergence indicator in purple.

The convergence indicator is the complete log-likelihood. It can help to follow convergence.

Note that the complete likelihood is not the same as the log-likelihood computed as separate task.

 

Log-likelihood

The likelihood is the probability density of the data given the population parameter, so the log-likelihood is defined as:

\(\sum_{i=1}^{N_{\text{ind}}}\log\left(p(y_i; \theta)\right)\)

The likelihood is the objective function, therefore it is the relevant quantity to compare model, but unfortunately it cannot be computed in closed form because the individual parameters \(\phi_i\) are unobserved. It requires to integrate over all possible values of the individual parameters. Thus, to estimate the log-likelihood an importance sampling Monte Carlo method is used in a separate task (or an approximation is calculated via linearization of the model).

 

Complete log-likelihood

On the contrary, the complete likelihood refers to the joint probability distribution of the data and the individual parameters. The convergence indicator (complete log-likelihood) is then defined as:

\(\sum_{i=1}^{N_{\text{ind}}}\log\left(p(y_i, \phi_i; \theta)\right)\)

The joint probability distribution can be decomposed using Bayes law as:

\(p(y_i, \psi_i; \theta)=p(y_i| \psi_i; \theta)p(\psi_i; \theta)\)

Those two terms have an analytical expression and are easy to calculate, using as \(\phi_i\) the individual parameters drawn by MCMC for the current iteration of SAEM. This quantity is calculated at each SAEM step and is useful to assess the convergence of the SAEM algorithm.

 

Typical shape of the convergence indicator

Typically, the convergence indicator decreases progressively and then stabilizes. The convergence indicator aggregates the information from all parameters and can serve to detect if the SAEM algorithm has already converged or not. When the indicator is stable, that is it oscillates around the same value without drifting, then we can be pretty confident that the maximum likelihood has been achieved.

5.1.2.2.Simulated annealing

The estimation of the population parameters with SAEM includes a method of simulated annealing. It is possible to disable this option in the settings of SAEM. The option is enabled by default.







Purpose

The simulated annealing option permits to keep the explored parameter space large for a longer time (compared to without simulated annealing). This allows to escape local maximums and improve the convergence towards the global maximum.

Calculations

In practice, the simulated annealing option constrains the variance of the random effects and the residual error parameters to decrease by maximum 5% (by default, the setting “Decreasing rate” can be changed) from one iteration to the next one. As a consequence, the variances decrease more slowly:

The size of the parameter space explored by SAEM depends on individual parameters sampled from their conditional distribution via Markov Chain Monte Carlo. If the standard deviation of the conditional distributions is large, the individual parameters sampled at iteration k can be quite far away from those at iteration (k-1), meaning a large exploration of the parameter space. The standard deviation of the conditional distribution depends on the standard deviation of the random effects (population parameters ‘omega’). Indeed, the conditional distribution is \(p(\psi_i|y_i;\hat{\theta})\) with \(\psi_i\) the individual parameters for individual \(i\), \(\hat{\theta}\) the estimated population parameters, and \(y_i\) the data (observations) for individual \(i\). The conditional distribution thus depends on the population parameters, and the larger the population parameters ‘omega’, the larger the standard deviation of the conditional distribution. That’s why we want to keep large ‘omega’ values during the first iterations.

Settings

The simulated annealing option can be disabled or enabled in the “Population parameters” task settings. In addition, the settings allow to change the default decreasing rates for the standard deviations of the random effects and the residual errors.

Choosing to enable or disable the simulated annealing

As the simulated annealing option permits to more surely find the global maximum, it should be used during the first runs, when the initial values may be quite far from the final estimates.
On the other side, the simulated annealing option may keep the omega values artificially high, even after a large number of SAEM iterations. This may prevent the identification of parameters for which the variability is in fact zero and lead to NaN in the standard errors. So once good initial values have been found and there is no risk to fall in a local maximum, the simulated annealing option can be disabled. Below we show an example where removing the simulated annealing permits to identify parameters for which the inter-individual variability can be removed.

Example: identifying parameters with no variability

The dataset used in the tobramycin case study is quite sparse. In these conditions, we expect that estimating the inter-individual variability for all parameters will be difficult. In this case, the estimation can be done in two steps, as shown below for a two-compartments model on this dataset:

  • First, we run SAEM with the simulated annealing option (default setting), which facilitates the convergence towards the global maximum. All four parameters V, k, k12 and k21 have random effects. The estimated parameters are shown below:

The parameters omega_k12 and omega_k21 have high standard errors, suggesting that the variability is difficult to estimate. The omega_k12 and omega_k21 values themselves are also high (100% inter-individual variability), suggesting that they may have been kept too high due to the simulated annealing.

  • As a second step, we use the last estimates as new initial values (as shown here), and run SAEM again after disabling the simulated annealing option. On the plot showing the convergence of SAEM, we can see omega_V, omega_k12 and omega_k21 decreasing to very low values. The data is then too sparse to correctly identify the inter-individual variability for V, k12 and k21. Thus, their random effects can be removed, but the random effect of k can be kept.

Note that because the omega_V, omega_k12 and omega_k21 parameters decrease without stabilizing, the convergence indicator does the same.

5.1.2.3.Bayesian estimation

In the tab “Initial estimates”, clicking on the wheel icon next to a population parameters opens a window to choose among three estimation methods (see image below). “Maximum Likelihood Estimation” corresponds to the default method using SAEM, detailed on this page. “Maximum A Posteriori Estimation” corresponds to Bayesian estimation, and “Fixed” to a fixed parameter.

 

 

 

Bayesian estimation

Purpose

Bayesian estimation allows to take into account prior information in the estimation of parameters. It is called in Monolix Maximum A Posteriori estimation, and it corresponds to a penalized maximum likelihood estimation, based on a prior distribution defined for a parameter. The weight of the prior in the estimation is given by the standard deviation of the prior distribution.

 


Objectives: learn how to combine maximum likelihood estimation and Bayesian estimation of the population parameters.


Projects: theobayes1_project, theobayes2_project,


Introduction

The Bayesian approach considers the vector of population parameters \(\theta\) as a random vector with a prior distribution \(\pi_\theta\). We can then define the *posterior distribution* of \(\theta\):

\(\begin{aligned} p(\theta | y ) &= \frac{\pi_\theta( \theta )p(y | \theta )}{p(y)} \\ &= \frac{\pi_\theta( \theta ) \int p(y,\psi |\theta) \, d \psi}{p(y)} . \end{aligned} \)

We can estimate this conditional distribution and derive statistics (posterior mean, standard deviation, quantiles, etc.) and the so-called maximum a posteriori (MAP) estimate of \(\theta\):

\(\begin{aligned} \hat{\theta}^{\rm MAP} &=\text{arg~max}_{\theta} p(\theta | y ) \\ &=\text{arg~max}_{\theta} \left\{ {\cal LL}_y(\theta) + \log( \pi_\theta( \theta ) ) \right\} . \end{aligned} \)

The MAP estimate maximizes a penalized version of the observed likelihood. In other words, MAP estimation is the same as penalized maximum likelihood estimation. Suppose for instance that \(\theta\) is a scalar parameter and the prior is a normal distribution with mean \(\theta_0\) and variance \(\gamma^2\). Then, the MAP estimate is the solution of the following minimization problem:

\(\hat{\theta}^{\rm MAP} =\text{arg~min}_{\theta} \left\{ -2{\cal LL}_y(\theta) + \frac{1}{\gamma^2}(\theta – \theta_0)^2 \right\} .\)

This is a trade-off between the MLE which minimizes the deviance, \(-2{\cal LL}_y(\theta)\), and \(\theta_0\) which minimizes \((\theta – \theta_0)^2\). The weight given to the prior directly depends on the variance of the prior distribution: the smaller \(\gamma^2\) is, the closer to \(\theta_0\) the MAP is. In the limiting case, \(\gamma^2=0\); this means that \(\theta\) is fixed at \(\theta_0\) and no longer needs to be estimated. Both the Bayesian and frequentist approaches have their supporters and detractors. But rather than being dogmatic and following the same rule-book every time, we need to be pragmatic and ask the right methodological questions when confronted with a new problem.
All things considered, the problem comes down to knowing whether the data contains sufficient information to answer a given question, and whether some other information may be available to help answer it. This is the essence of the art of modeling: find the right compromise between the confidence we have in the data and our prior knowledge of the problem. Each problem is different and requires a specific approach. For instance, if all the patients in a clinical trial have essentially the same weight, it is pointless to estimate a relationship between weight and the model’s PK parameters using the trial data. A modeler would be better served trying to use prior information based on physiological knowledge rather than just some statistical criterion.
Generally speaking, if prior information is available it should be used, on the condition of course that it is relevant. For continuous data for example, what does putting a prior on the residual error model’s parameters mean in reality? A reasoned statistical approach consists of including prior information only for certain parameters (those for which we have real prior information) and having confidence in the data for the others. Monolix allows this hybrid approach which reconciles the Bayesian and frequentist approaches. A given parameter can be

  • a fixed constant if we have absolute confidence in its value or the data does not allow it to be estimated, essentially due to lack of identifiability.
  • estimated by maximum likelihood, either because we have great confidence in the data or no information on the parameter.
  • estimated by introducing a prior and calculating the MAP estimate or estimating the posterior distribution.

 

Computing the Maximum a posteriori (MAP) estimate

  • demo project: theobayes1_project (data = ‘theophylline_data.txt’ , model = ‘lib:oral1_1cpt_kaVCl.txt’)

We want to introduce a prior distribution for \(ka_{\rm pop}\) in this example. Click on the option button

and select Maximum A Poteriori Estimation


We propose a typical value, here 2 and standard deviation 0.1 for \(ka_{\rm pop}\) and to compute the MAP estimate for \(ka_{\rm pop}\). The parameter is then colored in purple.

 

Starting from the 2021 version, it is possible to select maximum a posteriori estimation also for the omega parameters (standard deviations of the random effects). In this case, an inverse Wishart is set as a prior distribution for the omega matrix.

The following distributions for the priors are used:

  • typical value (*_pop): the distribution of the prior is the same as the distribution of the parameter. For instance, if ka has been set with a lognormal distribution in the “Statistical model & Tasks” tab, a lognormal distribution is also used for the prior on ka_pop. When a lognormal distribution is used, setting sd=0.1 roughtly corresponds to 10% uncertainty in the provided prior value for ka_pop.
  • covariate effects (beta_*): a normal distribution is used, to allow betas to be either positive or negative.
  • standard deviations (omega_*) [starting version 2021]: an inverse Wishart distribution is used. Inverse Wisharts are common prior distributions for variance-covariance matrices as they allow to fulfill the positive-definite matrix requirement. The weight of the prior in the estimation is based on the number of degrees of freedom (df) of the inverse Wishart, instead of a standard deviation. More degrees of freedom correspond to a higher constrain of the prior in the omega estimation. In Monolix, each omega parameter is handled as a 1×1 matrix with its own degree of freedom, independently from the other omegas. The univariate inverse Wishart \( W^{-1}(df, (df+2)\omega_{typ}) \) simplifies to an inverse gamma distribution with shape parameter \(\alpha=df/2 \) and scale parameter \(\beta=\omega_{typ}*(df+2)/2\). The coefficient of variation is thus \(CV=\frac{1}{\sqrt{\frac{df}{2}-2}}\). To obtain a 20% uncertainty on omega (CV=0.2), the user can set df=50.
  • correlations: it is currently not possible to set a prior on the correlation parameters

It is common to set the initial value of the parameter to be the same as the typical value of the prior. Note that the default value for the typical value of the prior is set to the initial value of the parameter. However, if the initial value of the parameter is modified afterwards, the typical value of the prior is not updated automatically.

Fixing the value of a parameter

Population parameters can be fixed to their initial values, in this case they are not estimated. It is possible to fix one, several or all population parameters, among the fixed effects, standard deviations of random effects, and error model parameters. In Monolix2021, it is also possible to fix correlation parameters.

To fix a population parameter, click on the wheel next to the parameter in the tab “Initial estimates” and select “Fixed”, like on the image below:

Fixed parameters appear on this tab in red. In Monolix2021, they are also colored in red in the subtab “Check initial estimates”.

 

  • theobayes2_project (data = ‘theophylline_data.txt’ , model = ‘lib:oral1_1cpt_kaVCl.txt’)

We can combine different strategies for the population parameters: Bayesian estimation for \(ka_{\rm pop}\), fixed value for \(V_{\rm pop}\) and maximum likelihood estimation for \(Cl_{\rm pop}\), for instance.

Remark:

  • The parameter \(V_{\rm pop}\) is fixed and then colored in red.
  • \(V_{\rm pop}\) is not estimated (it’s s.e. is not computed) but the standard deviation \(\omega_{V}\) is estimated as usual.

5.1.3.EBEs







Purpose

EBEs stands for Empirical Bayes Estimates. The EBEs are the most probable value of the individual parameters (parameters for each individual), given the estimated population parameters and the data of each individual. In a more mathematical language, they are the mode of the conditional parameter distribution for each individual.

These values are useful to compute the most probable prediction for each individual, for comparison with the data (for instance in the Individual Fits plot).

Calculation of the EBEs (conditional mode)

When launching the “EBEs” task, the mode of the conditional parameter distribution is calculated.

Conditional distribution

The conditional distribution is \( p(\psi_i|y_i;\hat{\theta})\) with \(\psi_i\) the individual parameters for individual i, \(\hat{\theta}\) the estimated population parameters, and \(y_i\) the data (observations) for individual i. The conditional distribution represents the uncertainty of the individual’s parameter value, taking into account the information at hand for this individual: the observed data for that individual, the covariate values for that individual and the fact that the individual belongs to the population for which we have already estimated the typical parameter value (fixed effects) and the variability (standard deviation of the random effects). It is not possible to directly calculate the probability for a given \(\psi_i\) (no closed form), but is possible to obtain samples from the distribution using a Markov-Chain Monte-Carlo procedure (MCMC). This is detailed more on the Conditional Distribution page.

Mode of the conditional distribution

The mode is the parameter value with the highest probability:

$$ \hat{\psi}_i^{mode} = \underset{\psi_i}{\textrm{arg max }}p(\psi_i|y_i;\hat{\theta})$$

To find the mode, we thus need to maximize the conditional probability with respect to the individual parameter value \(\psi_i\).

Individual random effects

Once the individual parameters values \(\psi_i\) are known, the corresponding individual random effects can be calculated using the population parameters and covariates. Taking the example of a parameter \(\psi\) having a normal distribution within the population and that depends on the covariate \(c\), we can write for individual \(i\):

$$ \psi_i = \psi_{pop} + \beta \times c_i + \eta_i$$

As \(\psi_i\) (estimated conditional mode), \(\psi_{pop}\) and \(\beta\) (population parameters) and \(c_i\) (individual covariate value) are known, the individual random effect \(\eta_i\) can easily be calculated.

Algorithm

For each individual, to find the \(\psi_i\) values that maximizes the conditional distribution, we use the Nelder-Mead Simplex algorithm [1].

As the conditional distribution does not have a closed form solution (i.e \(p(\psi_i|y_i;\hat{\theta})\) cannot be directly or easily calculated for a given \(\psi_i\)), we use the Bayes law to rewrite it in the following way (leaving the population parameters \(\hat{\theta}\) out for clarity):

$$p(\psi_i|y_i)=\frac{p(y_i|\psi_i)p(\psi_i)}{p(y_i)}$$

The conditional density function of  the data when knowing the individual parameter values (i.e \(p(y_i|\psi_i)\)) is easy to calculate, as well as the density function for the individual parameters (i.e \(p(\psi_i)\)), because they have closed form solutions. On the opposite, the likelihood \(p(y_i)\) has no closed form solution. But as it does not depend on \(\psi_i\), we can leave it out of the optimization procedure and only optimize \(p(y_i|\psi_i)p(\psi_i)\).

The initial value used for the Nelder-Mead simplex algorithm is the conditional mean (estimated during the conditional distribution task) if avaible (typical case of a full scenario), or the approximate conditional mean calculated at the end of SAEM otherwise.

Parameters without variability are not estimated, they are set to \( \psi_i = \psi_{pop} + \beta \times c_i \).

Running the EBEs task

When running the EBEs task, the progress is displayed in the pop-up window:

Dependencies between tasks:

Outputs

In the graphical user interface

In the Indiv.Param section of the Results tab, a summary of the individual parameters is proposed (min, max, median, quartiles, and shrinkage [in Monolix 2024 and later]) as shown in the figure below. The elapsed time for this task is also shown.

