Adaptive grid modelling for cancer cells in the early stage of invasion

Modelling the interaction of the urokinase plasminogen activator system with a model for cancer cells dynamics lead to a system of five coupled partial differential equations the solution of which, over a one-dimensional domain, can be obtained using the finite element method. As the discretization of the integration domain is crucial, particularly in the presence of strong variations of the domain variables, we tackle the problem solution implementing an adaptive grid numerical technique. Our results improve previous numerical simulations performed on uniformly discretized domains, allowing to capture finer spatial details characterizing the interface between cancer and healthy cells, which can be related to malignancy: low values of the cancer cells motility induce a high spatial heterogeneity at the cancer/healthy cells interface, while the dynamical evolution over the whole domain shows branched spatio-temporal patterns; on the contrary, higher motility values smooth the cancer cells spatial profile, letting cancer cell concentration to evolve according to a less heterogeneous pattern.