Numerical simulation of a thermodynamically consistent four-species tumor growth model

A. Hawkins-Daarud, K.G. Zee, van der, J.T. Oden

Research output: Contribution to journalArticleAcademicpeer-review

76 Citations (Scopus)

Abstract

In this paper, we develop a thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures. Unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models. A mixed finite element spatial discretization is developed and implemented to provide numerical results demonstrating the range of solutions this model can produce. A time-stepping algorithm is then presented for this system, which is shown to be first order accurate and energy gradient stable. The results of an array of numerical experiments are presented, which demonstrate a wide range of solutions produced by various choices of model parameters.
Original languageEnglish
Pages (from-to)3-24
Number of pages22
JournalInternational Journal for Numerical Methods in Biomedical Engineering
Volume28
DOIs
Publication statusPublished - 2011

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Tumor Growth
Growth Model
Tumors
Numerical Simulation
Computer simulation
Growth
Neoplasms
Food
Diffuse Interface
Auxiliary equation
Surface Effects
Model
Mixed Finite Elements
Time Stepping
Reaction-diffusion Equations
Nutrients
Nonlinear Partial Differential Equations
Range of data
Partial differential equations
Continuum

Cite this

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Numerical simulation of a thermodynamically consistent four-species tumor growth model. / Hawkins-Daarud, A.; Zee, van der, K.G.; Oden, J.T.

In: International Journal for Numerical Methods in Biomedical Engineering, Vol. 28, 2011, p. 3-24.

Research output: Contribution to journalArticleAcademicpeer-review

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AU - Hawkins-Daarud, A.

AU - Zee, van der, K.G.

AU - Oden, J.T.

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AB - In this paper, we develop a thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures. Unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models. A mixed finite element spatial discretization is developed and implemented to provide numerical results demonstrating the range of solutions this model can produce. A time-stepping algorithm is then presented for this system, which is shown to be first order accurate and energy gradient stable. The results of an array of numerical experiments are presented, which demonstrate a wide range of solutions produced by various choices of model parameters.

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