To see the estimated parameter value for each individual, the user can click on the [INDIV. ESTIM.] section. Notice that the user can also see them in the output files, which can be accessed via the folder icon at the bottom left. Notice that there is a “Copy table” icon on the top of each table to copy them in Excel, Word, … The table format and display will be kept.

In the output folder

After having run the EBEs task, the following files are available:

  • summary.txt: contains the summary statistics (as displayed in the GUI).
  • IndividualParameters/estimatedIndividualParameters.txt: the individual parameters for each subject-occasion are displayed. In addition to the already present approximation conditional mean from SAEM (*_SAEM), the conditional mode (*_mode) is added to the file.
  • IndividualParameters/estimatedRandomEffects.txt: the individual random effects for each subject-occasion are displayed (*_mode), in addition to the already present value based on the approximate conditional mean from SAEM (*_SAEM).
  • IndividualParameters/shrinkage.txt: starting with Monolix 2024, the shrinkage for each parameter for the conditional mode as shrinkage_mode.

More details about the content of the output files can be found here.

Settings

The settings are accessible through the interface via the button next to the EBEs task.

 

  • Maximum number of iterations (default: 200): maximum number of iterations for the Nelder-Mead Simplex algorithm, for each individual. Even if the tolerance criteria is not met, the algorithm stops after that number of iterations.
  • Tolerance (default: 1e-6): absolute tolerance criteria. The algorithm stops when the change of the conditional probability value between two iterations is less than the tolerance.

 

 

Calculate EBEs for a new data set using an existing model





5.1.4.Conditional distribution







Purpose

The conditional distribution represents the uncertainty of the individual parameter values. The conditional distribution estimation task permits to sample from this distribution. The samples are used to calculate the condition mean, or directly as estimators of the individual parameters in the plots to improve their informativeness [1]. They are also used to compute the statistical tests.

Calculation of the conditional distribution

Conditional distribution

The conditional distribution is \(p(\psi_i|y_i;\hat{\theta})\) with \(\psi_i\) the individual parameters for individual i, \(\hat{\theta}\) the estimated population parameters, and \(y_i\) the data (observations) for individual i. The conditional distribution represents the uncertainty of the individual’s parameter value, taking into account the information at hand for this individual:

  • the observed data for that individual,
  • the covariate values for that individual,
  • and the fact that the individual belongs to the population for which we have already estimated the typical parameter value (fixed effects) and the variability (standard deviation of the random effects).

It is not possible to directly calculate the probability for a given \(\psi_i\) (no closed form), but is possible to obtain samples from the distribution using a Markov-Chain Monte-Carlo procedure (MCMC).

MCMC algorithm

MCMC methods are a class of algorithms for sampling from probability distributions for which direct sampling is difficult. They consist of constructing a stochastic procedure which, in its stationary state, yields draws from the probability distribution of interest. Among the MCMC class, we use the Metropolis-Hastings (MH) algorithm, which has the property of being able to sample probability distributions which can be computed up to a constant. This is the case for our conditional distribution, which can be rewritten as:

$$p(\psi_i|y_i)=\frac{p(y_i|\psi_i)p(\psi_i)}{p(y_i)}$$

\(p(y_i|\psi_i)\) is the conditional density function of the data when knowing the individual parameter values and can be computed (closed form solution). \(p(\psi_i)\) is the density function for the individual parameters and can also be computed. The likelihood \(p(y_i)\) has no closed form solution but it is constant.

In brief, the MH algorithm works in the following way: at each iteration k, a new individual parameter value is drawn from a proposal distribution for each individual. The new value is accepted with a probability that depends on \(p(\psi_i)\) and \(p(y_i|\psi_i)\). After a transition period, the algorithm reaches a stationary state where the accepted values follow the conditional distribution probability \(p(\psi_i|y_i)\). For the proposal distribution, three different distributions are used in turn with a (2,2,2) pattern (setting “Number of iterations of kernel 1/2/3” in Settings > Project Settings): the population distribution, a unidimensional Gaussian random walk, or a multidimensional Gaussian random walk. For the random walks, the variance of the Gaussian is automatically adapted to reach an optimal acceptance ratio (“target acceptance ratio” setting in Settings > Project Settings).

Conditional mean

The draws from the conditional distribution generated by the MCMC algorithm can be used to estimate any summary statistics of the distribution (mean, standard deviation, quantiles, etc). In particular we calculate the conditional mean by averaging over all draws:

$$ \hat{\psi}_i^{mean} = \frac{1}{K}\sum_{k=1}^{K}\psi_i^{k}$$

The standard deviation of the conditional distribution is also calculated. The number of samples used to calculate the mean and standard deviation corresponds to the number of chains times the total number of iterations of the conditional distribution task (not only during the convergence interval length). The mean is calculated over the transformed individual parameters (in the gaussian space), and back-transformed to the non-gaussian space.

Samples from the conditional distribution

Among all samples from the conditional distribution, a small number (between 1 and 10, see “Simulated parameters per individual” setting) is kept to be used in the plots. These samples are unbiased estimators and they present the advantage of not being affected by shrinkage, as shown for example on the documentation of the plot “distribution of the individual parameters“.

Shrinkage and the use of random samples from the conditional distribution are explained in more details here.

Stopping criteria

At iteration k, the conditional mean is calculated for each individual by averaging over all k previous iterations. The average conditional means over all individuals (noted E(X|y)), and the standard deviation of the conditional means over all individuals (noted sd(X|y)) are calculated and displayed in the pop-up window. The algorithm stops when, for all parameters, the average conditional means and standard deviations of the last 50 iterations (“Interval length” setting) do not deviate by more than 5% (2.5% in each direction, “relative interval” setting) from the average and standard deviation values at iteration k.

In some very specific cases (for example with a parameter with a normal distribution and a value very close to 1), it can take many iterations to reach the convergence criteria because the criteria is defined as a percentage. In that case, the toggle “enable maximum number of iterations” can be used to limit the number iterations of this task. If the limit is reached, a warning message will be displayed in the interface.

Running the conditional distribution estimation task

During the evaluation of the conditional distribution, the following plot pop-ups, displaying the average conditional means over all individuals (noted E(X|y)), and the standard deviation of the conditional means over all individuals (noted sd(X|y)) for each iteration of the MCMC algorithm.

The convergence criteria described above means that the blue line, which represents the average over all individuals of the conditional mean, must be within the tube. The tube is centered around the last value of the blue line and spans over 5% of that last value. The algorithm stops when all blue lines are in their tube.

Dependencies between tasks:

  • The “Population parameters” task must be run before launching the conditional distribution task.
  • The conditional distribution task is recommended before calculating the log-likelihood task without the linearization method (i.e log-likelihood via importance sampling).
  • The conditional distribution task is necessary for the statistical tests.
  • The samples generated during the conditional distribution task will be reused for the Standard errors task (without linearization).

Outputs

In the graphical user interface

In the Indiv.Param section of the Results tab, a summary of the estimated conditional mean is given (min, max, quartiles, and shrinkage [in Monolix 2024 and later]), as shown in the figure below. Starting from Monolix2021R1, the number of iterations is also displayed, along with a message indicating whether convergence has been reached (“auto-stop”) or if the task was stopped by the user or reached the maximum number of iterations.

To see the estimated parameter value for each individual, the user can click on the [INDIV. ESTIM.] section. Notice that the user can also see them in the output files, which can be accessed via the folder icon at the bottom left. Notice that there is a “Copy table” icon on the top of each table to copy them in Excel, Word, … The table format and display will be kept.

In the output folder

After having run the conditional distribution task, the following files are available:

  • summary.txt: contains the summary statistics (as displayed in the GUI)
  • IndividualParameters/estimatedIndividualParameters.txt: the individual parameters for each subject-occasion are displayed. The conditional mean (*_mean) and the standard deviation (*_sd) of the conditional distribution are added to the file. The number of samples used to calculate the mean and standard deviation corresponds to the number of chains times the total number of iterations of the conditional distribution task (not only during the convergence interval length).
  • IndividualParameters/estimatedRandomEffects.txt: the individual random effects for each subject-occasion are displayed. Those corresponding to the conditional mean (*_mean) are added to the file, together with the standard deviation (*_sd).
  • IndividualParameters/simulatedIndividualParameters.txt: several simulated individual parameters (draws from the conditional distribution) are recorded for each individual. The rep column permits to distinguish the several simulated parameters for each individual.
  • IndividualParameters/simulatedRandomEffects.txt: the random effects corresponding to the simulated individual parameters are recorded.
  • IndividualParameters/shrinkage.txt: starting with Monolix 2024, the shrinkage for each parameter for the conditional mean as shrinkage_mean and for the conditional distribution as shrinkage_condDist.

More details about the content of the output files can be found here.

Settings

To change the settings, you can click on the settings button next the conditional distribution task.

  • Interval length (default: 50): number of iterations over which the convergence criteria is checked.
  • Relative interval (default: 0.05): size of the interval (relative to the current average or standard deviation) in which the last “interval length” iterations must be for the stopping criteria to be met. A value at 0.05 means that over the last “interval length” iterations, the value should not vary by more than 5% (2.5% in each direction).
  • Simulated parameters per individual (default: via calculation): number of draws from the conditional distribution that will be used in the plots. The number is calculated as min(10, idealNb) with idealNb = max(500 / number of subject , 5000 / number of observations). This means that the maximum number is 10 (which is usually the case for small data sets). For large data sets, the number may be reduced, but the number of individual times the number of simulated parameters should be at least 500, and the number of observations times the number of simulated parameters should be at least 5000. This ensures to have a sufficiently large but not unnecessarily large number of dots in the plots such as Observations versus predictions or Correlation between random effects.
    If the user sets the number of simulated parameters to a value larger than the number of chains (project settings) times the total number of iterations of the conditional distribution task (maximum number of iterations or when convergence criteria are reached), the number will be restricted to the number of available samples.
    If the user sets the number of simulated parameters to a value smaller than interval length times number of chains, the simulated parameters are picked evenly from the interval length and the chains. If the requested number of simulated parameters is larger, the last n (n=number of requested simulated parameters) samples are picked.
  • Enable maximum iterations limit (default: toggle off) [from version 2020 on]: When the toggle in “on”, a maximum number of iterations can be defined.
  • Maximum number of iterations (default: 500, needs to be larger than the interval length, available if “enable maximum iterations limit’ is on): maximum number of iterations for the conditional distribution task. Even if the convergence criteria are not fulfilled, the algorithm stops after this maximum number of iterations. If the maximum number of iterations is reached, a warning message will be displayed in the interface.

5.1.4.1.Understanding shrinkage and how to circumvent it







Introduction

Shrinkage is a phenomenon that appears when the data is insufficient to precisely estimate the individual parameters (EBEs). In that case, the EBEs “shrink” towards the center of the population distribution and do not properly represent the inter-individual variability. This leads to diagnostic plots that may be misleading, either hiding true relationships or inducing wrong ones.

In the diagnostic plots, Monolix uses samples from the conditional distribution as individual parameters, which lead to reliable plots even when shrinkage is present in the model [1]. This method is based on the calculation of the conditional distribution.

 

Conditional distribution

The conditional distribution is defined for each individual. It represents the uncertainty of the individual’s parameter value, taking the information at hand for this individual into account:

  • the observed data for that individual,
  • the covariate values for that individual,
  • the fact that the individual belongs to the population for which we have already estimated the typical parameter value (fixed effects) and the inter-individual variability (standard deviation of the random effects).

In a mathematical formalism, the conditional distribution is written \(p(\psi_i|y_i;\hat{\theta})\) with \(\psi_i\) the individual parameters for individual \(i\), \(\hat{\theta}\) the estimated population parameters, and \(y_i\) the data (observations) for individual \(i\).

It is not possible to directly calculate the probability for a given \(\psi_i\) (no closed form), but it is possible to obtain samples from the distribution using a Markov-Chain Monte-Carlo procedure (MCMC). This is what is done in the Conditional distribution task.

With the following conditional distribution for the volume V of individual i, we see that the most probable value is around 25 L but there is quite some uncertainty: the value could also be 15 or 40 for instance. For visual purpose, we have drawn the distribution as a smooth curve, but remember that the conditional distribution has no explicit expression. One can only obtain samples from this distribution using MCMC.

Conditional mode (EBEs) and conditional mean

It is often convenient to work with a single value for the individual parameters (called an estimator), instead of a probability distribution. Several “summary” values can be used, such as the mode or the mean of the conditional distribution.

The mode is also called maximum a posteriori or EBE (for empirical bayes estimate). It is often preferred over the mean, because the mode represents the most likely value, i.e the value which has the highest probability.

In Monolix, the mode is calculated via the EBEs task, while the mean is calculated via the Conditional distribution task, as the average of all samples drawn from the conditional distribution.

Once the value of the individual estimator (conditional mode or conditional mean) is known for each individual, one can easily calculate the individual random effects for each individual. For instance for the volume V, that depends on the covariate weight WT:

$$\begin{array}{rl} V_i& = V_{pop}\left(\frac{\textrm{WT}_i}{70}\right)^{\beta}e^{\eta_i} \\  \Rightarrow \quad \eta_i& = \log (V_i) – \log(V_{pop}) – \beta \log \left(\frac{\textrm{WT}_i}{70}\right) \end{array} $$

The individual parameters and individual random effects are used in diagnostic plots. They are used either directly, such as in the Correlation between random effects or Individual parameter versus covariate plots, or indirectly to generate individual predictions, such as in Individual Fits, Observations versus Predictions or Scatter plot of the residuals. But these diagnostic plots can be biased in presence of shrinkage.

 

Shrinkage

When the individual data brings only few information about the individual parameter value, the conditional distribution is large, reflecting the uncertainty of the individual parameter value. In that case, the mode of the conditional distribution is close to (or “shrinks” to) the mode of the population distribution. If this is the case for all or most of the individuals, all individual parameters end up concentrated around the mode of the population distribution and do not correctly represent the inter-individual variability which has been estimated via the standard deviation parameters (omega parameters in Monolix). This is the shrinkage phenomenon. Shrinkage typically occurs when the data is sparse.

Below we present the example of a parameter V which has a lot of shrinkage and a parameter k with almost no shrinkage. We consider a data set with 10 individuals. In the upper plots, the conditional distributions of each of the 10 individuals are shown. For the volume V, the individual parameter values are uncertain and their conditional distributions are large. When reporting the mode (closed circles) of the conditional distributions on the population distribution (black curve, bottom plots), the modes appear shrunk compared to the population distribution. On the opposite, for k, the conditional distributions are narrow and the modes are well spread over the population distribution. There is shrinkage for V, but not for k.

Pulling the individual parameters of all individuals together, one can overlay the population distribution (black line) with the histogram of individual parameters (i.e conditional modes) (blue bars). This is displayed in the Distribution of the individual parameters plot in Monolix:

The shrinkage phenomenon can be quantified via a shrinkage value for each parameter. In Monolix, the formula for shrinkage has been updated in version 2024 to use the standard deviation instead of the variance in accordance with industry standards.

Starting with Monolix version 2024, shrinkage is calculated from the empirical standard deviation of the random effects \( \textrm{sd}(\eta_i) \) and the estimated standard deviation (the omega population parameter \(\omega\)). The random effects \( \textrm{sd}(\eta_i) \) can be calculated from the EBEs, conditional mean or samples from the conditional distribution. Typically, the shrinkage is reported using the EBEs.

$$\eta\textrm{-sh}=1-\frac{\textrm{sd}(\eta_i)}{\omega}$$

In the case of inter occasion variability, shrinkage includes both inter individual and inter occasion variability (the gamma population parameter  \(\gamma\))

$$\eta\textrm{-sh}=1-\frac{\textrm{sd}(\eta_i)}{\sqrt{\omega^2 + \gamma^2}}$$

In Monolix versions 2023 and earlier, shrinkage is calculated using the ratio of the empirical variance and the estimated variance as:

$$\eta\textrm{-sh}=1-\frac{\textrm{var}(\eta_i)}{\omega^2}$$

Results

In Monolix versions 2023 and earlier, the shrinkage can be displayed in the Distribution of the individual parameters plot, by selecting the “information” toggle.

Starting in Monolix version 2024, shrinkage information is available in several ways:

  • Results / Indiv. Results / Cond. Mean [summary] – shrinkage is reported if the Conditional Distribution task has been run
  • Results / Indiv. Results / Cond. Mode [summary] – shrinkage is reported if the EBEs task has been run
  • In the IndividualParameters folder inside the results folder, there is a file shrinkage.txt
  • In reports, using the metric SHRINKAGE in the population or individual parameters table placeholder
  • In the plot Distribution of the individual parameters by switching on “information”
  • In the plot Distribution of the standardized random effects by switching on “information”

Calculating the shrinkage in R

Starting with Monolix version 2024, there is a method available to directly return the shrinkage information: getEtaShrinkage().

