- Introduction
- Fitting first a PK model to the PK data
- Simultaneous PKPD modeling
- Sequential PKPD modelling
- Fitting a PKPD model to the PD data only
- Case studies

**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’)

Individual PK parameters are set to their estimated values and used as constants in the PKPD model to fit the PD data. In this example, individual PK parameters \((\psi_i)\) were estimated as the modes of the conditional distributions\((p(\psi_i | y_i, \hat{\theta}))\). An additional column `IGNORED OBSERVATION` is necessary in the datafile in order to ignore the PK data. For that, we use MDV=1 for the line where YTYPE=1 (PK data), and MDV=0 on the line where YTYPE=2 (PD data).

In addition, the estimated individual PK parameters (purple frames) are defined as regression variables, using the reserved keyword `REGRESSOR`. The covariates used for defining the distribution of the individual PK parameters are not mandatory as all the information is already in the individual parameters.

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.**

## 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’)