Mathematical modelling of spatio-temporal cell dynamics in colonic crypts following irradiation

Abstract

Objectives: Modelling the apoptotic process is essential for simulating and understanding tumour growth, as most tumour tissues carry mutations in apoptotic signalling pathways. Thus here, we have aimed to construct a mathematical model of colonic crypts that explicitly incorporates the apoptotic mechanism.

Methods: A murine colonic crypt was described as being a two-dimensional rectangular surface model. In this system, three types of cells with different proliferating and differentiating potentials migrate. Apoptosis was described as a process activated by irradiation that progresses in a stepwise manner. Parameter values in the model were determined to be consistent with experimental data for changes in the apoptotic cell ratio within murine transverse colonic crypts following irradiation.

Results: First, we constructed a model reproducing cell proliferation dynamics in normal murine colonic crypts; next, we applied the apoptotic mechanism to this model. As a result, we succeeded in simultaneous reproduction of both spatial and temporal changes in distribution of apoptotic cells in murine colonic crypts by determining parameter values in numerical simulations. Through this adjustment process, we were able to predict that stem cells and transit amplifying (TA) cells in each generation must react distinctly from each other, to apoptosis-inducing stimuli.

Conclusions: We constructed a mathematical model with which we could quantitatively describe cell proliferative and apoptotic dynamics in a murine colonic crypt. Using this model, we were able to make novel predictions that sensitivity to apoptosis-inducing stimuli is dependent on cell type.

Figure 1

Model of murine colonic crypt. (a) Modelling of the crypt. A crypt is modelled as a cylindrical surface, which is unfolded to a two‐dimensional surface. Cells are spatially arranged on this rectangular surface whose width and height are given by x and y, respectively. (b) Scenario of cell proliferation and differentiation. Two regions are considered in a crypt. Lower portion of the crypt, whose height is b, is defined as the self‐replication region of stem cells. Within this region, a stem cell divides symmetrically and produces two stem cells with a probability P (0 ≤ P ≤ 1). When it divides asymmetrically, it produces a stem cell and a first‐generation transit amplifying (TA) cell with a probability 1 − P. Outside this region, a stem cell divides symmetrically and produces two‐first‐generation TA cells. Regardless of the region, a TA cell divides symmetrically and differentiates after g‐th divisions.

Figure 2

Modelling of the apoptotic process. (a) Schematic drawing of the apoptotic pathway, starting from DNA damage persisting after induction. This progresses in a stepwise manner. (b) Cells at each stage are assigned with a variable α. After irradiation, cells (α = 1) induce apoptosis with probability Q. Following this, the apoptotic process progresses sequentially with given time durations; between each stage these are assigned as t ₁, t ₂, t ₃, and t _del.

Figure 3

Labelling indices plotted against cell position. Simulation results (red closed diamonds and solid curve) were in good agreement with the experimental data (black closed circles and dashed curve) 13. To obtain these results, parameter values shown in Table 1 were used.

Figure 4

Existing ratio of each cell type. Existing ratios of stem (black circles), transit amplifying (TA) (first generation, red squares), TA (second generation, blue triangles), TA (third generation, green diamonds), and differentiated cells (pink inverted triangles) were plotted against cell position. To obtain these results, the parameter values shown in Table 1 were used.

Figure 5

Spatial and temporal changes of apoptotic indices. (a) Apoptotic indices plotted against cell position (4.5 h after irradiation). (b) Apoptotic indices plotted against time after irradiation. These simulation results (red closed diamonds) were in good agreement with the experimental data 11 (black closed circles). Parameter values shown in Table 1 were used for the simulations.