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 $\alpha \in \mathbb{R}$ 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 $\alpha$‘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.

化学剂的可利用性和演变在肿瘤的生长中起着重要作用，因此，对其消耗的数学描述尤为重要。通常，采用菲克扩散定律来描述化学物质演化的局部特征。但是，在高度复杂的异质性介质（如肿瘤）中，化学物种的进程可能会受到非局部相互作用的影响。在这方面，我们的目标是研究这种类型的扩散对无血管期肿瘤生长的影响。出于我们的目的，我们考虑化学试剂演化的扩散方程，该方程解释了非菲尼克斯方式中非局部相互作用的存在，并且涉及分数微积分的概念。特别是，$\alpha \in \mathbb{[R}$已被证明是描述各种非局部现象的有效数学工具。为了实现我们的目标，我们采用了先前工作中概述的部分建模假设，其中以肿瘤成分之间的质量转移和响应于生长的肿瘤本身中发生的结构变化来描述肿瘤的生长。后者的特征在于变形梯度张量的Bilby-Kröner-Lee分解。我们进行数值模拟，其结果表明在肿瘤生长建模中采用分数框架的相关性。具体来说，实参$\alpha$“支配”肿瘤的生长方式，因为它可以模拟从标准生长到完全不生长的多种生长模式。