Abstract
Immunotherapy is a newly emerging approach to cancer treatment that seeks to stimulate a body’s immune defenses, especially T cells, to combat and potentially eliminate tumors. Relevant tumor–immune interactions depend on stochasticity, since the dynamics involve a small and decreasing number of cells, and spatiotemporal heterogeneity, since the dynamics occur in a localized tumor environment. To account for these two aspects of the system, we develop mathematical models of an anti-tumor immune response using a cellular automaton and a system of partial differential equations. We explicitly model immune cell recruitment to the tumor via cytokine secretion and chemotaxis of immune cells. Our models exhibit three types of behavior: tumor elimination, oscillation, and uncontrolled tumor growth that depend substantially on the strength of immune cell chemotaxis, or recruitment, to the tumor site.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aguda, B.D., Marsh, C.B., Thacker, M., Crouser, E.D.: An in silico modeling approach to understanding the dynamics of sarcoidosis. PLoS One 6(5), e19,544 (2011)
Alarcón, T., Byrne, H.M., Maini, P.K.: A cellular automaton model for tumour growth in inhomogeneous environment. J. Theor. Biol. 225, 257–274 (2003)
Banerjee, S., Sarkar, R.R.: Delay-induced model for tumor-immune interaction and control of malignant tumor growth. BioSystems 91(1), 268–288 (2008)
Barbarossa, M.V., Kuttler, C., Zinsl, J.: Delay equations modeling the effects of phase-specific drugs and immunotherapy on proliferating tumor cells. Math. Biosci. Eng. 9(2), 241–257 (2012)
Bunimovich-Mendrazitsky, S., Byrne, H., Stone, L.: Mathematical model of pulsed immunotherapy for superficial bladder cancer. Bull. Math. Biol. 70(7), 2055–2076 (2008)
Buzea, C.G., Agop, M., Moraru, E., Stana, B.A., Gir?u, M., Iancu, D.: Some implications of Scale Relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response. J. Theor. Biol. 282(1), 52–64 (2011)
Cabrero, J.R., Serrador, J.M., Barreiro, O., Mittelbrunn, M., Naranjo-Suarez, S., Martin-Cofreces, N., Vicente-Manzanares, M., Mazitschek, R., Bradner, J.E., Avila, J., Valenzuela-Fernandez, A., Sanchez-Madrid, F.: Lymphocyte chemotaxis is regulated by histone deacetylase 6, independently of its deacetylase activity. Mol. Biol. Cell 17(8), 3435–3445 (2006)
Castiglione, F., Piccoli, B.: Cancer immunotherapy, mathematical modeling and optimal control. J. Theor. Biol. 247(4), 723–732 (2007)
Catron, D.M., Itano, A.A., Pape, K.A., Mueller, D.L., Jenkins, M.K.: Visualizing the first 50 h of the primary immune response to a soluble antigen. Immunity 21(3), 341–347 (2004)
De Boer, R.J., Homann, D., Perelson, A.S.: Different dynamics of CD4+ and CD8+ T cell responses during and after acute lymphocytic choriomeningitis virus infection. J Immunol. 171(8), 3928–3935 (2003)
DeConde, R., Kim, P.S., Levy, D., Lee, P.P.: Post-transplantation dynamics of the immune response to chronic myelogenous leukemia. J. Theor. Biol. 236(1), 39–59 (2005)
Depillis, L., Gallegos, A., Radunskaya, A.: A model of dendritic cell therapy for melanoma. Front. Oncol. 3, 56 (2013)
Eftimie, R., Bramson, J.L., Earn, D.J.: Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. Bull. Math. Biol. 73(1), 2–32 (2011)
Eikenberry, S., Thalhauser, C., Kuang, Y.: Tumor-immune interaction, surgical treatment, and cancer recurrence in a mathematical model of melanoma. PLoS Comput. Biol. 5, e1000,362 (2009)
Erjaee, G.H., Ostadzad, M.H., Amanpour, S., Lankarani, K.B.: Dynamical analysis of the interaction between effector immune and cancer cells and optimal control of chemotherapy. Nonlinear Dyn. Psychol. Life Sci. 17(4), 449–463 (2013)
Evans, L.C.: Partial Differential Equations, 2ndedn. American Mathematical Society, Providence (2010)
Finn, O.J.: Cancer vaccines: between the idea and the reality. Nat. Rev. Immunol. 3, 630–641 (2003)
Friedl, P., Gunzer, M.: Interaction of T cells with APCs: the serial encounter model. Trends Immunol. 22(2), 187–191 (2001)
