Mathematical Models For Colorectal Cancer
The first models for crypt dynamics were discrete 2D-grid models, in which the cylindrical crypt structure is rolled open to provide a two dimensional surface (Figure 4). At discrete time steps, cells move between predefined rows and columns on the surface. The main problem with this kind of approach is that the insertion of each newborn cell causes a whole column of cells to shift upwards, breaking many cell-cell contacts. This is unrealistic as epithelial cells are known to be linked by tight cell-cell junctions, forming a continuous barrier that prevents the lumen contents from entering the body.
The existing mathematical models for crypt dynamics and colorectal can be classified as follows:
- Spatial models: Models that describe the spatial location of each individual cell. The 2D-grid and 2D-lattice-free model fall in this category.
- Compartmental models: Models that only describe the transition between cell types, independently of their relative position in the crypt.
- Non-spatial stochastic models: Includes a wide range of models developed for specific purposes, such as investigate the consequences of APC hits, analyse the role of genetic instability or predict colorectal cancer epidemiological data.
Stochasticity has been incorporated in several ways:
- Variable cell cycle times
- Mortality rates
- Insertion of newborn cells
- Dedifferentiation probability for transit cells
- Mutation rates
- Probabilities of symmetric (giving two stem cells) and asymmetric division (giving one stem cell and one transit cell)
Figure 4: Two-dimensional models for a crypt from the small intestine (distinguisable from a colonic crypt by the presence of Paneth cells at its very bottom).
- Left figure: Typical 2D-grid model in which cells are forced to move within rows and columns. Cell types represented as in Figure 2.
- Rigth figure: Snap-shot from a simulation with the lattice-free model developed by Meineke et al. (2001). Cells highlighted with blue dots belong to the same lineage. Transit cells appear in yellow, differentiated cells in green, stem cells in red and Paneth cells in dark red. In lattice-free models, cell proliferation causes local cell rearrangements only.