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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: [email protected]
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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 

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