Topological Data Analysis of Vascular Disease: A Theoretical Framework - Nicponski, John; Jung, Jae-Hun (Frontiers in Applied Mathematics and Statistics, 2020)
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. In this paper, we propose a new methodology using topological data analysis, which can serve as a supplementary way of diagnosis to currently existing methods. We show that we may use persistent homology as a tool to measure stenosis levels for many different types of stenotic vessels. We first propose the critical failure value, which is an application of the $1$-dimensional homology to stenotic vessels as a generalization of the percent stenosis. We then propose the spherical projection method, which is meant to allow for future classification of different types and levels of stenosis. We use the $2$-dimensional homology of the spherical projection and showed that it can be used as a new index of vascular characterization. The 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.