## Abstract

In this work, we consider the problem of estimating the probability for a specific random genetic mutation to be present in a tumor of a given size. Previous mathematical models have been based on stochastic methods where the tumor was assumed to be homogeneous and, on average, growing exponentially. In contrast, we are able to obtain analytical results for cases where the exponential growth of cancer has been replaced by other, arguably more realistic types of growth of a heterogeneous tumor cell population. Our main result is that the probability that a given random mutation will be present by the time a tumor reaches a certain size, is independent of the type of curve assumed for the average growth of the tumor, at least for a general class of growth curves. The same is true for the related estimate of the expected number of mutants present in a tumor of a given size, if mutants are indeed present.

Original language | English (US) |
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Pages (from-to) | 1379-1395 |

Number of pages | 17 |

Journal | Bulletin of Mathematical Biology |

Volume | 74 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2012 |

Externally published | Yes |

## Keywords

- Branching processes
- Drug resistance
- Genetic mutations
- Ordinary differential equations
- Stem cells
- Tumor growth

## ASJC Scopus subject areas

- Neuroscience(all)
- Immunology
- Mathematics(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Environmental Science(all)
- Pharmacology
- Agricultural and Biological Sciences(all)
- Computational Theory and Mathematics