Influence of non-local diffusion in avascular tumour growth

Abstract

The availability and evolution of chemical agents play an important role in the growth of a tumour and, therefore, the mathematical description of their consumption is of special interest. Usually, Fick’s law of diffusion is adopted for describing the local character of the evolution of chemicals. However, in a highly complex, heterogeneous medium, as is a tumour, the progression of chemical species could be influenced by non-local interactions. In this respect, our goal is to investigate the influence of such types of diffusion on the growth of a tumour in the avascular stage. For our purposes, we consider a diffusion equation for the evolution of the chemical agents that accounts for the existence of non-local interactions in a non-Fickean manner, and that involves notions of fractional calculus. In particular, the introduction of derivatives or integrals of fractional type of order [Formula: see text] has proven to be an effective mathematical tool in the description of various non-local phenomena. To achieve our goals, we adopt part of the modelling assumptions outlined in previous works, in which the growth of a tumour is described in terms of mass transfer among the tumour’s constituents and structural changes that occur in the tumour itself in response to growth. The latter ones are characterised by means of the Bilby–Kröner–Lee decomposition of the deformation gradient tensor. We perform numerical simulations, whose results indicate the relevance of embracing a fractional framework in modelling tumour growth. Specifically, the real parameter [Formula: see text]‘dominates’ the way in which the tumour grows, since it permits the modelling of a variety of growth patterns ranging from the standard growth to no growth at all.