The physiopathological role of Wall Shear Stress (WSS) in intracranial aneurysm development/rupture and the action of contraction/expansion played by shear forces on vessel wall make topological skeleton analysis of the WSS vector field of great interest. Here we present a practical way to analyze WSS topological skeleton through the identification and classification of WSS fixed points and manifolds. The method is based on the calculation of the WSS vector field divergence and Poincarè index, and it is here successfully applied to a dataset computational hemodynamic models of intracranial aneurysms

Calò, K.

Gallo, D.

Khan, M. O.

Mazzi, V.

Morbiducci, U.

Steinman, D. A.

English

The physiopathological role of Wall Shear Stress (WSS) in intracranial aneurysm development/rupture and the action of contraction/expansion played by shear forces on vessel wall make topological skeleton analysis of the WSS vector field of great interest. Here we present a practical way to analyze WSS topological skeleton through the identification and classification of WSS fixed points and manifolds. The method is based on the calculation of the WSS vector field divergence and Poincarè index, and it is here successfully applied to a dataset computational hemodynamic models of intracranial aneurysms

V. Mazzi

D. Gallo

K. Calò

M. O. Khan

D. A. Steinman

U. Morbiducci

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

04 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEA Practical Approach for Wall Shear Stress Topological Skeleton Analysis Applied to Intracranial AneurysmHemodynamics / Mazzi, V.; Gallo, D.; Calò, K.; Khan, M. O.; Steinman, D. A.; Morbiducci, U.. - ELETTRONICO. - (2019),pp. 9-10. ((Intervento presentato al convegno ESB-ITA 2019 IX Annual Meeting tenutosi a Bologna nel 30/09/2019-01/10/2019.OriginalA Practical Approach for Wall Shear Stress Topological Skeleton Analysis Applied to IntracranialAneurysm HemodynamicsPublisher:PublishedDOI:Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/2763273 since: 2019-10-23T16:40:18Z2019 Italian Chapter of the European Society of BiomechanicsESB-ITA Meeting 2019, 30 September – 1 October 2019, Bologna, Italy 1 Abstract— The physiopathological role of Wall Shear Stress (WSS) in intracranial aneurysm development/rupture and the action of contraction/expansion played by shear forces on vessel wall make topological skeleton analysis of the WSS vector field of great interest. Here we present a practical way to analyse WSS topological skeleton through the identification and classification of WSS fixed points and manifolds. The method is based on the calculation of the WSS vector field divergence and Poincarè index, and it is here successfully applied to a dataset computational hemodynamic models of intracranial aneurysms.   Keywords—Wall shear stress, topological skeleton, divergence. I. INTRODUCTION large body of literature has demonstrated the importance of  WSS in the onset and progression of cardiovascular diseases [1]. In particular, WSS has been proposed as an indicator of rupture risk in intracranial aneurysms, characterized by a complex hemodynamics. However, the ability of WSS-based indicators to predict aneurysm wall pathophysiology and remodelling, and to discriminate aneurysms rupture risk, is still debated due to large variations and frequent contradictory results from different studies [2]. An in-depth analysis is needed, to close the gap of knowledge currently limiting the use of WSS as a biomarker for diagnostic and prognostic purposes. In this context, recent studies have highlighted the relevance of WSS fixed points, and the stable and unstable manifolds that connect them [3]. The relevance of these WSS topological features lies in their ability to highlight the complex and highly dynamic features of the WSS field, as well as their strong link with flow features like flow stagnation, separation, and recirculation. Technically, a fixed point of a vector field is a point where the vector field vanishes, while unstable/stable vector field manifolds identify contraction/expansion regions linking fixed points. The presence of WSS fixed points and of WSS contraction/expansion regions, highlighted by WSS manifolds, might induce focal vascular responses relevant for aneurysm rupture [3]. For these reasons, the topological skeleton analysis of the WSS vector field is of great interest and motivates the study present herein.  