International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 16, Pages 2655-2667

Exact representations for tumour incidence for some density-dependent models

P. R. Parthasarathy1 and Klaus Dietz2

1Department of Mathematics, Indian Institute of Technology, Madras, Chennai 600036, India
2Department of Medical Biometry, University of Tuebingen, Tuebingen 72070, Germany

Received 10 March 2005

Copyright © 2005 P. R. Parthasarathy and Klaus Dietz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Carcinogenesis is a multistage random process involving generic changes and stochastic proliferation and differentiation of normal cells and genetically altered stem cells. In this paper, we present the probability of time to tumour onset for a carcinogenesis model wherein the cells grow according to a birth and death process with density-dependent birth and death rates. This is achieved by transforming the underlying system of difference equations which results in a continued fraction. This continued fraction approach helps us to find the complete solutions. The popular Moolgavkar-Venzon-Knudson (MVK) model assumes constant birth, death, and transition rates.