# A simple algebraic cancer equation: calculating how cancers may arise with normal mutation rates

Peter Calabrese1 and Darryl Shibata2*

Author Affiliations

1 Program in Molecular and Computational Biology, Department of Biological Sciences, University of Southern California, Los Angeles, CA 90089, USA

2 Departments of Pathology, University of Southern California Keck School of Medicine, Los Angeles, CA 90033, USA

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BMC Cancer 2010, 10:3  doi:10.1186/1471-2407-10-3

Published: 5 January 2010

### Abstract

#### Background

The purpose of this article is to present a relatively easy to understand cancer model where transformation occurs when the first cell, among many at risk within a colon, accumulates a set of driver mutations. The analysis of this model yields a simple algebraic equation, which takes as inputs the number of stem cells, mutation and division rates, and the number of driver mutations, and makes predictions about cancer epidemiology.

#### Methods

The equation [p = 1 - (1 - (1 - (1 - u)d)k)Nm ] calculates the probability of cancer (p) and contains five parameters: the number of divisions (d), the number of stem cells (N × m), the number of critical rate-limiting pathway driver mutations (k), and the mutation rate (u). In this model progression to cancer "starts" at conception and mutations accumulate with cell division. Transformation occurs when a critical number of rate-limiting pathway mutations first accumulates within a single stem cell.

#### Results

When applied to several colorectal cancer data sets, parameter values consistent with crypt stem cell biology and normal mutation rates were able to match the increase in cancer with aging, and the mutation frequencies found in cancer genomes. The equation can help explain how cancer risks may vary with age, height, germline mutations, and aspirin use. APC mutations may shorten pathways to cancer by effectively increasing the numbers of stem cells at risk.

#### Conclusions

The equation illustrates that age-related increases in cancer frequencies may result from relatively normal division and mutation rates. Although this equation does not encompass all of the known complexity of cancer, it may be useful, especially in a teaching setting, to help illustrate relationships between small and large cancer features.