In Monolix version 2023 and earlier, there is no function from the lixoftConnectors to directly get the shrinkage, but you can easily calculate it from the estimated parameters like this for example for parameter V:

population_params <- read.csv("monolix_project/populationParameters.txt")
eta_estimated <- read.csv("monolix_project/IndividualParameters/estimatedRandomEffects.txt")
omega_V <- population_params$value[population_params$parameter=="V"]
eta_V_estimated_mode <- eta_estimated$eta_V_mode
shrinkage_eta_V_estimated_mode <- (1-var(eta_V_estimated_mode)/omega_V^2)*100

Comparison to Nonmem: the Nonmem definition of shrinkage is based on a ratio of standard deviations. This is also the case in Monolix 2024 and above. However, Monolix 2023 and below uses a ratio of variances (which is more common in statistics). Below we provide a “conversion table” which should be read in the following way: a situation that would in Nonmem and Monolix 2024 lead to a shrinkage calculation of 30%, would in Monolix 2023 lead to a calculation of shrinkage of around 50%. The Nonmem and Monolix 2024 version of the shrinkage can be calculated from the Monolix 2023 shrinkage using the following formula:

$$\eta\textrm{-shNM}=\eta\textrm{-shMLX24}=1-\sqrt{1-\eta\textrm{-shMLX23}}$$

Is it OK to get a negative shrinkage? Yes. In case of no shrinkage, \(var(\eta_i)= \omega^2\) when \(var(\eta_i)\) is calculated on an infinitely large sample. In practice, \(var(\eta_i)\) is calculated on a limited sample related to the number of individuals. Its value can be by chance a little bigger than \(\omega^2\), leading to a slightly negative shrinkage.

 

 

 

Consequences of shrinkage

In case of shrinkage the individual parameters (conditional mode/EBEs or conditional mean) are biased because they do not correctly reflect the population distribution.

As these individual parameters are used in diagnostic plots (in particular the Correlation between random effects and the Individual parameters versus covariates plots) the diagnostic plots can become misleading in presence of shrinkage, either hiding relations or suggesting wrong ones. This complicates the identification of mis-specifications and burdens the modeling process.

Note that the shrinkage of the EBEs has no consequences on the population parameter estimation via SAEM (which doesn’t use EBEs, contrary to FOCE for instance). However the lack of informative data may lead to large standard errors for the population parameters and a slower convergence.

 

 

How to circumvent shrinkage

Monolix provides a very efficient solution to circumvent the shrinkage problem, i.e the bias in the diagnostic plots induced by the use of shrunk individual parameters. Instead of using the shrunk conditional mode/EBEs or conditional mean, Monolix uses parameter values randomly sampled from the conditional distribution:

The fact of pooling the random samples of the conditional distribution of all individuals allows us to look at them as if they where sampled from the population distribution. And this is exactly what we want: to have individual parameter values (i.e the samples) that correctly reflect the population distribution.

From a mathematical point of view, one can show that the random samples are an unbiased estimator:

$$p(\psi_i)=\int p(\psi_i|y_i)p(y_i)dy_i=\mathbb{E}_{y_i}(p(\psi_i|y_i))$$

The improvement brought by the random samples from the conditional distributions can be visualized in the following way: while the mode (closed circles) are shrunk, the random samples (stars) spread over the entire population distribution (in black). One can even draw several random sample per individual to increase the informativeness of the diagnostic plots. This is what is done in the MonolixSuite2018R1 version (while MonolixSuite2016R1 uses one sample per individual).

In [1], the authors warrant the use of sampled individual parameters. They demonstrate their usefulness in diagnostic plots via numerical experiments with simulated data. They also show that statistical tests based on these sampled individual parameters are unbiased, the type I error rate is the desired significance level of the test and the probability to detect a mis-specification in the model increases with the magnitude of this mis-specification.

 

 

Example

Fitting the sparse Tobramycin data with a (V,k) model leads to a high shrinkage (75%) of the volume V when using the EBEs. On the opposite, when using samples from the conditional distributions of each individual, there is no shrinkage anymore.

The usefulness of using the samples from the conditional distribution can be seen in the Correlation between the random effects plot. Using the EBEs, the plot suggests a positive correlation of about 30% between the volume and the elimination rate. Using the random samples, the plot does not suggest this correlation any more. If the correlation is added to the model, it is estimated small and not significant.

Another example of shrinkage can be seen for the parameter ka in the warfarin data set. In this example, the data is sparse during the absorption phase leading to a large uncertainty of the individual parameter values.

 

Conclusion

The use of samples from the conditional distribution is a powerful way to avoid the bias due to the shrinkage in the diagnostic plots. This method has been validated mathematically and with numerical experiments.

In Monolix, the random samples are used by default in all diagnostic plots, if the Conditional distribution task has been run. The choice of the estimator for the individual parameters can be changed in the Settings tab:

5.1.4.2.Statistical tests

Several statistical tests may be automatically performed to test the different components of the model. These tests use individual parameters drawn from the conditional distribution, which means that you need to run the task “Conditional distribution” in order to get these results. In addition, the tests for the residuals require to have first generated the residuals diagnostic plots (scatter plot or distribution).
The tests are all performed using the individual parameters sampled from the conditional distribution (or the random effects and residuals derived thereof). They are thus not subject to bias in case of shrinkage. For each individual, several samples from the conditional distribution may be used. The used tests include a correction to take into account that these samples are correlated among each other.

Results of the tests are available in the tab “Results”  and selecting “Tests” in the left menu





The model for the individual parameters

Consider a PK example (warfarin data set from the demos) with the following model for the individual PK parameters (ka, V, Cl):

In this example, the different assumptions we make about the model are:

  • The 3 parameters are lognormally distributed
  • ka is function of sex
  • V is function of sex and weight. More precisely, the log-volume log(V) is a linear function of the log-weight \({\rm lw70 }= \log({\rm wt}/70)\).
  • Cl is not function of any of the covariates.
  • The random effects \(\eta_V\) and \(\eta_{Cl}\) are linearly correlated
  • \(\eta_{ka}\) is not correlated with \(\eta_V\) and \(\eta_{Cl}\)

Let’s see how each of these assumptions are tested:

Covariate model

Individual parameters vs covariates – Test whether covariates should be removed from the model

If an individual parameter is function of a continuous covariate, the linear correlation between the transformed parameter and the covariate is not 0 and the associated \(\beta\) coefficient is not 0 either. To detect covariates bringing redundant information, we check if these beta coefficients are different from 0. For this, we perform two different tests: a correlation test based on betas coefficients estimated with a linear regression, and a Wald test relying on the estimated population parameters and their standard error.

In both cases, a small p-value indicates that the null hypothesis can be rejected and thus that the estimated beta parameter is significantly different from zero. If this is the case, the covariate should be kept in the model. On the opposite, if the p-value is large, the null hypothesis cannot be rejected and this suggests to remove the covariate from the model. Note that if beta is equal to zero, then the covariate has no impact on the parameter. High p-values are colored in yellow (p-value in [0.01-0.05]), orange (p-value in [0.05-0.10]) or red (p-value in [0.10-1]) to draw attention on parameter-covariate relationships that can be removed from the model from a statistical point of view.

Correlation test

Briefly, we perform a linear regression between the covariates and the transformed parameters and test if the resulting beta coefficient are different from 0.

More precisely: for each individual \(i\), let \(z_i^l\) be the transformed individual parameters (e.g log(V) for log-normally distributed parameters or logit(F) for logit-distributed parameters) sampled from the conditional distribution (called replicates, index l). Here we will call covariates all continuous covariates and all non-referent categories of categorical covariates \(cov^{(c)}, c = 1..n_C\). For each individual \(i\), \(cov_i^{(c)}\) is the value of the c\(^{th}\) covariate, equal to 0 or 1 if the covariate is a category.

The transformed individual parameters are first averaged over replicates for each individual:

$$z_i^{(L)}=\frac{1}{L} \sum_{l=1}^{L} z_i^l$$

We then perform the following linear regression:

$$z_i^{(L)}=\alpha_0 + \sum_{c=1}^{n_c} \beta_c \text{cov}_i^{(c)} + e_i$$

If two covariates \(cov^{(1)}\) and \(cov^{(2)}\) (for example WT and BMI) are strongly correlated with a parameter \(z^{(L)}\) (for example the volume), only one of them is needed in the model because they are redundant. In the linear regression, only one of the estimated \(\hat{\beta_1}\) and \(\hat{\beta_2}\) will be significantly different from zero.

For each covariate \(c\) we conduct a t-test on the \(\hat{\beta_c}\) with the null hypothesis:

H0:  \(\beta_c = 0\).

The test statistic is

$$T_0 = \frac{\hat{\beta_c}}{se(\hat{\beta_c})}$$

where \(se(\hat{\beta_c})\) is the estimated standard error of \(\hat{\beta_c}\) (obtained by least squares estimation during the regression). If the null hypothesis is true, \(T_0\) follows a t distribution with \(N-n_C-1\) degrees of freedom (where N is the number of individuals).

In our example, the correlation test suggests to remove sex from ka:

Wald test

The Wald test relies on the standard errors. Thus the task “Standard errors” must have been calculated to see the test results. The test can be performed using the standard errors calculated using either the “linearization method” (indicated as “linearization”) or not (indicated as “stochastic approximation” in the tests).

The Wald test tests the following null hypothesis:

H0: the beta parameter estimated by SAEM is equal to zero.

The math behind: Let’s note \( \hat{\beta} \) the estimated beta value (which is a population parameter) and \(se(\hat{\beta}) \) the associated standard error calculated during the task “Standard errors”. The Wald test statistic is:

$$W=\frac{\hat{\beta}}{se(\hat{\beta})} $$

The test statistic is compared to a normal distribution (z distribution).

In our example, the Wald test suggests to remove sex from ka and V:

Remark: the Wald test and the correlation test may suggest different covariates to keep or remove. Note that the null hypothesis tested is not the same.

Random effects vs covariates – Test whether covariates should be added to the model

Pearson’s correlation tests and ANOVA are performed to check if some relationships between random effects and covariates not yet included in the model should be added to the model.

For continuous covariates, the Pearson’s correlation test tests the following null hypothesis:

H0: the person correlation coefficient between the random effects (calculated from the individual parameters sampled from the conditional distribution) and the covariate values is zero

For categorical covariates, the one-way ANOVA tests the following null-hypothesis:

H0: the mean of the random effects (calculated from the individual parameters sampled from the conditional distribution) is the same for each category of the categorical covariate

A small p-value indicates that the null hypothesis can be rejected and thus that the correlation between the random effects and the covariate values is significant. If this is the case, it is probably worth considering to add the covariate in the model. Note that the decision of adding a covariate in the model should not only be driven by statistical considerations but also biological relevance. Note also that for parameter-covariate relationships already included in the model, the correlation between the random effects and covariates is not significant (while the correlation between the parameter and the covariate can be – see above). Small p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]) to draw attention on parameter-covariate relationships that can be considered for addition in the model from a statistical point of view.

In our example, we already have sex on ka and V, and lw70 on V in the model. The only remaining relationship  that could possibly be worth investigating is between weight (or the log-transformed weight “lw70”) and clearance.

The math behind:

Continuous covariate: Let \(\eta_i^l\) the random effects corresponding to the \(L\) individual parameters sampled from the conditional distribution (called replicates) for individual \(i\), and \(cov_i\) the covariate value for individual \(i\). The random effects are first averaged over replicates for each individual:

$$ \eta_i^{(L)}=\frac{1}{L} \sum_{l=1}^{L} \eta_i^l $$

We note \(\overline{cov} = \sum_{i=1}^N cov_i \) the average covariate value over the N subjects and \(\overline{\eta}=\sum_{i=1}^N \eta_i^{(L)} \) the average random effect. The Pearson correlation coefficient is calculated as:

$$r=\frac{\sum_{i=1}^N(cov_i – \overline{cov})(\eta_i^{(L)} – \overline{\eta})}{\sqrt{ \sum_{i=1}^N(cov_i – \overline{cov})^2 \sum_{i=1}^N(\eta_i^{(L)} – \overline{\eta})^2}}$$

The test statistic is:

$$t=\frac{r}{\sqrt{1-r^2}}\sqrt{N-2}$$

and it is compared to a t-distribution with \(N-2\) degrees of freedom with \(N\) the number of individuals.

Categorical covariates: The random effects are first averaged over replicates for each individual and a one-way analysis of variance is performed (simplified to a t-test when the covariate has only two categories).

The model for the random effects

Distribution of the random effects – Test if the random effects are normally distributed

In the individual model, the distributions for the parameters assume that the random effects follow a normal distribution. Shapiro-Wilk tests are performed to test this hypothesis. The null hypothesis is:

H0: the random effects are normally distributed

If the p-value is small, there is evidence that the random effects are not normally distributed and this calls the choice of the individual model (parameter distribution and covariates) into question. Small p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).

In our example, there is no reason to reject the null-hypothesis and no reason to question the chosen log-normal distributions for the parameters.

The math behind: Let \(\eta_i^l\) the random effects corresponding to the \(L\) individual parameters sampled from the conditional distribution (called replicates) for individual \(i\). The Shapiro-Wilk test statistic is calculated for each replicate \(l\) (i.e the first sample from all individuals, then the second sample from all individuals, etc):

$$W^l=\frac{\left( \sum_{i=1}^N a_i \eta_i^l \right)^2}{ \sum_{i=1}^N (\eta_i^l – \overline{\eta}^l)^2}$$

with \(a_i\) tabulated coefficient and \(\overline{\eta}^l=\frac{1}{N}\sum_{i=1}^N \eta_i^l \) the average over all individuals, for each replicate.

The statistic displayed in Monolix corresponds to the average statistic over all replicates \(W=\frac{1}{L}\sum_{l=1}^L W^l \). For the p-values, one p-value is calculated for each replicate, using the Shapiro-Wild table with \(N\) (number of individuals) degrees of freedom. The Benjamini-Hochberg (BH) procedure is then applied: the p-values are ranked by ascending order and the BH critical value is calculated for each as \( \frac{\textrm{rank}}{L}Q \) with \(\textrm{rank}\) the individual p-value’s rank, \(L\) the total number of p-values (equal to the number of replicates) and \(Q=0.05\) the false discovery rate. The largest p-value that is smaller than the corresponding critical value is selected.

Joint distribution of the random effects – Test if the random effects are correlated

Correlation tests are performed to test if the random effects (calculated from the individual parameters sampled from the conditional distribution) are correlated. The null-hypothesis is:

H0: the expectation of the product of the random effects of the first and second parameter is zero

The null-hypothesis is assessed using a t-test.

Remark: In the 2018 version, a Pearson correlation test was used.

For correlations not yet included in the model, a small p-value indicates that there is a significant correlation between the random effects of two parameters and that this correlation should be estimated as part of the model (otherwise simulations from the model will assume that the random effects of the two parameters are not correlated, which is not what is observed for the random effects estimated using the data). Small p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).
For correlations already included in the model, a large p-value indicates that one cannot reject the hypothesis that the correlation between the random effects is zero. If the correlation is not significantly different from zero, it may not be worth estimating it in the model. High p-values are colored in yellow (p-value in [0.01-0.05]), orange (p-value in [0.05-0.10]) or red (p-value in [0.10-1])

In our example, we have assumed in the model that \(\eta_V\) and \(\eta_{Cl}\) are correlated. The high p-value (0.033, above the 0.01 threshold, see above) indicates that the correlation between the random effects of V and Cl is not significantly different from zero and suggests to remove this correlation from the model.

Remark: as correlations can only be estimated by groups (i.e if a correlation is estimated between (ka, V) and between (V, Cl), then one must also estimate the correlation between (ka, Cl)), it may happen that it is not possible to remove a non-significant correlation without removing also a significant one.

The math behind: Let \(\eta_{\psi_1,i}^l\) and \(\eta_{\psi_2,i}^l\) the random effects corresponding to the \(L\) individual parameters \(\psi_1\) and \(\psi_2\) sampled from the conditional distribution (called replicates) for individual \(i\). First we calculate the product of the random effects averaged over the replicates:

$$p_i^{(L)} = \frac{1}{L} \sum_{l=1}^{L} \eta_{\psi_1,i}^l \eta_{\psi_2,i}^l $$

We note \( \overline{p}=\sum_{i=1}^{N} p_i^{(L)} \) the average of the product over the individuals and \(s\) their standard deviation. The test statistic is:

$$ T=\frac{\overline{p}}{\frac{s}{\sqrt{N}}}$$

and it is compared to a t-distribution with \(N-1\) degrees of freedom with \(N\) the number of individuals.

