Multiphase modelling of vascular tumour growth in two spatial dimensions

JOURNAL OF THEORETICAL BIOLOGY

卷:316

Article

Hubbard, M. E.¹  Byrne, H. M.^2,3 

[1] Univ Leeds, Sch Comp, Leeds LS2 9JT, W Yorkshire, England

[2] Univ Oxford, Inst Math, Oxford Ctr Collaborat Appl Math, Oxford OX1 3LB, England

[3] Dept Comp Sci, Oxford OX1 3QD, England

关键词: Cancer modelling;    Partial differential equations;    Continuum mechanics;    Numerical simulations;    Nonlinear coupling;   

DOI  :  10.1016/j.jtbi.2012.09.031

来源: Elsevier

【 摘 要 】

In this paper we present a continuum mathematical model of vascular tumour growth which is based on a multiphase framework in which the tissue is decomposed into four distinct phases and the principles of conservation of mass and momentum are applied to the normal/healthy cells, tumour cells, blood vessels and extracellular material. The inclusion of a diffusible nutrient, supplied by the blood vessels, allows the vasculature to have a nonlocal influence on the other phases. Two-dimensional computational simulations are carried out on unstructured, triangular meshes to allow a natural treatment of irregular geometries, and the tumour boundary is captured as a diffuse interface on this mesh, thereby obviating the need to explicitly track the (potentially highly irregular and ill-defined) tumour boundary. A hybrid finite volume/finite element algorithm is used to discretise the continuum model: the application of a conservative, upwind, finite volume scheme to the hyperbolic mass balance equations and a finite element scheme with a stable element pair to the generalised Stokes equations derived from momentum balance, leads to a robust algorithm which does not use any form of artificial stabilisation. The use of a matrix-free Newton iteration with a finite element scheme for the nutrient reaction-diffusion equations allows full nonlinearity in the source terms of the mathematical model. Numerical simulations reveal that this four-phase model reproduces the characteristic pattern of tumour growth in which a necrotic core forms behind an expanding rim of well-vascularised proliferating tumour cells. The simulations consistently predict linear tumour growth rates. The dependence of both the speed with which the tumour grows and the irregularity of the invading tumour front on the model parameters is investigated. (C) 2012 Elsevier Ltd. All rights reserved.

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