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# PK model: 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)
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=2, target=Ac)

EQUATION:
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.