The nonstandard finite difference method applied to pharmacokinetic models

Abstract

A good understanding of pharmacokinetic-pharmacodynamic can shed light on situations where one or the other needs to be optimized in drug discovery and development. As a result of this, pharmaceutical companies aim to develop new tools to support drug discovery and e cacious dose for clinical use. Drugs take a complicated journey through the body before they produce their desired therapeutic e ects. Ultimately, these processes are usually best described by compartment pharmacokinetic-pharmacodynamics models. Pharmacokinetic (PK) models are commonly used to predict drug concentrations that drive controlled intravenous (I.V.) transfers (or infusion and oral transfers) while pharmacokinetic and pharmacodynamic (PD) interaction models are used to provide predictions of drug concentrations a ecting the response of these clinical drugs. These PK/PD models leads to di erential equations. Few of these di erential equations can be solved exactly. Therefore, a lack of exact solutions to many of these di erential equations leads to numerical approximation been used to determine the possible solution or behaviour of the di erential equations. The aim of this thesis is to apply standard nite di erence (SFD) and nonstandard nite di erence (NSFD) methods to continuous-time pharmacokinetic- pharmacodynamic models. Another aim of this thesis is to provide a rigorous analysis of these models to gain insight into the dynamical features. This will allow us to also comment on the impact of certain key parameters. The NSFD method was shown to be dynamically consistent with the original continuous-time models. Also, the NSFD method is able to predict the concentration{time pro le of a drug when there are alterations in the dosing regimen|this would not be possible were one to consider non-compartment analysis. Furthermore, the NSFD method preserves signi cant properties of the analogous models and consequently gives reliable numerical results even when analytical solutions are not possible. The standard approaches to multi-compartment models assume linear dynamics over the duration of each time step, whereas the NSFD method assumes exponential dynamics. Thus, in the case of a linear model the NSFD method recovers the model dynamics exactly. This thesis illustrates the ability of the NSFD method to solve compartment PK models in a stable and robust fashion.