Sparse optimal control of a phase field tumour model with mechanical effects

TL;DR

This paper develops a sparse optimal control framework for a complex phase field tumour model incorporating mechanical effects, aiming to minimize a cost functional involving nutrient supply, drug concentrations, and mechanical stresses.

Contribution

It introduces a novel sparse control approach for a coupled phase field tumour model with mechanical effects, utilizing convex regularization to induce sparsity in drug administration.

Findings

Optimal drug concentrations can vanish on certain time intervals due to sparsity regularization.

The control framework effectively manages nutrient supply and drug delivery in the tumour model.

The approach leverages previous well-posedness results for the coupled PDE system.

Abstract

In this paper, we study an optimal control problem for a macroscopic mechanical tumour model based on the phase field approach. The model couples a Cahn--Hilliard type equation to a system of linear elasticity and a reaction-diffusion equation for a nutrient concentration. By taking advantage of previous analytical well-posedness results established by the authors, we seek optimal controls in the form of a boundary nutrient supply, as well as concentrations of cytotoxic and antiangiogenic drugs that minimise a cost functional involving mechanical stresses. Special attention is given to sparsity effects, where with the inclusion of convex non-differentiable regularisation terms to the cost functional, we can infer from the first-order optimality conditions that the optimal drug concentrations can vanish on certain time intervals.