A Difference Scheme for Solving the Equations of Tumor Growth Subject to the Restricted Flow of Motile Cells

The article investigates one-dimensional mathematical model of tumor growth represented by a system of quasi-linear parabolic equations. We assume certain restrictions
on the full flow of the motile tumor cells, leading to the possible degeneration of the system into a hyperbolic type and emergence of discontinuous (weak) solution. To find weak solution we consider tumor growth as the emergence of a new phase. Thus we have a generalized (nonlinear) Stefan problem. The authors propose and implement a difference scheme with the explicit statement of the phase-change moving boundary to solve the problem. It is shown that this approach allows to describe different regimes of tumor growth.

Keywords

difference scheme; substrate taxis; the problem with movable boundary; break allocation.

References

1. Astanin S.A., Kolobov A.V., Lobanov A.I., Pimenova T.P., Polezhaev A.A., Solynik G.I. [An Effect of Spatial Heterogeneity of the Environment on the Tumor Growth and Invasion. Analysis Using Methods of Mathematical Modelling]. Meditsina v zerkale informatiki [Medicine in the Mirror of Informatics]. Moscow, Nauka, 2008, pp. 188-223.
2. Albu A.F., Zubov V.I. [On a Melting Process with Restriction on a Cooling Velocity]. Mathematical Models and Computer Simulations, 2002, vol. 14, no. 8, pp. 119-123. (in Russian)
3. Volosevich P.P., Zmitrenko N.V., Levanov E.I., Severina E.V. [Dynamic and Heating of Plasma Subject to Heat Flux Relaxation]. Mathematical Models and Computer Simulations, 2008, vol. 20, no. 4, pp. 57-68. (in Russian)
4. Hsiao A.Y., Torisawaa Y., Tunga Y.-C., Sudb S., Taichmanc R.S., Pientab K.J., Takayamaet S. Microfluidic System for Formation of PC-3 Prostate Cancer Co-Culture Spheroids. Biomaterials, 2009, vol. 30, pp. 3020-3027. DOI: 10.1016/j.biomaterials.2009.02.047
5. Freeman A.E., Hoffman R.M. In Vivo-Like Growth of Human Tumors in Vitro. Proceedings of the National Academy of Sciences of the United States of America, 1986, vol. 83, pp. 2694-2698. DOI: 10.1073/pnas.83.8.2694
6. Fedorenko R.P. A Difference Scheme for Stefan's Problem. USSR Computational Mathematics and Mathematical Physics, 1975, vol. 15, no. 5, pp. 246-251. DOI: 10.1016/0041-5553(75)90122-6
7. Niziev V.G., Koldoba A.V., Mirzade F.Kh., Panchenko V.Ya., Poveschenko Yu.A., Popov M.V. [Numerical Modelling of Laser Sintering of Two-Component Powder Mixtures]. Mathematical Models and Computer Simulations, 2011, vol. 23, no. 4, pp. 90-102. (in Russian)
8. Scharfetter D.L., Gummel H.K. Large-Signal Analysis of Silicon Read Diode Oscillator. IEEE Transactions on Electron Devices, 1969, vol. 16, no. 1, pp. 64-77. DOI: 10.1109/T-ED.1969.16566