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.In 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.We 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.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.2 new analytic tool in data analysis, inspiring researchers in various applications [4,27,28]. 3The analysis with TDA is based on persistent homology driven by the given topological 4 space. Various forms of data from various applications are actively being used by 5 researchers via TDA for possibly finding new knowledges out of the given data set.6

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 various 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.

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