Closed form solutions and dominant elimination pathways of simultaneous first-order and Michaelis–Menten kinetics

Abstract

The current study aims to provide the closed form solutions of one-compartment open models exhibiting simultaneous linear and nonlinear Michaelis-Menten elimination kinetics for single- and multiple-dose intravenous bolus administrations. It can be shown that the elimination half-time ([Formula: see text]) has a dose-dependent property and is upper-bounded by [Formula: see text] of the first-order elimination model. We further analytically distinguish the dominant role of different elimination pathways in terms of model parameters. Moreover, for the case of multiple-dose intravenous bolus administration, the existence and local stability of the periodic solution at steady state are established. The closed form solutions of the models are obtained through a newly introduced function motivated by the Lambert W function.

Morphism classification of a family of transcendent functions arising from pharmacokinetic modelling

“…where k el is the constant coefficient of first-order elimination for the linear elimination pathway, and other parameters are defined as in Equation (7). Then, using the transcendent X function introduced in Wu et al, 10 the closed-form expression of C(t) can be written as…”

“…for all x > 0, we conclude that the implicit solution of Equation 2exists and could be expressed within a closed-form through introducing some new transcendent functions. In order to explicitly express drug plasma concentration, 10 we introduced a new transcendent X function, which is the multivalued inverse of f(x) in Equation 3. Indeed, we refer X function to a family of functions X(z, p, q) with respect to the variable z ∈ , parametrized by p and q.…”

“…To be in accordance with the pharmacological meanings in the model, we take p and q as positive real numbers in the definition of X. Thus, Equation 6gives the closed-form solution of drug plasma concentration time course of the pharmacokinetic model that has simultaneous first-order and Michaelis-Menten elimination as discussed in Wu et al 10 Since then, more pharmacokinetic models were found with their solutions that can be analytically expressed through X functions. 11,12 Lambert W function has been applied in various fields involving mathematics, physics, engineering, and risk theory.…”

“…Consider the differentiability of the function for all x >0, we conclude that the implicit solution of Equation () exists and could be expressed within a closed‐form through introducing some new transcendent functions. In order to explicitly express drug plasma concentration, 10 we introduced a new transcendent X function, which is the multivalued inverse of f ( x ) in Equation ().…”

Impact of saturable distribution in compartmental PK models: dynamics and practical use

“…Such multi-compartmental models typically assume that drug enters the blood stream in the central compartment, is distributed from there via linear first order processes to the peripheral compartments, and finally is eliminated again from the central compartment via either a linear first order process or a saturable Michaelis–Menten process (see e.g. Wagner et al [ 1 ] and more recently, Wu et al [ 2 ], Brocks et al [ 3 ] and Scheerens et al [ 4 ]). While linear distribution from central to peripheral may often provide an adequate description of the observed PK, very few processes in biology are truly linear.…”