Hodge Decomposition of the wall shear stress vector fields characterizing biological flows

TL;DR

This paper introduces a novel discrete Hodge decomposition method for analyzing wall shear stress vector fields in biological flows, enabling better differentiation between healthy and diseased aortic conditions.

Contribution

It develops a boundary-sensitive Hodge decomposition framework on manifolds with boundary, providing new insights into blood flow patterns and disease classification.

Findings

Decomposition distinguishes harmonic blood flow from interior topology effects.

Significant differences in WSS components between CoA patients and controls.

Method enhances classification of physiological and pathological blood flows.

Abstract

A discrete boundary-sensitive Hodge decomposition is proposed as a central tool for the analysis of wall shear stress (WSS) vector fields in aortic blood flows. The method is based on novel results for the smooth and discrete Hodge-Morrey-Friedrichs decomposition on manifolds with boundary and subdivides the WSS vector field into five components: gradient (curl-free), co-gradient (divergence-free), and three harmonic fields induced from the boundary, which are called the center, Neumann and Dirichlet fields. First, an analysis of WSS in several simulated simplified phantom geometries (duct and idealized aorta) was performed in order to understand the impact of the five components. It was shown that the decomposition is able to distinguish harmonic blood flow arising from the inlet from harmonic circulations induced by the interior topology of the geometry. Finally, a comparative analysis of 11 patients with coarctation of the aorta (CoA) before and after treatment as well as 10 controls patient was done. The study shows a significant difference between the CoA patients and the healthy controls before and after the treatment. This means a global difference between aortic shapes of diseased and healthy subjects, thus leading to a new type of WSS-based analysis and classification of pathological and physiological blood flow.