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

Modelling the interaction of the urokinase plasminogen activator system with a modelfor cancer cells dynamics lead to a system of five coupled partial differential equationsthe solution of which, over a one-dimensional domain, can be obtained using the finiteelement method. As the discretization of the integration domain is crucial, particularlyin the presence of strong variations of the domain variables, we tackle the problemsolution implementing an adaptive grid numerical technique. Our results improve previousnumerical simulations performed on uniformly discretized domains, allowing to capturefiner spatial details characterizing the interface between cancer and healthy cells, whichcan be related to malignancy: low values of the cancer cells motility induce a high spatialheterogeneity at the cancer/healthy cells interface, while the dynamical evolution over thewhole domain shows branched spatio-temporal patterns; on the contrary, higher motilityvalues smooth the cancer cells spatial profile, letting cancer cell concentration to evolveaccording to a less heterogeneous pattern.