Stochastic dynamics of cancer initiation

Abstract

Most human cancer types result from the accumulation of multiple genetic and epigenetic alterations in a single cell. Once the first change (or changes) have arisen, tumorigenesis is initiated and the subsequent emergence of additional alterations drives progression to more aggressive and ultimately invasive phenotypes. Elucidation of the dynamics of cancer initiation is of importance for an understanding of tumor evolution and cancer incidence data. In this paper, we develop a novel mathematical framework to study the processes of cancer initiation. Cells at risk of accumulating oncogenic mutations are organized into small compartments of cells and proliferate according to a stochastic process. During each cell division, an (epi)genetic alteration may arise which leads to a random fitness change, drawn from a probability distribution. Cancer is initiated when a cell gains a fitness sufficiently high to escape from the homeostatic mechanisms of the cell compartment. To investigate cancer initiation during a human lifetime, a 'race' between this fitness process and the aging process of the patient is considered; the latter is modeled as a second stochastic Markov process in an aging dimension. This model allows us to investigate the dynamics of cancer initiation and its dependence on the mutational fitness distribution. Our framework also provides a methodology to assess the effects of different life expectancy distributions on lifetime cancer incidence. We apply this methodology to colorectal tumorigenesis while considering life expectancy data of the US population to inform the dynamics of the aging process. We study how the probability of cancer initiation prior to death, the time until cancer initiation, and the mutational profile of the cancer-initiating cell depends on the shape of the mutational fitness distribution and life expectancy of the population.

Figure 1. A schematic representation of the stochastic process governing cellular fitness

We consider a population of N cells residing in a compartment or niche of fixed size. These cells replicate according to a stochastic Moran process. During each elementary time step of the process, a cell is chosen at random proportional to fitness to divide and its offspring replaces another randomly chosen cell. During each cell division, a genetic or epigenetic alteration arises with probability u. Each alteration may confer a random additive fitness change to the cell. The parent cell, on the left, thus gives rise to a daughter cell with the same fitness, x, with probability 1 − u (upper right). With probability u, the parent cell produces a mutated offspring with fitness x + ψ, where ψ is a random additive fitness change selected according to the mutational fitness distribution f_ψ (lower right). If a cell within the compartment gains a sufficiently large fitness value, then tumorigenesis is initiated.

Figure 2. Sample path simulations of the two-dimensional stochastic process

We consider a stochastic process governing the evolution of fitness values of cells within the compartment (see Fig. 1) coupled with a stochastic process representing aging and death of the patient. The figure shows two sample path simulations of this two-dimensional process. The fitness of cells within the compartment is shown on the horizontal axis, and the lifetime of the patient is displayed on the vertical axis. If the trajectory hits the right boundary before reaching the upper border, then cancer initiation occurs before death of the patient (sample path (1)); if the trajectory hits the top boundary before reaching the right border, then death occurs before cancer initiation (sample path (2)). The fitness values bounding the area of interest are dictated by the cutoffs for neutral evolution [56].

Figure 3. The mutational fitness distribution

The stochastic process governing the evolution of fitness values of cells within a compartment depends on the shape of the fitness distribution conferred by mutations. For simplicity, we consider a general family of fitness distributions determined by the shape parameters α and β, which govern the decay of advantageous and deleterious mutations, respectively. This flexible setup allows us to study the effects of varying the mutational fitness distribution on the dynamics of cancer initiation. As specific examples, the figure displays the probability density function of f_ψ for α = 0.5 and β = 0.4, 0.5, 0.6.

Figure 4. The probability of and time until cancer initiation prior to death

(a) and (b) The panels display the probability of cancer initiation from a single compartment of cells, p_i, before death of the patient due to causes other than the cancer type of interest. The panels are plotted on a semilogarithmic plane for clarity. The probability of cancer initiation increases with α and decreases with β. (c) and (d) The panels display the expected time of cancer initiation, conditioned on initiation occurring before death of the patient. (a) and (c) We vary the shape parameter α, which governs the decay rate on the right of the distribution (i.e. advantageous mutations). (b) and (d) We vary the shape parameter β, which governs the decay rate on the left of the distribution (i.e. deleterious mutations). The aging process L is fit to US life expectancy data [57] and parameters are u = 0.001 and N = 10.

Figure 5. Advantageous mutations in the cancer-initiating cell

(a) The panel displays the expected number of advantageous mutations in the cell that reaches the fitness threshold 1 + 1/N and initiates tumorigenesis; these numbers of mutations are conditioned on cancer initiation occurring before death of the patient due to causes other than the cancer type of interest. We vary α, for β = 0.4, 0.5, 0.6, and display the expected number of advantageous mutations in the cancer initiating cell. The aging process L is fit to US life expectancy data [57] and parameters are u = 0.001 and N = 10. (b) The panel shows the empirical density (histogram) of the average fitness change of advantageous mutations that reach fixation in the compartment of cells prior to cancer initiation, conditioned on initiation occurring before death. Parameter values are α = 0.15 and β = 0.3(top), α = 0.15 and β = 0.5(middle), and α = 0.15 and β = 0.7(bottom). The otherparameters are as in (a). (c) The panel showsthe conditional expected number of advantageous, neutral and disadvantageous mutations that have reached fixation in the compartment prior to cancer initiation. The mode of the lifetime distribution ϕ_mode is varied. Parameters are u = 0.001, α = 0.25, β = 0.5 and N = 10.

Figure 6. The effects of the mutation rate on cancer initiation

The figure shows the probability of cancer initiation prior to death of the patient for varying mutation rate u. Two values for the mode of the lifetime distribution are shown: ϕ_mode = 50, 80. Parameters are α = 0.4, β = 0.6 and N = 10.