Lagrangian-based approaches have been recently proposed to identify WSS manifolds but have certain practical limitations [4]. A Eulerian approach has also been suggested, but only for 2D analytical fields [5]. Here we propose and demonstrate the use of a simple Eulerian approach for identifying WSS topological skeleton on 3D surfaces. The proposed method is applied to eight personalized computational hemodynamic models of intracranial aneurysm. II. METHODS A. Computational Hemodynamics Eight ICA aneurysm computational hemodynamics models enrolled in a broader study of 3D phase contrast MRI (PC-MRI) were considered. Details on geometry reconstruction, personalized conditions at boundaries and CFD simulations are reported elsewhere [6].  B. Practical way for WSS topological skeleton identification The Volume Contraction theory demonstrates that the divergence of vector fields gives practical information on the associated dynamical systems, avoiding numerical integration for manifolds identification (as required by Lagrangian-based approaches), thus reducing the computational effort. In particular, the divergence is able to (1) encase stable/unstable manifolds, and (2) identify the basins of attractions of each attractor. For those reasons, a divergence-based approach for WSS manifolds identification at the luminal surface of intracranial aneurysms is proposed. As WSS divergence depends upon the algebraic summation of the magnitude of single gradients of WSS vector components, it might fail in properly identifying expansion/contraction regions. To overcome this limitation, here the divergence of the normalized WSS vector field was considered: ∇ ⋅ (𝛕𝐮) = ∇ ⋅ (𝝉∥𝝉∥2),                                   (1) where 𝛕𝐮 is the WSS unit vector. Eq. (1), neglecting the vector field magnitude variation but taking into account variation of directions of the vector field only, correctly identifies WSS manifolds and therefore is suitable for practical WSS topological analysis. To complete the analysis, we propose a robust method for WSS fixed points identification. The Poincarè index is considered here for WSS fixed points identification because of its mesh-independent and topologically invariant proprieties. Then, a Jacobian analysis of WSS fixed points allows to classify the fixed point attractive or repelling nature. The proposed practical approach for the WSS topological skeleton identification is applied to both cycle-average WSS vector field ?̅?(𝒙):  ?̅?(𝒙) =1𝑇∫ 𝛕(𝑡, 𝒙)𝑑𝑡𝑇0                                 (2) where T is the cardiac cycle duration, and instantaneous WSS vector fields at the luminal surface. It was previously suggested that cycle-average WSS vector field ?̅? fixed points and their associated manifolds influence the near-wall intravascular transport [4]. However, it is clear from Eq. (2) A Practical Approach for Wall Shear Stress Topological Skeleton Analysis Applied to Intracranial Aneurysm Hemodynamics V. Mazzi1, D. Gallo1, K. Calò1, M. O. Khan2, D. A. Steinman3, and U. Morbiducci1 1𝑃𝑜𝑙𝑖𝑡𝑜𝐵𝐼𝑂 Med Lab, Department of Mechanical and Aerospace Engineering Politecnico di Torino, Turin, Italy  2Cardiovascular Biomechanics Computation Lab, Department of Pediatrics, Stanford University Stanford, CA, USA 3Biomedical Simulation Laboratory, Department of Mechanical & Industrial Engineering University of Toronto Toronto, ON, Canada A ESB-ITA Meeting 2019, 30 September – 1 October 2019, Bologna, Italy 2 that it could be possible by construction that a ?̅? fixed point would have never been a real WSS fixed point all along the cardiac cycle, thus stimulating a more in-depth investigation focusing on the time dependence of WSS fixed points and manifolds location and nature along the cardiac cycle.  Hence, WSS fixed points analysis is applied here to instantaneous WSS vector field and a measure to quantify the fraction of cardiac cycle spent by instantaneous WSS fixed points at a specific location at the luminal surface is proposed: 𝑅𝑇𝑥𝑓𝑝(𝑒) =?̅?𝐴𝑒1𝑇∫ 𝕀𝑒(𝒙𝑓𝑝 , 𝑡)𝑇0 𝑑𝑡                           (4) where 𝒙𝒇𝒑(𝑡) is the WSS fixed point position at time t ∈[0, T], e is the generic triangular element of the superficial mesh of area 𝐴𝑒, ?̅? the average surface area of all triangular elements of the superficial mesh and 𝕀 is the indicator function.  Moreover, a modified version of Eq. (4) is proposed here, where the residence time of a fixed point is weighted by the absolute value of the instantaneous WSS vector divergence, intended as a measure of the strength of the local contraction/expansion action of local shear forces:  𝑅𝑇𝛻xfp(e) =?̅?𝐴𝑒1T∫ 𝕀𝑒(𝒙fp, t)T0|(∇ · 𝛕)𝑒| dt              (5) where (∇ · 𝛕)𝑒 is the instantaneous WSS divergence computed in the surface triangle e containing the instantaneous WSS fixed point.  