Abstract
Mathematical concepts of graph theory and game theory both influence models of biological systems. We combine these two approaches to understand how game-like interactions influence the cellular topology of a planar tissue. We review the literature on the role of cell to cell interactions in tumourigenesis and survey the mathematical approaches that have been used to simulate such cell-cell interactions. We present how this game-graph approach can be used to simulate epithelial tissue growth and how it can foster our understanding of the role of cell-cell communication in the early stages of cancer development. We present computational models that we use to test how cooperating and non-cooperating cells build planar tissues and compare the simulated tissue topologies with literature data. We further discuss how such system could be used to model microenviromental communications between cancer cells and the surrounding tissue.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Adra, S., Sun, T., MacNeil, S., Holcombe, M., Smallwood, R.: Development of a Three Dimensional Multiscale Computational Model of the Human Epidermis. PLoS ONE 5, e8511 (2010)
Aegerter-Wilmsen, T., Smith, A.C., Christen, A.J., Aegerter, C.M., Hafen, E., Basler, K.: Exploring the effects of mechanical feedback on epithelial topology. Development 137, 499506 (2010)
Aguirre-Ghiso, J.A.: Models, mechanisms and clinical evidence for cancer dormancy. Nat Rev Cancer 7, 834–846 (2007)
Alarcon, T., Byrne, H.M., Maini, P.K.: A cellular automaton model for tumour growth in inho- mogeneous environment. J Theor Biol 225, 257–274 (2003)
Alarcon, T., Byrne, H.M., Maini, P.K.: A Multiple Scale Model for Tumor Growth. Mult Mod Sim 3, 440–475 (2005)
Anderson, A.R.A., Chaplain, M.A.J., Rejniak, K.A.: Single-Cell-Based Models in Biology and Medicine. Birkhauser, Basel, Switzerland (2007)
Anderson, A.R.A., Hassanein, M., Branch, K.M., Lu, J., Lobdell, N.A., Maier, J., Basanta, D., Weidow, B., Narasanna, A., Arteaga, C.L., Reynolds, A.B., Quaranta, V., Estrada, L., Weaver, A.M.: Microenvironmental Independence Associated with Tumor Progression. Cancer Res 69, 8797-8806(2009)
Anderson, A.R.A., Rejniak, K., Gerlee, P., Quaranta, V.: Microenvironment driven invasion: a multiscale multimodel investigation. J Math Biol 58, 579–624 (2009)
Anderson, A.R.A., Weaver, A.M., Cummings, P.T., Quaranta, V.: Tumor morphology and phenotypic evolution driven by selective pressure from the microenvironment. Cell 127, 905915 (2006)
Anderson, A.R.A., Quaranta, V.: Integrative mathematical oncology. Nat Rev Cancer 8, 227234 (2008)
Araujo, R., McElwain, D.: A history of the study of solid tumour growth: The contribution of mathematical modelling. Bull Math Biol 66, 1039–1091 (2004)
Attolini, C.S.O., Michor, F.: Evolutionary Theory of Cancer. Ann NY Acad Sci 1168, 23–51 (2009)
Axelrod, R., Axelrod, D.E., Pienta, K.J.: Evolution of cooperation among tumor cells. PNAS 103, 13474–13479 (2006) 10.1073/pnas.0606053103
Baker, N.E., Li, W.: Cell Competition and Its Possible Relation to Cancer. Cancer Res 68, 5505–5507 (2008)
Barabasi, A.L., Albert, R.: Emergence of Scaling in Random Networks. Science 286, 509–512 (1999)
Beerenwinkel, N., Antal, T., Dingli, D., Traulsen, A., Kinzler, K.W., Velculescu, V.E., Vogelstein, B., Nowak, M.A.: Genetic Progression and the Waiting Time to Cancer. PLoS Comput Biol 3, e225 (2007)
