Explicit reformulations of the Lambert W-omega function for calculations of the solutions to one-compartment pharmacokinetic models with Michaelis–Menten elimination kinetics

Abstract

The exact closed-form solutions to the integrated rate equations for one-compartment pharmacokinetic models that obey Michaelis-Menten elimination kinetics were derived recently (Tang and Xiao in J Pharmacokin Pharmacodyn 34:807-827, 2007). These solutions are expressed in terms of the Lambert W(x)-omega function; however, unfortunately, most of the available computer programs are not set up to handle equations that involve the W(x) function. Therefore, in this article, I provide alternative explicit analytical equations expressed in terms of elementary mathematical functions that accurately approximate exact solutions and can be simply calculated using any optional standard software.

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…”

“…which corresponds to the explicit solution of model (7). Compare Equations (16) and 22, then use lim…”

“…Over the past decades, Lambert W function has found its utility in the field of pharmaceutical sciences. [4][5][6][7][8] For drugs showing saturate elimination, where Michaelis-Menten kinetics is commonly used to model this pharmacokinetic feature, it has been proven a very efficient tool to obtain the closed-form solution representing the time course of drug plasma concentration. 9 Although most of pharmaceutical modeling practices chose numerical solutions from a pragmatic point of view, the research interest for the analytical solutions of such nonlinear differential equations has never lost ground as it directly points to the inner mathematical structure behind.…”

“…As a special function, W ( z ) cannot be directly expressed through elementary functions; however, it is very helpful for representing solutions of various algebraic equations having exponential terms or differential equations and even with that of delay terms as y ′ ( t )= ay ( t −1). Over the past decades, Lambert W function has found its utility in the field of pharmaceutical sciences 4‐8 . For drugs showing saturate elimination, where Michaelis–Menten kinetics is commonly used to model this pharmacokinetic feature, it has been proven a very efficient tool to obtain the closed‐form solution representing the time course of drug plasma concentration 9 .…”

Theory on the rate equations of Michaelis-Menten type enzyme kinetics with competitive inhibition

“…In general, sQSSA yields expressions which can be directly used by the experimentalists to obtain various enzyme kinetic parameters [20]. Recently, explicit closed form expressions of integrated rate equation corresponding to sQSSA were obtained in terms of Lambert's W functions [21][22][23][24][25]. The total QSSA (tQSSA) assumes that the amount of product formed near the steady state is much negligible compared to the total substrate concentration [26,27].…”