  Complex biological processes involve collective behavior of entities
(bacteria, cells, animals) over many length and time scales and can be
described by discrete models that track individuals or by continuum models
involving densities and fields. We consider hybrid stochastic agent-based
models of branching morphogenesis and angiogenesis (new blood vessel creation
from pre-existing vasculature), which treat cells as individuals that are
guided by underlying continuous chemical and/or mechanical fields. In these
descriptions, leader (tip) cells emerge from existing branches and follower
(stalk) cells build the new sprout in their wake. Vessel branching and fusion
(anastomosis) occur as a result of tip and stalk cell dynamics. Coarse-graining
these hybrid models in appropriate limits produces continuum partial
differential equations (PDEs) for endothelial cell densities that are more
analytically tractable. While these models differ in nonlinearity, they produce
similar equations at leading order when chemotaxis is dominant. We analyze this
leading order system in a simple quasi-one-dimensional geometry and show that
the numerical solution of the leading order PDE is well described by a soliton
wave that evolves from vessel to source. This wave is an attractor for
intermediate times until it arrives at the hypoxic region releasing the growth
factor. The mathematical techniques used here thus identify common features of
discrete and continuum approaches and provide insight into general biological
mechanisms governing their collective dynamics.