Bellomo, N., Li, N.K., Maini, P.K.: On the foundations of cancer modelling: Selected topics, speculations, and perspectives. Math Model Meth Appl Sci 18, 593–646 (2008)
Bidard, F.C., Pierga, J.Y., Vincent-Salomon, A., Poupon, M.-F.: A “class action” against the microenvironment: do cancer cells cooperate in metastasis? Cancer and Metastasis Rev 27, 5-10(2008)
Bock, M., Tyagi, A., Kreft, J.U., Alt, W.: Generalized Voronoi Tessellation as a Model of Two-dimensional Cell Tissue Dynamics. Bull Math Biol 72, 1696–1731 (2010)
Byrne, H.M.: Dissecting cancer through mathematics: from the cell to the animal model. Nat Rev Cancer 10, 221–230 (2010)
Cairns, J.: Mutation selection and the natural history of cancer. Nature 255, 197–200 (1975)
Cavaliere, M., Sedwards, S., Tarnita, C.E., Nowak, M.A., Csikasz-Nagy, A.: Prosperity is associated with instability in dynamical networks. J Theor Biol 299, 126–38 (2012)
Chaplain, M.A.J.: Multiscale mathematical modelling in biology and medicine. IMA J Appl Math 76, 371-388(2011)
Craig Maclean, R., Brandon, C.: Stable public goods cooperation and dynamic social interactions in yeast. J Evol Biol 21, 1836–1843 (2008)
Csikasz-Nagy, A.: Computational systems biology of the cell cycle. Brief Bioinform 10, 424434 (2009)
d’Onofrio, A.: A general framework for modeling tumor-immune system competition and immunotherapy: Mathematical analysis and biomedical inferences. Physica D 208, 220–235 (2005)
d’Onofrio, A., Gandolfi, A.: A family of models of angiogenesis and anti-angiogenesis anticancer therapy. Math Med Biol 26, 63–95 (2009)
d’Onofrio, A., Gatti, F., Cerrai, P., Freschi, L.: Delay-induced oscillatory dynamics of tumour-immune system interaction. Math Comp Modell 51, 572–591 (2010)
d’Onofrio, A., Tomlinson, I.P.M.: A nonlinear mathematical model of cell turnover, differentiation and tumorigenesis in the intestinal crypt. J Theor Biol 244, 367–374 (2007)
Deisboeck, T.S., Stamatakos, G.S.: Multiscale cancer modeling. CRC Press, Boca-Raton (2010)
Deutsch, A., Dormann, S.: Cellular automaton modeling of biological pattern formation: characterization, applications, and analysis. Birkhauser, Basel (2005)
Dingli, D., Chalub, F.A.C.C., Santos, F.C., Van Segbroeck, S., Pacheco, J.M.: Cancer pheno- type as the outcome of an evolutionary game between normal and malignant cells. Br J Cancer 101, 1130–1136 (2009)
Drasdo, D., Hoehme, S.: A single-cell-based model of tumor growth in vitro : monolayers and spheroids. Phys Biol 2, 133 (2005)
Drasdo, D., Hoehme, S., Block, M.: On the Role of Physics in the Growth and Pattern Formation of Multi-Cellular Systems: What can we Learn from Individual-Cell Based Models? J Stat Phys 128, 287–345 (2007)
Dubertret, B., Aste, T., Ohlenbusch, H.M., Rivier, N.: Two-dimensional froths and the dynamics of biological tissues. Phys Rev E 58, 6368 (1998)
Duvdevani-Bar, S., Segel, L.: On Topological Simulations in Developmental Biology. J Theor Biol 166, 33-50(1994)
Edelman, L.B., Eddy, J.A., Price, N.D.: In silico models of cancer. WIRE Syst Biol Med 2, 438-459(2010)
Enderling, H., Anderson, A.R.A., Chaplain, M.A.J., Beheshti H., Hlatky, L., Hahnfeldt, P.: Paradoxical Dependencies of Tumor Dormancy and Progression on Basic Cell Kinetics. Cancer Res 69, 8814–8821 (2009)
Enderling, H., Hahnfeldt, P.: Cancer stem cells in solid tumors: Is ‘evading apoptosis’ a hallmark of cancer? Prog Biophys Mol Biol 106, 391–399 (2011)
