The effects of non-local diffusion on the growth of an avascular tumour #

Ariel Ramirez Torres, Mariam Al Mudarra, Alfio Grillo

11:30 Tuesday in 4Q56.

Part of the Chemo-mechanical couplings in growing tissues session.

Abstract #

Chemical agents, such as nutrients, play a crucial role in tumour progression as they are essential for cell proliferation. However, the heterogeneous and extremely complex tumour microenvironment may call for generalisations of the classical laws of diffusion, e.g., Fick’s law. In this talk, we investigate the effect of anomalous diffusion of nutrients, influenced by the intrinsic, microstructural properties of the tumour, such as tortuosity, on the evolution of avascular tumour growth. For this purpose, we characterise the nutrients’ non-local spatial interactions by means of a mass flux vector involving derivatives and integrals of fractional order and describe the tumour’s volumetric growth in terms of mass transfer among its constituents and structural transformations in response to growth [1]. To account for these transformations, our approach is based on the Bilby-Kr"{o}ner-Lee (BKL) multiplicative decomposition of the deformation gradient tensor, which introduces a growth tensor, treated as an unknown kinematic variable of the model, and an elastic accommodating tensor [2,3]. Our results show the relevance of embracing a fractional framework that ``dominates’’ the tumour’s growth and illustrate the role of the tumour’s microstructural characteristics. Although our approach provides valuable information on the tumour’s evolution, further research is required to assess the relationship of fractional-order parameters with microstructural features.

[1] Ramírez-Torres A, Di Stefano S, Grillo A. Influence of non-local diffusion in avascular tumour growth. Mathematics and Mechanics of Solids. 2021;26(9):1264-1293. [2] Micunovic M. Thermomechanics of viscoplasticity. New York: Springer; 2009. [3] Grillo A, Di Stefano S. A formulation of volumetric growth as a mechanical problem subjected to non-holonomic and rheonomic constraint. Mathematics and Mechanics of Solids. 2022 (Accepted).