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

了解氧气是如何通过血管网络供应的，在一系列医疗应用中至关重要。这些网络的结构可能差异很大。例如，与健康的脉管系统相比，肿瘤附近的血管生成的特征在于具有许多环的弯曲血管。血管靶向剂攻击肿瘤周围的血管，以限制血管连通性，从而（希望）限制氧气供应。血管生成和靶向剂都改变血管网络的拓扑结构。因此，重要的是要了解和量化网络中拓扑结构的变化如何影响氧气流向组织。Stolz等人[1]最近的工作表明，拓扑数据分析提供了一个强大的工具，用于“关联血管网络的形式和功能”。这个哲学博士项目将尝试在这个领域的现有分析中加入方向性。拓扑数据分析是代数拓扑学和数据科学交叉的一个不断发展的领域。诸如持久同源性之类的工具可以定量地捕获数据集中的重要特征，例如循环、弯曲度和聚类。至关重要的是，持久同源性的输出是稳定的基础数据集的扰动下。重要的功能，目前在广泛的规模，从瞬态，嘈杂的功能区分。这提供了数据集的拓扑摘要，取决于应用，可以进一步统计或机器学习技术。此外，这种技术可以应用于许多形式的数据集，包括点云，图形和空间网络。然而，血液是一种非牛顿流体，因此必须修改应力张量以考虑这一点。更糟糕的是，血管不能准确地表示为刚性管，因此经典流体动力学经常福尔斯不足。这就需要使用替代模型来理解血管系统的大规模行为。给定典型脉管系统的网络结构，自然的方法是将血管网络建模为空间网络。动态系统，捕捉传统的流体动力学术语，如对流和扩散，然后可以施加在这个网络模型的血流。然后可以适当地改变底层网络，底层拓扑结构的变化可以通过TDA的技术来测量，对流动的影响可以通过运行动力学系统来测量。由于血管网络中的流动是定向的，一个自然的问题是如何通过底层网络中的不对称性来增加方向性，从而影响流动。此外，在这种定向设置中，当底层网络结构发生变化时，流如何响应？也不清楚什么是测量有向网络拓扑的最合适方法。可能的技术包括欧拉特征线和Lutgehetmann的flagser包[2]，它计算有向旗复形的持久同调。另外，Chowdhury和Mémoli [3]最近的工作引入了持久路径同源性，能够区分重要的有向图基序，这是无法区分的标志。这些方法的计算要求也是重要的考虑因素。与持久同源性的所有应用一样，正确的过滤选择对于实现期望的结果至关重要。可以使用血管长度和血管直径作为过滤参数，这可能需要使用多参数持续同源性。这个博士项目将试图回答其中的一些问题，并开发一个血管网络中血流和氧气输送的方向模型。这个项目福尔斯EPSRC几何和拓扑研究领域，更具体地说，是“应用驱动的拓扑数据分析”。

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'.