Escudero, L.M., da F. Costa, L., Kicheva, A., Briscoe, J., Freeman, M., Babu, M.M.: Epithelial organisation revealed by a network of cellular contacts. Nat Commun 2, 526 (2011)
Farhadifar, R., Röper, J.C., Aigouy, B., Eaton, S., Jülicher, F.: The influence of cell mechanics, cell-cell interactions, and proliferation on epithelial packing. Curr Biol 17, 2095–2104 (2007)
Fozard, J., Kirkham, G., Buttery, L., King, J., Jensen, O., Byrne, H.: Techniques for analysing pattern formation in populations of stem cells and their progeny. BMC Bioinf 12, 396 (2011)
Galle, J., Aust, G., Schaller, G., Beyer, T., Drasdo, D.: Individual cell-based models of the spatial-temporal organization of multicellular systems—Achievements and limitations. Cytometry A 69, 704–710 (2006)
Galle, J., Loeffler, M., Drasdo, D.: Modeling the Effect ofDeregulated Proliferation and Apoptosis on the Growth Dynamics of Epithelial Cell Populations In Vitro. Biophys J 88, 62–75 (2005)
Gatenby, R.A., Gillies, R.J.: A microenvironmental model of carcinogenesis. Nat Rev Cancer 8, 56–61 (2008)
Gatenby, R.A., Gillies, R.J., Brown, J.S.: The evolutionary dynamics of cancer prevention. Nat Rev Cancer 10, 526–527 (2010)
Gatenby, R.A., Vincent, T.L.: An Evolutionary Model Of Carcinogenesis. Cancer Res 63, 6212–6220 (2003)
Gibson, M.C., Patel, A.B., Nagpal, R., Perrimon, N.: The emergence of geometric order in proliferating metazoan epithelia. Nature 442, 1038–1041 (2006)
Gibson, W.T., Veldhuis, J.H., Rubinstein, B., Cartwright, H.N., Perrimon, N., Brodland, G.W., Nagpal, R., Gibson, M.C.: Control of the Mitotic Cleavage Plane by Local Epithelial Topology. Cell 144, 427–438 (2011)
Grant, M.R., Mostov, K.E., Tlsty, T.D., Hunt, C.A.: Simulating Properties of In Vitro Epithelial Cell Morphogenesis. PLoS Comput Biol 2, e129 (2006)
Greaves, M., Maley, C.C.: Clonal evolution in cancer. Nature 481, 306–313 (2012)
Gross, T., Blasius, B.: Adaptive coevolutionary networks: a review. J R Soc Interface 5, 259271 (2008)
Guillaud, M., Clem, C., MacAulay, C.: An in silico platform for the study of epithelial preinvasive neoplastic development. Biosystems 102, 22–31 (2010)
Hanahan, D., Weinberg, R.A.: The hallmarks of cancer. Cell 100, 57–70 (2000)
Hanahan, D., Weinberg, R.A.: Hallmarks of Cancer: The Next Generation. Cell 144, 646–674 (2011)
Hofbauer, J., Sigmund, K.: Evolutionary games and population dynamics. London Mathematical Society Student Texts. Cambridge University Press, Cambridge (1988)
Hofbauer, J., Sigmund, K.: Evolutionary game dynamics. Bull Am Math Soc 40, 479 (2003)
Honda, H., Tanemura, M., Nagai, T.: A three-dimensional vertex dynamics cell model of space-filling polyhedra simulating cell behavior in a cell aggregate. J Theor Biol 226, 439453 (2004)
Hsu, Y.C., Fuchs, E.: A family business: stem cell progeny join the niche to regulate homeostasis. Nat Rev Mol Cell Biol 13, 103–114 (2012)
Hu, M., Polyak, K.: Microenvironmental regulation of cancer development. Curr Opin Gen Dev 18, 27–34 (2008)
Hunter, T.: Signaling—2000 and Beyond. Cell 100, 113–127 (2000)
Izaguirre, J.A., Chaturvedi, R., Huang, C., Cickovski, T., Coffland, J., Thomas, G., Forgacs, G., Alber, M., Hentschel, G., Newman, S.A., Glazier, J.A.: CompuCell, a multi-model framework for simulation of morphogenesis. Bioinformatics 20, 1129–1137 (2004)
Jackson, M.O.: Social and Economic Networks. Princeton University Press, Princeton (2008)
