A Eulerian method to analyze wall shear stress fixed points and manifolds in cardiovascular flows
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ORIGINAL PAPER
A Eulerian method to analyze wall shear stress fixed points and manifolds in cardiovascular flows Valentina Mazzi1,2 · Diego Gallo1,2 · Karol Calò1,2 · Mehdi Najafi4 · Muhammad Owais Khan3 · Giuseppe De Nisco1,2 · David A. Steinman4 · Umberto Morbiducci1,2 Received: 5 June 2019 / Accepted: 8 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract Based upon dynamical systems theory, a fixed point of a vector field such as the wall shear stress (WSS) at the luminal surface of a vessel is a point where the vector field vanishes. Unstable/stable manifolds identify contraction/expansion regions linking fixed points. The significance of such WSS topological features lies in their strong link with “disturbed” flow features like flow stagnation, separation and reversal, deemed responsible for vascular dysfunction initiation and progression. Here, we present a Eulerian method to analyze WSS topological skeleton through the identification and classification of WSS fixed points and manifolds in complex vascular geometries. The method rests on the volume contraction theory and analyzes the WSS topological skeleton through the WSS vector field divergence and Poincaré index. The method is here applied to computational hemodynamics models of carotid bifurcation and intracranial aneurysm. An in-depth analysis of the time dependence of the WSS topological skeleton along the cardiac cycle is provided, enriching the information obtained from cycle-average WSS. Among the main findings, it emerges that on the carotid bifurcation, instantaneous WSS fixed points co-localize with cycle-average WSS fixed points for a fraction of the cardiac cycle ranging from 0 to 14.5% ; a persistent instantaneous WSS fixed point confined on the aneurysm dome does not co-localize with the cycle-average low-WSS region. In conclusion, the here presented approach shows the potential to speed up studies on the physiological significance of WSS topological skeleton in cardiovascular flows, ultimately increasing the chance of finding mechanistic explanations to clinical observations. Keywords Topological skeleton · Wall shear stress divergence · Computational fluid dynamics · Vascular disease · Carotid bifurcation · Intracranial aneurysm
1 Introduction
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10237-019-01278-3) contains supplementary material, which is available to authorized users. * Umberto Morbiducci [email protected] 1
Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Turin, Italy
2
PolitoBIOMed Lab, Politecnico di Torino, Turin, Italy
3
Department of Pediatrics, Stanford University, Stanford, CA, USA
4
Biomedical Simulation Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada
A large body of literature has demonstrated the importance of Wall Shear Stress (WSS) in the onset and progression of cardiovascular di
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