The distribution of the individual parameters

Distribution of the individual parameters not dependent on covariates – Test if transformed individual parameters are normally distributed

When an individual parameter doesn’t depend on covariates, its distribution (normal, lognormal, logit or probit) can be transformed into the normal distribution. Then, a Shapiro-Wilk test can be used to test the normality of the transformed parameter. The null hypothesis is:

H0: the transformed individual parameter values (sampled from the conditional distribution) is normally distributed

If the p-value is small, there is evidence that the transformed individual parameter values are not normally distributed and this calls the choice of the parameter distribution into question. Small p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).

In our example, there is no reason to reject the null hypothesis of lognormality for Cl.

Remark:  testing the normality of a transformed individual parameter that does not depend on covariates is equivalent to testing the normality of the associated random effect. We can check in our example that the  Shapiro-Wilk tests for \(\log(Cl)\) and \(\eta_{Cl}\) are equivalent.

The math behind: Let \(z_i^l\) the transformed individual parameters (e.g log(V) for log-normally distributed parameters and logit(F) for logit-distributed parameters) sampled from the conditional distribution (called replicates, index \(l\) ) for individual \(i\). The Shapiro-Wilk test statistic is calculated for each replicate \(l\) (i.e the first sample from all individuals, then the second sample from all individuals, etc):

$$W^l=\frac{\left( \sum_{i=1}^N a_i z_i^l \right)^2}{ \sum_{i=1}^N (z_i^l – \overline{z}^l)^2}$$

with \(a_i\) tabulated coefficient and \(\overline{z}^l=\frac{1}{N}\sum_{i=1}^N z_i^l \) the average over all individuals, for each replicate.

The statistic displayed in Monolix corresponds to the average statistic over all replicates \(W=\frac{1}{L}\sum_{l=1}^L W^l \). For the p-values, one p-value is calculated for each replicate, using the Shapiro-Wild table with \(N\) (number of individuals) degrees of freedom. The Benjamini-Hochberg (BH) procedure is then applied: the p-values are ranked by ascending order and the BH critical value is calculated for each as \( \frac{\textrm{rank}}{L}Q \) with \(\textrm{rank}\) the individual p-value’s rank, \(L\) the total number of p-values (equal to the number of replicates) and \(Q=0.05\) the false discovery rate. The largest p-value that is smaller than the corresponding critical value is selected.

Distribution of the individual parameters dependent on covariates – test the marginal distribution of each individual parameter

Individual parameters that depend on covariates are not anymore identically distributed. Each transformed individual parameter is normally distributed, with its own mean that depends on the value of the individual covariate. In other words, the distribution of an individual parameter is a mixture of (transformed) normal distributions. A Kolmogorov-Smirnov test is used for testing the distributional adequacy of these individual parameters. The null-hypothesis is:

H0: the individual parameters are samples from the mixture of transformed normal distributions (defined by the population parameters and the covariate values)

A small p-value indicates that the null hypothesis can be rejected. Small p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).

With our example, we obtain:

The model for the observations

A combined1 error model with a normal distribution is assumed in our example:

Distribution of the residuals

Several tests are performed for the individual residuals (IWRES), the NPDE and for the population residuals (PWRES).

Test if the distribution of the residuals is symmetrical around 0

A Miao, Gel and Gastwirth (2006) test (or Van Der Waerden test in the 2018 release) is used to test the symmetry of the residuals. Indeed, symmetry of the residuals around 0 is an important property that deserves to be tested, in order to decide, for instance, if some transformation of the observations should be done. The null hypothesis tested is:

H0: the median of the residuals is equal to its mean

A small p-value indicates that the null hypothesis can be rejected. Small p-values are colored in yellow (p-value in [0.05-0.10]), orange (p-value in [0.01-0.05]) or red (p-value in [0.00-0.01]).

With our example, we obtain:

The math behind: Let \(R_i\) the residuals (NPDE, PWRES or IWRES) for each individual \(i\), \(\overline{R}\) the mean of the residuals, and \(M_R\) their median. The MGG test statistic is:

$$T=\frac{\sqrt{n}}{0.9468922}\frac{\overline{R}-M_R}{ \sum_{i=1}^{n}|R_i-M|}$$

with \(n\) the number of residuals. The test statistic is compared to a standard normal distribution.

The formula above is valid for i.i.d (independent and identically distributed) residuals. For the IWRES, the residuals corresponding to a given time and given id are not independent (they ressemble each other). To solve the problem, we estimate an effective number of residuals. The number of residuals \(n\) can be split into the number of replicates \(L\) times the number of observations \(m\). We look for the effective number of replicates \(\tilde{L}\) such that:

$$ \frac{\tilde{L}}{L} \sum_{l=1}^L (R_i^l)^2 \approx \chi^2(\tilde{L})$$

using a maximum likelihood estimation. The number of residuals is then calculated as \(n=\tilde{L} \times m \).

Test if the residuals are normally distributed

A Shapiro Wilk test is used for testing the normality of the residuals. The null hypothesis is:

H0: the residuals are normally distributed

If the p-value is small, there is evidence that the residuals are not normally distributed. The Shapiro Wilk test is known to be very powerful. Then, a small deviation of the empirical distribution from the normal distribution may lead to a very significant test (i.e. a very small p-value), which does not necessarily means that the model should be rejected. Thus, no color highlight is made for this test.

In our example, we obtain:

The math behind: Let \(R_i^l\) the residuals (NPDE, PWRES or IWRES) for individual \(i\). NPDE and PWRES have one values per time points and per individual. IWRES have one value per time point, per individual and per replicate (corresponding to the \(L\) individual parameters sampled from the conditional distribution). The Shapiro-Wilk test statistic is calculated for each replicate \(l\) (i.e the first sample from all individuals, then the second sample from all individuals, etc):

$$W^l=\frac{\left( \sum_{i=1}^N a_i R_i^l \right)^2}{ \sum_{i=1}^N (R_i^l – \overline{R}^l)^2}$$

with \(a_i\) tabulated coefficient and \(\overline{R}^l=\frac{1}{N}\sum_{i=1}^N R_i^l \) the average over all individuals, for each replicate.

The statistic displayed in Monolix corresponds to the average statistic over all replicates \(W=\frac{1}{L}\sum_{l=1}^L W^l \). For the p-values, one p-value is calculated for each replicate, using the Shapiro-Wild table with \(N\) (number of individuals) degrees of freedom. The Benjamini-Hochberg (BH) procedure is then applied: the p-values are ranked by ascending order and the BH critical value is calculated for each as \( \frac{\textrm{rank}}{L}Q \) with \(\textrm{rank}\) the individual p-value’s rank, \(L\) the total number of p-values (equal to the number of replicates) and \(Q=0.05\) the false discovery rate. The largest p-value that is smaller than the corresponding critical value is selected.

5.1.4.3.Proposal

Starting from the 2019 version, the section Proposal in the tab Results includes automatic proposals of improvements for the statistical model, based on comparisons of many correlation, covariate and error models.





Model selections are performed by using a BIC criteria (called Criteria in the interface), based on the current simulated individual parameters. This is why the proposal is computed by the task Conditional distribution.

Note that the BIC criteria is not the same as the BIC computed for the Monolix project with the task log-likelihood: it does not characterize the whole model but only each part of the statistical model evaluated in the proposal, and thus yields different values for each type of model. Each criteria is given by the formula:

\(BIC = -2\log({\cal L’})/nRep+\log(N) * k\),

where:

  • \(\cal L’\) = likelihood of the linear regression
  • N = number of individuals
  • k = number of estimates betas
  • nRep = number of replicates (samples per individual)

The number of estimated parameters k characterizes the part of the statistical model that is evaluated. The likelihood \(\cal L’\) is not the same as the one computed by the log-likelihood task, it is based on the joint distribution:

\(p(y_i, \phi_i; \theta)=p(y_i| \phi_i; \theta)p(\phi_i; \theta)\)

where all the parameters are fixed to the values estimated by Monolix except the parameters characterizing the model that is evaluated: error parameters (in \(\theta\)) for the error models, individual parameters or random effects (in \(\phi_i\)) for the covariate models or the correlation models.

 

Proposed model

The section Proposal is organized in 4 tabs. The first tab summarizes the best proposal for the statistical model, that is the combination of best proposals for the error, covariate and correlation models.

The current statistical model is displayed below the proposed model, and the differences are highlighted in light blue.

The proposed model can be applied automatically with the button “Apply”. This modifies the current project to include all elements of the proposed statistical model. It is then recommended to save the project under a new name to avoid overwriting previous results.

Error model

The error model selection is done by computing the criterion for each possible residual error model (constant, proportional, combined1, combined2), where the error parameters are optimized based on the data and the current predictions, for each observation model.
The evaluated models are displayed for each observation mode in increasing order of criterion. The current error model is highlighted in blue.

 

Covariate model

The covariate model selection is based on the evaluation of the criterion for each individual parameter independently.
For each individual parameter, all covariate models obtained by adding or removing one covariate are evaluated: beta parameters are estimated with linear regression, and the criterion is computed. The model with the best criterion is retained to continue the same procedure.
All evaluated models are displayed for each parameter, in increasing order of criterion. The current covariate model is highlighted in blue.
Since possibly many covariate models can be evaluated for each parameter, the maximum number of displayed models per parameter is 4 by default and can be changed with a slider, as below. Additional evaluated models can still be displayed by clicking on “more entries” below each table.

Beta parameters are computed for each evaluated model by linear regression. They are not displayed by default, but can be displayed with a toggle as shown below. For categorical covariates, the categories corresponding to the beta parameters are also displayed.

 

 

Correlation model

The correlation model selection is done by computing the criterion for each possible correlation block at each dimension, starting with the best solution from the previous dimension. The correlation models are displayed by increasing value of criterion. The current correlation model is highlighted in blue.

 

In Monolix2021, the correlation model includes also random effects at the inter-occasion level, like in the example below, while they were excluded from the proposal in previous versions.

 

 

5.1.5.Standard error using the Fisher Information Matrix

Purpose

The standard errors represent the uncertainty of the estimated population parameters. In Monolix, they are calculated via the estimation of the Fisher Information Matrix. They can for instance be used to calculate confidence intervals or detect model overparametrization.





Calculation of the standard errors

Several methods have been proposed to estimate the standard errors, such as bootstrapping or via the Fisher Information Matrix (FIM). In the Monolix GUI, the standard errors are estimated via the FIM. Bootstrapping can be accessed either through the Rsmlx R package or is now incorporated directly into the Monolix interface starting from version 2024R1.

The Fisher Information Matrix (FIM)

The observed Fisher information matrix (FIM) \(I \) is minus the second derivatives of the observed log-likelihood:

$$ I(\hat{\theta}) = -\frac{\partial^2}{\partial\theta^2}\log({\cal L}_y(\hat{\theta})) $$

The log-likelihood cannot be calculated in closed form and the same applies to the Fisher Information Matrix. Two different methods are available in Monolix for the calculation of the Fisher Information Matrix: by linearization or by stochastic approximation.

Via stochastic approximation

A stochastic approximation algorithm using a Markov chain Monte Carlo (MCMC) algorithm is implemented in Monolix for estimating the FIM. This method is extremely general
and can be used for many data and model types (continuous, categorical, time-to-event, mixtures, etc.).

Via linearization

This method can be applied for continuous data only. A continuous model can be written as:

$$\begin{array}{cl} y_{ij} &= f(t_{ij},z_i)+g(t_{ij},z_i)\epsilon_{ij} \\ z_i &= z_{pop}+\eta_i \end{array}$$

with \( y_{ij} \) the observations, f the prediction, g the error model, \( z_i\) the individual parameter value for individual i, \( z_{pop}\) the typical parameter value within the population and \(\eta_i\) the random effect.
Linearizing the model means using a Taylor expansion in order to approximate the observations \( y_{ij} \) by a normal distribution. In the formulation above, the appearance of the random variable \(\eta_i\) in the prediction f in a nonlinear way leads to a complex (non-normal) distribution for the observations \( y_{ij} \).
The Taylor expansion is done around the EBEs value, that we note \( z_i^{\textrm{mode}} \).

Standard errors

Once the Fisher Information Matrix has been obtained, the standard errors can be calculated as the square root of the diagonal elements of the inverse of the Fisher Information Matrix. The inverse of the FIM \(I(\hat{\theta})\) is the variance-covariance matrix \(C(\hat{\theta})\):

$$C(\hat{\theta})=I(\hat{\theta})^{-1}$$

The standard error for parameter \( \hat{\theta}_k \) can be calculated as:

$$\textrm{s.e}(\hat{\theta}_k)=\sqrt{\tilde{C}_{kk}(\hat{\theta})}$$

Note that in Monolix, the Fisher Information Matrix and variance-covariance matrix are calculated on the transformed normally distributed parameters. MonolixSuite version 2024R1 incorporates a change in the calculation of the variance covariance matrix \( \tilde{C} \). In version 2023 and before, \( \tilde{C} \) was obtained for untransformed parameters using the Jacobian matrix \(J\). The Jacobian matrix is a first-order approximation.

$$\tilde{C}=J^TC J$$

Starting from version 2024R1 and later, the Jacobian matrix has been replaced by exact formulas to compute the variance, dependent on the distribution of the parameters. Parameters are distingushed into those with normal distribution, lognormal distribution, as well as logit and probitnormal distribution:

  • Normal distributed parameters: No transformation is required.
  • Lognormal distributed parameters: The variance in the Gaussian domain (e.g variance of log(V_pop)) are obtained via the FIM. To obtain the variance of the untransformed parameyers (e.g variance of V_pop), the following formula is applied: \( Var(\hat{\theta}_{k}) = (\exp(\sigma ^{2}-1))\exp(2\mu + \sigma ^{2})\), with \( \mu = \ln(\hat{\theta}_{k}) \) and \( \sigma ^{2}=Var(\ln(\hat{\theta}_{k}))\)
  • Logitnormal and probitnormal distributed parameters: There are no explicit formula to obtain the variance of the untransformed parameters (e.g bioavailability F) from the transformed parameter (e.g logit(F)). Therefore, a Monte Carlo sampling approach is used. 100000 samples are drawn from the covariance matrix in gaussian domain. Then the samples are transformed from gaussian to non-gaussian domain. For instance in the case of a logitnormal distributed parameter within bounds \( a\) and \( b\), the \(i\)-th sample of parameter \( \theta_{k,i}\): \(\mu_{k,i}=logit(\theta_{k,i})\) is transformed into non-gaussian space by the inverse logit, i.e. \(\theta_{k,i}=\frac{b \times \exp(\mu_{k,i})+ a}{(1 + \exp(\mu_{k,i}))}\). Then the empirical variance \(\sigma^2\) over all transformed samples \(\theta_{k}\) is calculated.

This transformation applies only to the typical values (fixed effects) “pop”. For the other parameters (standard deviation of the random effects “omega”, error model parameters, covariate effects “beta” and correlation parameters “corr”), we obtain directly the variance of these parameters from the FIM.

Correlation matrix

The correlation matrix is calculated from the variance-covariance matrix as:

$$\text{corr}(\theta_i,\theta_j)=\frac{\tilde{C}_{ij}}{\textrm{s.e}(\theta_i)\textrm{  s.e}(\theta_j)}$$

Wald test

For the beta parameters characterizing the influence of the covariates, the relative standard error can be used to perform a Wald test, testing if the estimated beta value is significantly different from zero.

 

Running the standard errors task

When running the standard error task, the progress is displayed in the pop-up window. At the end of the task, the correlation matrix is also shown, along with the elapsed time and number of iterations.

 

Dependencies between tasks:

The “Population parameters” task must be run before launching the Standard errors task. If the Conditional distribution task has already been run, the first iterations of the Standard errors (without linearization) will be very fast, as they will reuse the same draws as those obtained in the Conditional distribution task.

 

Output

In the graphical user interface

In the Pop.Param section of the Results tab, three additional columns appear in addition to the estimated population parameters:

  • S.E: the estimated standard errors
  • R.S.E: the relative standard error (standard error divided by the estimated parameter value)

To help the user in the interpretation, a color code is used for the p-value and the RSE:

  • For the p-value: between .01 and .05, between .001 and .01, and less than .001.
  • For the RSE: between 50% and 100%, between 100% and 200%, and more than 200%.

When the standard errors were estimated both with and without linearization, the S.E and R.S.E are displayed for both methods.

In the STD.ERRORS section of the Results tab, we display:

  • R.S.E: the relative standard errors
  • Correlation matrix: the correlation matrix of the population parameters
  • Eigen values: the smallest and largest eigen values, as well as the condition number (max/min)
  • The elapsed time and, starting from Monolix2021R1, the number of iterations for stochastic approximation, as well as a message indicating whether convergence has been reached (“auto-stop”) or if the task was stopped by the user or reached the maximum number of iterations.

