# Game Theoretical Model of Cancer Dynamics with Four Cell Phenotypes

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Classical Game Theory

**Theorem**

**1.**

## 3. Replicator Dynamics

## 4. Spatial Model

- a cell is chosen to be replaced by one of its neighbors weighted by the mortality propensity; then
- one neighboring cell is chosen to replace the removed cell weighted by the reproduction propensity.

## 5. Discussion

## Supplementary Materials

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Proof of Theorem 1

**Proof.**

- {A−, P, A+}: If $g>0$, then the strategy P strictly dominates A−, eliminating that strategy from the playable set. Thus a rock-paper-scissors cycle cannot exist.
- {A−, P, C}: If $g>0$ and $c<1$, then strategy P strictly dominates A−, thus eliminating that strategy from the playable set and preventing a rock-paper-scissors cycle.If instead $c>1$ then one strategy emerges as the best response to two different strategies. When $g+b<e$, the cytotoxic strategy C is the best response to both A− and P, while P is the best response to A− and itself when $g+b>e$. The former scenario generates either the Nash equilibrum (A−,C) or (C,C), depending upon the comparative sizes of b and c. The latter scenario guarantees the Nash equilibrium (P,P). Both instances preclude potential cycles of best response of length 3.
- A−→ A+ → C → A−: The first best response (A−→ A+) requires $d>a$ for A+ to be the best response to A−; however, the third best response statement (C → A−) requires the opposite condition, $a>d$. The parametric contradiction precludes this cycle.
- A−→ C → A+ → A−: The first best response (A−→ C) requires $e>b$ for C to be the best response to A−; however, the third best response statement (A+ → A−) requires $b>e$. The parametric contradiction precludes this cycle.
- A+ → C → P → A+: The second best response (C → P) requires $a+g(1-c)>d$ for P to be the best response to C; however, the third best response statement (P → A+) requires $d>a+g$. Because we assume that $c>0$, these two conditions cannot be simultaneously satisfied as $a+g>a+g(1-c)$. The parametric contradiction precludes this cycle.
- A+ → P → C → A+: The first best response (A+ → P) requires $g+b>e$ for P to be the best response to A+; however, the second best response statement (P → C) requires $e>g+b$. The parametric contradiction precludes this potential cycle.

