Inverse problems in tumor modelling

In this project, we deepen the analysis of a tumour growth model, recently proposed by Garcke et al. in [1]. This model describes tumour and healthy cells evolution as well as tumour cells’ nutrients, mixture velocity and pressure in the domain. Furthermore, it takes into account chemotaxis and apoptosis death of tumour cells, through a system of parabolic nonlinear PDE, that is a Cahn-Hilliard Darcy model, together with an advection-diffusion-reaction equation describing the evolution of nutrients. We perform a dimensional analysis and we build a numerical solver by use of the finite element method in space, a Backward Euler in time and a Newton method to tackle the nonlinearity. We perform several numerical simulations in order to recover results obtained in the article and to catch a general growth of the tumour depending on parameters of interest. Finally, a PDE-constrained optimization problem is formulated and solved, aiming at determining the shape of the tumur after a fixed time from an initial guess of its location. From the numerical simulations we obtained for the nutrients, we notice that the concentration of nutrients in an observable zone around the tumor region could possibly bring enough information to achieve this goal. Therefore, a previous numerical simulation of nutrients will be taken as a target, in order to recover the controlled tumor function, previously simulated numerically. In this respect, preliminary numerical results show that, to some extents, it is possible to identify the general shape of the tumor, even if the exact result of the numerical simulation could not be recovered.