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# An example of individual parameter model

### Introduction

The subsection here present an example of a one compartmental model with a first order absorption model. This is a dynamical model with two states represented by the following equations $\left\{\begin{array}{l}\frac{dA_d}{dt} = -k_a A_d + Adm \\ \frac{dA_c}{dt} = k_a A_d -\frac{Cl}{V}A_c \\ C_c = \frac{A_c}{V} \end{array}\right.$

where $(A_d,A_c)$ represent the administration and central compartments respectively.

### Example : inter individual variability on the volume

In that case, we define the inter individual variability (iiv) on the volume to see the impact on the output. Notice that we only put iiv on the volume on this example for sake of simplicity and clarity. One could add iiv on the administration rate $k_a$ and the clearance $Cl$. In our case, we define the volume as lognormally distributed with a typical value $V_{pop} = 10$, and a standard deviation $\omega_V=.2$. In Mlxtran, this distribution writes

[INDIVIDUAL]
input = {V_pop, omega_V}

DEFINITION:
V = {distribution = lognormal, typical = V_pop, sd = omega_V}


Notice that, in the presented case, no EQUATION: was needed. Elsewise, a block EQUATION: would have been added to define for example the typical value. If we sum up all the model, it is implemented in the file mlxt_iiv.txt as the following

DESCRIPTION: Dynamical model with inter-individual variability on the volume. The volume varies according to a lognormal distribution law
[LONGITUDINAL]
input = {ka, V, Cl}

EQUATION:
t0 = 0
Ac_0 = 0

ddt_Ac = ka*Ad -Cl/V*Ac
Cc = Ac/V

[INDIVIDUAL]
input = {V_pop, omega_V}

DEFINITION:
V = {distribution = lognormal, typical = V_pop, sd = omega_V}


The simulation is proposed using Mlxplore using $(ka=1, V_pop=10, omega_V=.2, Cl=.5)$. The associated Mlxplore project can be found here : mlxplore_iiv.txt. Based on this distribution definition, Mlxplore draws $N$ values and then simulates the $N$individuals based on their volumes. This is shown in the following figure with $N=200$. On this figure, the median value at each time is displayed in solid line. As it is the median value over a certain number of draws, it does not perfectly corresponds to the reference value without iiv, the median curve converge toward the reference one when $N$ is sufficiently large. Moreover, the percentile are displayed to explore the impact of the iiv on the output. A color bar is displayed on the right to see which part of the curve corresponds to each percentile. In the presented figure, we used 8 bands for a level of 90, which means that we look at the percentiles between 5% and 95%.