A study on discrete and discrete fractional pharmacokinetics-pharmacodynamics models for tumor growth and anti-cancer effects

Ferhan M Atıcı,Ngoc Nguyen,Tilekbek Zhoroev,Mustafa Atıcı,Gilbert Koch

doi:10.1515/cmb-2019-0002

Abstract

Abstract We study the discrete and discrete fractional representation of a pharmacokinetics - pharmacodynamics (PK-PD) model describing tumor growth and anti-cancer effects in continuous time considering a time scale h ℕ 0 h $h\mathbb{N}_0^h$ , where h > 0. Since the measurements of the tumor volume in mice were taken daily, we consider h = 1 and obtain the model in discrete time (i.e. daily). We then continue with fractionalizing the discrete nabla operator to obtain the model as a system of nabla fractional difference equations. The nabla fractional difference operator is considered in the sense of Riemann-Liouville definition of the fractional derivative. In order to solve the fractional discrete system analytically we state and prove some theorems in the theory of discrete fractional calculus. For the data fitting purpose, we use a new developed method which is known as an improved version of the partial sum method to estimate the parameters for discrete and discrete fractional models. Sensitivity analysis is conducted to incorporate uncertainty/noise into the model. We employ both frequentist approach and Bayesian method to construct 90 percent confidence intervals for the parameters. Lastly, for the purpose of practicality, we test the discrete models for their efficiency and illustrate their current limitations for application.

Highlights

Mathematical modeling of pharmacokinetics (PK) and pharmacodynamics (PD) deals with (i) the distribution and elimination of a drug, the pharmacokinetics [17], and (ii) the therapeutic e ect of a drug on a speci c target, the pharmacodynamics [24]
We study the discrete and discrete fractional representation of a pharmacokinetics - pharmacodynamics (PK-PD) model describing tumor growth and anti-cancer e ects in continuous time considering a time scale hNh, where h >
We continue with fractionalizing the discrete nabla operator to obtain the model as a system of nabla fractional di erence equations

Summary

Introduction

Mathematical modeling of pharmacokinetics (PK) and pharmacodynamics (PD) deals with (i) the distribution and elimination of a drug, the pharmacokinetics [17], and (ii) the therapeutic e ect of a drug on a speci c target, the pharmacodynamics [24]. During anticancer treatment it is assumed that the growth dynamics of the tumor will be perturbed by the anticancer drug e ect described with the model parameter k. For the rst equation in the above system, we use some properties of the nabla-di erence operator and the Gamma function, we obtain the following iteration formula for u(t). In our study we used the improved partial sum method in Mathematica to estimate the parameters for the discrete and discrete fractional models. We employ both the frequentist approach and Bayesian method to construct 90% con dence intervals for the model parameters. The reason is that the construction of con dence intervals using frequentist approach is based upon the asymptotic theory of estimators which assumes that the sample size has to be su ciently large.

Limitations in Discrete Models

A Appendix