# Spatial Moran models, II: cancer initiation in spatially structured tissue

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## Abstract

We study the accumulation and spread of advantageous mutations in a spatial stochastic model of cancer initiation on a lattice. The parameters of this general model can be tuned to study a variety of cancer types and genetic progression pathways. This investigation contributes to an understanding of how the selective advantage of cancer cells together with the rates of mutations driving cancer, impact the process and timing of carcinogenesis. These results can be used to give insights into tumor heterogeneity and the “cancer field effect,” the observation that a malignancy is often surrounded by cells that have undergone premalignant transformation.

## Keywords

Biased voter model Shape theorem Asymptotics for waiting times Cancer field effect## Mathematics Subject Classification

60K35 92C50## Notes

### Acknowledgments

RD is partially supported by NIH grant 5R01GM096190, JF by NSF grants DMS-1224362 and DMS-1349724, and KL by NSF Grants DMS-1224362 and CMMI-1362236. We would like to thank Marc Ryser for helpful comments on previous versions of this paper.

## References

- Bozic I et al (2010) Accumulation of driver and passenger mutations during tumor progression. Proc Natl Acad Sci 107:18545–18550CrossRefGoogle Scholar
- Bramson M, Cox T, Le Gall J (2001) Super-Brownian Limits of Voter Model Clusters. Ann Probab 29:1001–1032MathSciNetzbMATHGoogle Scholar
- Bramson M, Griffeath D (1980a) Asymptotics for Interacting Particle Systems on \(\mathbb{Z}^d\). Z fur Wahr 53:183–196CrossRefMathSciNetzbMATHGoogle Scholar
- Bramson M, Griffeath D (1980b) On the Williams-Bjerknes tumour growth model. II Math Proc Cambridge Philos Soc 88:339–357CrossRefMathSciNetzbMATHGoogle Scholar
- Bramson M, Griffeath D (1981) On the Williams-Bjerknes tumour growth model. I Ann Probab 9:173–185CrossRefMathSciNetzbMATHGoogle Scholar
- Chatterjee S, Durrett R (2011) Asymptotic Behavior of Aldous’ Gossip Process. Ann Appl Probab 21:2447–2482CrossRefMathSciNetzbMATHGoogle Scholar
- Cox JT, Durrett R, Perkins EA (2000) Rescaled voter models converge to super-Brownian motion. Ann Probab 28:185–234CrossRefMathSciNetzbMATHGoogle Scholar
- Durrett R, Moseley S (2014) Spatial Moran models. I. tunneling in the neutral case. Ann Appl Probab (to appear)Google Scholar
- Durrett R, Zähle I (2007) On the width of hybrid zones. Stoch Proc Appl 117:1751–1763CrossRefzbMATHGoogle Scholar
- Durrett R (1995) Ten lectures on particle systems. In: St. Flour lecture notes. Lecture notes in math 1608. Springer-Verlag, New York, pp 97–201Google Scholar
- Durrett R (2008) Probability models for DNA sequence evolution. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Durrett R, Schmidt D (2008) Waiting for two mutations: with applications to regulatory sequence evolution and the limits of Darwinian evolution. Genetics 180:1501–1509CrossRefGoogle Scholar
- Ethier S, Kurtz T (1986) Markov processes: characterization and convergence. Wiley, New YorkCrossRefzbMATHGoogle Scholar
- Foo J, Leder K, Ryser MD (2014) Multifocality and recurrence risk: a quantitative model of field cancerization. J Theor Biol 355:170–184CrossRefMathSciNetGoogle Scholar
- Griffeath DS (1978) Additive and cancellative interacting particle systems. In: Lecture notes in mathematics, vol 724. Springer, New YorkGoogle Scholar
- Harris TE (1976) On a class of set-valued Markov processes. Ann Probab 4:175–194CrossRefzbMATHGoogle Scholar
- Jones S et al (2008) Comparative lesion sequencing provides insights into tumor evolution. Prov Natl Acad Sci 105:4283–4288CrossRefGoogle Scholar
- Kimura M, Weiss GH (1964) The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 49:561–576Google Scholar
- Komarova NL (2007) Spatial stochastic models of cancer initiation and progression. Bull Math Biol 68:1573–1599CrossRefMathSciNetGoogle Scholar
- Komarova NL (2013) Spatial stochastic models of cancer: fitness, migration, invasion. Math Biosci Eng 10:761–775CrossRefMathSciNetzbMATHGoogle Scholar
- Lieberman E, Hauert C, Nowak MA (2005) Evolutionary dynamics on graphs. Nature 433:312–316CrossRefGoogle Scholar
- Martens EA, Hallatschek O (2011) Interfering waves of adaptation promote spatial mixing. Genetics 189:1045–1060CrossRefGoogle Scholar
- Martens EA, Kostadinov R, Maley CC, Hallatschek O (2011) Spatial structure increases the waiting time for cancer. New J Phys 13:115014Google Scholar
- Maruyama T (1970) On the fixation probability of mutant genes in a subdivided population. Genet Res 15:221–225CrossRefMathSciNetGoogle Scholar
- Maruyama T (1974) A simple proof that certain quantities are independent of the geographical structure of population. Theor Pop Biol 5:148–154CrossRefGoogle Scholar
- Merle M (2008) Hitting probability of a distant point for the voter model started with a single 1. Ann Probab 36:807–861CrossRefMathSciNetzbMATHGoogle Scholar
- Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. Belknap Press, CambridgeGoogle Scholar
- Parzen E (1999) Stochastic processes. In: Classics in Applied Mathematics, vol 24. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (Reprint of the 1962 original)Google Scholar
- Revuz D, Yor M (1991) Continuous Martingales and Brownian motion. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Sprouffske K, Merlo L, Gerrish P, Maley C, Sniegowski P (2012) Cancer in light of experimental evolution. Curr Biol 22:R762–R771CrossRefGoogle Scholar
- Weinberg R (2013) The biology of cancer. Garland Science, New YorkGoogle Scholar
- Williams T, Bjerknes R (1972) Stochastic model for abnormal clone spread through epithelial basal layer. Nature 235:19–21CrossRefGoogle Scholar
- Wodarz D, Komarova NL (2014) Dynamics of cancer: mathematical foundations of oncology. World Scientific Publishing Company, SingaporeCrossRefGoogle Scholar