On a Mathematical Model Arising from an Optimal Chemotherapeutic Drug Treatment for Tumor Cells

TL;DR

This paper develops a mathematical model for chemotherapeutic treatment of tumor cells using a nonlinear reaction-diffusion system, establishing well-posedness, analyzing long-term behavior, and designing an optimal drug dosage under clinical constraints.

Contribution

It introduces a new nonlinear reaction-diffusion model for tumor-immune-drug interactions and formulates an optimal control problem with clinical constraints, including a minimum normal cell level.

Findings

Existence and uniqueness of solutions for the model

Long-term stability analysis of the system

Existence of an optimal drug dosage satisfying constraints

Abstract

In this paper we consider an optimal control problem arising from a chemotherapeutic drug treatment for tumor cells in a living tissue. The mathematical model for the interaction of chemotherapeutic drug and the normal, tumor and immune cells are governed by a nonlinear reaction-diffusion system. We first establish the well-posedness for the nonlinear system. Then we study the long-time behavior of the solution for the model problem. Finally, we design an optimal drug dosage, which leads to an optimal control problem under certain constrains. A complicated factor for the optimal control problem is that a minimum level of normal cells in patients must be maintained during a treatment. It is shown that there is an optimal drug dosage for the chemotherapeutic treatment for patients.