Friedman, A., Tian, J.P., Fulci, G., Chiocca, E.A., Wang, J.: Glioma virotherapy: effects of innate immune suppression and increased viral replication capacity. Cancer Res. 66(4), 2314–2319 (2006)
Goodhill, G.J.: Diffusion in axon guidance. Eur. J. Neurosci. 9(7), 1414–1421 (1997)
Jaini, R., Kesaraju, P., Johnson, J.M., Altuntas, C.Z., Jane-Wit, D., Tuohy, V.K.: An autoimmune-mediated strategy for prophylactic breast cancer vaccination. Nat. Med. 16, 799–803 (2010)
Janeway Jr., C.A., Travers, P., Walport, M., Shlomchik, M.J.: Immunobiology: The Immune System in Health and Disease, 6th edn. Garland Science Publishing, New York (2005)
Kareva, I., Berezovskaya, F., Castillo-Chavez, C.: Myeloid cells in tumour-immune interactions. J. Biol. Dyn. 4(4), 315–327 (2010)
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26, 399–415 (1970)
Kim, P.S., Lee, P.P.: Modeling protective anti-tumor immunity via preventative cancer vaccines using a hybrid agent-based and delay differential equation approach. PLoS Comput. Biol. 8(10), e1002,742 (2012)
Kim, P.S., Lee, P.P., Levy, D.: Dynamics and potential impact of the immune response to chronic myelogenous leukemia. PLoS Comput. Biol. 4(6), e1000,095 (2008)
Kirschner, D., Panetta, J.C.: Modeling immunotherapy of the tumor-immune interaction. J. Math. Biol. 37, 235–252 (1998)
Kuroishi, T., Tominaga, S., Morimoto, T., Tashiro, H., Itoh, S., Watanabe, H., Fukuda, M., Ota, J., Horino, T., Ishida, T.: Tumor growth rate and prognosis of breast cancer mainly detected by mass screening. Jpn. J. Cancer Res. 81(5), 454–462 (1990)
Kuznetsov, V.A., Makalkin, I.A., Taylor, M.A., Perelson, A.S.: Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56(2), 295–321 (1994)
León, K., Lage, A., Carneiro, J.: How regulatory CD25+CD4+ T cells impinge on tumor immunobiology? on the existence of two alternative dynamical classes of tumors. J. Theor. Biol. 247(1), 122–137 (2007)
León, K., Lage, A., Carneiro, J.: How regulatory CD25+CD4+ T cells impinge on tumor immunobiology: the differential response of tumors to therapies. J. Immunol. 179(9), 5659–5668 (2007)
Lin, A.: A model of tumor and lymphocyte interactions. Discrete and Continuous Dynamical Systems B 4(1), 241–266 (2004)
Mackay, C.R.: Chemokine receptors and T cell chemotaxis. J. Exp. Med. 184(3), 799–802 (1996)
Mallet, D.G., De Pillis, L.G.: A cellular automata model of tumor-immune system interactions. J. Theor. Biol. 239, 334–350 (2006)
Matzavinos, A., Chaplain, M.A.: Travelling-wave analysis of a model of the immune response to cancer. C. R. Biol. 327(11), 995–1008 (2004)
Matzavinos, A., Chaplain, M.A., Kuznetsov, V.A.: Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour. Math. Med. Biol. 21(1), 1–34 (2004)
Maurer, M., von Stebut, E.: Macrophage inflammatory protein-1. Int. J. Biochem. Cell Biol. 36(10), 1882–1886 (2004)
Merrill, S.J.: A model of the role of natural killer cells in immune surveillance I. J. Math. Biol. 12, 363–373 (1981)
Michaelson, J., Satija, S., Moore, R., Weber, G., Halpern, E., Garland, A., Kopans, D.B.: Estimates of breast cancer growth rate and sojourn time from screening database information. J. Women Imaging 5(1), 11–19 (2003)
Moore, H., Li, N.K.: A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction. J. Theor. Biol. 225(4), 513–523 (2004)
Murray, J.D.: Mathematical Biology: I. An Introduction, 3rd edn. Springer, New York (2002)
Nestle, F.O., Tonel, G., Farkas, A.: Cancer vaccines: the next generation of tools to monitor the anticancer immune response. PLoS Med. 2, e339 (2005)
Okubo, A., Levin, S.A.: Diffusion and Ecological Problems: Modern Perspectives, 2rd edn. Springer, New York (2010)
Penington, C.J., Hughes, B.D., Landman, K.A.: Building macroscale models from microscale probabilistic models: a general probabilistic approach for nonlinear diffusion and multispecies phenomena. Phys. Rev. E 84(4 Pt 1), 041,120 (2011)
Pennisi, M.: A mathematical model of immune-system-melanoma competition. Comput. Math. Methods Med. 2012, 850,754 (2012)
