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Interfaces and Free Boundaries

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Volume 21, Issue 4, 2019, pp. 463–493
DOI: 10.4171/IFB/428

Published online: 2019-12-18

A tractable mathematical model for tissue growth

Joe Eyles[1], John R. King[2] and Vanessa Styles[3]

(1) University of Sussex, Brighton, UK
(2) University of Nottingham, UK
(3) University of Sussex, Brighton, UK

Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a ‘kinetic under-cooling’ regularisation) where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We perform linear stability analysis and derive a diffuse-interface approximation of the model. Finite-element discretisations of two closely related models are presented, together with computational results comparing the approximate solutions.

Keywords: Tissue growth, forced mean curvature flow, formal asymptotic methods, finite elements, diffuse interface model

Eyles Joe, King John, Styles Vanessa: A tractable mathematical model for tissue growth. Interfaces Free Bound. 21 (2019), 463-493. doi: 10.4171/IFB/428