Modelling Tumour Dynamics with Variable-Order Non-Local Diffusion

Mariam Almudarra (University of Glasgow, UK)

Tuesday 21st May 11:40-12:20 Lecture Theatre 116

Abstract

We investigates the dynamics of tumour growth with a particular emphasis on the mass exchange between tumour constituents and the resulting inelastic distortions due to growth. A central aspect of our investigation is the role of non-local diffusion, influenced by the tumour's complex microenvironment, which we propose significantly affects the growth process. We introduce a variable-order fractional operator to depict the spatial variation in nutrient diffusivity within the tumour tissue, a critical component of our analysis [1].

Inspired by previous studies [2], we aim to establish a governing law for the evolution of growth-induced inelastic distortions in tumours. By linking generalised forces with kinematic descriptors related to the growth tensor, we seek to clarify the relationship between these forces and the tumour's growth behaviour. This formulation is grounded in the dissipation inequality, shedding light on the interaction between inelastic distortions and the mass balance's source/sink terms.

The initial phase of our research evaluates how the growth law influences key variables dictating tissue evolution in tumours. We then examine how limited diffusion, as depicted by the variable-order fractional operator, impacts tumour growth. Through this exploration, our study provides new insights into tumour growth mechanisms and emphasises the crucial role of non-Fickian diffusion in tumour development.

References:

[1] Almudarra MM, Ramírez-Torres A. Examining avascular tumour growth dynamics: A variable-order non-local modelling perspective. Mathematics and Mechanics of Solids. 2024;0(0). doi:10.1177/10812865241230269
[2] Grillo A, Di Stefano S (2023) An a posteriori approach to the mechanics of volumetric growth. Mathematics and Mechanics of Complex Systems, 11: 57–86.

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