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A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane
Author(s) -
Chen Ying,
Wise Steven M.,
Shenoy Vivek B.,
Lowengrub John S.
Publication year - 2014
Publication title -
international journal for numerical methods in biomedical engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.741
H-Index - 63
eISSN - 2040-7947
pISSN - 2040-7939
DOI - 10.1002/cnm.2624
Subject(s) - scheme (mathematics) , nonlinear system , membrane , materials science , mechanics , control theory (sociology) , mathematics , computer science , physics , mathematical analysis , chemistry , artificial intelligence , quantum mechanics , biochemistry , control (management)
SUMMARY In this paper, we extend the 3D multispecies diffuse‐interface model of the tumor growth, which was derived in Wise et al . (Three‐dimensional multispecies nonlinear tumor growth‐I: model and numerical method, J. Theor. Biol. 253 (2008) 524–543), and incorporate the effect of a stiff membrane to model tumor growth in a confined microenvironment. We then develop accurate and efficient numerical methods to solve the model. When the membrane is endowed with a surface energy, the model is variational, and the numerical scheme, which involves adaptive mesh refinement and a nonlinear multigrid finite difference method, is demonstrably shown to be energy stable. Namely, in the absence of cell proliferation and death, the discrete energy is a nonincreasing function of time for any time and space steps. When a simplified model of membrane elastic energy is used, the resulting model is derived analogously to the surface energy case. However, the elastic energy model is actually nonvariational because certain coupling terms are neglected. Nevertheless, a very stable numerical scheme is developed following the strategy used in the surface energy case. 2D and 3D simulations are performed that demonstrate the accuracy of the algorithm and illustrate the shape instabilities and nonlinear effects of membrane elastic forces that may resist or enhance growth of the tumor. Compared with the standard Crank–Nicholson method, the time step can be up to 25 times larger using the new approach. Copyright © 2014 John Wiley & Sons, Ltd.