**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.