## References

- Bussard, K.D.; Mutkus, L.; Stumpf, K.; Gomez-Manzano, C.; Marini, F.C. Tumor-associated stromal cells as key contributors to the tumor microenvironment. Breast Cancer Res.
**2016**, 18, 84. [Google Scholar] [CrossRef] [PubMed] - Basanta, D.; Hatzikirou, H.; Deutsch, A. Studying the emergence of invasiveness in tumours using game theory. Eur. Phys. J. B
**2008**, 63, 393–397. [Google Scholar] [CrossRef][Green Version] - Crespi, B.; Summer, K. Evolutionary biology of cancer. Trends Ecol. Evol.
**2005**, 20, 545–552. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gerlinger, M. Intratumor heterogeneity and branched evolution revealed by multiregion sequencing. N. Engl. J. Med.
**2012**, 366, 883–892. [Google Scholar] [CrossRef] [PubMed] - Navin, N. Tumour evolution inferred by single-cell sequencing. Nature
**2011**, 472, 90–94. [Google Scholar] [CrossRef] [PubMed][Green Version] - Attolini, C.S.O.; Michor, F. Evolutionary theory of cancer. Ann. N. Y. Acad. Sci.
**2009**, 1168, 23–51. [Google Scholar] [CrossRef] [PubMed] - Basanta, D.; Scott, J.G.; Fishman, M.N.; Ayala, G.; Hayward, S.W.; Anderson, A.R.A. Investigating prostate cancer tumour-stroma interactions: Clinical and biological insights from an evolutionary game. Br. J. Cancer
**2012**, 106, 174–181. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hanahan, D.; Weinberg, R.A. Hallmarks of cancer: The next generation. Cell
**2016**, 144, 646–674. [Google Scholar] [CrossRef] [PubMed] - Mao, Y.; Keller, E.T.; Garfield, D.H.; Shen, K.; Wang, J. Stroma cells in tumor microenvironment and breast cancer. Cancer Metastasis Rev.
**2013**, 32, 303–315. [Google Scholar] [CrossRef] [PubMed] - Basanta, D.; Gatenby, R.A.; Anderson, A.R.A. Exploiting evolution to treat drug resistance: Combination therapy and the double blind. Mol. Pharm.
**2012**, 9, 914–921. [Google Scholar] [CrossRef] [PubMed] - Nowell, P.C. The clonal evolution of tumor cell populations. Science
**1976**, 194, 23–28. [Google Scholar] [CrossRef] [PubMed] - Merlo, L.; Pepper, J.; Reid, B.; Maley, C. Cancer as an evolutionary and ecological process. Nat. Rev. Cancer
**2006**, 6, 924–935. [Google Scholar] [CrossRef] [PubMed] - Hanahan, D.; Weinberg, R.A. The hallmarks of cancer. Cell
**2000**, 100, 57–70. [Google Scholar] [CrossRef] - Negrini, S.; Gorgoulis, V.G.; Halazonetis, T.D. Genomic instability—An evolving hallmark of cancer. Nat. Rev. Mol. Cell Biol.
**2010**, 11, 220–228. [Google Scholar] [CrossRef] [PubMed] - Colotta, F.; Allavena, P.; Sica, A.; Garlanda, C.; Mantovani, A. Cancer-related inflammation, the seventh hallmark of cancer: Links to genetic instability. Carcinogenesis
**2009**, 30, 51–57. [Google Scholar] [CrossRef] [PubMed] - Pavlova, N.N.; Thompson, C.B. The emerging hallmarks of cancer metabolism. Cell Metab.
**2016**, 23, 27–47. [Google Scholar] [CrossRef] [PubMed] - Byrne, H.M. Dissecting cancer through mathematics: From the cell to the animal model. Nature
**2010**, 10, 221–230. [Google Scholar] [CrossRef] [PubMed] - Egeblad, M.; Nakasone, E.S.; Werb, Z. Tumors as organs: Complex tissues that interface with the entire organism. Dev. Cell
**2010**, 18, 884–901. [Google Scholar] [CrossRef] [PubMed] - Orlando, P.A.; Gatenby, R.A.; Brown, J.S. Cancer treatment as a game: Integrating evolutionary game theory into the optimal control of chemotherapy. J. Theor. Biol.
**2012**, 243, 065007. [Google Scholar] [CrossRef] [PubMed] - Pietras, K.; Östman, A. Hallmarks of cancer: Interactions with the tumor stroma. Exp. Cell Res.
**2010**, 316, 1324–1331. [Google Scholar] [CrossRef] [PubMed] - Dvorak, H.F. Tumors: Wounds that do not heal. Similarities between tumor stroma generation and wound healing. N. Engl. J. Med.
**1986**, 315, 1650–1659. [Google Scholar] [PubMed] - Bremnes, R.M.; Dønnem, T.; Al-Saad, S.; Al-Shibli, K.; Andersen, S.; Sirera, R.; Camps, C.; Marinez, I.; Busund, L.T. The role of tumor stroma in cancer progression and prognosis: Emphasis on carcinoma-associated fibroblasts and non-small cell lung cancer. J. Thorac. Oncol.
**2011**, 6, 209–217. [Google Scholar] [CrossRef] [PubMed] - Ohlund, D.; Elyada, E.; Tuveson, D. Fibroblast heterogeneity in the cancer wound. J. Exp. Med.
**2014**, 211, 1503–1523. [Google Scholar] [CrossRef] [PubMed][Green Version] - Marusyk, A.; Polyak, K. Tumour heterogeneity: Causes and consequences. Biochim. Biophys. Acta
**2010**, 1805, 105–117. [Google Scholar] [PubMed] - Archetti, M. Evolutionary game theory of growth factor production: Implications for tumour heterogeneity and resistance to therapies. Br. J. Cancer
**2013**, 109, 1056–1062. [Google Scholar] [CrossRef] [PubMed][Green Version] - Maynard-Smith, J.; Price, G.R. The logic of animal conflict. Nature
**1973**, 246, 16–18. [Google Scholar] - Maynard-Smith, J. Evolution and the Theory of Games, 1st ed.; Cambridge University Press: Cambridge, UK, 1982. [Google Scholar]
- Basanta, D.; Deutsch, A. A game theoretical perspective on the somatic evolution of cancer. Sel. Top. Cancer Model.
