Partial differential equation modelling in cancer and development

Abstract

This thesis explores various partial differential equation (PDE) models of the spatiotemporal and evolutionary dynamics of cell populations in different problems in cancer and development. In particular, these models are used to investigate: (i) the emergence of intratumour phenotypic heterogeneity and the development of chemotherapeutic resistance in vascularised tumours; (ii) the formation of endothelial progenitor cell clusters during the early stages of vasculogenesis; (iii) mechanical pattern formation under different linear viscoelasticity assumptions for the extracellular matrix. The mathematical models proposed for these problems are formulated as systems of nonlinear and nonlocal PDEs, which provide a mean-field representation of the underlying cellular dynamics and pose a series of interesting analytical and numerical challenges. These are tackled by means of formal asymptotic methods, linear stability analyses and appropriate numerical schemes preventing the emergence of spurious oscillations. Numerical simulations, relying on parameter values drawn from the extant literature, complement the analytical results and are employed for in silico investigations qualitatively testing the model assumptions against empirical observations. The obtained results help us shed light on the hidden mechanisms responsible for the emergence of the studied phenomena in biology and medicine, suggesting promising research perspectives.