  Advances in imaging techniques enable high resolution 3D visualisation of
vascular networks over time and reveal abnormal structural features such as
twists and loops, and their quantification is an active area of research. Here
we showcase how topological data analysis (TDA), the mathematical field that
studies `shape' of data, can characterise the geometric, spatial and temporal
organisation of vascular networks. We propose two topological lenses to study
vasculature, which capture inherent multi-scale features and vessel
connectivity, and surpass the single scale analysis of existing methods. We
analyse images collected using intravital and ultramicroscopy modalities and
quantify spatio-temporal variation of twists, loops, and avascular regions
(voids) in 3D vascular networks. This topological approach validates and
quantifies known qualitative trends such as dynamic changes in tortuosity and
loops in response to antibodies that modulate vessel sprouting; furthermore, it
quantifies the effect of radiotherapy on vessel architecture.
