Skip to main page content
Access keys NCBI Homepage MyNCBI Homepage Main Content Main Navigation
. 2017 Feb;79(2):237-276.
doi: 10.1007/s11538-016-0234-5. Epub 2016 Nov 30.

Spatial Measures of Genetic Heterogeneity During Carcinogenesis

Affiliations
Free PMC article

Spatial Measures of Genetic Heterogeneity During Carcinogenesis

K Storey et al. Bull Math Biol. .
Free PMC article

Abstract

In this work we explore the temporal dynamics of spatial heterogeneity during the process of tumorigenesis from healthy tissue. We utilize a spatial stochastic model of mutation accumulation and clonal expansion in a structured tissue to describe this process. Under a two-step tumorigenesis model, we first derive estimates of a non-spatial measure of diversity: Simpson's Index, which is the probability that two individuals sampled at random from the population are identical, in the premalignant population. We next analyze two new measures of spatial population heterogeneity. In particular we study the typical length scale of genetic heterogeneity during the carcinogenesis process and estimate the extent of a surrounding premalignant clone given a clinical observation of a premalignant point biopsy. This evolutionary framework contributes to a growing literature focused on developing a better understanding of the spatial population dynamics of cancer initiation and progression.

Keywords: Carcinogenesis; Genetic heterogeneity; Spatial evolution.

Figures

Figure 6
Figure 6
Regions
Figure 7
Figure 7
Z1(x1, t1): The region in which the occurrence of a second mutation would make the cells located at a and b different, given that the first mutation occurred in R1.
Figure 8
Figure 8
Z2(x1, t1): The region in which the occurrence of a second mutation would make the cells located at a and b different, given that the first mutation occurred in R2.
Figure 9
Figure 9
Z4(x1, t1): The region in which the occurrence of a second mutation would make the cells located at a and b the same, given that the first mutation occurred in R4.
Figure 10
Figure 10
A1(x1, t1): The region inside VaVb that is affected by a mutation at (x, t) ∈ R1, and thus is not susceptible to subsequent mutation.
Figure 11
Figure 11
A4(x1, t1): The region inside VaVb that is affected by a mutation at (x, t) ∈ R4, and thus is not susceptible to subsequent mutation.
Figure 12
Figure 12
Overlap of space time cones at time s.
Figure 13
Figure 13
When two mutation circles collide, they will continue to expand along the line perpendicular to the line segment joining the two mutation origins.
Figure 14
Figure 14
The cross-section of the cones Va(t0), Vb(t0), Ca(t1), and Cb(t1) at the moment when a mutation occurs in the intersection, M(r, t0). If a second mutation occurs in the shaded mutation, then the cells located at a and b will be different.
Figure 15
Figure 15
The cross-section of the cones Va(t0), Vb(t0), and Ca(t1) at the moment when a mutation occurs in D(r, t0). If a second mutation occurs in the shaded mutation, then the cells located at a and b will be the same.
Figure 16
Figure 16
Plot of .5 in 2D as a function of (A) selection strength, s, and (B) time of sampling, t. In all panels N = 2e5, and 1e4 Monte Carlo simulations are performed. Unless varied, s = 0.1, u1 = 7.5e − 7, and t is the median of the detection time τ with μ = 2e − 6.
Figure 1
Figure 1. Time-dependence of non-spatial Simpson’s Index R(t)
The temporal evolution of the expected value of the non-spatial Simpson’s Index is shown (A) for varying values of the mutation rate u1, and (B) for varying values of the fitness advantage s of preneoplastic cells over normal cells. In both simulations: M1 = M2 = 500, N = 104, and u1 = 7.5 × 10−7, s = 0.1 unless specified.
Figure 2
Figure 2
I1 in 2D as a function of sampling time t0. We vary the sampling radius r and set s = 0.01 and u1 = 1e − 5, so the mutation rate is 1e − 7. We also set the mutant growth rate cd = 0.25.
Figure 3
Figure 3
I1 in 2D as a function of u1, which contributes to the mutation rate. Mutations arise according to a Poisson process with rate u1s, and we set s = 0.01. We vary the sampling radius r and set the sampling time t0 = 300 and the mutant growth rate cd = 0.25.
Figure 4
Figure 4
Plot of I2 (r, t) in 2D as a function of sampling radius for (A) varying selection strength, s, (B) varying u1, and (C) varying t. In all panels N = 2e5, and 1e4 Monte Carlo simulations are performed. Unless varied, s = 0.1, u1 = 7.5e − 7, and t is the median of the detection time τ with μ = 2e − 6.
Figure 5
Figure 5
Plot of I2 (r, τ) in 2D as a function of sampling radius. In panel (A) we vary the selection strength, s, and in panel B we vary u1. In all panels N = 2e5, we use 1e4 Monte Carlo simulations, and for the random detection time τ we use μ = 2e − 6. If not mentioned we set s = 0.1, u1 = 7.5e − 7.

Similar articles

See all similar articles

Cited by 1 article

Publication types

LinkOut - more resources

Feedback