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

用于建模血管网络的定向空间网络中的流的拓扑数据分析

• 批准号: 2423011
• 金额: --
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2423011
• 关键词: Topological; data; analysis; flows; directed

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'.

了解如何通过血管网络供应氧气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，与健康相比，肿瘤附近的血管生成的特点是具有许多环的弯曲血管。血管靶向剂攻击肿瘤周围的血管，以限制血管连接，从而（希望）血管生成和靶向剂都改变血管网络的拓扑结构。 Stolz 等人最近的工作 [1] 量化了网络拓扑的变化如何影响组织中的氧气流动，拓扑数据分析为“关联血管网络的形式和功能”提供了强大的工具。该项目将尝试在该领域的现有分析中添加方向性。拓扑数据分析是代数拓扑和数据科学交叉领域的一个不断发展的领域，同源等工具可以定量捕获数据集中持久的重要特征，例如循环、至关重要的是，持久同源性的输出在基础数据集的扰动下是稳定的，这提供了数据集的拓扑总结。根据应用，可以采用进一步的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、图形和空间网络。纳维-斯托克斯方程用于然而，血液是非牛顿流体，因此必须修改应力张量以考虑到这一点，更糟糕的是，血管无法准确地表示为刚性管，因此经典流体动力学常常会失败。简而言之，这需要使用替代模型来理解血管系统的大规模行为，考虑到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一个动态系统，捕获传统的流体动力学。术语如对流和扩散，然后可以施加到该网络上以模拟血流，然后可以适当地改变底层网络，可以通过 TDA 的技术来测量底层拓扑的变化，并且可以通过运行来测量对血流的影响。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增加方向性来影响流动。此外，在这种定向设置中，当底层网络结构发生变化时，流动如何响应？也不清楚什么是最测量有向网络拓扑的适当方法包括欧拉特征和 Lutgehetmann [2] 的 flagser 包，它计算有向标记持久复合体的同源性，或者，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，而这些基序对于 flagser 来说是无法区分的，与持久性的所有应用一样，这些方法的计算要求也是一个重要的考虑因素。同源性，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，可能需要使用多参数持久同源性。血管网络中血流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，属于“应用驱动的拓扑数据分析”。