To help the user in the interpretation, a color code is used:

  • For the correlation: between .5 and .8, between .8 and .9, and higher than .9.
  • For the RSE: between 50% and 100%, between 100% and 200%, and more than 200%.

When the standard errors were estimated both with and without linearization, both results appear in different subtabs.

If you hover on a specific value with the mouse, both parameters are highlighted to know easily which parameter you are looking at:

 

In the output folder

After having run the Standard errors task, the following files are available:

  • summary.txt: contains the s.e, r.s.e, p-values, correlation matrix and eigenvalues in an easily readable format, as well as elapsed time and number of iterations for stochastic approximation (starting from Monolix2021R1).
  • populationParameters.txt: contains the s.e, r.s.e and p-values in csv format, for the method with (*_lin) or without (*_sa) linearization
  • FisherInformation/correlationEstimatesSA.txt: correlation matrix of the population parameter estimates, method without linearization (stochastic approximation)
  • FisherInformation/correlationEstimatesLin.txt: correlation matrix of the population parameter estimates, method with linearization
  • FisherInformation/covarianceEstimatesSA.txt: variance-covariance matrix of the transformed normally distributed population parameter, method without linearization (stochastic approximation)
  • FisherInformation/covarianceEstimatesLin.txt: variance-covariance matrix of the transformed normally distributed population parameter, method with linearization

 

Interpreting the correlation matrix of the estimates

The color code of Monolix’s results allows to quickly identify population parameter estimates that are strongly correlated. This often reflects model overparameterization and can be further investigated using Mlxplore and the convergence assessment. This is explained in details in this video:





 

Settings

The settings are accessible through the interface via the button next to the Standard errors task:

  • Minimum number of iterations: minimum number of iterations of the stochastic approximation algorithm to calculate the Fisher Information Matrix.
  • Maximum number of iterations: maximum number of iterations of the stochastic approximation algorithm to calculate the Fisher Information Matrix. The algorithm stops even if the stopping criteria are not met.

 

Good practices and tips

When to use “use linearization method”?

Firstly, it is only possible to use the linearization method for continuous data. For the linearization is available, this method is generally much faster than without linearization (i.e stochastic approximation) but less precise. The Fisher Information Matrix by model linearization will generally be able to identify the main features of the model. More precise– and time-consuming – estimation procedures such as stochastic approximation will have very limited impact in terms of decisions for these most obvious features. Precise results are required for the final runs where it becomes more important to rigorously defend decisions made to choose the final model and provide precise estimates and diagnosis plots.

I have NANs as results for standard errors for parameter estimates. What should I do? Does it impact the likelihood?

NaNs as standard errors often appear when the model is too complex and some parameters are unidentifiable. They can be seen as an infinitely large standard error.
The likelihood is not affected by NaNs in the standard errors. The estimated population parameters having a NaN as standard error are only very uncertain (infinitely large standard error and thus infinitely large confidence intervals).

5.1.6.Log Likelihood estimation

Purpose

The log-likelihood is the objective function and a key information. The log-likelihood cannot be computed in closed form for nonlinear mixed effects models. It can however be estimated.

Log-likelihood estimation

Performing likelihood ratio tests and computing information criteria for a given model requires computation of the log-likelihood

$$ {\cal L}{\cal L}_y(\hat{\theta}) = \log({\cal L}_y(\hat{\theta})) \triangleq \log(p(y;\hat{\theta})) $$

where \(\hat{\theta}\) is the vector of population parameter estimates for the model being considered, and \(p(y;\hat{\theta})\) is the probability distribution function of the observed data given the population parameter estimates. The log-likelihood cannot be computed in closed form for nonlinear mixed effects models. It can however be estimated in a general framework for all kinds of data and models using the importance sampling Monte Carlo method. This method has the advantage of providing an unbiased estimate of the log-likelihood – even for nonlinear models – whose variance can be controlled by the Monte Carlo size.

Two different algorithms are proposed to estimate the log-likelihood:

  • by linearization,
  • by Importance sampling.

 

Log-likelihood by importance sampling





The observed log-likelihood \({\cal LL}(\theta;y)=\log({\cal L}(\theta;y))\) can be estimated without requiring approximation of the model, using a Monte Carlo approach. Since

$${\cal LL}(\theta;y) = \log(p(y;\theta)) = \sum_{i=1}^{N} \log(p (y_i;\theta))$$

we can estimate \(\log(p(y_i;\theta))\) for each individual and derive an estimate of the log-likelihood as the sum of these individual log-likelihoods. We will now explain how to estimate \(\log(p(y_i;\theta))\) for any individual i. Using the \(\phi\)-representation of the model (the individual parameters are transformed to be Gaussian), notice first that \(p(y_i;\theta)\) can be decomposed as follows:

$$p(y_i;\theta) = \int p(y_i,\phi_i;\theta)d\phi_i = \int p(y_i|\phi_i;\theta)p(\phi_i;\theta)d\phi_i = \mathbb{E}_{p_{\phi_i}}\left(p(y_i|\phi_i;\theta)\right)$$

Thus, \(p(y_i;\theta)\) is expressed as a mean. It can therefore be approximated by an empirical mean using a Monte Carlo procedure:

  1. Draw M independent values \(\phi_i^{(1)}\), \(\phi_i^{(2)}\), …, \(\phi_i^{(M)}\) from the marginal distribution \(p_{\phi_i}(.;\theta)\).
  2. Estimate \(p(y_i;\theta)\) with \(\hat{p}_{i,M}=\frac{1}{M}\sum_{m=1}^{M}p(y_i | \phi_i^{(m)};\theta)\)

By construction, this estimator is unbiased, and consistent since its variance decreases as 1/M:

$$\mathbb{E}\left(\hat{p}_{i,M}\right)=\mathbb{E}_{p_{\phi_i}}\left(p(y_i|\phi_i^{(m)};\theta)\right) = p(y_i;\theta) ~~~~\mbox{Var}\left(\hat{p}_{i,M}\right) = \frac{1}{M} \mbox{Var}_{p_{\phi_i}}\left(p(y_i|\phi_i^{(m)};\theta)\right)$$

We could consider ourselves satisfied with this estimator since we “only” have to select M large enough to get an estimator with a small variance. Nevertheless, it is possible to improve the statistical properties of this estimator.

For any distribution \(\tilde{p_{\phi_i}}\) that is absolutely continuous with respect to the marginal distribution \(p_{\phi_i}\), we can write

$$ p(y_i;\theta) = \int p(y_i|\phi_i;\theta) \frac{p(\phi_i;\theta)}{\tilde{p}(\phi_i;\theta)} \tilde{p}(\phi_i;\theta)d\phi_i = \mathbb{E}_{\tilde{p}_{\phi_i}}\left(p(y_i|\phi_i;\theta)\frac{p(\phi_i;\theta)}{\tilde{p}(\phi_i;\theta)} \right).$$

We can now approximate \(p(y_i;\theta)\) using an importance sampling integration method with \(\tilde{p}_{\phi_i}\) as the proposal distribution:

  1. Draw M independent values \(\phi_i^{(1)}\), \(\phi_i^{(2)}\), …, \(\phi_i^{(M)}\) from the proposal distribution \(\tilde{p_{\phi_i}}(.;\theta)\).
  2. Estimate \(p(y_i;\theta)\) with \(\hat{p}_{i,M}=\frac{1}{M}\sum_{m=1}^{M}p(y_i | \phi_i^{(m)};\theta)\frac{p(\phi_i^{(m)};\theta)}{\tilde{p}(\phi_i^{(m)};\theta)}\)

By construction, this estimator is unbiased, and its variance also decreases as 1/M:

$$\mbox{Var}\left(\hat{p}_{i,M}\right) = \frac{1}{M} \mbox{Var}_{\tilde{p_{\phi_i}}}\left(p(y_i|\phi_i^{(m)};\theta)\frac{p(\phi_i^{(m)};\theta)}{\tilde{p}(\phi_i^{(m)};\theta)}\right)$$

There exist an infinite number of possible proposal distributions \(\tilde{p}\) which all provide the same rate of convergence 1/M. The trick is to reduce the variance of the estimator by selecting a proposal distribution so that the numerator is as small as possible.

For this purpose, an optimal proposal distribution would be the conditional distribution \(p_{\phi_i|y_i}\). Indeed, for any \(m = 1,2, …, M,\)

$$ p(y_i|\phi_i^{(m)};\theta)\frac{p(\phi_i^{(m)};\theta)}{p(\phi_i^{(m)}|y_i;\theta)} = p(y_i;\theta) $$

which has a zero variance, so that only one draw from \(p_{\phi_i|y_i}\) is required to exactly compute the likelihood \(p(y_i;\theta)\).

The problem is that it is not possible to generate the \(\phi_i^{(m)}\) with this exact conditional distribution, since that would require computing a normalizing constant, which here is precisely \(p(y_i;\theta)\).

Nevertheless, this conditional distribution can be estimated using the Metropolis-Hastings algorithm and a practical proposal “close” to the optimal proposal \(p_{\phi_i|y_i}\) can be derived. We can then expect to get a very accurate estimate with a relatively small Monte Carlo size M.

The mean and variance of the conditional distribution \(p_{\phi_i|y_i}\) are estimated by Metropolis-Hastings for each individual i. Then, the \(\phi_i^{(m)}\) are drawn with a noncentral student t-distribution:

$$ \phi_i^{(m)} = \mu_i + \sigma_i \times T_{i,m}$$

where \(\mu_i\) and \(\sigma^2_i\) are estimates of \(\mathbb{E}\left(\phi_i|y_i;\theta\right)\) and \(\mbox{Var}\left(\phi_i|y_i;\theta\right)\), and \((T_{i,m})\) is a sequence of i.i.d. random variables distributed with a Student’s t-distribution with \(\nu\) degrees of freedom (see section Advanced settings for the log-likelihood for the number of degrees of freedom).

Remark: The standard error of the LL on all the draws is proposed. It represents the impact of the variability of the draws on the LL uncertainty, given the estimated population parameters, but it does not take into account the uncertainty of the model that comes from the uncertainty on the population parameters.

Remark: Even if \(\hat{\cal L}_y(\theta)=\prod_{i=1}^{N}\hat{p}_{i,M}\) is an unbiased estimator of \({\cal L}_y(\theta)\), \(\hat{\cal LL}_y(\theta)\) is a biased estimator of \({\cal LL}_y(\theta)\). Indeed, by Jensen’s inequality, we have :

$$\mathbb{E}\left(\log(\hat{\cal L}_y(\theta))\right) \leq \log \left(\mathbb{E}\left(\hat{\cal L}_y(\theta)\right)\right)=\log\left({\cal L}_y(\theta)\right)$$

Best practice: the bias decreases as M increases and also if \(\hat{\cal L}_y(\theta)\) is close to \({\cal L}_y(\theta)\). It is therefore highly recommended to use a proposal as close as possible to the conditional distribution \(p_{\phi_i|y_i}\), which means having to estimate this conditional distribution before estimating the log-likelihood (i.e. run task  “Conditional distribution” before).

 

Log-likelihood by linearization

The likelihood of the nonlinear mixed effects model  cannot be computed in a closed-form. An alternative is to approximate this likelihood by the likelihood of the Gaussian model deduced from the nonlinear mixed effects model after linearization of the function f (defining the structural model) around the predictions of the individual parameters \((\phi_i; 1 \leq i \leq N)\).
Notice that the log-likelihood can not be computed by linearization for discrete outputs (categorical, count, etc.) nor for mixture models.

Best practice: We strongly recommend to compute the conditional mode before computing the log-likelihood by linearization. Indeed, the linearization should be made around the most probable values as they are the same for both the linear and the nonlinear model.

 

Best practices: When should I use the linearization and when should I use the importance sampling?

Firstly, it is only possible to use the linearization algorithm for the continuous data. In that case, this method is generally much faster than importance sampling method and also gives good estimates of the LL. The LL calculation by model linearization will generally be able to identify the main features of the model. More precise– and time-consuming – estimation procedures such as stochastic approximation and importance sampling will have very limited impact in terms of decisions for these most obvious features. Selection of the final model should instead use the unbiased estimator obtained by Monte Carlo.

 

Display and outputs

In case of estimation using the importance sampling method, a graphical representation is proposed to see the valuation of the mean value over the Monte Carlo iterations as on the following:

The final estimations are displayed in the result frame as below. Notice that there is a “Copy table” icon on the top of each table to copy them in Excel, Word, … The table format and display will be kept.

The log-likelihood is given in Monolix together with the Akaike information criterion (AIC) and Bayesian information criterion (BIC):
$$ AIC = -2 {\cal L}{\cal L}_y(\hat{\theta}) +2P $$
$$ BIC = -2 {\cal L}{\cal L}_y(\hat{\theta}) +log(N)P $$
where P is the total number of parameters to be estimated and N the number of subjects.

The new BIC criterion penalizes the size of \(\theta_R\) (which represents random effects and fixed covariate effects involved in a random model for individual parameters) with the log of the number of subjects (\(N\)) and the size of \(\theta_F\) (which represents all other fixed effects, so typical values for parameters in the population, beta parameters involved in a non-random model for individual parameters, as well as error parameters) with the log of the total number of observations (\(n_{tot}\)), as follows:
$$ BIC_c = -2 {\cal L}{\cal L}_y(\hat{\theta}) + \dim(\theta_R)\log N+\dim(\theta_F)\log n_{tot}$$

If the log-likelihood has been computed by importance sampling, the number of degrees of freedom used for the proposal t-distribution (5 by default) is also displayed, together with the standard error of the LL on the individual parameters drawn from the t-distribution.

 

In terms of output, a folder called LogLikelihood is created in the result folder where the following files are created

  • logLikelihood.txt: containing for each computed method, the -2 x log-likelihood, the Akaike Information Criteria (AIC), the Bayesian Information Criteria (BIC), and the corrected Bayesian Information Criteria (BICc).
  • individualLL.txt: containing the -2 x log-likelihood for each individual for each computed method.

 

Advanced settings for the log-likelihood

Monolix uses a t-distribution as proposal. By default, the number of degrees of freedom of this distribution is fixed to 5. In the settings of the task, it is also possible to optimize the number of degrees of freedom. In such a case, the default possible values are 1, 2, 5, 10 and 20 degrees of freedom. A distribution with a small number of degree of freedom (i.e. heavy tails) should be avoided in case of stiff ODE’s defined models.

5.2.Algorithms convergence assessment

Monolix includes a convergence assessment tool. It allows to execute a workflow of estimation tasks several times, with different, randomly generated, initial values of fixed effects, as well as different seeds. The goal is to assess the robustness of the convergence.







Running the convergence assessment

For that, click on the shortcut button in the “Tasks” part.


A dedicated panel opens as in the figure below. The first shortcut button next to Run can be used to go back to the estimation.

The user can define

    • the number of runs, or replicates
    • the type of assessment:
    • the initial parameters. By default, initial values are uniformly drawn from intervals defined around the estimated values if population parameters have been estimated, the initial estimates otherwise. Notice that it is possible to set one initial parameter constant while generating the others. The minimum and maximum of the generated parameters can be modified by the user.

All settings used are saved and reloaded with the run containing the convergence assessment results.

Notice that

  • In the case of estimation of the standard errors and log-likelihood by linearization, the individual parameters with the conditional mode method are computed as well to have more relevant linearization.
  • In the case of estimation of the standard errors and log-likelihood without the linearization, the conditional distribution method is computed too to have more relevant estimation.
  • The workflow is the same between the runs and is not the one defined in the interface.

Click on Run to execute the tool. Thus you are able to estimate the population parameters using several initial seeds and/or several initial conditions.

Display and outputs

Several kinds of plots are given as a summary of the results.
First of all, the SAEM convergence assessment is proposed. The convergence of each parameter on each run is proposed. It allows to see if the convergence for each run is ok.

Then, a plot showing the estimated values for each replicate is proposed. If the estimation of the standard errors was included in the scenario, the estimated standard errors are also displayed as horizontal bars. It allows to see if all parameters converge statistically to the same values.

Starting from the 2019 version, it is possible to export manually all the plots in the Assessment folder in your result folder by clicking on the “export” icon (purple box on the previous figure).
Finally, if log-likelihood without linearization is used, the curves for convergence of importance sampling are proposed.

