Existence, Stability and Optimal Drug Dosage for a Reaction-Diffusion System Arising in a Cancer Treatment

TL;DR

This paper analyzes a reaction-diffusion model for cancer treatment, establishing well-posedness, long-term behavior, and optimal drug injection strategies to eliminate tumor cells.

Contribution

It introduces a well-posedness framework, studies long-term tumor dynamics, and formulates an optimal control problem for drug injection in cancer therapy.

Findings

Cancer cells can be eliminated if their reproduction rate is sufficiently small.

Existence of an optimal boundary drug injection rate is proven.

First-order optimality conditions for drug dosage are derived.

Abstract

In this paper, a reaction-diffusion system modeling injection of a chemotherapeutic drug on the surface of a living tissue during a treatment for cancer patients is studied. The system describes the interaction of the chemotherapeutic drug and the normal, tumor and immune cells. We first establish well-posedness for the nonlinear reaction-diffusion system, then investigate the long-time behavior of solutions. Particularly, it is shown that the cancer cells will be eliminated assuming that its reproduction rate is sufficiently small in a short time period in each treatment interval. The analysis is then essentially exploited to study an optimal drug injection rate problem during a chemotherapeutic drug treatment for tumor cells, which is formulated as an optimal boundary control problem with constraints. For this, we show that the existence of an optimal drug injection rate through the boundary, and derive the first-order optimality condition.