A Stochastic Cellular Automata Model for Avascular Tumor Growth with Surveillance of an Immune Response Against the Tumorous Cells : on a Cubic Lattice
Abstract
A kinetic model for avascular tumor growth is presented. The model includes an immune surveillance mechanism that recognizes the cancerous cell and makes the cell susceptible to certain binding reactions. The particular binding interactions of interest are those that lead to the formation of tumor-immune complexes consisting of the cancerous cells and cytotoxic agents (effectors) such as cells or biochemical which can cause the apoptosis of the cancerous cells. The model allows for the possibility of the cancerous cells escaping the immune activity after the binding reactions have occurred, or dying but not undergoing apoptosis when the immune agents are ineffectual. A stochastic cellular automata model on a three-dimensional cubic lattice is used to implement the kinetic model. The simulation results, such as the growth curves are explained at a kinetic microscopic scale. The sensitivity analysis of the effects on parameters from the morphologies of simulated tumors by measuring the spatial distribution of proliferating cells will be presented. The model shows that an increase in the dormancy rate leads to an increase in the density of the proliferating cells in the outermost region. Keywords : stochastic model, in silico model of tumor growth, model on a cubic lattice, tumor growth with immune responseReferences
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six tumor cell lines in vitro and in vivo. Estimate of the transition point from exponential to
gompertzian growth and potential clinical applications. Tumori, 77(3), 189–195.
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immune response to tumor growth. Cancer. Res., 65(17), 7950 - 7958.
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Kataki, A., Scheid, P., Piet, M., Marie, B., Martinet, N., Martinet, Y. & Vignaud, J. M. (2002). Tumor infiltrating
lymphocytes and macrophages have a potential dual role in lung cancer by supporting both host-
defense and tumor progression. J. Lab. Clin. Med., 140(5), 320-328.
Kuznetsov, V. A., Makalkin, I. A., Taylor, M. A. & Perelson, A. S. (1994). Nonlinear dynamics of immunogenic
tumors: Parameter estimation and global bifurcation analysis. Bull. Math. Biol., 56(2), 295-321.
Le Mire, L., Hollowood, K., Gray, D., Bordea, C. & Wojnarowsk, F. (2006). Melanomas in renal transplant
recipients. Br. J. Dermatol., 154(3), 472-477.
Mallet, D. G. & de Pillis, L. G. (2006). A cellular automata model of tumor-immune system interactions.
J. Theor. Biol., 239(3), 334-350.
Matzavinos, A., Chaplain, M. A. & Kuznetsov, V. A. (2004). Mathematical modelling of the spatio-temporal
response of cytotoxic T-lymphocytes to a solid tumour. Math. Med. Biol., 21(1), 1-34.
Mocellin, S., Marincola, F. M. & Young, H. A. (2005). Interleukin-10 and the immune response against cancer: a
counterpoint. J. Leukoc. Biol., 8(5), 1043-1051.
Nieswandt, B., Hafner, M., Echtenacher, B. & Mannel, D. N. (1999). Lysis of tumor cells by natural killer cells in
mice is impeded by platelets. Cancer. Res. ,59, 1295–1300.
Norton, L. (1988). A Gompertzian Model of Human Breast Cancer Growth. Cancer Res. ,48, 7067–7071.
Parish, C. R. (2003). Cancer immunotherapy: The past, the present and the future. Immunol. Cell. Biol.,
81,106–113.
Preziosi, L. (2003). Cancer modeling and Simulation, Chapman and Hall: Boca Raton.
Qi, A. S., Zheng, X., Du, C. Y. & An, B. S. (1993). A cellular Automaton of cancerous growth.
J. Theor. Biol. ,161(1), 1-12.
Steel, G. G. (1977). Growth Kinetics of Tumors, Clarendon Press: Oxford.
System(3rded.). Saunders Elsevier: Philadelphia.
Boondirek, A., Lenbury, Y., Wong-ekkabut, J., Triampo, W., Tang, I. M. & Picha, P. (2006). A Stochastic
Model of Cancer Growth with Immune Response. J. Korean. Phys.Soc., 49(4), 1652-1666.
Boondirek, A., Teerapabolarn, K. & Triampo, W. (2011). A Stochastic Model of Tumor Growth with Immune
Response: Three Dimensional Cubic Lattice. Int. J. Open Problems Compt. Math., 4(3), 29-36.
