Comparative analysis of continuum angiogenesis models

While discrete approaches are increasingly used to model biological phenomena, it remains unclear in such frameworks how complex population-level behaviours emerge from the rules used to describe interactions between individuals. Insight may be gained by deriving coarse-grained continuum models, which describe the mean-field dynamics of a discrete model. Differential equations derived from such discrete-to-continuum approaches, however, often contain nonlinearities that depend on microscopic rules in the discrete model, and there has been little work done to analytically compare these coarse-grained equations with those constructed from simpler phenomenological frameworks. We address this problem in the context of angiogenesis (the creation of new blood vessels from existing vasculature). We compare asymptotic solutions of a classical, phenomenological 'snail-trail' partial differential equation (PDE) model for angiogenesis with those of a more complicated, fully nonlinear PDE system derived via a systematic coarse-graining procedure. For distinguished parameter regimes corresponding to chemotaxis-dominated cell movement and low branching rates, both continuum models reduce at leading order to an identical system of PDEs. Numerical and analytical results confirm that solutions to the two continuum models are in good agreement if these conditions hold, which allows us to determine when we can use the simpler model to capture the results of a more complicated coarse-grained system that describes the same biological process.