Evolutionary dynamics of tumor progression with random fitness values

Abstract

Most human tumors result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Mutations that confer a fitness advantage to the cell are known as driver mutations and are causally related to tumorigenesis. Other mutations, however, do not change the phenotype of the cell or even decrease cellular fitness. While much experimental effort is being devoted to the identification of the functional effects of individual mutations, mathematical modeling of tumor progression generally considers constant fitness increments as mutations are accumulated. In this paper we study a mathematical model of tumor progression with random fitness increments. We analyze a multi-type branching process in which cells accumulate mutations whose fitness effects are chosen from a distribution. We determine the effect of the fitness distribution on the growth kinetics of the tumor. This work contributes to a quantitative understanding of the accumulation of mutations leading to cancer.

Figure 1

Plot of the exact Laplace transform (LT) for t^(1+p) e^{−(λ₀+b)t} Z₁ (t) at times t = 60, 80,100,120, the approximations from Monte Carlo (MC) simulations at the corresponding times, and the asymptotic Laplace transform from Theorem 2. Parameter values: a₀ = 0.2, b₀ = 0.1, b = 0.01, and u₁ = 10⁻³. g is uniform on [0, .01].

Figure 2

Plot of the approximations to the Laplace transform of t^2+p₂e^{−(λ₀+2b)t} Z₂(t) from Monte Carlo (MC) simulations at times t = 80,100,120 along with the asymptotic Laplace transform from Theorem 5. Parameter values: a₀ = 0.2, b₀ = 0.1, b = 0.01, and u₁ = u₂ = 10⁻³. g is uniform on [0, 0.01].