Johnston, L.A.: Competitive Interactions Between Cells: Death, Growth, and Geography. Science 324, 1679–1682 (2009)
Johnston, M.D., Maini, P.K., Jonathan Chapman, S., Edwards, C.M., Bodmer, W.F.: On the proportion of cancer stem cells in a tumour. J Theor Biol 266, 708–711 (2010)
Kholodenko, B.N.: Cell-signalling dynamics in time and space. Nat Rev Mol Cell Biol 7, 165176 (2006)
Kim, S., Debnath, J., Mostov, K., Park, S., Hunt, C.A.: A computational approach to resolve cell level contributions to early glandular epithelial cancer progression. BMC Syst Biol 3, 122 (2009)
Lecuit, T., Lenne, P.-F.: Cell surface mechanics and the control of cell shape, tissue patterns and morphogenesis. Nat Rev Mol Cell Biol 8, 633–644 (2007)
Lewis, F.T.: The correlation between cell division and the shapes and sizes of prismatic cells in the epidermis of cucumis. Anat Rec 38, 341–376 (1928)
Lewis, J.: From Signals to Patterns: Space, Time, and Mathematics in Developmental Biology. Science 322, 399–403 (2008)
Lieberman, E., Hauert, C., Nowak, M.A.: Evolutionary dynamics on graphs. Nature 433, 312- 316(2005)
Liotta, L.A., Kohn, E.C.: The microenvironment of the tumour-host interface. Nature 411, 375-379(2001)
Mansury, Y., Diggory, M., Deisboeck, T.S.: Evolutionary game theory in an agent-based brain tumor model: Exploring the ‘Genotype-Phenotype’ link. J Theor Biol 238, 146–156 (2006)
Martin, F.A., Herrera, S.C., Morata, G.S.: Cell competition, growth and size control in the Drosophila wing imaginal disc. Development 136, 3747–3756 (2009)
Marusyk, A., Polyak, K.: Tumor heterogeneity: Causes and consequences. BBA-Rev Cancer 1805, 105-117(2010)
Maynard Smith, J.: The theory of games and the evolution of animal conflicts. J Theor Biol 47, 209–221 (1974)
Meinhardt, H.: The algorithmic beauty of sea shells. Springer-Verlag New York, Inc., (1995)
Merks, R.M.H., Glazier, J.A.: A cell-centered approach to developmental biology. Physica A 352, 113-130(2005)
Merks, R.M.H., Guravage, M., Inze, D., Beemster, G.T.S.: VirtualLeaf: an open-source framework for cell-based modeling of plant tissue growth and development. Plant Physiol 155, 656- 666(2011)
Merlo, L.M.F., Pepper, J.W., Reid, B.J., Maley, C.C.: Cancer as an evolutionary and ecological process. Nat Rev Cancer 6, 924–935 (2006)
Meyer-Hermann, M.: Delaunay-Object-Dynamics: Cell Mechanics with a 3D Kinetic and Dynamic Weighted Delaunay-Triangulation. Curr Top in Dev Biol 81, 373–399 (2008)
Michor, F., Iwasa, Y., Nowak, M.A.: Dynamics of cancer progression. Nat Rev Cancer 4, 197–205 (2004)
Moreira, J., Deutsch, A.: Cellular Automation Models of Tumor Development: A Critical Review. Adv Compl Sys 5, 247–268 (2002)
Moreno, E.: Is cell competition relevant to cancer? Nat Rev Cancer 8, 141–147 (2008)
Morrison, S.J., Spradling, A.C.: Stem Cells and Niches: Mechanisms That Promote Stem Cell Maintenance throughout Life. Cell 132, 598–611 (2008)
Murray, J.D.: Mathematical biology, vol. 2. Springer, (2002)
Nadell, C.D., Xavier, J.B., Foster, K.R.: The sociobiology of biofilms. FEMS Microbiol Rev 33, 206–224 (2009)
Nagpal, R., Patel, A., Gibson, M.C.: Epithelial topology. Bioessays 30, 260–266 (2008)
Nowak, M.A.: Evolutionary Dynamics: Exploring the Equations of Life. Harvard University Press, Cambridge, MA (2006)
Nowak, M.A., Michor, F., Komarova, N.L., Iwasa, Y.: Evolutionary dynamics of tumor suppressor gene inactivation. PNAS 101, 10635–10638 (2004)
