Intratumor heterogeneity in evolutionary models of tumor progression - PubMed (original) (raw)
Intratumor heterogeneity in evolutionary models of tumor progression
Rick Durrett et al. Genetics. 2011 Jun.
Abstract
With rare exceptions, human tumors arise from single cells that have accumulated the necessary number and types of heritable alterations. Each such cell leads to dysregulated growth and eventually the formation of a tumor. Despite their monoclonal origin, at the time of diagnosis most tumors show a striking amount of intratumor heterogeneity in all measurable phenotypes; such heterogeneity has implications for diagnosis, treatment efficacy, and the identification of drug targets. An understanding of the extent and evolution of intratumor heterogeneity is therefore of direct clinical importance. In this article, we investigate the evolutionary dynamics of heterogeneity arising during exponential expansion of a tumor cell population, in which heritable alterations confer random fitness changes to cells. We obtain analytical estimates for the extent of heterogeneity and quantify the effects of system parameters on this tumor trait. Our work contributes to a mathematical understanding of intratumor heterogeneity and is also applicable to organisms like bacteria, agricultural pests, and other microbes.
Figures
Figure 1.—
A sample cross-section of tumor heterogeneity. (A) The composition of a tumor cell population at t=150 time units after tumor initiation. Each “wave” of cells, defined as the set of cells harboring the same number of (epi)genetic alterations, is represented by a different color. Individual clones of cells with identical genotype are shown as circles, positioned on the horizontal axis according to fitness (i.e., birth rate) and on the vertical axis according to clone size. Note that later waves tend to have larger fitness values. (B) The time at which individual clones of cells were created during tumor progression. The color scale depicts the time of emergence of each clone. In A and B, parameters are a0=0.2, b0=0.1, v∼U([0,0.05]), and u=0.001.
Figure 2.—
The process for the small mutation limit. The limiting process is shown for the case in which the mutation rate goes to zero. (A) The time, t, on the horizontal axis vs. the log number of cells of wave k, zk(t), as given by Equation 3. Both time and space are given in units of L = log(1/u). This plot shows the first 20 waves started at t=10=1/λ0, i.e., the time that type-1 cells begin to be born. (B) A closer look at the first 7 waves from a, showing the changes in the dominant type. (C) The birth times of the first 20 generations as a function of the generation number. (D) The dominant type in the population as a function of time. The index m(t) of the largest generation at time t is defined as zm(t)(t)=max{zk(t) : k≥0}. In A–D, parameters are λ0=0.1 and b=0.05.
Figure 3.—
The size of the first four generations of cells. The log size of generations 1–4 is shown as a function of time t. Both time and space are plotted in units of L=log(1/u). (A) The average values of the log generation sizes over 106 sample simulations. (B) The limiting approximation from Equation 3 for the log size of the generations. (C) The approximation from Equation 6 using the mean of log Vd,k. (D) The approximation from Equation 6 using a value two standard deviations above the mean of log Vd,k to demonstrate an extreme scenario. Parameters are u=10−5, v∼δ(b), a0=0.2, b0=0.1, and b=0.05.
Figure 4.—
The distribution of log _V_d,1. The relative frequency histogram of 1000 random samples from the distribution of log Vd,1 is shown, as specified by Equation 1.
Figure 5.—
The expected value of Simpson's index for the first wave of cells. The sample mean of Simpson's index (dots) over time t and the expected value of Simpson's index for the limiting point process (line) are shown, for two different values of α. Parameters are λ0=0.1, a0=0.2, and v∼U([0,b]), where b=0.01 in a and b=0.05 in b.
Figure 6.—
The empirical distribution of Simpson's index for the first wave of cells. Individual plots of Simpson's index are shown for the branching process at times t=70, 90, 110, and 130 along with Simpson's index for the limiting point process (t=∞). The histograms show the average over 1000 sample simulations. For the limiting point process, we approximate Simpson's index by examining the largest 104 points in the process. Parameters are λ0=0.1, a0=0.2, and v∼U([0,0.01]).
Figure 7.—
The largest clones in the population of cells. (A) A comparison between Monte Carlo estimates for EVn−1 and the limit (1−α)−1. (B) A comparison of the Monte Carlo estimates for EVn and the curve (1−α)+. The Monte Carlo estimates are averaged over 100 sample simulations.
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