Hodge decomposition of wall shear stress vector fields characterizing biological flows.

Faniry H Razafindrazaka,Konrad Polthier,Leonid Goubergrits,Konstantin Poelke,Pavlo Yevtushenko

doi:10.1098/rsos.181970

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 centre, 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 nature 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 control patients was done. The study shows a significant difference between the CoA patients before and after the treatment, and the healthy controls. 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.

Highlights

Biological flows or haemodynamics of the cardiovascular system play an important role in the genesis, progress and treatment of& 2019 The Authors
The major haemodynamic parameter describing an interaction between haemodynamics and a vessel wall, which is covered by endothelial cells, is the wall shear stress (WSS)
We focus on a discretization by piecewise constant vector fields (PCVF) resulting from computational fluid dynamics (CFD)-based analyses of the blood flow

Summary

Introduction

Biological flows or haemodynamics of the cardiovascular system play an important role in the genesis, progress and treatment of& 2019 The Authors. Abnormal WSS is considered an important local risk factor for a set of diseases or pathological processes. These include, for example, atherosclerosis of carotid arteries [3] or coronary artery disease [4], rupture risk of cerebral aneurysms [5,6] or abdominal aortic aneurysms [7], aortic dilatation [8] and thrombus formation [9]. The WSS is a surface-bounded vector field with vector magnitudes and directions varying in space and time This allows for a definition of a set of parameters, which were proposed during the last years as haemodynamic risk parameters for endothelial dysfunction and related wall remodelling. The primary source of data for the WSS analysis, is computational fluid dynamics (CFD), since an accurate WSS assessment requires a high spatial resolution as shown by mesh independence studies for CFD solutions [16]

Methods

Results

Discussion

Conclusion