Inverse reconstruction of cell proliferation laws in cancer invasion modelling

The process of local cancer cell invasion of the surrounding tissue is key for the overall tumour growth and spread within the human body, the past 3 decades witnessing intense mathematical modelling efforts in these regards. However, for a deep understanding of the cancer invasion process these modelling studies require robust data assimilation approaches. While being of crucial importance in assimilating potential clinical data, the inverse problems approaches in cancer modelling are still in their early stages, with questions regarding the retrieval of the characteristics of tumour cells motility, cells mutations, and cells population proliferation, remaining widely open. This study deals with the identification and reconstruction of the usually unknown cancer cell proliferation law in cancer modelling from macroscopic tumour snapshot data collected at some later stage in the tumour evolution. Considering two basic tumour configurations, associated with the case of one cancer cells population and two cancer cells subpopulations that exercise their dynamics within the extracellular matrix, we combine Tikhonov regularisation and gaussian mollification approaches with finite element and finite differences approximations to reconstruct the proliferation laws for each of these sub-populations from both exact and noisy measurements. Our inverse problem formulation is accompanied by numerical examples for the reconstruction of several proliferation laws used in cancer growth modelling