Nutrient control for a viscous Cahn-Hilliard-Keller-Segel model

TL;DR

This paper develops a mathematical framework for controlling tumor growth modeled by a coupled Cahn-Hilliard and Keller-Segel system, focusing on nutrient regulation to influence tumor-healthy cell segregation.

Contribution

It introduces a novel control approach for a complex PDE system combining tumor segregation and chemotaxis, with existence and optimality conditions established.

Findings

Existence of an optimal control for the system

First-order necessary optimality conditions derived

Analysis limited to two-dimensional spatial domain

Abstract

In this paper, we address a distributed control problem for a system of partial differential equations describing the evolution of a tumor that takes the biological mechanism of chemotaxis into account. The system describing the evolution is obtained as a nontrivial combination of a Cahn-Hilliard type system accounting for the segregation between tumor cells and healthy cells, with a Keller-Segel type equation accounting for the evolution of a nutrient species and modeling the chemotaxis phenomenon. First, we develop a robust mathematical background that allows us to analyze an associated optimal control problem. This analysis forced us to select a source term of logistic type in the nutrient equation and to restrict the analysis to the case of two space dimensions. Then, the existence of an optimal control and first-order necessary conditions for optimality are established.