Exact solution of a two-type branching process: Models of tumor progression

TL;DR

This paper provides an exact solution for a two-type birth-death branching process modeling tumor progression, capturing mutation dynamics and their impact on tumor evolution over time.

Contribution

It introduces an explicit solution and generating function for a general two-type process with mutation, including novel large-time scaling limits and distribution properties.

Findings

Exact generating function derived for the process

Distribution of mutants exhibits power law tail in certain limits

Large time scaling limits reveal diverging averages and atypical behaviors

Abstract

An explicit solution for a general two-type birth-death branching process with one way mutation is presented. This continuous time process mimics the evolution of resistance to treatment, or the onset of an extra driver mutation during tumor progression. We obtain the exact generating function of the process at arbitrary times, and derive various large time scaling limits. In the simultaneous small mutation rate and large time scaling limit, the distribution of the mutant cells develops some atypical properties, including a power law tail and diverging average.