Optimal distributed control of a diffuse interface model of tumor growth

TL;DR

This paper develops an optimal control framework for a tumor growth model combining Cahn-Hilliard and reaction-diffusion equations, deriving conditions for controlling tumor cell fractions via nutrient or medication inputs.

Contribution

It establishes the differentiability of the control-to-state operator and derives first-order optimality conditions for the distributed control problem.

Findings

Control-to-state operator is Frechet differentiable.

First-order optimality conditions are derived.

Model effectively links tumor growth dynamics with control inputs.

Abstract

In this paper, a distributed optimal control problem is studied for a diffuse interface model of tumor growth which was proposed in [A. Hawkins-Daruud, K.G. van der Zee, J.T. Oden, Numerical simulation of a thermodynamically consistent four-species tumor growth model, Int. J. Numer. Math. Biomed. Engng. 28 (2011), 3-24]. The model consists of a Cahn-Hilliard equation for the tumor cell fraction coupled to a reaction-diffusion equation for a variable representing the nutrient-rich extracellular water volume fraction. The distributed control monitors as a right-hand side the reaction-diffusion equation and can be interpreted as a nutrient supply or a medication, while the cost function, which is of standard tracking type, is meant to keep the tumor cell fraction under control during the evolution. We show that the control-to-state operator is Frechet differentiable between appropriate Banach spaces and derive the first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables.