A Computational Framework for Simulating Multiphase Models of Tissue Growth.
by
Mr James Osborne
Oxford University Computing Laboratory, Oxford, OX1 3QD, UK
Coauthors: Helen M Byrne: Centre for Mathematical Medicine, Division of Theoretical Mechanics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. email: Helen.Byrne@maths.nottingham.ac.uk Sarah Waters: Oxford Center for Industrial & Applied Mathematics, Mathematical Institute, 24-29 St. Giles', Oxford, OX1 3LB, UK, email: waters@maths.ox.ac.uk Jonathan Whiteley: Oxford University Computing Laboratory, Oxford University, Oxford, OX1 3QD, UK, email: Jonathan.Whiteley@comlab.ox.ac.uk

Multiphase modelling is a natural framework for studying many biological systems, for example tissue engineering and cancer development, where different phases represent the constituents of the tissue of interest (e.g. extracellular matrix, cancer cells and interstitial fluid when studying solid tumour growth). The resulting models comprise non-standard mixed systems of nonlinear PDEs. For example, multiphase models used to describe tissue engineering applications and solid tumour growth may generate equations that consist of: (i) viscous fluid flow equations for each phase: (ii) hyperbolic PDEs for mass conservation; and (iii) elliptic or parabolic PDEs for chemical concentrations. Analytical progress with such systems is usually only possible if additional model assumptions are made (e.g. radial symmetry, or small aspect ratio). A complementary approach is to seek a numerical solution of the governing equations without making any such simplifications. The numerical solution of these equations presents numerous challenges: the numerical methods for solving fluid flow equations and hyperbolic PDEs are notoriously prone to complications such as instability and computational time. Further complexity may be introduced if the problem is posed on a growing domain (e.g. a growing tumour). Advanced numerical algorithms are required in order to guarantee an accurate and efficient solution.

We have developed a numerical and computational framework based upon the Galerkin Finite Element Method that allows the numerical solution of coupled systems of parabolic, elliptic and hyperbolic PDEs described above in two or three dimensions. This enables us to investigate the effect of interactions between constitutive phases. We have used this framework to investigate tissue growth in a bioreactor and also the development of a solid tumour, under non-uniform environmental conditions. We illustrate the versatility of our numerical method by presenting results for these two case studies.

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Date received: May 13, 2008