**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 and be, respectively, the amounts in the depot compartment (gut) and the central compartment (bloodtsream). Kinetics of and are described by the following system of ODEs

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

The target compartment is the depot compartment () for oral administrations and the central compartment () 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 , and . The model properly fits the data:

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