Biological backgroundCapillaries are made of endothelial cells, which play a vital role in supplying tissues with nutrients. Their growth is essentially driven by two processes: vasculogenesis and angiogenesis. Vasculogenesis consists of local differentiation of precursor cells to endothelial ones, that assemble into a vascular network by directed migration and cohesion. Angiogenesis is essentially characterized by sprouting of novel structures and their remodelling.
Mathematical ModelsI am working on two models which both can be used to produce the blood vessel network. The first model(left) is proposed by A.Gamba et al. in  , while the second one(right) is derived by M. Alber et al. , as the continuous limit from a two dimensional stochastic cellular Potts model(CPM). In both systems, n or p denotes the cell density, c denotes the concentration of chemo-attractant. The differences are: The first system involves the velovity of the cells while the nonlinear diffusion term showed up in the second system represents excluded volume interactions.
Discretization MethodsWe solved the first system by third order finite difference weno scheme and third order finite volume weno scheme(simulation results listed only), coupled with third order Runge-Kutta method. While classical central difference scheme, first order upwinding scheme and adaptive Runge-Kutta method are applied to the second one.
Some Simulation Results
The first Model
Here is the movie of the blood vessel network formation process, got by third order finite volume weno scheme on triangular mesh.
The second Model (from left to right as time goes on)
Reference G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly, the EMBO Journal, Vol. 22, No. 8, (2003), pp. 1771-1779.