Numerical simulations of the Gatenby–Gawlinski model with heterogeneous acid diffusion in one space dimension

In this work, we introduce a variant of the Gatenby–Gawlinski model for acid- mediated tumour invasion in the one-dimensional experimental setting, accounting for hetero- geneous diffusion of the lactic acid across the surrounding healthy tissues. Numerical simulations are performed by employing finite volume schemes on staggered cartesian grids, together with explicit time discretization. The effectiveness of such approach is proven by reproducing biolog- ically relevant results like the formation of propagating invasion fronts and the emergence of an interstitial gap between normal and cancerous cells, which is driven by the pH lowering strategy and depends significantly on the non-homogenous diffusion rates. By means of a comparison analysis, we infer that a homogenization phenomenon arises in the long run for appropriate values of the physical parameters, thus paving the way for complex applications to interface diffusion problems of invasive processes.