Fig 1.
Stochastic branching process model of tumor evolution.
(a) Stochastic branching process model for tumor expansion. Initiated tumor cells (blue) divide with birth rate b, die with death rate d, and accrue passenger mutations with mutation rate u. Type-1 cells, which carry the driver mutation, divide with birth rate b1, die with death rate d1, and accrue passenger mutations with mutation rate u. (b) The initiated tumor, or type-0, (blue) population growth is initiated from a single cell. A driver mutation occurs in a single type-0 cell at time t1, starting the type-1 population (red). The tumor sample is collected and bulk sequenced at times t1 + t and t1 + t + Δ, where the driver fraction is α1 and α2, respectively. Tumor size (in number of cells) is M1 and M2 at first and second sample collection dates. (c) By the time the tumor is observed, it has a high level of genetic heterogeneity due to the mutations that have accrued in both type-0 (blue) and type-1 populations (red). Each yellow star represents a different passenger mutation.
Fig 2.
Accuracy of parameter inferences from simulated data.
We simulated tumor growth by performing a Monte Carlo simulation, which simulates the birth, death, and accumulation of mutations in the individual cells that make up a tumor, and generates the mutation frequency and tumor size data used by the estimates. Simulations are of fast-growing tumors with (a) single driver subclone and mutation rate u = 1, (b) single driver subclone and u = 3, (c) two nested driver subclones with u = 1, and (d) two sibling driver subclones with u = 1. Mean percent errors (MPEs) of estimates are shown in black above the plots, and mean absolute percent errors (MAPEs) are shown in gray. Boxes contain 25th-75th quartiles, with median indicated by thick horizontal black line. Whiskers of boxplots indicate 2.5 and 97.5 percentiles. Violins are smoothed density estimates of the percent error data points. Complete parameter values and number of runs are included in S1 Table.
Fig 3.
Corrections for observed mutation counts.
(a) If passenger mutations (circles with stars) that occur after the driver reach fixation in the driver population (red), then they are indistinguishable from the passengers that were present in the first cell with the driver, which accrued in the type-0 population (blue). The estimate of when the driver occurred needs to account for these mutations (circled). In (b), we compare percent errors of parameter estimates for time from tumor initiation until appearance of a driver subclone, t1, with and without this correction (Eq (6)). Errors for estimate with correction are shown in blue, and for estimate without correction (Eq. (5)) in orange. Errors are plotted as a kernel density estimate for Monte Carlo simulations of slow-growing tumor with mutation rate u = 5. Mean percent errors (MPEs) and mean absolute percent errors (MAPEs) are listed. (c) Mutations present on two or fewer variant reads (red) are filtered out in post-processing. Mutations with more than two variant reads (black) are included. The number of subclonal mutations between frequencies f1 and f2, γ, which is used in the mutation rate estimate, must be corrected for mutations that are filtered out. In (d), the percent errors for the observed (orange) and corrected (blue) γ (Eq (7)) are plotted as kernel density estimates. Observed mutations are those that passed post-processing, i.e. those that have more than L = 2 mutant reads. True mutation frequencies were generated from 135 surviving runs of a Monte Carlo simulation of a fast-growing tumor with mutation rate u = 1, from which sequencing reads were simulated with 200x average coverage (see Materials and methods). Percent errors are calculated relative to the true γ measured from the true mutation frequencies.
Table 1.
Inferred parameters for CLL patients with exponential growth patterns, for which there are at least two longitudinal bulk sequencing measurements before treatment.
Estimates are computed from tumor size measurements and mutation frequencies from whole exome sequencing. Mutation rates are for the exome only. The time estimates are in terms of the patient’s age in years.
Table 2.
Confidence intervals for inferred parameters for CLL patients with exponential growth patterns, for which there are at least two longitudinal bulk sequencing measurements before treatment.
Estimates are computed from tumor size measurements and mutation frequencies from whole exome sequencing. Mutation rates are for the exome only. The time estimates are in terms of the patient’s age in years.
Fig 4.
Reconstructing the timeline of CLL evolution in patients.
We applied our methodology to estimate subclonal growth rates, mutation rates and evolutionary timelines in CLL tumors from Ref. [27]. Vertical height of a clone represents its log10-scaled size. Mutations were clustered into clones and phylogenetic trees were inferred using PhylogicNDT [43]. Tree edges are colored by clone number and are labeled with driver mutations, if any. For each patient, we show estimates for patient age at CLL initiation and times of appearance of CLL subclones. Dashed white line indicates when the patient was diagnosed. Solid black arrows indicate times of bulk sequencing measurements.