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2010, 7(2): 385-400. doi: 10.3934/mbe.2010.7.385

Diffusion-limited tumour growth: Simulations and analysis

Philip Gerlee 1, and  Alexander R. A. Anderson 2,

1. 

Center for Models of Life, Niels Bohr Institute, Blegdamsvej 17, 2200 Copenhagen O, Denmark
2. 

H. Lee Moffitt Cancer Center & Research Institute, Integrated Mathematical Oncology, 12902 Magnolia Drive, Tampa, FL 33612, United States

Received  June 2009 Revised  September 2009 Published  April 2010

The morphology of solid tumours is known to be affected by the background oxygen concentration of the tissue in which the tumour grows, and both computational and experimental studies have suggested that branched tumour morphology in low oxygen concentration is caused by diffusion-limited growth. In this paper we present a simple hybrid cellular automaton model of solid tumour growth aimed at investigating this phenomenon. Simulation results show that for high consumption rates (or equivalently low oxygen concentrations) the tumours exhibit branched morphologies, but more importantly the simplicity of the model allows for an analytic approach to the problem. By applying a steady-state assumption we derive an approximate solution of the oxygen equation, which closely matches the simulation results. Further, we derive a dispersion relation which reveals that the average branch width in the tumour depends on the width of the active rim, and that a smaller active rim gives rise to thinner branches. Comparison between the prediction of the stability analysis and the results from the simulations shows good agreement between theory and simulation.
Keywords: Tumour morphology., Diffusion-limited growth.
Mathematics Subject Classification: Primary: 82B24; Secondary: 37B1.
Citation: Philip Gerlee, Alexander R. A. Anderson. Diffusion-limited tumour growth: Simulations and analysis. Mathematical Biosciences & Engineering, 2010, 7 (2) : 385-400. doi: 10.3934/mbe.2010.7.385
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