**2008**, 63, 393–397. [Google Scholar] - Tomlinson, I.P.M.; Bodmer, W.F. Modelling the consequences of interactions between tumour cells. Br. J. Cancer
**1997**, 75, 157. [Google Scholar] [CrossRef] [PubMed] - 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.
**2006**, 238, 146–156. [Google Scholar] [CrossRef] [PubMed] - Tomlinson, I.P.M. Game-theory models of interactions between tumour cells. Eur. J. Cancer
**1997**, 33, 1495–1500. [Google Scholar] [CrossRef] - Gatenby, R.A.; Gillies, R.J. Why do cancers have high aerobic glycolysis? Nat. Rev. Cancer
**2004**, 4, 891–899. [Google Scholar] [CrossRef] - Basanta, D.; Scott, J.G.; Rockne, R.; Swanson, K.R.; Anderson, A.R. The role of IDH1 mutated tumour cells in secondary glioblastomas: An evolutionary game theoretical view. Phys. Biol.
**2011**, 8, 015016. [Google Scholar] [CrossRef] [PubMed] - Gatenby, R.A.; Vincent, T.L. An evolutionary model of carcinogenesis. Cancer Res.
**2003**, 63, 6212–6220. [Google Scholar] [PubMed] - Dingli, D.; Chalub, F.A.C.C.; Santos, F.C.; Van Segbroeck, S.; Pacheco, J.M. Cancer phenotype as the outcome of an evolutionary game between normal and malignant cells. Br. J. Cancer
**2009**, 101, 1130–1136. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nowak, M.A.; May, R.M.; Sigmund, G. Evolutionary games and spatial chaos. Nature
**1992**, 359, 826–829. [Google Scholar] [CrossRef] - Nowak, M.A.; May, R.M. The spatial dilemmas of evolution. Int. J. Bifurc. Chaos
**1993**, 3, 35–78. [Google Scholar] [CrossRef] - Bach, L.A.; Sumpter, D.J.T.; Alsnere, J.; Loeschckeb, V. Spatial evolutionary games of interaction among generic cancer cells. J. Theor. Med.
**2003**, 5, 47–58. [Google Scholar] [CrossRef] - Swierniak, A.; Krzeslak, M.; Student, S.; Rzeszowska-Wolny, J. Development of a population of cancer cells: Observation and modeling by a Mixed Spatial Evolutionary Games approach. J. Theor. Biol.
**2016**, 405, 94–103. [Google Scholar] [CrossRef] - Hofbauer, J.; Sigmund, K. Evolutionary Games and Population Dynamics; Cambridge University Press: Cambridge, UK, 1998. [Google Scholar]
- Capcarrere, M.S. Evolution of Asynchronous Cellular Automata. In Parallel Problem Solving from Nature—PPSN VII. PPSN 2002. Lecture Notes in Computer Science; Guervós, J.J.M., Adamidis, P., Beyer, H.G., Schwefel, H.P., Fernández-Villacañas, J.L., Eds.; Springer: Berlin/Heidelberg, Germany, 2002; Volume 2439, pp. 903–912. [Google Scholar]
- Folkman, J. Tumor angiogenesis factor. Cancer Res.
**1974**, 34, 2109–2113. [Google Scholar] [PubMed] - Bach, L.A.; Bentzen, S.; Alsner, J.; Christiansen, F.B. An evolutionary-game model of tumour–cell interactions: Possible relevance to gene therapy. Eur. J. Cancer
**2001**, 37, 2116–2120. [Google Scholar] [CrossRef] - Staňková, K.; Brown, J.S.; Dalton, W.S.; Gatenby, R.A. Optimizing cancer treatment using game theory: A review. JAMA Oncol.
**2018**. [Google Scholar] [CrossRef] [PubMed] - Archetti, M. Cooperation among cancer cells as public goods games on Voronoi networks. J. Theor. Biol.
**2016**, 396, 191–203. [Google Scholar] [CrossRef] [PubMed][Green Version] - You, L.; Brown, J.S.; Thuijsman, F.; Cunningham, J.J.; Gatenby, R.A.; Zhang, J.; Staňková, K. Spatial vs. non-spatial eco-evolutionary dynamics in a tumor growth model. J. Theor. Biol.
**2017**, 435, 78–97. [Google Scholar] [CrossRef] [PubMed] - Gillies, R.J.; Verduzco, D.; Gatenby, R.A. Evolutionary dynamics of carcinogenesis and why targeted therapy does not work. Nat. Rev. Cancer
**2012**, 12, 487–493. [Google Scholar] [CrossRef] [PubMed][Green Version] - Swanton, C. Intratumor heterogeneity: Evolution through space and time. Cancer Res.
**2012**, 72, 4876–4882. [Google Scholar] [CrossRef] [PubMed] - Kreso, A.; O’brien, C.A.; Van Galen, P.; Gan, O.I.; Notta, F.; Brown, A.M.; Ng, K.; Ma, J.; Wienholds, E.; Dunant, C.; et al. Variable clonal repopulation dynamics influence chemotherapy response in colorectal cancer. Science
**2013**, 339, 543–548. [Google Scholar] [CrossRef] [PubMed] - Gupta, G.P.; Minn, A.J.; Kang, Y.; Siegel, P.M.; Serganova, I.; Cordón-Cardo, C.; Olshen, A.B.; Gerald, W.L.; Massagué, J. Identifying site-specific metastatis genese and functions. Cold Spring Harb. Symp. Quant. Biol.
**2005**, 70, 149–158. [Google Scholar] [CrossRef] [PubMed] - Buonomo, B.; Della Marca, R.; d’Onofrio, A. Optimal public health intervention in a behavioural vaccination model: The interplay between seasonality, behaviour and latency period. Math. Med. Biol. J. IMA
**2018**, dqy011. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Parametric regions for pure Nash equilibria. Each parametric region admits a limited number of pure Nash equilibria. Panels (

