Optimal control of treatment time in a diffuse interface model of tumor growth

TL;DR

This paper formulates and analyzes an optimal control problem for a tumor growth model involving a coupled Cahn-Hilliard and reaction-diffusion system, optimizing drug concentration and treatment duration.

Contribution

It introduces a novel optimal control framework with a free treatment time variable for a complex tumor growth model, deriving first-order optimality conditions.

Findings

Derivation of necessary optimality conditions for control variables.

Inclusion of treatment time as an optimization variable.

Application to a coupled tumor growth model.

Abstract

We consider an optimal control problem for a diffuse interface model of tumor growth. The state equations couples a Cahn-Hilliard equation and a reaction-diffusion equation, which models the growth of a tumor in the presence of a nutrient and surrounded by host tissue. The introduction of cytotoxic drugs into the system serves to eliminate the tumor cells and in this setting the concentration of the cytotoxic drugs will act as the control variable. Furthermore, we allow the objective functional to depend on a free time variable, which represents the unknown treatment time to be optimized. As a result, we obtain first order necessary optimality conditions for both the cytotoxic concentration and the treatment time.