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Page 1: A haemodynamic predictor of intraluminal thrombus

, 20140163, published 8 October 2014470 2014 Proc. R. Soc. A P. Di Achille, G. Tellides, C. A. Figueroa and J. D. Humphrey thrombus formation in abdominal aortic aneurysmsA haemodynamic predictor of intraluminal  

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ResearchCite this article: Di Achille P, Tellides G,Figueroa CA, Humphrey JD. 2014 Ahaemodynamic predictor of intraluminalthrombus formation in abdominalaortic aneurysms. Proc. R. Soc. A 470: 20140163.http://dx.doi.org/10.1098/rspa.2014.0163

Received: 27 February 2014Accepted: 13 August 2014

Subject Areas:biomedical engineering, computer modellingand simulation

Keywords:platelet activation, endothelial dysfunction,computational fluid dynamics, thrombosis

Author for correspondence:J. D. Humphreye-mail: [email protected]

Electronic supplementary material is availableat http://dx.doi.org/10.1098/rspa.2014.0163 orvia http://rspa.royalsocietypublishing.org.

A haemodynamic predictorof intraluminal thrombusformation in abdominalaortic aneurysmsP. Di Achille1, G. Tellides2,3, C. A. Figueroa4

and J. D. Humphrey1,3

1Department of Biomedical Engineering, Yale University,New Haven, CT, USA2Department of Surgery, and 3Vascular Biology and TherapeuticsProgram, Yale School of Medicine, New Haven, CT, USA4Department of Surgery and Biomedical Engineering,University of Michigan, Ann Arbor, MI, USA

Intraluminal thrombus (ILT) is present in over 75%of all abdominal aortic aneurysms (AAAs) andprobably contributes to the complex biomechanicsand pathobiology of these lesions. A reliable predictorof thrombus formation in enlarging lesions couldthereby aid clinicians in treatment planning. Theprimary goal of this work was to identify a newphenomenological metric having clinical utility thatis motivated by the hypothesis that two basichaemodynamic features must coincide spatially andtemporally to promote the formation of a thrombuson an intact endothelium—platelets must be activatedwithin a shear flow and then be presented toa susceptible endothelium. Towards this end, wepropose a new thrombus formation potential (TFP)that combines information on the flow-induced shearhistory experienced by blood-borne particles thatcome in close proximity to the endothelium withinformation on both the time-averaged wall shearstress (WSS) and the oscillatory shear index (OSI)that locally affect the endothelial mechanobiology. Toillustrate the possible utility of this new metric, weshow computational results for 10 carotid arteriesfrom five patients where regions of low WSS andhigh OSI tend not to be presented with activatedplatelets (i.e. they have a low TFP), consistent withthe thrombo-resistance of the healthy carotid despiteits complex haemodynamics. Conversely, we show

2014 The Author(s) Published by the Royal Society. All rights reserved.

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results for three patients that high TFP co-localizes with regions of observed thin thrombusin AAAs, which contrasts with findings of low TFP for the abdominal aorta of three healthysubjects. We submit that these promising results suggest the need for further consideration ofthe TFP, or a similar combined metric, as a potentially useful clinical predictor of the possibleformation of ILT in AAAs.

1. IntroductionAbdominal aortic aneurysms (AAAs) are responsible for significant morbidity and mortality,particularly in older males. Over 75% of these lesions contain an intraluminal thrombus (ILT),which, in many cases, fills the lesion and yields a lumen of nearly normal size despite theextreme dilatation of the aortic wall. There remains a significant controversy over the preciseroles an ILT may play in the enlargement and potential rupture of these aneurysms, but it appearsthat thrombus is important both biomechanically and biochemically [1–3]. There is, therefore, apressing need to understand better the conditions that lead to the formation of an ILT within thesepotentially life-threatening lesions.

The natural history of an ILT can be thought to consist of three basic phases: an initial changein platelet activity, the formation of an associated insoluble fibrin clot and possible fibrinolysis.The first phase actually involves three steps as well: the activation, adhesion and aggregationof platelets. Many prior studies have focused on specific aspects of the mechanisms underlyingthrombus formation within a general haemodynamic field as revealed by multiple papers [4–6].In particular, there has been an appropriate move towards multiscale models wherein one cancapture contributions ranging from molecular-level chemical reaction kinetics to macroscopichaemodynamics. Notwithstanding these many advances, Cito et al. [6, p. 122] correctly concludethat ‘there is not any validated tool capable of informing clinicians in terms of predictivediagnostics and surgical planning of diseases wherein blood clotting plays a relevant role,such as aneurysm’.

