The impact of exclusion processes on angiogenesis models
The authors study angiogenesis models and derive a class of nonlinear continuous PDE models from a lattice-based cellular automaton model by a mean-field approximation. More precisely, by letting \(N\) and \(E\) denote the densities of tip cells and endothelial cells, respectively, as well as \(c\) the concentration of the tumor angiogenic factor, the authors obtain in the spatially one-dimensional setting the model \[ \begin{cases} N_t = D(1-a_n N - B a_e E) N_{xx} - \chi \left( (N(1-N)c_x)_x + a_n N N_x c_x - B a_e E (N c_x)_x \right) \\ \qquad - \mu \left( a_n N^2 + B a_e NE \right) + \lambda c N(1-N-BE)^2, \\ E_t = \mu N(1-N + 2a_n N) - D(1-2a_n)N N_{xx} - \chi N \left( (1- a_n) c_x N_x + a_n (N c_x)_x \right).\end{cases}\tag{1} \] Here, \(a_n , a_e \in [0,1]\) account for tip cell and endothelical cell volume exclusion, respectively, in the following sense. If a tip cell tries to move to a site which is occupied by a tip cell or endothelial cell, then with probability \(1-a_n\) or \(1-a_e\) the move is aborted. The authors study on the one hand the model with \(B=0\), accounting for tip cell volume exclusion only, and on the other hand the model with \(B=1\), accounting for both types of volume exclusion. By numerical simulations the authors compare the behavior of these models with the corresponding cellular automaton model and with the PDE model from [\textit{H. M. Byrne} and \textit{M. A. J. Chaplain}, Bull. Math. Biol. 57, No. 3, 461--486 (1995; Zbl 0812.92011)]. It is discussed in how far the two variants of model (1) are able to capture the effects of volume exclusion and how appropriate parameter values can be estimated.
scientific article; zbMATH DE number 6996573
scientific article