When treatments span over a long time with numerous repeated doses, the clearance can be observed to change over time, for instance due to a change in the disease status. One possible approach to handle this case is to define in a parametric way how the clearance changes over time. A common assumption is to use an Imax or Emax model. Below, we show how to implement such a model in the Mlxtran language.

Examples of time-varying clearances are for instance presented for PEGylated Asparaginase, and for mycophenolic acid here

### Mlxtran structural model

In the example below, we assume that the clearance is decreasing over time, with a sigmoidal shape. We consider a one-compartment model with first-order absorption. The Mlxtran code for the structural model reads:

[LONGITUDINAL] input = {ka, V, Clini, Imax, gamma, T50} PK: depot(target=Ac, ka) EQUATION: t_0 = 0 Ac_0 = 0 Clapp = Clini * (1 - Imax * t^gamma/(t^gamma + T50^gamma)) ddt_Ac = - Clapp/V * Ac Cc = Ac/V OUTPUT: output = {Cc}

The `Clini`

parameter described the initial clearance, `Imax`

the maximal reduction of clearance, `T50`

the time at which the clearance is reduced by half of the maximal reduction, and `gamma`

characterizes the sigmoidal shape.

The apparent clearance `Clapp`

is defined via an analytical formula depending on the time `t`

and is then used in the ODE system. The first-order absorption is directly defined using the depot macro, which here indicates that the doses of the data set must be applied to the target Ac via a first-order absorption with rate ka (ka is a reserved keyword).

### Model exploration with Mlxplore

To define an Mlxplore project, we in addition define the parameter values, the administration scheme and the output. The full code for the Mlxplore project reads:

<MODEL> [LONGITUDINAL] input = {ka, V, Cl, Imax, gamma, T50} PK: depot(target=Ac, ka) EQUATION: t_0 = 0 Ac_0 = 0 Clapp = Cl * (1 - Imax * t^gamma/(t^gamma + T50^gamma)) ddt_Ac = - Clapp/V * Ac Cc = Ac/V <PARAMETER> ka=1 V=10 Cl = 10 Imax = 0.8 gamma=3 T50 = 50 <DESIGN> [ADMINISTRATION] adm = {time=0:10:300, amount=100} <OUTPUT> list={Cc} grid=0.1:0.1:300

We decide to administrate the drug every 10 days during 300 days. The initial clearance is `Clini=10 L/day`

and progressively decreases to `Clend=Clini*(1-Imax)=2 L/day`

with a characteristic time of 50 days.

The simulation in Mlxplore shows how **the peak and trough concentrations increase over time due to the reduced clearance**:

One can compare the original simulation (light blue) with a simulation where the **clearance stays constant at 10 L/day** (dark blue, with Imax=0):

One can also compare the original simulation (light blue) with a simulation where **the clearance stays constant at the Clend value 2 L/day** (dark blue, with Imax=0 and Clini=2):