Mechanical model for tumour growth: mathematical analysis and optimal therapies

Abstract.
Over the last decades, great strides have been made by the mathematical and medical communities towards the understanding of solid tumor growth. In this talk, we aim at providing some mathematical insights for a macroscopic mechanical model for tumour growth in which cell-cell adhesion effects are taken into account with the help of a Ginzburg-Landau type energy. In the overall model an equation of Cahn-Hilliard type is coupled to the system of linear elasticity and a reaction-diffusion equation for a nutrient concentration.
After fixing some ideas concerning modelling, we address some selected well-posedness results, and lastly, we discuss an optimal control problem. 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.
This is a joint work with Harald Garcke (University of Regensburg) and Kei Fong Lam (Hong Kong Baptist University).