Michaelis–Menten pharmacokinetics based on uncertain differential equations

Michaelis–Menten kinetics are commonly used to represent enzyme-catalysed reactions in pharmacokinetics. Obviously, metabolizing organs and tissues are subject to various internal and external noises that change over time. However, both deterministic and stochastic modelling approaches can not account for these dynamic noises rationally. Motivated by system pharmacology, this paper deduces an uncertain Michaelis–Menten equation using uncertain differential equations under the framework of uncertainty theory to model dynamic noises in pharmacokinetics better. Based on belief reliability theory, several essential pharmacokinetic indexes are investigated. Furthermore, generalized moment estimations for unknown parameters in the uncertain Michaelis–Menten equations are given. A real data analysis using ethanol concentrations in six subjects illustrates our methods in details. Uncertain Michaelis–Menten equation can be updated with the initial time, and produces more elaborate results for pharmacokinetic indexes. Finally, a paradox of the stochastic Michaelis–Menten equation is pointed out.