We consider a non-local tumour growth model of phase-field type, describing the evolution of tumour cells through proliferation in the presence of a nutrient. The model consists of a coupled system, incorporating a non-local Cahn-Hilliard equation for the tumour phase variable and a reaction-diffusion equation for the nutrient. The optimal control problem aims to identify a suitable therapy, capable of guiding the evolution of the tumour towards a predefined target. We first establish novel regularity results for the PDE system by applying maximal regularity theory in weighted Lp spaces. Such a technique enables us to prove the local existence and uniqueness of a regular solution in a quite general framework, which also includes chemotaxis effects to some extent, but restricts us to regular double-well potentials. Then, by leveraging the time-regularisation properties of the weighted spaces and some global boundedness estimates, we further extend the solution to a global one. In a second version of the model, we add a viscous regularisation term which allows us to prove the existence and uniqueness of a global regular solution under more general hypotheses. Indeed, we can also include singular double-well potentials and cross-diffusive chemotactic effects, at the expense of some additional hypotheses on the controls. These results provide the foundation for addressing the optimal control problem in both cases. Specifically, we prove the existence of an optimal therapy and then, by studying the Fréchet-differentiability of the control-to-state operator and introducing the adjoint system, we derive first-order necessary optimality conditions. We finally discuss some open questions and future research directions.
This video was produced by the SITE Research Center at New York University, as part of their talk series.
