Topological data analysis of flows in directed spatial networks for modelling vascular networks

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Understanding how oxygen is supplied through vascular networks is of vital importance in a range of medical applications. The structure of these networks can vary greatly. For example, angiogenesis in the vicinity of tumours is characterised by bendy vessels with many loops, in contrast to healthy vasculature. Vascular targeting agents attack the blood vessels around tumours, to limit vessel connectivity and thus (hopefully) oxygen supply. Angiogenesis and targeting agents both alter the topology of the vascular network. It is therefore important to understand and quantify how changes to the topology in the network affect the flow of oxygen to the tissue. Recent work by Stolz et al [1] has shown that topological data analysis provides a robust tool for "relating the form and function of vascular networks". The DPhil project will attempt to add directionality into existing analysis in this area.Topological data analysis is a growing field at the intersection of algebraic topology and data science. Tools such as persistent homology can quantitatively capture important features in data sets, such as loops, tortuosity and clusters. Crucially, the outputs of persistent homology are stable under perturbation of the underlying data set. Important features, present at a wide range of scales, are distinguished from transient, noisy features. This provides a topological summary of a data set, amenable to further statistics or machine learning techniques, depending on the application. Moreover, this technique can be applied to data sets in many forms, including point clouds, graphs and spatial networks.xThe Navier-Stokes equations are used to model fluid flow in a variety of settings. However, blood is a non-Newtonian fluid, so the stress tensor must be modified to take this into account. Worse yet, vessels cannot be accurately represented as rigid tubes and thus classical fluid dynamics often falls short. This necessitates the use of alternative models for understanding the large-scale behaviour of vascular systems. Given the network structure of typical vasculature, a natural approach is to model the vessel network as a spatial network. A dynamical system, capturing traditional fluid dynamics terms such as convection and diffusion, can then be imposed on this network to model blood flow. The underlying network can then be altered appropriately, the change in the underlying topology can be measured through techniques from TDA and the effect to the flow can be measured by running the dynamical system.Since flow in vascular networks is directed, a natural question is how adding directionality, through asymmetry in the underlying network, affects the flow. Furthermore, in this directed setting, when the underlying network structure changes, how does the flow respond? It is also not clear what is the most appropriate method for measuring the topology of a directed network. Possible techniques include the Euler Characteristic and the flagser package by Lutgehetmann [2], which computes the persistent homology of a directed flag complex. Alternatively, recent work by Chowdhury and Mémoli [3] has introduced persistent path homology, capable of distinguishing between important digraph motifs, which are indistinguishable to flagser. The computational requirements of these methods is also an important consideration. As with all applications of persistent homology, correct filtration choice is vital to achieving desirable results. One could use both vessel length and vessel diameter as filtration parameters, potentially requiring the use of multi-parameter persistent homology. This DPhil project will attempt to answer some of these questions and develop a directional model for blood flow and oxygen delivery in vascular networks.This project falls within the EPSRC Geometry & Topology research area and, more specifically, 'Application driven Topological Data Analysis'.