Biomathematical model for simulating abnormal orifice patterns in colonic crypts

Mathematical model for simulation of morphological changes associated to crypt fission in the colon

“…Here p = p(x, t) is the cell-cell adhesion pressure exerted between cells located in the crypt walls and related to the mitotic activity [18,42,63]. This partial differential system (3) is similar to that one used in previous works [22,23,24,25] concerning colonic cell dynamics and classified as a transport/mass conservation model on the dynamics of two types of cells. Moreover, it is also used in the context of a general avascular tumour growth, when different types of cells are considered, see [56].…”

“…In fact, the rate β given in ( 5) is activated only at the top and transforms transit cells T into fully differentiated cells F . Note that, these rates are different to those used in previous works [22,23,24,25], since they are scaled on the dimension of the crypt height, allowing to model colonic cell dynamics independently of the crypt size and scale. The parameters γ > 0 and τ > 0 are obtained by using the property that, in the normal case, cell velocity at the orifice x 3 = h corresponds to the rate of 0.85 positions per hour [41,48,51,63].…”

“…Galerkin finite element method. Let us denote by Ω(t) the set of all the crypts generated throughout the crypt fission at time t. We are interested in solving numerically the equation system (23) in the discrete time interval [t n−1 , t n ], fixed a time step ∆ t and t n = t n−1 + ∆ t , by using the Galerkin finite element method [37] associated to the finite element space V h…”

“…where T h = {K l } l=1,...,Ntriang is a triangulation of Ω(t), N triang is the number of triangles of T h and {φ m } m=1,...,N nodes is the basis of the space V h , with N nodes the number of the internal nodes of T h . Now, the variational formulation of the equations system (23) considered in order to compute T and p in the time interval [t n−1 , t n ] for all basis functions φ m is given by…”

“…Numerical results. In this section we present some numerical results obtained by solving the differential equation system (23) coupled with the discrete crypt fission model of Sect. 4, hereafter called "differential fission model".…”