Examples of mathematical modeling: tales from the crypt

Abstract

Mathematical modeling is being increasingly recognized within the biomedical sciences as an important tool that can aid the understanding of biological systems. The heavily regulated cell renewal cycle in the colonic crypt provides a good example of how modeling can be used to find out key features of the system kinetics, and help to explain both the breakdown of homeostasis and the initiation of tumorigenesis. We use the cell population model by Johnston et al. to illustrate the power of mathematical modeling by considering two key questions about the cell population dynamics in the colonic crypt. We ask: how can a model describe both homeostasis and unregulated growth in tumorigenesis; and to which parameters in the system is the model most sensitive? In order to address these questions, we discuss what type of modeling approach is most appropriate in the crypt. We use the model to argue why tumorigenesis is observed to occur in stages with long lag phases between periods of rapid growth, and we identify the key parameters.

Figure 1

A schematic of a colonic crypt showing the compartmental structure used in the model by Johnston et al. The stem cells differentiate into semi-differentiated cells, which in turn differentiate into fully-differentiated cells. Each cell population can die, and the stem cells and semi-differentiated 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 (described in Example 1) are the rates of conversion α_i, β_i and γ measured in hours⁻¹. Note that these compartment sizes are not to scale and that, in reality, the number of stem cells is very much less than the number of transit cells.

Figure 2

A plot of the stem cell population using the discrete model (equation 2), age-structured model (equation 5) and continuous model (equation 7). The parameters are taken to be α = 0.02 hours⁻¹, t₀ = 24 hours and n̂₀ = 5. a₃ is chosen such that a₃ = e^αt₀/2 (as in ref. 5).

Figure 3

Regions of stability in the (α, β) parameter space for the system with saturating feedback in both stem and transit cell populations. If α < 0 and β < k₁ / m₁ (Region I), then the stem cells cannot sustain their number and the crypt becomes extinct. If 0 < α < k₀ / m₀ and β < k₁ / m₁ (Region II), a healthy stable crypt is achieved. In all other regions the growth saturation limit is exceeded and the cell populations grow without bound. In Region III (α > k₀ / m₀ and β < k₁ / m₁) the cancer stem cell driving the unbounded growth is a tissue stem cell, whereas in Region IV (β > k₁ / m₁ with α < k₀ / m₀) the cancer stem cell derives from a transit-amplifying cell. In Region V (α > k₀ / m₀ and β > k₁ / m₁) cancer stem cells originate from both tissue stem cells and transit cells.

Figure 4

Regions of stability in the (α, β) parameter space for the system with linear feedback for stem cells and saturating feedback for transit cells. If α < 0 and β < k₁ / m₁ (Region I), then the stem cells cannot sustain their number and the crypt becomes extinct. If α > 0 and β < k₁ / m₁ (Region II), a healthy stable crypt is achieved. If β > k₁ / m₁ (Region III) unbounded growth occurs, and is driven by a cancer stem cell derived from the transit cell population.

Figure 5

An illustrative sequence of mutations that occur every 100 days. The initial parameters are taken to be α₁ = 0.1, α₂ = 0.3, α₃ = 0.69, β₁ = 0.1, β₂ = 0.3, β₃ = 0.397, γ = 0.139, k₀ = m₀ = 0.1 k₁ = 0.0003 and m₁ = 0.0004, which gives α = 0.286, β = −0.0027. All the parameters are measured in hours⁻¹ apart from m₀ and m₁ which are dimensionless. The mutations cause, successively, β = 0.1, β = 0.2 and finally β = 0.8. After the last mutation, β > k₁ / m₁ which means there is no steady state and the cell populations grow without bound. The numbers 1–4 correspond to the different points in the parameter space marked on Figure 3.