Branching Process Models of Cancer

These notes originated as part of a lecture series on Stochastics in Biological Systems at the Mathematical Biosciences Institute in Ohio, USA.  In this contribution the author uses multitype branching processes with mutation to model cancer. With cancer progression, resistance to therapy, the time of the first type $k$ mutation, and $sigma_k$, the time of the first type $k$ mutation that founds a family line that does not die out, as well as the growth of the number of type $k$ cells. The last three sections apply these results to metastasis, ovarian cancer, and tumor heterogeneity.  Even though martingales and stable laws are mentioned, these notes with examples and applications should be accessible to students and researchers who are familiar with Poisson processes and continuous time Markov chains.

 Richard Durrett is Professor of Mathematics at Duke University, USA. He is the author of 8 books, over 200 journal articles, and has supervised more than 40 Ph.D. students. Most of his current research concerns the applications of probability to biology, ecology, genetics, and most recently cancer.

This volume develops results on continuous time branching processes and applies them to study rate of tumor growth, extending classic work on the Luria-Delbruck distribution. As a consequence, the author calculate the probability that mutations that confer resistance to treatment are present at detection and quantify the extent of tumor heterogeneity. As applications, the author evaluate ovarian cancer screening strategies and give rigorous proofs for results of Heano and Michor concerning tumor metastasis. These notes should be accessible to students who are familiar with Poisson processes and continuous time Markov chains.

Richard Durrett is a mathematics professor at Duke University, USA. He is the author of 8 books, over 200 journal articles, and has supervised more than 40 Ph.D students. Most of his current research concerns the applications of probability to biology: ecology, genetics and most recently cancer.

Multistage Theory of Cancer.- Mathematical Overview.- Branching Process Results.- Time for Z_0 to Reach Size M.- Time Until the First Type 1.- Mutation Before Detection?.- Accumulation of Neutral Mutations.- Properties of the Gamma Function.- Growth of Z_1(t).- Movements of Z_1(t).- Luria-Delbruck Distributions.- Number of Type 1's at Time T_M.- Gwoth of Z_k(t).- Transitions Between Waves.- Time to the First Type au_k, k ge 2.- Application: Metastasis.- Application: Ovarian Cancer.- Application: Intratumor Heterogeneity.