Nowak, M.A., Sigmund, K.: Evolutionary Dynamics of Biological Games. Science 303, 793799 (2004)
Nowell, P.C.: The clonal evolution of tumor cell populations. Science 194, 23–28 (1976)
Ohtsuki, H., Hauert, C., Lieberman, E., Nowak, M.A.: A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505 (2006)
Ohtsuki, H., Nowak, M.A.: Evolutionary stability on graphs. J Theor Biol 251, 698–707 (2008)
Orr, A.W., Helmke, B.P., Blackman, B.R., Schwartz, M.A.: Mechanisms of Mechanotrans- duction. Develop Cell 10, 11–20 (2006)
Osborne, J.M., Walter, A., Kershaw, S.K., Mirams, G.R., Fletcher, A.G., Pathmanathan, P., Gavaghan, D., Jensen, O.E., Maini, P.K., Byrne, H.M.: A hybrid approach to multi-scale modelling of cancer. Phil Trans Roy Soc A 368, 5013–5028 (2010)
Pacheco, J.M., Traulsen, A., Nowak, M.A.: Coevolution of Strategy and Structure in Complex Networks with Dynamical Linking. Phys Rev Lett 97, 258103–258104 (2006)
Patel, A.B., Gibson, W.T., Gibson, M.C., Nagpal, R.: Modeling and Inferring Cleavage Patterns in Proliferating Epithelia. PLoS Comput Biol 5, e1000412 (2009)
Perc, M., Szolnoki, A.: Coevolutionary games-A mini review. Biosystems 99, 109–125 (2010)
Pienta, K.J., McGregor, N., Axelrod, R., Axelrod, D.E.: Ecological therapy for cancer: defining tumors using an ecosystem paradigm suggests new opportunities for novel cancer treatments. Transl Oncol 1, 158–164 (2008)
Polyak, K., Haviv, I., Campbell, I.G.: Co-evolution of tumor cells and their microenvironment. Trends Genet 25, 30–38 (2009)
Preziosi, L.: Cancer modelling and simulation, vol. 3. CRC Press, (2003)
Preziosi, L., Tosin, A.: Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J Math Biol 58, 625–656 (2009)
Prusinkiewicz, P., Lindenmayer, A., Hanan, J.: The algorithmic beauty of plants. Springer Verlag, (1990)
Rainey, P.B., Kerr, B.: Cheats as first propagules: A new hypothesis for the evolution of individuality during the transition from single cells to multicellularity. BioEssays 32, 872- 880(2010)
Rankin, D.J., Bargum, K., Kokko, H.: The tragedy of the commons in evolutionary biology. Trends Ecol Evol 22, 643–651 (2007)
Rejniak, K.A.: An immersed boundary framework for modelling the growth of individual cells: An application to the early tumour development. J Theor Biol 247, 186–204 (2007)
Rejniak, K.A., Anderson, A.R.A.: Hybrid models of tumor growth. WIRE Syst Biol Med 3, 115-125(2011)
Rosen, R.: The theory of graphs and its applications. Bull Math Biol 24, 441–443 (1962)
Rozenberg, G.: Handbook of graph grammars and computing by graph transformation: volume I: Foundations. World Scientific, River Edge, NJ, USA (1997)
Santorelli, L.A., Thompson, C.R.L., Villegas, E., Svetz, J., Dinh, C., Parikh, A., Sucgang, R., Kuspa, A., Strassmann, J.E., Queller, D.C., Shaulsky, G.: Facultative cheater mutants reveal the genetic complexity of cooperation in social amoebae. Nature 451, 1107–1110 (2008)
Santos, F.C., Pacheco, J.M.: Scale-Free Networks Provide a Unifying Framework for the Emergence of Cooperation. Phys Rev Letters 95, 098104 (2005)
Schaller, G., Meyer-Hermann, M.: Kinetic and dynamic Delaunay tetrahedralizations in three dimensions. Comp Phys Comm 162, 9–23 (2004)
Schrader, J., Deuster, O., Rinn, B., Schulz, M., Kautz, A., Dodel, R., Meyer, B., Al-Abed, Y., Balakrishnan, K., Reese, J., Bacher, M.: Restoration of contact inhibition in human glioblastoma cell lines after MIF knockdown. BMC Cancer 9, 464 (2009)
