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

了解氧气如何通过血管网络供应在一系列医疗应用中至关重要。这些网络的结构变化很大。例如，与健康的血管系统相比，肿瘤附近的血管新生的特征是具有许多循环的弯曲血管。血管靶向药物攻击肿瘤周围的血管，限制血管连通性，从而（有希望）限制氧气供应。血管生成和靶向药物都能改变血管网络的拓扑结构。因此，理解和量化网络拓扑结构的变化如何影响氧气流向组织是很重要的。Stolz等人最近的研究表明，拓扑数据分析为“联系血管网络的形式和功能”提供了一个强大的工具。哲学博士项目将尝试在这一领域的现有分析中增加方向性。拓扑数据分析是代数拓扑和数据科学交叉的新兴领域。像持久同源这样的工具可以定量地捕捉数据集中的重要特征，比如循环、扭曲和聚类。关键是，在底层数据集的扰动下，持久同调的输出是稳定的。重要的特征，存在于广泛的尺度，从瞬态，噪声特征区分。这提供了数据集的拓扑摘要，可以根据应用程序进一步统计或机器学习技术。此外，该技术可以应用于多种形式的数据集，包括点云、图和空间网络。Navier-Stokes方程用于模拟各种情况下的流体流动。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到这一点。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往往是不足的。这就需要使用替代模型来理解血管系统的大规模行为。鉴于典型脉管系统的网络结构，一种自然的方法是将血管网络建模为空间网络。一个动态系统，捕捉传统的流体动力学术语，如对流和扩散，然后可以施加到这个网络来模拟血液流动。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技术来测量，对流量的影响可以通过运行动力系统来测量。既然血管网络中的流动是定向的，那么一个自然的问题是，通过潜在网络中的不对称性来增加方向性是如何影响流动的。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？测量有向网络拓扑结构最合适的方法是什么也不清楚。可能的技术包括欧拉特征和lugehetmann[2]的标志包，它计算定向标志复合体的持久同源性。另外，Chowdhury和m<s:1> moli[3]最近的研究引入了持续路径同源性（persistent path homology），能够区分重要的有向图基序，而旗子无法区分这些有向图基序。这些方法的计算要求也是一个重要的考虑因素。与所有持久同源性的应用一样，正确的过滤选择对于获得理想的结果至关重要。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久同源性。这个博士项目将尝试回答其中的一些问题，并开发血管网络中血液流动和氧气输送的定向模型。该项目属于EPSRC几何与拓扑研究领域，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'.