Pharmacokinetics and Pharmacodynamics Models of Tumor Growth and Anticancer Effects in Discrete Time

Ferhan M Atıcı,Sarah E Pedersen,Ngoc Nguyen,Gilbert Koch,Kamala Dadashova

doi:10.1515/cmb-2020-0105

Abstract

Abstract We study the h-discrete and h-discrete fractional representation of a pharmacokinetics-pharmacodynamics (PK-PD) model describing tumor growth and anticancer effects in continuous time considering a time scale h𝕅0, where h > 0. Since the measurements of the drug concentration in plasma were taken hourly, we consider h = 1/24 and obtain the model in discrete time (i.e. hourly). We then continue with fractionalizing the h-discrete nabla operator in the h-discrete model to obtain the model as a system of nabla h-fractional difference equations. In order to solve the fractional h-discrete system analytically we state and prove some theorems in the theory of discrete fractional calculus. After estimating and getting confidence intervals of the model parameters, we compare residual squared sum values of the models in one table. Our study shows that the new introduced models provide fitting as good as the existing models in continuous time.

Highlights

We study the h-discrete and h-discrete fractional representation of a pharmacokineticspharmacodynamics (PK-PD) model describing tumor growth and anticancer e ects in continuous time considering a time scale hN, where h >
In a typical pharmacokinetics and pharmacodynamics (PK-PD) model, which describes the impact of anticancer treatment on the dynamics of tumor growth, a transit compartment model consists of a system of rst order di erential equations
Since the concentration of each drug was measured hourly, we have hN as the domain of the discrete model

Summary

Introduction

In a typical pharmacokinetics and pharmacodynamics (PK-PD) model, which describes the impact of anticancer treatment on the dynamics of tumor growth, a transit compartment model consists of a system of rst order di erential equations. A rst order di erential equation is an equation that includes an unknown function of time t (call f ) and its rst order derivative, which stands for the instantaneous rate of change of f at time t. A fractional di erence equation is an equation which includes an unknown function of time (call f ); and is de ned as the rate of change of the ( − α)-th order sum of the function f at time t. From a practical point of view, the mentioned non-integer order sum can be considered as the history of the function f from the initial time t to t.

Objectives

Conclusion