Mathematical modeling and numerical simulation of a multiscale cancer invasion of host tissue

In this study, we develop an advection-reaction-diffusion system of partial differential equations (PDEs) to describe interactions between tumor cells and extracellular matrix (ECM) at the macroscopic level. At the subcellular level, we model the interaction of proteolytic enzymes and the ECM with a set of ordinary differential equations (ODEs). A contractivity function is used to couple the macroscopic and microscopic events. The model is supplemented with nutrients supply from the underlying tissue. These PDE-ODE systems of equations model the on-set of tumor cell invasion of the host extracellular matrix. The model accounts for different time and spatial scales at the macroscopic and microscopic levels. Contact inhibition between the tumor cells and the tumor micro-environment are also accounted for through a nonlinear density-dependent diffusion and haptotaxis coefficients. In the numerical simulations, we use a nonstandard finite difference method to illustrate the model predictions. Qualitatively, our results confirm the three distinct layers of proliferating, quiescent and necrotic cells as observed in multicellular spheroids experiments.