Time Scale Calculus: a new approach to multi-dose pharmacokinetic modeling

In this paper, we use Time Scale Calculus (TSC) to formulate and solve pharmacokinetic models exploring multiple dose dynamics. TSC is a mathematical framework that allows the modeling of dynamical systems comprising continuous and discrete processes. This characteristic makes TSC particularly suited for multi-dose pharmacokinetic problems, which inherently feature a blend of continuous processes (such as absorption, metabolization, and elimination) and discrete events (drug intake). We use this toolkit to derive analytical expressions for blood concentration trajectories under various multi-dose regimens across several flagship pharmacokinetic models. We demonstrate that this mathematical framework furnishes an alternative and simplified way to model and retrieve analytical solutions for multi-dose dynamics. For instance, it enables the study of blood concentration responses to arbitrary dose regimens and facilitates the characterization of the long-term behavior of the solutions, such as their steady state.