In addition, a Monolix project and its result folder is generated for each set of initial parameters. They are located in the Assessment subfolder of the main project’s result folder (located by default next to the .mlxtran project file). Along with all the runs, there is a summary of all the runs “assessment.txt” providing all the individual parameter estimates along with the -2LL, as in the following:

Parameters,Run_1,Run_2,Run_3,Run_4,Run_5
Cl_pop,0.03994527,0.04017999,0.04016216,0.04012077,0.0400175
V_pop,0.4575748,0.4556463,0.4560732,0.4557009,0.4569431
a,0.4239969,0.42482,0.4227559,0.4294611,0.435585
b,0.05653124,0.05684357,0.05700663,0.054965,0.05450724
ka_pop,1.527947,1.521184,1.5226,1.519333,1.519678
omega_Cl,0.2653109,0.2643172,0.268475,0.266199,0.2693083
omega_V,0.1293328,0.1274441,0.122951,0.1301242,0.1261098
omega_ka,0.6530206,0.6655251,0.643456,0.6425528,0.6424614
-2LL,339.387,339.417,339.429,339.444,339.462

Notice that, starting from the 2019 version, it is possible to reload all the results of a previous convergence if nothing has changed in the project. Starting from version 2021, the settings of the convergence assessment are also reloaded.

Best practices: what is the use the convergence assessment tool?

We cannot claim that SAEM always converges (i.e., with probability 1) to the global maximum of the likelihood. We can only say that it converges under quite general hypotheses to a maximum – global or perhaps local – of the likelihood. A large number of simulation studies have shown that SAEM converges with high probability to a “good” solution – hopefully the global maximum – after a small number of iterations. The purpose of this tool is to evaluate the SAEM algorithm with initial conditions and see if the estimated parameters are the “global” minimum.

The trajectory of the outputs of SAEM depends on the sequence of random numbers used by the algorithm. This sequence is entirely determined by the “seed.” In this way, two runs of SAEM using the same seed will produce exactly the same results. If different seeds are used, the trajectories will be different but convergence occurs to the same solution under quite general hypotheses. However, if the trajectories converge to different solutions, that does not mean that any of these results are “false”. It just means that the model is sensitive to the seed or to the initial conditions. The purpose of this tool is to evaluate the SAEM algorithm with several seeds to see the robustness of the convergence.

5.3.Model building

Starting from the 2019 version, a panel Model building provides automatic model building tools:

This panel is accessible via the button Model building next to Run in the interface of Monolix, or from the section Perspective in the tab Home.

 

5.3.1.Statistical tests for model assessment

5.3.2.Automatic statistical model building (covariate model, correlation model, error model)

5.3.3.Automatic complete statistical model building

5.3.4.Automatic covariate model building

5.3.5.Automatic variability model building

5.3.6.Statistical model building with SAMBA

Starting from the 2019 version, an automatic statistical model building algorithm is implemented in Monolix: SAMBA (Stochastic Approximation for Model Building Algorithm)
SAMBA is an iterative procedure to accelerate and optimize the process of model building by identifying at each step how best to improve some of the model components (residual error model, covariate effects, correlations between random effects). This method allows to find the optimal statistical model which minimizes some information criterion in very few steps. It is described in more details in the following publication:

Prague, M, Lavielle, M. SAMBA: A novel method for fast automatic model building in nonlinear mixed-effects models. CPT Pharmacometrics Syst Pharmacol. 2022; 00: 1- 12. doi:10.1002/psp4.12742

At each iteration, the best statistical model is selected with the same method as the Proposal. This step is very quick as it does not require to estimate new parameters with SAEM for all evaluated models. Initial estimates are set to the estimated values from the Proposal.
The population parameters of the selected model are then estimated with SAEM, individual parameters are simulated from the conditional distributions, and the log-likelihood is computed to evaluate the improvement of the model. The algorithm stops when no improvement is brought by the selected model or if it has already been tested.

 

Initialization

 

Settings

The linearization method is selected by default to compute the log-likelihood of the estimated model at each iteration. It can be unselected to use importance sampling.
The improvement can be evaluated with two different criteria based on the log-likelihood, that can be selected in the settings (available via the icon next to Run):

  • BICc (by default)
  • LRT (likelihood ratio threshold): by default the forward threshold is 0.01 and the backward threshold is 0.01. These values can be changed in the settings.

Starting from the 2021 version, all settings used are saved and reloaded with the run containing the model building results.

Selecting covariates and parameters

It is possible to select part of the covariates and individual parameters to be used in the algorithm:

Moreover, a panel “Locked relationships” can be opened to lock in or lock out some covariate-parameter relationships among the ones that are available:

Starting from the 2021 version, all relationships considered in model building are part of the settings which are saved and reloaded with the run containing the model building results.

Results

The results of the model building are displayed in a tab Results, with the list of model run at each iteration (see for example the figure below) and the corresponding -2LL and BICc values.

All resulting runs are also located in the ModelBuilding subfolder of the project’s result folder (located by default next to the .mlxtran project file).

In the Results tab, by default runs are displayed by order of iteration, except the best model which is displayed in first position, highlighted in blue.

Note that the table of iterations can be sorted by iteration number or criteria (see green marks below). Buttons “export and load” (see blue mark below) can also be used to export the model estimated at this iteration as a new Monolix project with a new name and open it in the current Monolix session.

While the algorithm is running, the progress of the estimation tasks at each iteration is displayed on a white pop-up window, and temporary green messages confirm each successful task.

5.4.Bootstrap

[Bootstrap is not available in versions prior to 2024R1]





Introduction

The Bootstrap module in Monolix provides a robust method to assess parameter uncertainty, offering an alternative to calculating standard errors via inversion of the Fisher Information Matrix. This approach becomes particularly valuable when facing issues such as NaNs in standard errors due to numerical errors in matrix inversion or biases in results caused by incorrect assumptions of asymptotic normality for parameter estimates. Bootstrap overcomes these challenges by sampling many replicate datasets and re-estimating parameters on each replicate.

While powerful, bootstrap comes with certain drawbacks:

  • Running many replicates for population parameter estimation can be time-consuming. Bootstrap in Monolix can be used from command line and with distributed calculation (parallelization on a grid via MPI).
  • Saving a large number of new datasets and results may raise storage issues. You can choose in the settings whether to save sampled datasets and results.

Accessing the bootstrap module

Bootstrap can be accessed under the “Perspective” menu or with a shortcut next to “Run” in the “Statistical model & Tasks” tab:

   

Available bootstrap settings

Users can customize the following settings:

  • Number of runs: Number of bootstrap replicates
  • Estimation tasks: Estimation tasks to run among population pramaters, Standard Errors, and Log-Likelihood
  • Initial values: Whether bootstrap runs should start their estimation from the same initial values as the initial run, or from the final estimated values from the initial run.
  • Sampling method: Type of bootstrap:
    • Nonparametric Bootstrap (case resampling): New datasets are sampled from the initial dataset for each replicate. Each individual of the new dataset is sampled randomly with replacement from the initial dataset. This means that some individuals from the original dataset will appear several times in the resampled dataset, while some individuals will not appear at all. The generated datasets are thus similar but different compared to the original dataset. This method makes no assumption on the model.
    • Parametric Bootstrap (SSE): This method is also called SSE for stochastic simulation and estimation. New datasets are simulated from the model for each replicate. Individual parameters are sampled from the population distribution and individuals are simulated using the same design as in the original dataset (same treatments, covariates, regressors, and observation times). Residual error is added on top of the model predictions to obtain a realistic dataset. If the initial dataset has censored data, censoring limits to apply to the simulated datasets can be specified by the user. This method assumes that the model correctly captures the original data.
  • Sample size: Number of individuals in the bootstrap datasets. By default it is set equal to the number of individuals of the original dataset but can be modified if required.
  • Stratified resampling: Selection of categorical covariates which distribution should be preserved when re-sampling individuals to create datasets for non-parametric bootstrap. When the original dataset contains few individuals, or when a categorical covariate distribution is highly imbalanced with only few individuals belonging to one of the categories, resampled bootstrap datasets can results in quite different distributions (e.g only one individual left in one of the categories) and this can lead to a bias, in particular on the covariate effects (beta) parameters. To avoid this situation, the sampling can be done such that the number of individuals in each category of the covariates selected in “stratified resampling” remains the same as in the initial dataset.
  • Confidence interval level: Level of the confidence interval bounds displayed in the results to summarize the uncertainty on the population parameters.
  • Save bootstrap runs: Option to save or not save bootstrap datasets and results folder of each bootstrap run. Both options need to be selected if you want to be able to reload each bootstrap run including the results.
  • Replace runs with failed convergence: Option to replace or not bootstrap replicates that have a failed convergence. This option allows to replace runs for which the “auto-stop” criteria of the exploratiry phase of the population parameter estimation have not been reached. You can also set a maximal number of runs to replace, so that bootstrap will not go on forever if many runs do not converge. Bootstrap will stop once the maximal number is reached.

Running bootstrap

After clicking on “Run”, bootstrap is launched and the population parameters estimates from all runs already done are shown as a table and as a plot of their medians and their confidence intervals with respect to the bootstrap iterations. The table and plot updates as soon as new bootstrap run finishes. An estimation of the remaining time to complete all bootstrap runs is available at the top.

If you have stopped bootstrap before the last run, or you want to add more runs to your bootstrap results, you can resume bootstrap so that the runs already done will be reused instead of being rerun:

Results

Bootstrap results reported in the interface and the output files include the following. Depending on the choice of estimation tasks in the bootstrap settings, results for population parameters, RSE (if Standard error task was selected), and LL (if log-likelihood task was selected) are available.

  • Results > [Estimates]: Estimates from each bootstrap run displayed in tables.

When the options to save both the results folder and the datasets have been selected, it is possible to load each bootstrap run via the “load” button in the “Pop [Estimates]” tab:

  • Plots: Estimates from each bootstrap run displayed as distribution plots. Distribution plots can be displayed as boxplots or as histogramms (pdf).

   

  • Results > [Summary]: Summary table of bootstrap estimates including value of the initial run (reference), mean and median over bootstrap runs, standard error (SE) over bootstrap runs, relative standard error (RSE) over bootstrap runs, confidence intervals (according to chosen confidence level). For population parameter, the bias of the mean over bootstrap runs compared ot the reference value is calculated as bias = (mean – reference)/reference*100.
    If some runs have not reached convergences, a toggle appears to filter the summary table and keep only runs which have converged.

 

  • Main window (Estimation) > Results > Pop Param: The bootstrap results also appear in the table of population parameter estimates of the parent run. This is convenient to compare the estimated values and their confidence intervals estimated via the Fisher Information Matrix, to the median and confidence intervals estimated via bootstrap. Bootstrap results can be included in the pop param table when generating reports from Monolix.

  • Saved datasets: In the case of non-parametric bootstrap, resampled datasets used in bootstrap runs have new subject identifiers defined as integers in the column tagged as ID, and include an additional column named “original_id” with the corresponding subject identifiers from the initial dataset.

5.5.Output files

Monolix generates a lot of different output files depending on the tasks done by the user. Here is a complete listing of the files, along with the condition for their creation and their content.

Population parameter estimation

summary.txt

Description: summary file.
Outputs:

  • Header: project file name, date and time of run, Monolix version
  • Estimation of the population parameters: Estimated population parameters & computation time

populationParameters.txt

Description: estimated population parameters (with SAEM).
Outputs:

  • First column (no name): contains the parameter names (e.g ‘V_pop’ and ‘omega_V’).
  • value: contains the estimated parameter values.

Individual parameters estimation

All the files are in the IndividualParameters folder of the result folder

estimatedIndividualParameters.txt

Description: Individual parameters (from SAEM, mode, and mean of the conditional distribution)
Outputs:

  • ID: subject name and occasion (if applicable). If there is one type of occasion, there will be an additional(s) column(s) defining the occasions.
  • parameterName_SAEM: individual parameter estimated by SAEM, it corresponds to the average of the individual parameters sampled by MCMC during all iterations of the smoothing phase. When several chains are used (see project settings), the average is also done over all chains. This value is an approximation of the conditional mean.
  • parameterName_mode (if conditional mode was computed): individual parameter estimated by the conditional mode task, i.e mode of the conditional distribution \(p(\psi_i|y_i;\hat{\theta})\).
  • parameterName_mean (if conditional distribution was computed) : individual parameter estimated by the conditional distribution task, i.e mean of the conditional distribution \(p(\psi_i|y_i;\hat{\theta})\) . The average of samples from all chains and all iterations is computed in the gaussian space (eg mean of the log values in case of a lognormal distribution), and back-transformed.
  • parameterName_sd (if conditional distribution was computed): standard deviation of the conditional distribution \(p(\psi_i|y_i;\hat{\theta})\) calculated during the conditional distribution task.
  • COVname: continuous covariates values corresponding to all data set columns tagged as “Continuous covariate” and all the associated transformed covariates.
  • CATname: modalities associated to the categorical covariates (including latent covariates and the bsmm covariates) and all the associated transformed covariates.

estimatedRandomEffects.txt

Description: individual random effect, calculated using the population parameters, the covariates and the conditional mode or conditional mean. For instance if we have a parameter defined as \(k_i=k_{pop}+\beta_{k,WT}WT_i+\eta_i\), we calculate \(\eta_i=k_i – k_{pop}-\beta_{k,WT}WT_i\) with \(k_i\) the estimated individual parameter (mode or mean of the conditional distribution), \(WT_i\) the individual’s covariate, and \(k_{pop}\) and \(\beta_{k,WT}\) the estimated population parameters.
Outputs:

  • ID: subject name and occasion (if applicable). If there is one type of occasion, there will be an additional(s) column(s) defining the occasions.
  • eta_parameterName_SAEM: individual random effect estimated by SAEM, it corresponds to the last iteration of SAEM.
  • eta_parameterName_mode (if conditional mode was computed): individual random effect estimated by the conditional mode task, i.e mode of the conditional distribution \(p(\psi_i|y_i;\hat{\theta})\).
  • eta_parameterName_mean (if conditional distribution was computed) : individual random effect estimated by the conditional distribution task, i.e mean of the conditional distribution \(p(\psi_i|y_i;\hat{\theta})\) . The average of random effect samples from all chains and all iterations is computed.
  • eta_parameterName_sd (if conditional distribution was computed): standard deviation of the conditional distribution \(p(\psi_i|y_i;\hat{\theta})\) calculated during the conditional distribution task.
  • COVname: continuous covariates values corresponding to all data set columns tagged as “Continuous covariate” and all the associated transformed covariates.
  • CATname: modalities associated to the categorical covariates (including latent covariates and the bsmm covariates) and all the associated transformed covariates.

simulatedIndividualParameters.txt

Description: Simulated individual parameter (by the conditional distribution)
Outputs:

  • rep: replicate of the simulation
  • ID: subject name and occasion (if applicable). If there is one type of occasion, there will be an additional(s) column(s) defining the occasions.
  • parameterName: simulated individual parameter corresponding to the draw rep.
  • COVname: continuous covariates values corresponding to all data set columns tagged as “Continuous covariate” and all the associated transformed covariates.
  • CATname: modalities associated to the categorical covariates (including latent covariates and the bsmm covariates) and all the associated transformed covariates.

simulatedRandomEffects.txt

Description: Simulated individual random effect (by the conditional distribution)
Outputs:

  • rep: replicate of the simulation
  • ID: subject name and occasion (if applicable). If there is one type of occasion, there will be an additional(s) column(s) defining the occasions.
  • eta_parameterName: simulated individual random effect corresponding to the draw rep.
  • COVname: continuous covariates values corresponding to all data set columns tagged as”Continuous covariate” and all the associated transformed covariates.
  • CATname: modalities associated to the categorical covariates (including latent covariates and the bsmm covariates) and all the associated transformed covariates.

Fisher Information Matrix calculation

summary.txt

Description: summary file.
Outputs:

  • Header: project file name, date and time of run, Monolix version (outputted population parameter estimation task)
  • Estimation of the population parameters: Estimated population parameters & computation time (outputted population parameter estimation task). Standard errors and relative standard errors are added.
  • Correlation matrix of the estimates: correlation matrix by block, eigenvalues and computation time

populationParameters.txt

Description: estimated population parameters, associated standard errors and p-value.
Outputs:

  • First column (no name): contains the parameter names (outputted population parameter estimation task)
  • Column ‘parameter’: contains the estimated parameter values (outputted population parameter estimation task)
  • se_lin / se_sa: contains the standard errors (s.e.) for the (untransformed) parameter, obtained by linearization of the system (lin) or stochastic approximation (sa).
  • rse_lin / rse_sa: contains the parameter relative standard errors (r.s.e.) in % (param_r.s.e. = 100*param_s.e./param), obtained by linearization of the system (lin) or stochastic approximation (sa).
  • pvalues_lin / pvalues_sa: for beta parameters associated to covariates, the line contains the p-value obtained from a Wald test of whether beta=0. If the parameter is not a beta parameter, ‘NaN’ is displayed.