III. RESULTS A. Analysis of the Cycle-Average WSS Vector Field An analytical vector field was used for benchmarking purposes. The proposed method was compared to the classical vector field integration approach [7] and to the recent trajectory-free method [5], providing excellent results. Then, the cycle-average WSS vector field at the luminal surface of the 8 intracranial aneurysm models was analyzed. The cycle-average WSS topological skeleton of one explanatory intracranial aneurysm model is presented in Figure 1.  Figure 1: Topological skeleton of cycle-average WSS vector. The contraction and expansion patterns, identifying unstable and stable manifolds, represent the basins of attraction for the stable fixed points associated with the manifolds. Notably, all cycle-average WSS stable fixed points at the luminal surface of the 8 intracranial aneurysm models were located within contraction regions, while the unstable node identified was located within an expansion region, thus confirming the appropriateness of the proposed method. B. Analysis of instantaneous WSS vector field  For an in-depth characterization of the WSS fixed points, quantities 𝑅𝑇xfp(𝑒) and 𝑅𝑇𝛻xfp(e) were computed on the surface of all models and an explanatory example is presented in Figure 2. For visualization purposes, regions of interest 𝑅𝑓𝑝 were identified at the luminal surface around high 𝑅𝑇xfp(𝑒) areas and including the identified cycle-average WSS fixed points locations (labeled from 𝐴𝐴 to 𝐺𝐴). For the explanatory model, the results of the WSS fixed points residence times analysis clearly show that: (1) in regions 𝑅𝐸𝐴  and 𝑅𝐷𝐴 fixed points residence times were up to 70% of the cardiac cycle; (2) instantaneous WSS fixed points resided for moderate fractions of the cardiac cycle (range 3.6-58.2%) in cycle-average WSS fixed points identified locations; (3) the instantaneous WSS fixed points are always of the same nature as cycle-average WSS fixed points; (4) elevated values of the strength of the expansion/contraction of the WSS vector field around instantaneous WSS fixed points are observed on the aneurysmal dome and close to the neck region, reflecting the intricate hemodynamics characterizing these regions.  Figure 2: Maps of fixed points residence time 𝑹𝑻𝒙𝒇𝒑(𝒆) and of measure 𝑹𝑻𝜵𝒙𝒇𝒑(𝒆). IV. CONCLUSION A practical approach to analyze WSS topological skeleton was introduced and applied to computational hemodynamic models of intracranial aneurysm. The proposed approach requires WSS vector field and its divergence only and it can be easily applied to 3D vector fields defined on complex geometries. The analysis of instantaneous WSS fixed points along the cardiac cycle allows to evaluate their residence time and is the strength of local contraction/expansion using WSS divergence. Our findings on intracranial aneurysms underline the importance of focusing on instantaneous WSS fixed points analysis. The approach proposed here could contribute to facilitate and speed up studies on the significance of WSS topological skeleton in cardiovascular flows. REFERENCES [1] U. Morbiducci et. al, “Atherosclerosis at arterial bifurcations: evidence for the role of haemodynamics and geometry”, Thrombosis and Haemostasis 115(03), 2016, pp. 484-492. [2] L. Liang et al., “Towards the Clinical utility of CFD for assessment of intracranial aneurysm rupture - a systematic review and novel parameter-ranking tool.” J. Neurointervent Surg. 11, 2018. pp:153-158. [3] A. Arzani and S.C. Shadden, “Wall shear stress fixed points in cardiovascular fluid mechanics.” J Biomech. 73, 2018, pp: 145-152. [4] A. Arzani, A.M. Gambaruto, G. Chen, S.C. Shadden, “Lagrangian wall shear stress structures and near-wall transport in high-Schmidt-number aneurysmal flows.” J Fluid Mech.  790, 2016, pp: 158-172. [5] G.K. Nave Jr, P.J. Nolan, S.D. Ross, “Trajectory-free approximation of phase space structures using the trajectory divergence rate,” Nonlinear Dynamics, 2018 [6] C. Chnafa et al., “Errors in power-law estimations of inflow rates for intracranial aneurysm CFD,” J Biomech. 80, 2018, pp: 159-165. [7] A.M. Gambaruto, D.J. Doorly, T. Yamaguchi, “Flow structures in cerebral aneurysms,” Computer & Fluids, 2012, pp: 56-65. 

A Practical Approach for Wall Shear Stress Topological Skeleton Analysis Applied to Intracranial Aneurysm Hemodynamics

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A Practical Approach for Wall Shear Stress Topological Skeleton Analysis Applied to Intracranial Aneurysm Hemodynamics

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