Topological data analysis of vascular disease: A theoreticalframework

Abstract

Vascular disease is a leading cause of death world wide and therefore the treatment thereof is critical. Understanding and classifying the types and levels of stenosis can lead to more accurate and better treatment of vascular disease. Some clinical techniques to measure stenosis from real patient data are invasive or of low accuracy.\n\nIn this paper, we propose a new methodology, which can serve as a supplementary way of diagnosis to existing methods, to measure the degree of vascular disease using topological data analysis. We first proposed the critical failure value, which is an application of the 1-dimensional homology group to stenotic vessels as a generalization of the percent stenosis. We demonstrated that one can take important geometric data including size information from the persistent homology of a topological space. We conjecture that we may use persistent homology as a general tool to measure stenosis levels for many different types of stenotic vessels.\n\nWe also proposed the spherical projection method, which is meant to allow for future classification of different types and levels of stenosis. We showed empirically using the spectral approximation of different vasculatures that this projection could provide a new medical index that measures the degree of vascular disease. Such a new index is obtained by calculating the persistence of the 2-dimensional homology of flows. We showed that the spherical projection method can differentiate between different cases of flows and reveal hidden patterns about the underlying blood flow characteristics, that is not apparent in the raw data. We showed that persistent homology can be used in conjunction with this technique to classify levels of stenosis.\n\nThe main interest of this paper is to focus on the theoretical development of the framework for the proposed method using a simple set of vascular data.