**a**–

**d**) show the equilibria for games with strategies A−, A+, and C, while (

**e**,

**f**) use A+, C, and P. (

**a**) $b>c$, e, cytotoxic cells excluded; (

**b**) $b<c$, e. A− cells excluded; (

**c**) $c<b<e$, A− and C are naturally polymorphic with one another; (

**d**) $e<b<c$, A− and C are exclusionary in isolation; (

**e**) $(e-b)<g<(c-b)/(1-c)$ and $g>(c-b)$, P and C are exclusionary in isolation; (

**f**) $(c-b)/(1-c)<g<(e-b)$ and $g(1-c)<(e-b)$, P and C are naturally polymorphic with one another.

**Figure 2.**(

**a**) Replicator dynamics in discrete time where proliferative cell dominates other strategies. Parameters: a, b, c, d, e, f, $g=0.1$; (

**b**) Replicator dynamics in continuous time where proliferative cell dominates other strategies. Parameters: $a=0.02$, $b=0.02$, $c=0.11$, $d=0.1$, $e=0.1$, $f=0.1$, $g=0.15$.

**Figure 3.**Equilibria between phenotypes under continuous replicator dynamics. In all figures: $a=0.02$, $b=0.04$. (

**a**) Cytotoxic and proliferative equilibrium. Parameters: $c=0.08$, $d=0.06$, $e=0.15$, $f=0.1$, $g=0.06$; (

**b**) Angiogenic and cytotoxic equilibrium. Parameters: $c=0.08$, $d=0.1$, $e=0.15$, $f=0.1$, $g=0.05$; (

**c**) Proliferative and angiogenic equilibrium. Parameters: $c=0.01$, $d=0.1$, $e=0.06$, $f=0.05$, $g=0.06$; (

**d**) Angiogenic, cytotoxic, and proliferative equilibrium. Parameters: $c=0.01$, $d=0.1$, $e=0.12$, $f=0.05$, $g=0.05$.

**Figure 4.**The evolution of the spatial game with A+, P, and C cells coexisting in equilibrium. Color code: white A−, green A+, blue P, red C. Parameters: $a=b=c=d=f=0.1$, $e=0.01$, $g=0.3$. (

**a**) Initial environment with 95% A− cells; (

**b**–

**f**) Snapshot of the environment after every 200,000 steps.

**Figure 5.**Final phenotype frequency distribution in the spatial game for various parameter combinations. All parameters except the focal one are fixed at the following values: $a=b=c=d=f=0.1$, $e=0.01$, $g=0.3$. (

