Computational approaches and analysis for a spatio-structural-temporal invasive carcinoma model

Spatio-temporal models have long been used to describe biological systems of cancer, but it has not been until very recently that increased attention has been paid to structural dynamics of the interaction between cancer populations and the molecular mechanisms associated with local invasion. One system that is of particular interest is that of the urokinase plasminogen activator (uPA) wherein uPA binds uPA receptors on the cancer cell surface, allowing plasminogen to be cleaved into plasmin, which degrades the extracellular matrix and this way leads to enhanced cancer cell migration. In this paper, we develop a novel numerical approach and associated analysis for spatio-structuro-temporal modelling of the uPA system for up to two-spatial and two-structural dimensions. This is accompanied by analytical exploration of the numerical techniques used in simulating this system, with special consideration being given to the proof of stability within numerical regimes encapsulating a central differences approach to approximating numerical gradients. The stability analysis performed here reveals instabilities induced by the coupling of the structural binding and proliferative processes. The numerical results expound how the uPA system aids the tumour in invading the local stroma, whilst the inhibitor to this system may impede this behaviour and encourage a more sporadic pattern of invasion.