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A Hierarchical Probability Model of Colon Cancer

Part of: Markov processes

Published online by Cambridge University Press:  04 January 2016

Michael Kelly
Michael Kelly*
Affiliation:
University of California, San Diego
*
∗ Current address: School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. SE, Minneapolis, MN 55455, USA. Email address: mbkelly@math.ucsd.edu
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Abstract

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We consider a model of fixed size N = 2l in which there are l generations of daughter cells and a stem cell. In each generation i there are 2i−1 daughter cells. At each integral time unit the cells split so that the stem cell splits into a stem cell and generation 1 daughter cell and the generation i daughter cells become two cells of generation i+1. The last generation is removed from the population. A stem cell acquires first and second mutations at rates u1 and u2, and a daughter cell acquires first and second mutations at rates v1 and v2. We find the distribution for the time it takes to acquire two mutations as N goes to ∞ and the mutation rates go to 0. The mutation rates may tend to 0 at different speeds. We also find the distribution for the locations of the mutations. In particular, we determine whether or not the mutations occur on a stem cell and if not, at what generation in the daughter cells they occur. Several outcomes are possible, depending on how fast the rates go to 0. The model considered has been proposed by Komarova (2007) as a model for colon cancer.

Keywords

CancermutationPoisson processpopulation model

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 92D99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 4 , December 2012 , pp. 1052 - 1083
Copyright
© Applied Probability Trust 

References

Armitage, P. and Doll, R. (1954). The age distribution of cancer and a multi-stage theory of carcinogenesis. Br. J. Cancer 8, 1–12.CrossRefGoogle Scholar
Cairns, J. (2006). Cancer and the immortal strand hypothesis. Genetics 174, 1069–1072.Google Scholar
Dingli, D. and Michor, F. (2006). Successful therapy must eradicate cancer stem cells. Stem Cells 24, 2603–2610.Google Scholar
Durrett, R. and Moseley, S. (2010). A spatial model for tumor growth. Preprint.Google Scholar
Durrett, R., Schmidt, D. and Schweinsberg, J. (2009). A waiting time problem arising from the study of multi-stage carcinogenesis. Ann. Appl. Prob. 19, 676–718.Google Scholar
Frank, S. A., Iwasa, Y. and Nowak, M. A. (2003). Patterns of cell divisions and the risk of cancer. Genetics 163, 1527–1532.CrossRefGoogle ScholarPubMed
Fristedt, B. and Gray, L. (1997). A Modern Approach to Probability Theory. Birkhäuser, Boston, MA.CrossRefGoogle Scholar
Iwasa, Y., Michor, F., Komarova, N. L. and Nowak, M. A. (2005). Population genetics of tumor suppressor genes. J. Theoret. Biol. 233, 15–23.Google Scholar
Kingman, J. F. C. (1993). Poisson Processes. Oxford University Press, New York.Google Scholar
Knudson, A. G. Jr. (1971). Mutation and cancer: statistical study of retinoblastoma. Proc. Nat. Acad. Sci. USA 68, 820–823.Google Scholar
Knudson, A. G. Jr. (1978). Retinoblastoma: a prototypic hereditary neoplasm. Semin. Oncol. 5, 57–60.Google ScholarPubMed
Komarova, N. L. (2007). Loss- and gain-of-function mutations in cancer: mass-action, spatial and hierarchical models. J. Statist. Phys. 128, 413–446.Google Scholar
Komarova, N. L. and Cheng, P. (2006). Epithelial tissue architecture protects against cancer. Math. Biosci. 200, 90–117.CrossRefGoogle ScholarPubMed
Komarova, N. L. and Wang, L. (2004). Initiation of colorectal cancer: where do the two hits hit? Cell Cycle 3, 1558–1565.Google Scholar
Komarova, N. L., Sengupta, A. and Nowak, M. A. (2003). Mutation-selection networks of cancer initiation: tumor suppressor genes and chromosomal instability. J. Theoret. Biol. 233, 433–450.CrossRefGoogle Scholar
Michor, F. (2007). Chronic myeloid leukemia blast crises arises from progenitors. Stem Cells 25, 1114–1118.CrossRefGoogle ScholarPubMed
Nowak, M. A. (2006). Evolutionary Dynamics: Exploring the Equations of Life. Belknap Press, Cambridge, MA.Google Scholar
Schweinsberg, J. (2008). The waiting time for m mutations. Electron. J. Prob. 13, 1442–1478.CrossRefGoogle Scholar
Wodarz, D. and Komarova, N. L. (2005). Computational Biology of Cancer: Lecture Notes and Mathematical Modeling. World Scientific Publishing, London.Google Scholar