Abstract
We propose probabilistic cellular automata on a square lattice for simulating the dynamic of cancer growth in a reaction-diffusion frame. In the reaction step each cancerous cell can proliferate, be quiescent, or die due to apoptosis or necrosis phenomenon. The three-state Potts model is used for calculating the probabilities in the reaction step. We consider the effect of nutrient in the tumor growth in order to improve the precision of the model. We use a simple and suitable method for the diffusion step to simplify movement of cells and nutrient in the model. In the diffusion step the lattice is partitioned by 3×3 blocks. In each block we count the number of different types of cells and redistribute them in the block. In the next time step, each block will be shifted one row down and one column to the right and the operation will be continued. The redistribution step for nutrient molecules is same as cells. It is shown tumor growths asymmetrically toward nutrient source. It has been shown such a simple model could simulate tumor growth with good accuracy, which is based on the well known physical ground i.e. the three-state Potts model.
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References
Knowles, M.A., Selby, P.J.: Introduction to the Cellular and Molecular Biology of Cancer, 4th edn. Oxford University Press, Oxford (2005)
Matzavinos, A., Chaplain, M.A.J.: Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour. Mathematical Medicine and Biology 21, 1–34 (2004)
Gerlee, P., Anderson, A.R.A.: An evolutionary hybrid cellular automaton model of solid tumor growth. J. Theoretical Biology 246(4), 583–603 (2007)
Khain, E., Sander, L.M.: Dynamics and pattern formation in invasive tumor growth. Physical Review Letters 96, 188103 (2006)
Marciniak-Czochraa, A., Kimmel, M.: Reaction–diffusion approach to modeling of the spread of early tumors along linear or tubular structures. J. Theoretical. Biology 244, 375–387 (2007)
Ferreira, S.C., Martins, M.L., Vilela, M.J.: Reaction-diffusion model for the growth of avascular tumor. Phys. Rev. E 65, 021907 (2002)
Adam, J.A., Bellomo, N.: A Survey of Models for Tumor-Immune System Dynamics, Birkhäuser, Boston (1997)
Dormann, S., Deutsch, A.: Modeling of self-organized avascular tumor growth with a hybrid cellular automaton. Silico Biology 2, 0035 (2002)
Wolfram, S.: A new kind of science. Wolfram Media Inc., Champaign (2001)
Wolf-Gladrow, D.A.: Lattice-Gas Cellular Automata and Lattice Boltzmann Models: An Introduction. Springer, Heidelberg (2000)
Ghaemi, M., Shahrokhi, A.: Combination of the Cellular Potts Model and Lattice Gas Cellular Automata for Simulating the Avascular Cancer Growth. In: El Yacoubi, S., Chopard, B., Bandini, S. (eds.) ACRI 2006. LNCS, vol. 4173, pp. 297–303. Springer, Heidelberg (2006)
Glauber, R.J.: Time-Dependent Statistics of the Ising Model. J. Math. Phys. 4, 294 (1963)
Wu, F.: The Potts-model. Rev. Mod. Phys. 54, 235–268 (1982)
Vanag, V.K.: Study of spatially extended dynamical systems using probabilistic cellular automata. J. Pysics-Uspekhi 42(5), 413–434 (1999)
Piotrowska, M.J., Angus, S.D.: A quantitative cellular automaton model of in vitro multicellular spheroid tumour growth. J. Theoretical Biology 258(2), 165–178 (2009)
Calabresi, P., Schein, P.E.: Medical oncology, 2nd edn. Mc Graw-Hill, New York (1993)
Ganong, W.F.: Review of medical physiology, 19th edn. Appleton and Lang, New York (1993)
Christley, S., Zhu, X., Newman, S.A., Alber, M.S.: Multiscale Agent-Based Simulation for Chondrogenic Pattern Formation In Vitro. Cybernetics and Systems: An International Journal 38, 707–727 (2007)
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Ghaemi, M., Naderi, O., Zabihinpour, Z. (2010). A Novel Method for Simulating Cancer Growth . In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds) Cellular Automata. ACRI 2010. Lecture Notes in Computer Science, vol 6350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15979-4_15
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DOI: https://doi.org/10.1007/978-3-642-15979-4_15
Publisher Name: Springer, Berlin, Heidelberg
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