### Multiscale approaches for analysing vascular networks

Helen Byrne

Mathematical Institute

University of Oxford

Although discrete approaches are increasingly being used to describe biological phenomena such as tumour angiogenesis, it remains unclear how population-level behaviours emerge from the rules used to define the discrete models. Discrete-to-continuum approaches can be used to derive coarse-grained equations that describe the mean-field dynamics of a microscopic model and, as such, can provide insight into emergent behaviours. The resulting continuum models are often analytically intractable due to the appearance of nonlinearities. By contrast, phenomenological continuum models, such as the classical snail-trail model of angiogenesis, are typically easier to analyse but their relationship to microscopic descriptions is unclear. In this talk, I will introduce approaches for coarse-graining discrete models of angiogenesis and compare the resulting continuum models with the classical snail-trail model. I will use asymptotic techniques to identify parameter regimes in which the continuum models are equivalent, at leading order. If time permits, I will also explain how we are using techniques from topological data analysis to generate new insight into the structure of synthetic and biological vascular networks.