A model for avascular tumour growth with non-Fickean diffusion

Abstract

Chemical agents, such as nutrients, play a crucial role in tumour progression as they are essential for tumour cell proliferation. In this work, we investigate the influence of the diffusion of chemical species on the mechanisms of avascular tumour growth. As suggested in the scientific literature, the heterogeneous and extremely complex environment, such as that in a tumour, where diffusion taking place may lead to non-Fickean diffusion laws. In particular, derivatives and integrals of fractional order are often considered to explain this kind of behaviour, and we adopt this approach here to investigate the growth of an avascular tumour. For this purpose, we characterise the growth in terms of mass transfer between the tumour’s constituents and the growth-induced structural transformations. Besides, we account for non-local interactions through a convolution-type law featuring the chemical species’ mass fraction gradient and identifiable with classical fractional derivatives. Additionally, we investigate the impact of considering the fractional-order parameter as a function of space and time and connected to intrinsic properties of the system under study. Further research is required to assess the influence of variable order fractional parameters and their relationship with other geometric or physical characteristics that may affect tumour progression.