Comparative analysis of angiogenesis models: MATLAB data files

This data set contains the MATLAB files that were used to generate figures
located in the article "Comparative analysis of angiogenesis
models" (J. Math. Biol, in press). The
article's abstract may be found below.  Although
discrete approaches are increasingly employed to model biological
phenomena, it remains unclear how complex, population-level behaviours in
such frameworks arise from the rules used to represent interactions
between individuals. Discrete-to-continuum approaches, which are used to
derive systems of coarse-grained equations describing the mean-field
dynamics of a microscopic model, can provide insight into such emergent
behaviour. Coarse-grained models often contain nonlinear terms that depend
on the microscopic rules of the discrete framework, however, and such
nonlinearities can make a model difficult to mathematically analyse. By
contrast, models developed using phenomenological approaches are typically
easier to investigate but have a more obscure connection to the underlying
microscopic system. To our knowledge, there has been little work done to
compare solutions of phenomenological and coarse-grained models. Here 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" model for
angiogenesis to solutions of a nonlinear system of partial differential
equations (PDEs) derived via a systematic coarse-graining procedure
(Pillay, 2017). For distinguished parameter regimes corresponding to
chemotaxis-dominated cell movement and low branching rates, both continuum
models reduce at leading order to identical PDEs within the domain
interior. Numerical and analytical results confirm that pointwise
differences between solutions to the two continuum models are small if
these conditions hold, and demonstrate how perturbation methods can be
used to determine when a phenomenological model provides a good
approximation to a more detailed coarse-grained system for the same
biological process.