Percolative analysis of 3D vasculogenesis

Vascular networks form by the spontaneous aggregation of individual cells migrating toward vascularization sites (vasculogenesis). The study of this fascinating process is performed by biologists using in vitro and in vivo assays, both in twodimensional and threedimensional settings. A succesfull theoretical model of twodimensional experimental vasculogenesis has been recently proposed, showing the relevance of percolation concepts and of cell cross-talk (chemotactic autocrine loop) to the understanding of the self-aggregation process. Here we study the natural 3D extension of the earlier proposed computational model, which we take as a starting point for the investigation of the genuinely threedimensional process of vasculogenesis in vertebrate embryos. The computational model is based on a multidimensional Burgers equation, which is a well studied paradigm in the research on spontaneous pattern formation, integrated with a feedback term describing the chemotactic autocrine loop. The numerical approximation of the computational model poses several technical problems which are here briefly discussed. Starting from initial conditions mimicking the experimentally observed ones the numerical simulations produce network-like structures qualitatively similar to those observed in the early stages of in vivo vasculogenesis. Following the lesson learned in the twodimensional case, we develop the computation of critical percolative indices as a robust measure of the network geometry.