Boondirek, A. & Triampo, W. (2009), Cancer Research: Computer Simulation of Tumor Growth with Immune
Response. Naresuan University Journal, 17(3), 196 – 200.
Bru, A., Albertos, S., Subiza, J. Garcia-Asenjo, J.L. & Bru, I. (2004). The universal dynamics of tumor growth.
Biophys. J. ,85(5), 2948-2961.
Cameron, D. A ., Ritchie, A. A. & Miller, W. R. (2001). The relative importance of proliferation and cell death in
breast cancer growth and response to tamoxifen. Eur. J. Cancer. ,37(12), 1545-1553.
Demicheli, R., Pratesi, G. & Foroni, R. (1991). The exponential-gompertzian growth model: data form
six tumor cell lines in vitro and in vivo. Estimate of the transition point from exponential to
gompertzian growth and potential clinical applications. Tumori, 77(3), 189–195.
De Pillis, L. G., Radunskaya, A. E. & Wiseman, C. L. (2005). A validated mathematical model of cell-mediated
immune response to tumor growth. Cancer. Res., 65(17), 7950 - 7958.
Eftimie, R., Bramson, J. L. & Earn, D. J. (2011). Interactions Between the Immune System and Cancer: A Brief
Review of Non-spatial Mathematical Models. Bull. Math. Biol., 73(1), 2–32.
Gajewski, T. F., Schreiber, H. & Fu, Y. X. (2013). Innate and adaptive immune cells in the tumor microenvironment.
Nat. Immunol., 14, 1012-1022.
Hanahan, D. & Weinberg, R. A. (2011). Hallmarks of Cancer: The Next Generation. Cell., 144(5), 646-674.
Kasiske, B. L., Snyder, J. J., Gilbertson, D. T. & Wang, C. (2004). Cancer after kidney transplantation in the
United States. Am. J. Transplant., 4(6), 905-913.
Kataki, A., Scheid, P., Piet, M., Marie, B., Martinet, N., Martinet, Y. & Vignaud, J. M. (2002). Tumor infiltrating
lymphocytes and macrophages have a potential dual role in lung cancer by supporting both host-
defense and tumor progression. J. Lab. Clin. Med., 140(5), 320-328.
Kuznetsov, V. A., Makalkin, I. A., Taylor, M. A. & Perelson, A. S. (1994). Nonlinear dynamics of immunogenic
tumors: Parameter estimation and global bifurcation analysis. Bull. Math. Biol., 56(2), 295-321.
Le Mire, L., Hollowood, K., Gray, D., Bordea, C. & Wojnarowsk, F. (2006). Melanomas in renal transplant
recipients. Br. J. Dermatol., 154(3), 472-477.
Mallet, D. G. & de Pillis, L. G. (2006). A cellular automata model of tumor-immune system interactions.
J. Theor. Biol., 239(3), 334-350.
Matzavinos, A., Chaplain, M. A. & Kuznetsov, V. A. (2004). Mathematical modelling of the spatio-temporal
response of cytotoxic T-lymphocytes to a solid tumour. Math. Med. Biol., 21(1), 1-34.
Mocellin, S., Marincola, F. M. & Young, H. A. (2005). Interleukin-10 and the immune response against cancer: a
counterpoint. J. Leukoc. Biol., 8(5), 1043-1051.
Nieswandt, B., Hafner, M., Echtenacher, B. & Mannel, D. N. (1999). Lysis of tumor cells by natural killer cells in
mice is impeded by platelets. Cancer. Res. ,59, 1295–1300.
Norton, L. (1988). A Gompertzian Model of Human Breast Cancer Growth. Cancer Res. ,48, 7067–7071.
Parish, C. R. (2003). Cancer immunotherapy: The past, the present and the future. Immunol. Cell. Biol.,
81,106–113.
Preziosi, L. (2003). Cancer modeling and Simulation, Chapman and Hall: Boca Raton.
Qi, A. S., Zheng, X., Du, C. Y. & An, B. S. (1993). A cellular Automaton of cancerous growth.
J. Theor. Biol. ,161(1), 1-12.
Steel, G. G. (1977). Growth Kinetics of Tumors, Clarendon Press: Oxford.
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2020-01-28
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