de Pillis, L.G., Mallet, D.G., Radunskaya, A.E.: Spatial tumor-immune modeling. Comput. Math. Methods Med. 7(2–3), 159–176 (2006)
de Pillis, L.G., Radunskaya, A.E., Wiseman, C.L.: A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res. 65(17), 7950–7958 (2005)
Qi, A.S., Zheng, X., Du, C.Y., An, B.S.: A cellular automaton model of cancerous growth. J. Theor. Biol. 161(1), 1–12 (1993)
Robert-Tissot, C., Nguyen, L.T., Ohashi, P.S., Speiser, D.E.: Mobilizing and evaluating anticancer T cells: pitfalls and solutions. Expert Rev. Vaccines 12(11), 1325–1340 (2013)
Simpson, M.J., Baker, R.E.: Corrected mean-field models for spatially dependent advection-diffusion-reaction phenomena. Phys. Rev. E 83(5 Pt 1), 051,922 (2011)
Simpson, M.J., Landman, K.A., Hughes, B.D.: Multi-species simple exclusion processes. Phys. A 388(4), 399–406 (2009)
Soiffer, R., Hodi, F.S., Haluska, F., Jung, K., Gillessen, S., Singer, S., Tanabe, K., Duda, R., Mentzer, S., Jaklitsch, M., Bueno, R., Clift, S., Hardy, S., Neuberg, D., Mulligan, R., Webb, I., Mihm, M., Dranoff, G.: Vaccination with irradiated, autologous melanoma cells engineered to secrete granulocyte-macrophage colony-stimulating factor by adenoviral-mediated gene transfer augments antitumor immunity in patients with metastatic melanoma. J. Clin. Oncol. 21, 3343–3350 (2003)
Soiffer, R., Lynch, T., Mihm, M., Jung, K., Rhuda, C., Schmollinger, J.C., Hodi, F.S., Liebster, L., Lam, P., Mentzer, S., Singer, S., Tanabe, K.K., Cosimi, A.B., Duda, R., Sober, A., Bhan, A., Daley, J., Neuberg, D., Parry, G., Rokovich, J., Richards, L., Drayer, J., Berns, A., Clift, S., Cohen, L.K., Mulligan, R.C., Dranoff, G.: Vaccination with irradiated autologous melanoma cells engineered to secrete human granulocyte-macrophage colony-stimulating factor generates potent antitumor immunity in patients with metastatic melanoma. Proc. Natl. Acad. Sci. USA 95, 13,141–13,146 (1998)
Spratt, J.A., von Fournier, D., Spratt, J.S., Weber, E.E.: Decelerating growth and human breast cancer. Cancer 71, 2013–2019 (1993)
Villasana, M., Radunskaya, A.: A delay differential equation model for tumor growth. J. Math. Biol. 47(3), 270–294 (2003)
Wang, W., Epler, J., Salazar, L.G., Riddell, S.R.: Recognition of breast cancer cells by CD8+ cytotoxic T-cell clones specific for NY-BR-1. Cancer Res. 66, 6826–6833 (2006)
Wang, Z., Hillen, T.: Classical solutions and pattern formation for a volume filling chemotaxis model. Chaos 17(3), 037,108 (2007)
Wang, Z.A.: On chemotaxis models with cell population interactions. Math. Model Nat. Phenom. 5(3), 173–190 (2010)
Weedon-Fekjaer, H., Lindqvist, B.H., Vatten, L.J., Aalen, O.O., Tretli, S.: Breast cancer tumor growth estimated through mammography screening data. Breast Cancer Res. 10, R41 (2008)
Wiedemann, A., Depoil, D., Faroudi, M., Valitutti, S.: Cytotoxic T lymphocytes kill multiple targets simultaneously via spatiotemporal uncoupling of lytic and stimulatory synapses. Proc. Natl. Acad. Sci. USA 103, 10,985–10,990 (2006)
Wilkinson, P.C., Komai-Koma, M., Newman, I.: Locomotion and chemotaxis of lymphocytes. Autoimmunity 26(1), 55–72 (1997)
Acknowledgements
AKC was supported by the KE Bullen Scholarship III awarded to honours students in the School of Mathematics and Statistics at the University of Sydney, and PSK was supported by the Australian Research Council Discovery Early Career Research Award (DE120101113).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Science+Business Media New York
About this paper
Cite this paper
Cooper, A.K., Kim, P.S. (2014). A Cellular Automata and a Partial Differential Equation Model of Tumor–Immune Dynamics and Chemotaxis. In: Eladdadi, A., Kim, P., Mallet, D. (eds) Mathematical Models of Tumor-Immune System Dynamics. Springer Proceedings in Mathematics & Statistics, vol 107. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1793-8_2
Download citation
DOI: https://doi.org/10.1007/978-1-4939-1793-8_2
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-1792-1
Online ISBN: 978-1-4939-1793-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)