Seluanov, A., Hine, C., Azpurua, J., Feigenson, M., Bozzella, M., Mao, Z., Catania, K.C., Gorbunova, V.: Hypersensitivity to contact inhibition provides a clue to cancer resistance of naked mole-rat. PNAS 106, 19352–19357 (2009)
Shirinifard, A., Gens, J.S., Zaitlen, B.L., Poplawski, N.J., Swat, M., Glazier, J.A.: 3D Multi- Cell Simulation of Tumor Growth and Angiogenesis. PLoS ONE 4, e7190 (2009)
Shraiman, B.I.: Mechanical feedback as a possible regulator of tissue growth. PNAS 102, 3318-3323(2005)
Skyrms, B., Pemantle, R.: A dynamic model of social network formation. PNAS 97, 93409346 (2000)
Smallwood, R.: Computational modeling of epithelial tissues. WIRE Syst Biol Med 1, 191- 201(2009)
Smith, R.S., Guyomarc’h, S., Mandel, T., Reinhardt, D., Kuhlemeier, C., Prusinkiewicz, P.: A plausible model of phyllotaxis. PNAS 103, 1301–1306 (2006)
Spencer, S.L., Gerety, R.A., Pienta, K.J., Forrest, S.: Modeling Somatic Evolution in Tumori- genesis. PLoS Comput Biol 2, e108 (2006)
Szabo, G., Fath, G.: Evolutionary games on graphs. Phys Rep 446, 97–216 (2007)
Thorne, B.C., Bailey, A.M., Peirce, S.M.: Combining experiments with multi-cell agent- based modeling to study biological tissue patterning. Brief Bioinform 8, 245–257 (2007)
Tomlinson, I.P.M.: Game-theory models of interactions between tumour cells. Eur J Cancer 33, 1495-1500(1997)
Vogel, V., Sheetz, M.: Local force and geometry sensing regulate cell functions. Nat Rev Mol Cell Biol 7, 265–275 (2006)
Walker, D., Georgopoulos, N., Southgate, J.: From pathway to population - a multiscale model of juxtacrine EGFR-MAPK signalling. BMC Sys Biol 2, 102 (2008)
Wang, C.C., Jamal, L., Janes, K.A.: Normal morphogenesis of epithelial tissues and progression of epithelial tumors. WIRE Sys Biol Med 4, 51–78 (2011)
Wang, L.D., Wagers, A.J.: Dynamic niches in the origination and differentiation of haematopoietic stem cells. Nat Rev Mol Cell Biol 12, 643–655 (2011)
West, S.A., Griffin, A.S., Gardner, A.: Social semantics: altruism, cooperation, mutualism, strong reciprocity and group selection. J Evol Biol 20, 415–432 (2007)
Xavier, J.B.: Social interaction in synthetic and natural microbial communities. Mol Syst Biol 7, 483 (2011)
Zhang, L., Athale, C.A., Deisboeck, T.S.: Development of a three-dimensional multiscale agent-based tumor model: Simulating gene-protein interaction profiles, cell phenotypes and multicellular patterns in brain cancer. J Theor Biol 244, 96–107 (2007)
Zhang, L., Wang, Z., Sagotsky, J.A., Deisboeck, T.S.: Multiscale agent-based cancer modeling. J Math Biol 58, 545–559 (2009)
Acknowledgements
The authors are grateful to Peter Csermely for comments on the manuscript and acknowledge support from the Italian Research Fund FIRB (RBPR0523C3) and from program JAEDoc15 (“Programa junta para la ampliacion de estudios”).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Italia
About this chapter
Cite this chapter
Csikász-Nagy, A., Cavaliere, M., Sedwards, S. (2012). Combining Game Theory and Graph Theory to Model Interactions between Cells in the Tumor Microenvironment. In: d’Onofrio, A., Cerrai, P., Gandolfi, A. (eds) New Challenges for Cancer Systems Biomedicine. SIMAI Springer Series. Springer, Milano. https://doi.org/10.1007/978-88-470-2571-4_1
Download citation
DOI: https://doi.org/10.1007/978-88-470-2571-4_1
Publisher Name: Springer, Milano
Print ISBN: 978-88-470-2570-7
Online ISBN: 978-88-470-2571-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)