Mathematical modeling of cell population dynamics in the colonic crypt and in colorectal cancer

Abstract

Colorectal cancer is initiated in colonic crypts. A succession of genetic mutations or epigenetic changes can lead to homeostasis in the crypt being overcome, and subsequent unbounded growth. We consider the dynamics of a single colorectal crypt by using a compartmental approach [Tomlinson IPM, Bodmer WF (1995) Proc Natl Acad Sci USA 92:], which accounts for populations of stem cells, differentiated cells, and transit cells. That original model made the simplifying assumptions that each cell population divides synchronously, but we relax these assumptions by adopting an age-structured approach that models asynchronous cell division, and by using a continuum model. We discuss two mechanisms that could regulate the growth of cell numbers and maintain the equilibrium that is normally observed in the crypt. The first will always maintain an equilibrium for all parameter values, whereas the second can allow unbounded proliferation if the net per capita growth rates are large enough. Results show that an increase in cell renewal, which is equivalent to a failure of programmed cell death or of differentiation, can lead to the growth of cancers. The second model can be used to explain the long lag phases in tumor growth, during which new, higher equilibria are reached, before unlimited growth in cell numbers ensues.

Fig. 1.

Schematic representation of a colonic crypt. (Left) A schematic diagram of a crypt, with stem, semidifferentiated (transit-amplifying), and fully differentiated cell populations. The dimensions given are for a human colonic crypt according to Halm and Halm (23). (Right) A diagram showing the compartmental structure used in the model by Tomlinson and Bodmer (22). The stem cells differentiate into semidifferentiated cells, which in turn differentiate into fully differentiated cells. Each cell population can die, and the stem cells and semidifferentiated cells can renew. The parameters for the age-structured model are the proportions of the populations a_i, b_i, and c that are leaving the compartments, and the parameters for the continuous model are rates of conversion α_i, β_i, and γ.

Fig. 2.

An illustrative sequence of mutations (occurring every 100 days) in the saturating feedback model (feedback model 2). The initial parameters are taken to be α = 0.286, α₂ = 0.3, β = 0.432, β₂ = 0.3, γ = 0.323, k₀ = 0.1, m₀ = 0.1, k₁ = 0.01, and m₁ = 0.01. The mutations cause, successively, β = 0.512, α = 0.5, β = 0.697, β = 1.1. For β = 1.1 there is no steady state and unbounded growth occurs.

Fig. 3.

Plots of the regions of stability of the cell population models for no feedback (i), linear feedback (ii), and saturating feedback (iii). In the case of no feedback, there are only stable solutions on the line α = 0, β < 0; otherwise, there is either extinction or unbounded growth. In the case of the linear feedback, there is never unbounded growth in cell numbers, and there is a steady state if α > 0, and extinction if α < 0. For the saturating feedback, there is a strip of the parameter space that permits stable solutions, when 0 < α < k₀/m₀ and β < k₁/m₁, and outside this region there is either extinction or unbounded growth. The crosses correspond to the five states in the (α, β)-parameter space from Fig. 2.