Optimal medication for tumors modeled by a Cahn-Hilliard-Brinkman equation

Abstract

In this paper, we study a distributed optimal control problem for a diffuse interface model for tumor growth. The model consists of a Cahn-Hilliard type equation for the phase field variable coupled to a reaction diffusion equation for the nutrient and a Brinkman type equation for the velocity. The system is equipped with homogeneous Neumann boundary conditions for the tumor variable and the chemical potential, Robin boundary conditions for the nutrient and a "no-friction" boundary condition for the velocity. The control acts as a medication by cytotoxic drugs and enters the phase field equation. The cost functional is of standard tracking type and is designed to track the variables of the state equation during the evolution and the distribution of tumor cells at some fixed final time. We prove that the model satisfies the basics for calculus of variations and we establish first-order necessary optimality conditions for the optimal control problem.

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• Penalisation of Long Treatment Time and Optimal Control of a Tumour Growth Model of Cahn-Hilliard
• Optimal Distributed Control for a Cahn-Hilliard-Darcy System with Mass Sources, Unmatched Viscosities and Singular Potential
• Existence of weak solutions to multiphase Cahn-Hilliard-Darcy and Cahn-Hilliard-Brinkman models for stratified tumor growth with chemotaxis and general source terms
• Vanishing parameter for an optimal control problem modeling tumor growth
• Second-order analysis of an optimal control problem in a phase field tumor growth model with singular potentials and chemotaxis