Notice that if the Fisher Information Matrix is difficult to invert, some parameter’s standard error can maybe not be computed leading to NaN in the corresponding columns.

All the more detailed files are in the FisherInformation folder of the result folder.

covarianceEstimatesSA.txt and/or covarianceEstimatesLin.txt

Description: variance-covariance matrix of the estimates for the (untransformed) parameters or transformed normally distributed parameters depending on the MonolixSuite version (see below)
Outputs: matrix with the project parameters as lines and columns. First column contains the parameter names.

The Fisher information matrix (FIM) is calculated for the transformed normally distributed parameters (i.e log(V_pop) if V has a lognormal distribution). By inverting the FIM, we obtain the variance-covariance matrix \(\Gamma\) for the transformed normally distributed parameters \(\zeta\). This matrix is then multiplied by the jacobian J (which elements are defined by \(J_{ij}=\frac{\partial\theta_i}{\partial\zeta_j}\)) to obtain the variance-covariance matrix \(\tilde{\Gamma}\) for the untransformed parameters \(\theta\):

$$\tilde{\Gamma}=J^T\Gamma J$$

The diagonal elements of the variance-covariance matrix \(\tilde{\Gamma}\) for the untransformed parameters are finally used to calculate the standard errors.

correlationEstimatesSA.txt and/or correlationEstimatesLin.txt

Description: correlation matrix for the (untransformed) parameters
Outputs: matrix with the project parameters as lines and columns. First column contains the parameter names.

The correlation matrix is calculated as:

$$\text{corr}(\theta_i,\theta_j)=\frac{\text{covar}(\theta_i,\theta_j)}{\sqrt{\text{var}(\theta_i)}\sqrt{\text{var}(\theta_j)}}$$

This implies that the diagonal is unitary. The variance-covariance matrix for the untransformed parameters \(\theta\) is obtained from the inverse of the Fisher Information Matrix and the jacobian. See above for the formula.

Log-Likelihood calculation

summary.txt

Description: summary file.
Outputs:

  • Header: project file name, date and time of run, Monolix version (outputted population parameter estimation task)
  • Estimation of the population parameters: Estimated population parameters & computation time (outputted population parameter estimation task). Standard errors and relative standard errors are added.
  • Correlation matrix of the estimates: correlation matrix by block, eigenvalues and computation time
  • Log-likelihood Estimation: -2*log-likelihood, AIC and BIC values, together with the computation time

All the more detailed files are in the LogLikelihood folder of the result folder

logLikelihood.txt

Description: Summary of the log-likelihood calculation with the two methods.
Outputs:

  • criteria: OFV (Objective Function Value), AIC (Akaike Information Criteria), and BIC (Bayesian Information Criteria )
  • method: ImportanceSampling and/or linearization

individualLL.txt

Description: -2LL for each individual. Notice that we only have one by individual even if there are occasions.
Outputs:

  • ID: subject name
  • method: ImportanceSampling and/or linearization

Tests

Tables

predictions.txt

Description: predictions at the observation times
Outputs:

  • ID: subject name. If there are occasions, additional columns will be added to describe the occasions.
  • time: Time from the data set.
  • MeasurementName: Measurement from the data set.
  • RegressorName: Regressor value.
  • popPred_medianCOV: prediction using the population parameters and the median covariates.
  • popPred: prediction using the population parameters and the covariates, e.g \(V_i=V_{pop}\left(\frac{WT_i}{70}\right)^{\beta}\) (without random effects).
  • indivPred_SAEM: prediction using the mean of the conditional distribution, calculated using the last iterations of the SAEM algorithm.
  • indPred_mean (if conditional distribution was computed): prediction using the mean of the conditional distribution, calculated in the Conditional distribution task.
  • indPred_mode (if conditional mode was computed): prediction using the mode of the conditional distribution, calculated in the EBEs task.
  • indWRes_SAEM: weighted residuals \(IWRES_{ij}=\frac{y_{ij}-f(t_{ij}, \psi_i)}{g(t_{ij}, \psi_i)}\) with \(\psi_i\) the mean of the conditional distribution, calculated using the last iterations of the SAEM algorithm.
  • indWRes_mean (if conditional distribution was computed): weighted residuals \(IWRES_{ij}=\frac{y_{ij}-f(t_{ij}, \psi_i)}{g(t_{ij}, \psi_i)}\) with \(\psi_i\) the mean of the conditional distribution, calculated in the Conditional distribution task.
  • indWRes_mode (if conditional mode was computed): weighted residuals \(IWRES_{ij}=\frac{y_{ij}-f(t_{ij}, \psi_i)}{g(t_{ij}, \psi_i)}\) with \(\psi_i\) the mode of the conditional distribution, calculated in the EBEs task.

Notice that in case of several outputs, Monolix generates predictions1.txt, predictions2.txt, …

Below is a correspondence of the terms used for predictions in Nonmem versus Monolix:

Charts data

All plots generated by Monolix can be exported as a figure or as text files in order to be able to plot it in another way or with other software for more flexibility. The description of all generated text files is described here.

5.6.Reproducibility of MonolixSuite results

Consistent reload of MonolixSuite projects and results

MonolixSuite applications ensure full reproducibility and consistency of analysis results. Here is how it works:
  • MonolixSuite applications save in each project all the elements necessary to run the analysis and replicate results from submitted analyses. In practice, the path to the dataset file, the path to the model file (if custom model not from library), and all settings chosen in the Monolix GUI (tagging of the data set columns, initial parameter values, statistical model definition, task settings, seed of the random number generator, etc) are saved in the .mlxtran file. When an existing Monolix project is loaded and re-run, the results will be exactly the same as for the first run.
  • In addition, the results of the analysis are loaded in the interface from the results folder as valid results only if nothing has changed in the project since the time of the run. In practice, when launching a run, the content of the mlxtran file is saved in the result folder. When a Monolix project is reloaded, the content of the loaded mlxtran file and the information in the result folder is compared. If they are the same, it means that the results present in the result folder are consistent with the loaded mlxtran and that therefore they are valid. If anything that could affect the results has changed (e.g initial values indicated in the loaded mlxtran file are different from the initial values used to generate the results), the results are not loaded and the warning message “Results have not been loaded due to an old inconsistent project” appears.
  • Starting with the 2021R1 version, in Monolix and PKanalix a fingerprint of the dataset is also generated and checked when the project is loaded to ensure that not only the path to the data set file but also the content of the dataset has not been modified.
  • Plot settings are saved in a different file (not the .mlxtran file) and do not impact the reloading of the results.

Rerunning a project with the same MonolixSuite version and the same OS (because random number generators are different in Windows, Linux and Mac) yields the exact same results if nothing has changed in the project. The software version used to generate the results is displayed in the file summary.txt in the results folder.









History of runs

In addition, in Monolix and PKanalix there is a timestamping option in the Preferences of the software called “Save History”. When it is enabled, the project and its results is saved in the results folder after each run, in a subfolder named “History”, thus making sure that previous runs are not lost even if they have been overwritten by a new run.
Below is an example of “History” subfolder in which 3 runs have been saved. Each subfolder named with the time of the run contains the project and its results as they were at the time of the run.

5.7.Comments





Starting from the 2019 version, it is possible to add comments to the project with a dedicated part in the interface in the frame “Comments” as in the following figure.

It allows to put information on the project and save it within the project. Notice that you can put information before running the project and/or after running the project and the plots to comment the VPC for example. The text can be formatted with HTML markup or markdown syntax. A preview window is also proposed in order to see the formatted text.

The comments are displayed in Sycomore.

6.Plots

6.1.Overview

6.1.1.Generating and exporting plots

 

Generating plots

Diagnostic plots can be generated by clickin on the task “Plots” or as part of the scenario when clicking on Run. By default, when reloading a project which has already run, only the plots Observed data and Covariate viewer are displayed. To generate the other diagnostic plots, click on the Plot task. Generating plots automatically at project load can be enabled via a Preference.

The set of plots to compute can be selected by clicking on the button next to the task as shown below, prior to running the task.

By default, only a subset of plots are selected (see below), with one occurence for each plot. The number of occurences can be selected via the number selection box. Having several occurences of the same plot allows to choose different settings for each occurence, for instance one on linear scale and and one on log scale. The plot selection and number of occurences is saved as part of the mlxtran file upon save, and reapplied at project reload.

The green arrow can be used to generate only one plot type with the given number of occurences without re-generating all other plots. The “+” button allows to add one additional occurence. If the information necessary to generate a plot is not available, these buttons are hidden.

List of available plots

Data

  • Observed data: This plot displays the original data w.r.t. time as a spaghetti plot, along with some additional information.

Model for the observations

  • Individual fits: This plot displays the individual fits: individual predictions using the individual parameters and the individual covariates w.r.t. time on a continuous grid, with the observed data overlaid.
  • Observations vs predictions: This plot displays observations w.r.t. the predictions computed using the population parameters or the individual parameters.
  • Scatter plot of the residuals: This plot displays the PWRES (population weighted residuals), the IWRES (individual weighted residuals), and the NPDE (Normalized Prediction Distribution Errors) w.r.t. the time and the prediction.
  • Distribution of the residuals: This plot displays the distributions of PWRES, IWRES and NPDE as histograms for the probability density function (PDF) or as cumulative distribution functions (CDF).

Diagnosis plots based on individual parameters

Predictive checks and predictions

Convergence diagnosis

Tasks results

 

Exporting plots and setting plot-related preferences

Several features can be used to export plots or plot data or create plots automatically:

  • Saving a single plot as image (icon button)
  • Create plots at project load (Preference)
  • Save charts data as binary file (Preference)
  • Export VPC simulations (Preference and Export menu)
  • Export charts datasets (Preference and Export menu)
  • Export plots (Preference and Export menu)
  • Export charts data (Preference and Export menu)

Each feature is detailed below in this section.

Most of them are located:

  • in the Export menu: to export the plots as image file or a text files for a single project.
  • in the Preferences: to export the plots as image file or a text files automatically at the end of each plot generation task. Preferences are user-specific and apply to all projects. The preferences are saved in C:\Users\<username>\lixoft\monolix\monolixXXXXRX\config\config.ini.

Saving a single plot as image (icon button)

The user can choose to export each plot as an image with an icon on top of the plot, choosing between png and svg format. The files are saved in <result folder>/ChartsFigures with the name of the plot, the observation name (e.g y1 or y2) and a postfix indicating the plot occurence, and page number. Note that information frames are not exported. The icon becomes available only when a structural model has been selected.

Create plots at project load (Preference)

By default, when a project which has already run is re-opened, only the plots “Observed data” and “Covariate viewer” are displayed. To generate the other diagnostic plots, it is necessary to click on the “Plots” task. Note that it is not necessary to rerun the entire scenario of all tasks.

Starting with version 2024, it is possible to choose whether plots should be created automatically at project load. This option is available in Settings > Preferences > Create plots at project load. When this option is “on”, plots are generated when opening a project with results. Note that this increases the load time, in particular if the charts data have not been saved as binary (see below). By default, the option is “off”.

Save charts data as binary file (Preference)

Generating the plots can take several minutes depending on the model and dataset, because it requires to redo the simulations used in the VPC for instance. Starting with version 2024, it is possible to save the charts data (in particular the simulations) as a binary file, which is re-used when the plots are re-generated. This greatly speeds up the time necessary to generate the diagnostic plots. This option is available in Settings > Preferences > Save charts data as binary file. When this option is “on” (default), the charts data are saved in a non-human-readable format in <result folder>/ChartsData/.Internals/chartsData.dat each time the plots are generated. At project reload, if the file chartsData.dat exists and is valid, it is used to generate the plots faster.

Export VPC simulations (Preference and Export menu)

Simulations used to generate the VPC plot can be exported as txt file. This allows to replot the VPC with an external software for instance. If this export is required for a single project, the user can click Export > Export VPC simulations. The simulations are saved in <result folder>\ChartsData\VisualPredictiveCheck\XXX_simulations.txt. If the user wishes to do this export each time the VPC plot is generated, the option Settings > Preferences > Export VPC simulations can be set to “on”.

Export charts datasets (Preference and Export menu)

Simulations used for the VPC and the Individual fits (prediction on a fine time grid) can be exported as MonolixSuite-formatted datasets (including dose records, regressors, etc). This is useful if these simulations need to be loaded in another MonolixSuite application such as PKanalix for instance. To export these formatted simulations once use Export > Export Charts Datasets > VPC or Individual fits. A pop-up window will let you choose the name and location of the saved files.

To export the simulations as formatted dataset systematically each time the plots are generated, the option Settings > Preferences > Export charts datasets can be set to “on”. The files are saved in <result folder>/DataFile/ChartsData/indfits_data.csv and vpc_data.csv.

Export plots (Preference and Export menu)

Plots can be saved as image files. To save a single plot, use the icon on the top of the plot (see section above). To save all plots for a single project, use Export > Export plots. To systematically save all generated plots as image file, set the the option Settings > Preferences > Explot plots to “on”. The files are saved in <result folder>/ChartsFigures with the name of the plot, the observation name (e.g y1 or y2) and a postfix indicating the plot occurence, and page number. The files are saved at the end of the plot generation task. The file format (png or svg) can be chosen in the Preferences. Note that information frames are not exported.

Export charts data (Preference and Export menu)

The values used to draw the plots can be saved as txt files. This can be useful to regenerate the plots using an external tool. To export the charts data for a single project, use Export > Export charts data. To systematically export the charts data at the end of the plot generation, set the the option Settings > Preferences > Explot charts data to “on”. The files are saved in <result folder>/ChartsData with one subfolder per plot. The content of each file is described on the dedicated page. The file separator can be chosen in the Preferences.

 

Plot settings

A few plot settings impact the simulations required for the plots. They can be chosen in the Settings panel for the Plots task.

  • Number of simulations: number of replicates simulated for the VPC, NPC and Prediction distribution plot (default: 500)
  • Grid: number of points of the fine time grid used for the individuals fits (default: 250). See the video below for more details.

Customizing time grid in plots

6.1.2.Interacting with the plots

 

Interactive diagnostic plots



In the PLOTS tab, the right panel has several sub – tabs (at the bottom) to interact with the plots:

  • The tab “Settings” provides options specific to each plot, such as hiding or displaying elements of the plot, modifying some elements, or changing axes scales, ticks and limits.
  • The tab “Stratify” can be used to select one or several covariates for splitting, filtering or coloring the points of the plot. See below for more details.
  • The tab “Preferences” alows to customize graphical aspects such as colors, font size, dot radius, line width, …

These tabs are marked in purple on the following figure, which is the panel that is showed for observed data:

Highlight: tooltips and ID

In all the plots, when you hover a point or a curve with your mouse, some informations are provided as tooltips. For example, the ID is displayed when hovering a point or the curve of an individual in the observed data plot, the ID, the time and/or the prediction is displayed in the scatter plot of the residuals.
In addition, starting from the 2019 version, when hovering one point/ID in a plot, the same ID will be highlighted in all the plots with the same color.





 

Stratification: split, color, filter

The stratification panel allows to create and use covariates for stratification purposes. It is possible to select one or several covariates for splitting, filtering or coloring the data set or the diagnosis plots as exposed on the following video.



The following figure shows a plot of the observed data from the warfarin dataset, stratified by coloring individuals according to the continuous covariate wt: the observed data is divided into three groups, which were set to equal size with the button “rescale”. It is also possible to set groups of equal width, or to personalize dividing values.
In addition, the bounds of the continuous covariate groups can be changed manually.

Moreover, clicking on a group highlights only the individuals belonging to this group, as can be seen below:

Values of categorical covariates can also be assigned to new groups, which can then be used for stratification.
In addition, the number of subjects in each categorical covariate groups is displayed.

Starting from the 2021R1 version, the list of subplots obtained by split is displayed below the split selection. It is possible to reorder the items in this list with drag-and-drop, which also reorders the subplot layout. Moreover, an “edit” icon next to each subplot name in the list allows to change the subplot titles.

 

 

Preferences: customizing the plot appearance

In the “preferences” tab, the user can modify the different aspects of the plot: colors, line style and width, fonts and label position offsets, …







The following figures show on the warfarin demo the choices for the plot content and the choices for the labels and titles (in the “Plotting region” section). The sizes of all elements of the plots can also be changed with the single “zoom” preference in “Plotting region”.

Layout




The layout can be modified with buttons on top of each plot.

The first button can be used to select a set of subplots to display in the page. For example, as shown below, it is possible to display 9 individual fits per page instead of 12 (default number). The layout is then automatically adapted to balance the number of rows and columns.

The second button can be used to choose a custom layout (number of rows and columns). On the example figures below, the default layout with 3 subplots (left) is modified to arrange them on a single column (right).

