Model with saturation of the absorption rate

In some cases, for instance in case of transporter-mediated uptake, the absorption from the depot compartment into the central compartment can saturate. To model this phenomenon, one can replace the first-order rate of absorption by a Michaelis-Menten term.

Examples of drugs displaying a saturating absorption include Phenylbutazone, Naproxen, Chlorothiazide, beta-lactam antibiotics and are reviewed in:

Wood, J. H., & Thakker, K. M. (1982). Michaelis-menten absorption kinetics in drugs: Examples and implications. European Journal of Clinical Pharmacology, 23(2), 183–188. 

Mlxtran model

To describe a saturable absorption, the depot compartment must be explicitly described and the model must be written as an ODE system. Below we present a one-compartment model with linear elimination and saturable absorption.
The depot macro permits to add the doses defined in the data set to the amount in the depot compartment. The Michaelis-Menten term is written using the amount of the depot compartment instead of the concentration as the volume needed to calculate the concentration is unidentifiable.

[LONGITUDINAL]
input={Vm, Km, V, Cl}

PK:
depot(target=Ad)

EQUATION:
t_0 = 0
Ad_0 = 0
Ac_0 = 0

ddt_Ad = -Vm*Ad/(Ad+Km)
ddt_Ac = Vm*Ad/(Ad+Km) - Cl/V*Ac

Cc = Ac/V

OUTPUT:
output = {Cc}

Exploration with Mlxplore

We explore the difference between a linear and a saturating absorption using the following Mlxplore script:

<MODEL>
[LONGITUDINAL]
input={Vm,Km,ka,V,Cl}

PK:
; depot for saturating absorption
depot(target=Ad)
; model for linear absorption
Cc_lin = pkmodel(ka,V,Cl)

EQUATION:
; model for saturating absorption
t_0 = 0
Ad_0 = 0
Ac_0 = 0

ddt_Ad = -Vm*Ad/(Ad+Km)
ddt_Ac = Vm*Ad/(Ad+Km) - Cl/V*Ac

Cc_sat = Ac/V

<PARAMETER>
Vm = 10
Km = 10
V = 15
Cl = 1
ka = 0.25

<DESIGN>
[ADMINISTRATION]
adm = {time=0, amount=100}

<OUTPUT>
list={Cc_sat, Cc_lin}
grid=0.1:0.1:50

<RESULTS>
[GRAPHICS]
p1 = {y={Cc_sat,Cc_lin}, ylabel='Concentration', xlabel='time'}

Below the linear absorption is shown in blue and the saturating in orange:

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