**a**) cost of exposure to cytotoxins; (

**b**) exploitation benefit for cytotoxic cells; (

**c**) synergistic activity among angiogenesis-factor producing cells; and (

**d**) proliferative cell reproductive advantage.

Parameter | Interpretation |
---|---|

a | Cost of producing angiogenesis factors |

b | Cost of producing cytotoxin |

c | Cost of interaction with cytotoxin |

d | Resource benefit when interacting with A+ |

e | Exploitation benefit for C when cytotoxin damages others |

f | Synergistic resource benefit when two A+ cells interact |

g | Reproductive advantage of P cell |

Player\Opponent | A− | A+ | P | C |
---|---|---|---|---|

A− | 1 | $1+d$ | 1 | $1-c$ |

A+ | $1-a+d$ | $1-a+d+f$ | $1-a+d$ | $1-c-a+d$ |

P | $1+g$ | $1+d+g$ | $1+g$ | $(1+g)(1-c)$ |

C | $1-b+e$ | $1-b+d+e$ | $1-b+e$ | $1-b$ |

**Table 3.**Pairwise contrast of strategies. Each pair of strategies is compared in a reduced two-strategy game, with the stated outcome given by the listed condition(s). If there are additional conditions that arise for strict dominance when all four strategies are available, these are listed under “Greater Context”.

Contrast | Outcome | Condition | Greater Context |
---|---|---|---|

A+ vs. A− | A+ dominates | f, $d>a$ | None |

A− dominates | f, $d<a$ | None | |

Exclusion | $d<a<f$ | – | |

Sympatry | $f<a<d$ | – | |

A+ vs. P | A+ dominates | f, $d>a+g$ | None |

P dominates | f, $d<a+g$ | $d<a+g(1-c)$ | |

Exclusion | $d<a+g<f$ | – | |

Sympatry | $f<a+g<d$ | – | |

A+ vs. C | A+ dominates | $f>a+(e-b)$, $d>a+(c-b)$ | $d>a+(e-b)$ |

C dominates | $f<a+(e-b)$, $d<a+(c-b)$ | $d<a+(e-b)$ | |

Exclusion | $f>a+(e-b)$, $d<a+(c-b)$ | – | |

Sympatry | $f<a+(e-b)$, $d>a+(c-b)$ | – | |

A− vs. P | A− dominates | $g<0$ | $c<1$ |

P dominates | $g>0$ | $c<1$ | |

Exclusion | N/A | – | |

Sympatry | $g=0$ | – | |

A− vs. C | A− dominates | $b>c$, e | None |

C dominates | $b<c$, e | None | |

Exclusion | $e<b<c$ | – | |

Sympatry | $c<b<e$ | – | |

P vs. C | P dominates | $b+g>c(1+g)$, e | None |

C dominates | $b+g<c(1+g)$, e | None | |

Exclusion | $e<b+g<c(1+g)$ | – | |

Sympatry | $c(1+g)<b+g<e$ | – |

**Table 4.**Mortality and reproduction propensities in the spatial multiplayer game. Notation: T is the number of C neighbors, A is the number of A+ neighbors.

Phenotype | Mortality Propensity | Reproduction Propensity with No A+ Neighbors | Reproduction Propensity with at Least One A+ Neighbor |
---|---|---|---|

A− | $1+Tc$ | 1 | $1+d$ |

A+ | $1+Tc$ | $1-a+d$ | $1-a+d+Af$ |

P | $1+Tc(1+g)$ | $1+g$ | $1+g+d$ |

C | 1 | $1-b+(8-T)e$ | $1-b+(8-T)e+d$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hurlbut, E.; Ortega, E.; Erovenko, I.V.; Rowell, J.T. Game Theoretical Model of Cancer Dynamics with Four Cell Phenotypes. *Games* **2018**, *9*, 61.
https://doi.org/10.3390/g9030061

**AMA Style**

Hurlbut E, Ortega E, Erovenko IV, Rowell JT. Game Theoretical Model of Cancer Dynamics with Four Cell Phenotypes. *Games*. 2018; 9(3):61.
https://doi.org/10.3390/g9030061

**Chicago/Turabian Style**

Hurlbut, Elena, Ethan Ortega, Igor V. Erovenko, and Jonathan T. Rowell. 2018. "Game Theoretical Model of Cancer Dynamics with Four Cell Phenotypes" *Games* 9, no. 3: 61.
https://doi.org/10.3390/g9030061