 

 

6.1.3.Transferring plot formatting

Note: the features described on this page are not available in MonolixSuite versions prior to 2024R1.

MonolixSuite offers the ability to transfer plot formatting from one plot to another within the same project or across projects. This can save you time and effort when you want to apply the same formatting options, such as stratification, preferences, or settings, to multiple plots, or when you want to use the same plot formatting for different analyses or datasets.

 

Transferring plot formatting within one project

To transfer plot formatting within one project, you need to use the “Apply…” button on the bottom left of the Plots tab and select “From plot” in the window that opens:

Selecting the source and target plots

The window allows you to select the source plot, which is the plot that has the formatting you want to transfer, and the target plots, which are the plots that you want to apply the formatting to. You can select all plots by clicking on the Select all button, or select all plots specific to some observation id with the “Fast selection”, or corresponding to an occurrence rank if some plots have been generated with several occurences.

 

Choosing the sections to transfer

You can also choose which sections of the configuration you want to transfer: Stratify, Settings, or Preferences.

By default, Stratify and Preferences are selected.

  • The Stratify section includes the options to stratify the plots by covariates or occasions.
  • The Settings section includes the options to customize the content of the plots: elements displayed or not on the plots, such as the legend, the grid, the observed data points, etc, and calculation settings such as the axes, bins etc.
  • The Preferences section includes the options to change the colors, fonts, sizes, etc.

Once you have selected the source plot, the target plots, and the sections to transfer, you can click on the Apply button to transfer the plot configurations.

 

Transferring plot formatting across projects with plot presets

To transfer plot formatting across different projects, you need to use the plot presets feature, which allows you to define and apply presets for plot configurations.

Defining a plot preset

To define a plot preset, you need to first set up your plot configuration as you would like by choosing the stratification, preferences, and settings options for each plot (you can use the transfer or plot formatting between plots within the same project, described in the first section, to help you). Then, you can save this configuration as a preset by clicking on the Save icon in the Apply from the Plot Formatting  panel:

You will be asked to give a name and a description to your preset, and to select which sections of the formatting you want to include in the preset among Stratify, Settings and Preferences. If the project has several types of observations, only the plots corresponding to one of them can be used to define the preset so it is necessary to select the observation id. You can also choose to set this preset as your default, which means that when creating a new project or when you click on Reset, the plot formatting from this preset will be applied instead of the system default.

Applying a plot preset

After saving your preset, it will be available in the list of presets that you can choose from when you click on “From preset” in the window that opens with “Apply…” in the Plot Formatting panel.

You can apply your preset to a list of plots in another project, by opening this project, clicking on “Apply…” and “From preset” and by selecting the preset and the target plots, and finally clicking on Apply. This will transfer the plot configurations from the preset to the target plots.

The target plots must be:

  • the same type of plots that were used to define the preset: for example if no VPC had been generated in the project used to define the preset, then the plot preset cannot be applied to a VPC in a new project, as there is no VPC formatting option to apply. The VPC will be greyed out in the list of target plots.
  • the same plot occurrence that was used to define the preset: for example if the project used to define the preset had only one VPC occurrence, and you want to apply that preset in a project where you have generated two VPC occurrences, then the second VPC occurrence will be greyed out in the list of target plots.

Note that axis limits values and bins settings are not saved as part of the preset because they are considered too project-specific.

Please note this major difference between “Apply formatting from plot” and “Apply formatting from preset”:

  • “Apply formatting from plot” applies all selected options from the source plot to all selected target plots. Thus, if you select a VPC stratified by STUDY as source plot and the Observed data plot as target plot, and “Stratify” section in plot formatting to be transfer, you will obtain the Observed data plot stratified by STUDY.
  • “Apply formatting from preset” applies the selected options from each plot saved in the preset to the corresponding plot in your project. Thus, if you saved a preset including the “Stratify” section based on a project where the VPC is stratified by STUDY but not the Observed data plot, applying that preset to another project will stratify the VPC by STUDY but not the Observed data plot.

 

Managing plot presets

You can also manage your presets by going to Settings > Manage presets > Plot formatting.

This will open a window where you can see all presets, modify, update or remove your custom presets, and export or import them as separate files with a lixpst extension. That way you can easily share your presets with your colleagues or collaborators to standardize your plot formatting.

 

 

Applying pre-defined plot formatting

In addition to your custom presets, MonolixSuite also provides two pre-defined presets that you can use to apply plot formatting. These presets are:

  • Greyscale_Preferences: This preset affects only the Preferences section of the plot configurations, and produces publication-ready greyscale plots with optimized size and spacing of the texts to improve readability of the plots (example shown below for the VPC).

  • Typical_Reporting_Settings: This preset affects only the Settings section of the plot configuration, and corresponds to plot settings typically chosen to report the analysis results. In addition to removing the grid in the background for all plots:
    • VPC: observed data points and predicted percentiles are displayed (see example below)
    • Observation vs Predictions: Observations vs Population Predictions are displayed in addition to Observations vs Individual Predictions
    • Scatterplot of the residuals: Population residuals and NPDE are displayed in addition to Individual residuals.

 

You can apply these presets to any plot in your project by selecting them from the list of presets and clicking on Apply. You can also combine these presets with your own presets or with the Apply from plot feature, to create the plot configuration that suits your needs.

 

Restoring default plot configuration

Using the Reset button

If you want to restore the default plot configuration, you can click on the Reset icon in the Apply from plot panel. This will reset all plots to the system default, or to your custom default if you have defined one.

Note that resetting the plot configuration will not affect the plot presets that you have defined or imported. You can still apply them to any plot after resetting.

 

Defining a custom default

To define a custom default, you need to create a preset and select the option “Use this preset as default” when saving it. This will make this preset the default configuration for all plots. You can also unset a preset as default by going to Settings > Manage presets > Plot formatting, and unchecking the option “Use this preset as default” for the preset.

 

 

6.1.4.Export charts



All plots generated by Monolix can be exported

All plots generated by Monolix can be exported as a figure or as text files in order to be able to plot it in another way or with other software for more flexibility.
All the files can be exported in R for example using the following command

 read.table("/path/to/file.txt", sep = ",", comment.char = "", header = T)

Remarks

  • The separator is the one defined in the user preferences. We set “,” in this example as it is the one by default.
  • The command comment.char = "" is needed for some files because to define groups or color, we use the character # that can be interpreted as a comment character by R.

The list of plots below corresponds to all the plots that Monolix can generate. They are computed with the task “Plots”, and the list of plots to compute can be selected by clicking on the button next to the task as shown below, prior to running the task.
Exporting the charts data can be made through the Export menu or through the preferences as described here.

In the following, we describe all the files generated by the export function

Charts concerning the Data

Observed data (continuous, categorical, and count)

xxx_observations.txt

Description: observation values

Full output file description

Column Description Comment
id Subject identifier
OCC Occasion value (optional) if there is IOV in the data set
time Observation time
y Observation value (loq) The name of the column is the observation name
censored 1 if the observation is censored, 0 otherwise
split Name of the split the subject belongs to
color Name of the color the observation is colored with
filter 1 if the subject is filtered, 0 otherwise

Observed data (event)

xxx_curves.txt

Description: observation values

Full output file description

Column Description Needed Task
time Observation times
survivalFunction Survival of first event
averageEventNumber Average number of event at that time
split Name of the split the subject occasion belongs to

xxx_censored.txt

Description: censored values

Full output file description

Column Description Needed Task
time Observation times
values Survival of first event
split Name of the split the subject occasion belongs to

Model for the observations

Individual Fits

xxx_observations.txt

Description: observation values

Full output file description

Column Description Comment
id Subject identifier
OCC Occasion value (optional) if there is IOV in the data set
time Observation time
y Observation value (loq)
median Prediction interval median
piLower Lower percentile of the individual prediction interval
piUpper Upper percentile of the individual prediction interval
censored 1 if the observation is censored, 0 otherwise

xxx_fits.txt

Description: individual fits based on population parameters and individual parameters

Full output file description

Column Description Needed Task
id Subject identifier
OCC Occasion value (optional) if there is IOV in the data set
time Continuous time grid used to compute fits
pop Prediction using population parameter values and average covariate value from the population (continuous) or reference covariate value (categorical).
popPred Prediction using population parameter values and individual covariates.
indivPredMean Prediction based on the individual parameter values estimated by conditional mean Conditional distribution need to be computed
indivPredMode Prediction based on the individual parameter values estimated by conditional mode EBEs need to be computed

Observation vs Prediction

xxx_obsVsPred.txt

Description: observation and prediction (pop & indiv) values

Full output file description

Column Description Needed Task
id Subject identifier
OCC Occasion value (optional) if there is IOV in the data set
time time of the observation
y Observation value (loq)
y_simBlq Observation value (simulated blq)
popPred Predictions based on population parameter values
indivPredMean Predictions based on the individual parameter values estimated by conditional mean Conditional distribution need to be computed
indivPredMode Predictions based on the individual parameter values estimated by conditional mode EBEs need to be computed
censored 1 if the observation is censored, 0 otherwise
split Name of the split the subject occasion belongs to
color Name of the color the observation is colored with
filter 1 if the subject is filtered, 0 otherwise

xxx_obsVsSimulatedPred.txt

Description: observation and simulated prediction values

Full output file description

Column Description Needed Task
rep Replicate id
id Subject identifier
OCC Occasion value (optional) if there is IOV in the data set
time time of the observation
y Observation value (loq)
y_simBlq Observation value (simulated blq)
indivPredSimulated Predictions based on the simulated individual parameter values estimated by conditional distribution Conditional distribution need to be computed
censored 1 if the observation is censored, 0 otherwise
split Name of the split the subject occasion belongs to
color Name of the color the observation is colored with
filter 1 if the subject is filtered, 0 otherwise

xxx_visualGuides.txt

Description: splines and confidence intervals for predictions

Full output file description

Column Description Needed Task
popPred Continuous grid over population prediction values
popPred_spline Spline ordinates for population predictions
popPred_piLower Lower percentile of prediction interval for population predictions
popPred_piUpper Upper percentile of prediction interval for population predictions
indivPred Continuous grid over individual prediction values
indivPred_spline Spline ordinates for individual predictions
indivPred_piLower Lower percentile of prediction interval for individual predictions
indivPred_piUpper Upper percentile of prediction interval for individual predictions
split Name of the split the visual guides belong to

Distribution of the residuals

xxx_pdf.txt

Description: probability density function of each residual type (pwres, iwres, npde)

Full output file description

Column Description Needed Task
pwRes_abscissa Abscissa for pwres pdf
pwRes_pdf Pdf of pwres
iwRes_abscissa Abscissa for iwres pdf
iwRes_pdf Pdf of the iwres
npde_abscissa Abscissa for npde pdf
npde_pdf Pdf of the npde
split

xxx_cdf.txt

Description: cumulative distribution function of each residual type (pwres, iwres, npde)

Full output file description

Column Description Needed Task
pwRes_abscissa Abscissa for pwres cdf
pwRes_cdf Cdf of pwres
iwRes_abscissa Abscissa for iwres cdf
iwRes_cdf Cdf of the iwres
npde_abscissa Abscissa for npde cdf
npde_cdf Cdf of the npde
split

theoreticalGuides.txt

Description: theoretical guides for the pdf and the cdf

Full output file description

Column Description Needed Task
abscissa,pdf,cdf Abscissa for the theoretical curves
pdf Theoretical value of the pdf
cdf Theoretical value of the cdf

Scatter plot of the residuals

xxx_prediction_percentiles_iwRes.txt

Description: prediction percentiles of the iwREs to plot iwRes w.r.t. the prediction. The same files exists with the pwres and the npde.

Full output file description

Column Description Needed Task
prediction Value of the prediction
empirical_median Empirical median of the iwRes
empirical_lower Empirical lower percentile of the iwRes
empirical_upper Empirical upper percentile of the iwRes
theoretical_median Theoretical median of the iwRes
theoretical_lower Theoretical lower of the iwRes
theoretical_upper Theoretical upper of the iwRes
theoretical_median_piLower Lower bound of the theoretical median prediction interval
theoretical_median_piUpper Upper bound of the theoretical median prediction interval
theoretical_lower_piLower Lower bound of the theoretical lower prediction interval
theoretical_lower_piUpper Upper bound of the theoretical lower prediction interval
theoretical_upper_piLower Lower bound of the theoretical upper prediction interval
theoretical_upper_piUpper Upper bound of the theoretical upper prediction interval
split Name of the split the subject occasion belongs to

xxx_time_percentiles_iwRes.txt

Description: time percentiles of the iwREs to plot iwRes w.r.t. the time. The same files exists with the pwres and the npde.

Full output file description

Column Description Needed Task
time Value of the time
empirical_median Empirical median of the iwRes
empirical_lower Empirical lower percentile of the iwRes
empirical_upper Empirical upper percentile of the iwRes
theoretical_median Theoretical median of the iwRes
theoretical_lower Theoretical lower of the iwRes
theoretical_upper Theoretical upper of the iwRes
theoretical_median_piLower Lower bound of the theoretical median prediction interval
theoretical_median_piUpper Upper bound of the theoretical median prediction interval
theoretical_lower_piLower Lower bound of the theoretical lower prediction interval
theoretical_lower_piUpper Upper bound of the theoretical lower prediction interval
theoretical_upper_piLower Lower bound of the theoretical upper prediction interval
theoretical_upper_piUpper Upper bound of the theoretical upper prediction interval
split Name of the split the subject occasion belongs to

xxx_residuals.txt

Description: residuals values (pwres, iwres, npde)

Full output file description

Column Description Needed Task
id Subject identifier
OCC Occasion value (optional) if there is IOV in the data set
time Observation times
prediction_pwRes Predictions based on population parameter values SAEM
pwRes PwRes (computed with observations) SAEM
pwRes_blq PwRes (computed with simulated blq)
prediction_iwRes_mean Predictions based on the individual parameter values estimated by conditional mean (INDIVESTIM) if available, SAEM either
iwRes_mean IwRes (computed with observations and individual parameter values estimated by conditional mean (INDIVESTIM) if available, SAEM either)
iwRes_mean_simBlq IwRes (computed with simulated blq and individual parameter values estimated by conditional mean (INDIVESTIM) if available, SAEM either)
prediction_iwRes_mode Predictions based on the individual parameter values estimated by conditional mode (INDIVESTIM)
iwRes_mode IwRes (computed with observations and the individual parameter values estimated by conditional mode (INDIVESTIM))
iwRes_mean_simBlq IwRes (computed with simulated blq and the individual parameter values estimated by conditional mode (INDIVESTIM))
prediction_npde Predictions based on population parameter values
npde Npde (computed with observations)
npde_simBlq Npde (computed with simulated blq) SAEM – If there are some censored data in the data set
censored 1 if the observation is censored, 0 otherwise
split Name of the split the subject occasion belongs to
color Name of the color the observation is colored with
filter 1 if the subject is filtered, 0 otherwise

xxx_simulatedResiduals.txt

Description: simulated residuals values

Full output file description

Column Description Needed Task
rep replicate
id Subject identifier
OCC Occasion value (optional) if there is IOV in the data set
time Observation times
prediction_iwRes Predictions based on the simulated individual parameter values based on the conditional distribution
iwRes_simulated IwRes (computed with observations and the simulated individual parameter values)
iwRes_simulated_simBlq IwRes (computed with simulated blq and the simulated individual parameter values)
censored 1 if the observation is censored, 0 otherwise
split Name of the split the subject occasion belongs to
color Name of the color the observation is colored with
filter 1 if the subject is filtered, 0 otherwise

xxx_spline.txt

Description: splines (residuals values against time and prediction)

Full output file description

Column Description Needed Task
time_pwRes Time grid for pwRes spline SAEM
time_pwRes_spline pwRes against time spline SAEM
time_iwRes Time grid for iwRes spline At least SAEM
time_iwRes_spline iwRes against time spline At least SAEM
time_npde Time grid for npde spline SAEM
time_npde_spline npde against time spline SAEM
prediction_pwRes Prediction grid for pwRes spline SAEM
prediction_pwRes_spline pwRes against population prediction spline SAEM
prediction_iwRes Prediction grid for iwRes spline At least SAEM
prediction_iwRes_spline iwRes against individual prediction spline At least SAEM
prediction_npde Prediction grid for npde spline SAEM
prediction_npde_npde npde against population prediction spline SAEM
split Name of the split the visual guides belong to If the chart is splitted

xxx_{time,population,individual}Bins.txt

Description: bins values for the corresponding axis.

Full output file description

Column Description Needed Task
binsValues Abscissa bins values
split Name of the split the bins refer to If the chart is splitted

Model for the individual parame