Journal Article

Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties

S. J. Franks and J. R. King

in Mathematical Medicine and Biology: A Journal of the IMA

Published on behalf of Institute of Mathematics and its Applications

Volume 20, issue 1, pages 47-89
Published in print March 2003 | ISSN: 1477-8599
Published online March 2003 | e-ISSN: 1477-8602 | DOI: http://dx.doi.org/10.1093/imammb/20.1.47
Interactions between a uniformly proliferating tumour and its surroundings: uniform material properties

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The stability of a planar tumour growing into neighbouring tissue is examined and, in particular, its dependence on the properties of the tumour and of the surrounding material studied. An abundant supply of nutrient is assumed, so the proliferation of cells is uninhibited (resulting in exponential growth). We consider two possible constitutive relations, Darcy's law and Stokes flow, in describing the deformation of the tissue and the resulting model takes the form of a coupled system comprising a nonlinear reaction–diffusion–convection equation for the tumour cell concentration and an elliptic system for the deformation and stress fields.

Using a combination of linear‐stability analysis, numerical methods and thin‐film approximations, the evolution of the advancing tumour boundary is determined. It is shown that when the tumour and surrounding material properties are the same, a planar interface is always linearly unstable, with the Stokes flow problem being reducible to the Darcy one. We treat the subsequent (nonlinear) evolution and suggest possible extensions to this work.

Keywords: tumour invasiveness and stability; tissue deformation; constitutive properties

Journal Article.  0 words. 

Subjects: Applied Mathematics ; Biomathematics and Statistics

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