A mathematical model for the nutrient distribution of a spheroidal avascular cancer tumour within an inhomogeneous environment

When a cancerous cell colony grows within a healthy environment, the entire structure can be modelled as a continuous two-phase fluid with five bounded compartments, governed by the laws of mass conservation, Fick’s diffusion law, and fluid mechanics principles. The interfaces of the five bounded compartments of the colony are defined by critical values of nutrient concentration. In studying the evolution of the exterior tumour boundary, nutrient concentration is the primary parameter. Although most existing research focuses on spherical tumours, significant implications for nutrient distribution emerge when spherical symmetry is abandoned, such as the occurrence of critical values at specific points rather than across the entire surface. In this work, we consider an oblate spheroidal tumour and investigate the effects of non-homogeneity in both nutrient supply and consumption rates. Our findings indicate that critical values are encountered within the interior of a thin layer, rather than at a single interface, although the interface is still included. We study the variation of nutrient concentration on the tumour’s interfaces through plots, highlighting the critical locations. The prolate spheroidal case can be derived via a simple transformation, and comparisons with similar spherical models are also discussed.