In this paper, we adopted a different, complementary, approach to understand conditions thatcan lead to thrombus development in AAAs. Rather than including complexities of the molecularmechanisms, we sought a phenomenological metric that could reveal a haemodynamic thresholdthat stratifies the risk of forming a thrombus within a complex unsteady flow field. Moreover,different from most prior studies that focus on stenoses, ruptured atherosclerotic plaques orimplanted devices wherein thrombus tends to form, we focused first on a region of the normalvasculature that is not susceptible to thrombus formation despite its complex geometry andhaemodynamics. That is, we suggest that one can unveil lower bounds of useful haemodynamicmetrics by identifying values for which thrombus does not occur and then contrast these valueswith those associated with clinical cases wherein thrombus has been observed. Towards thisend, we focused first on the non-thrombogenic human carotid artery, including the bifurcationand the sinus, which is characterized by a local dilatation of the wall, and then examined threecounter-examples based on AAAs harbouring a thin ILT. To complete the study, we also examinedthree models of healthy abdominal aorta free of thrombus. From these three cases, we founda new ‘thrombus formation potential’ (TFP) that appears to predict the location of thrombusformation within AAAs while explaining why normal regions of the arterial tree typically remainthrombus free.

2. Material and methodsRetrospective de-identified computed tomography (CT) or magnetic resonance (MR) images werecollected for 11 patients, five to study the paired carotid arteries, three to study non-thromboticinfrarenal aortas and three to study AAAs harbouring a hint of ILT.

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Table 1. Imaging modality and resolution of the 11 analysed datasets. C1–5 denote images from which the 10 carotid modelswere derived. IAA1–3 denote images from which the three healthy infrarenal abdominal aortic models were derived. AAA1–3denote images from which the three AAA models were derived.

patient imaging modality dimensions (voxels) resolution (mm)

C1 head CTA 512 × 512 × 130 0.506 × 0.506 × 2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C2 head CTA 512 × 512 × 561 0.438 × 0.438 × 0.625. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C3 head MRA 448 × 120 × 576 0.677 × 0.800 × 0.677. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C4 head CTA 512 × 512 × 289 0.391 × 0.391 × 0.625. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C5 head CTA 512 × 512 × 709 0.352 × 0.352 × 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IAA1 abdominal CTA 512 × 512 × 743 0.559 × 0.559 × 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IAA2 abdominal MRA 512 × 124 × 512 0.781 × 1.499 × 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IAA3 abdominal MRA 384 × 72 × 448 0.781 × 1.599 × 0.78. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AAA1 abdominal CTA 512 × 512 × 226 0.735 × 0.735 × 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AAA2 abdominal CTA 512 × 512 × 288 0.742 × 0.742 × 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AAA3 abdominal CTA 512 × 512 × 1534 0.781 × 0.781 × 0.301. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(a) Model construction: carotid arteriesLuminal geometries of 10 carotid arteries (two per subject) were extracted from four CT andone MR angiography scans (table 1) of patients under observation for conditions not directlyrelated to carotid disease (average common carotid radius approx. 3.09 ± 0.20 mm). Modest wallcalcifications were found in the right carotid artery of Patient C2, but all arteries were free ofstenosis. All patients presented a low degree of tortuosity in the carotid bifurcation except PatientC3, but thrombus was absent in all five patients.

The five pairs of images were first processed using a three-dimensional level-set segmentationalgorithm available in the open source Vascular Modeling Toolkit (VMTK) [7]. In order tominimize the impact of boundary effects in the haemodynamic simulations, we consistentlyincluded extended segments of the common and internal carotid arteries; the external carotidartery was segmented up to its first branch. VMTK was also used to measure select geometricfeatures such as the radius of the common carotid artery, defined as the average of multiplemeasurements taken along the vessel centreline, and the bifurcation angle between theinternal and external carotids [8]. Subsequently, the processed level-set images were importedinto a customized version of the open source code SimVascular [9], where two-dimensionalsegmentations were extracted at several cross sections via thresholding. Non-uniform rationalB-spline (NURB) surfaces were then lofted for each vessel through the segmented contours andfinally merged into a complete geometric model (figure 1). A blending procedure was then used tosmooth the junctions between the internal and external carotid arteries and thereby to minimizesegmentation artefacts in the simulations.

The geometric model was then meshed using tetrahedral elements of nominal size 0.3 mmusing the MeshSim library (Simmetrix Inc., Clifton Park, NY, USA) included in SimVascular. Tocapture the steep gradients in velocity close to the arterial wall, which was modelled as rigid,we meshed such regions at an increased resolution using four boundary layers of graduallydecreasing thickness. Curvature-based refinement was also used in regions of high curvaturesuch as the junctions between the internal and external carotids. On average, the meshesincluded about 1.7 million elements (1 724 027 ± 307 495 elements), with variations attributableto differences in vessel sizes and curvatures.

Given the similar sizes of the 10 carotid models and the lack of patient-specific measurementsof blood flow, we applied the same inflow boundary conditions to all models to simulate both rest

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CTA or MRA geometric modellofting

SimVascular

three-dimensionallevel-setVMTK

two-dimensionalsegmentationSimVascular

caro

tids

IAA

s or

AA

As

Figure 1. Patient-specific models of carotid arteries, including the bifurcation and sinus, healthy infrarenal abdominal aortas(IAAs) and AAAs were reconstructed from medical images (CTA and MRA). An automatic three-dimensional segmentationalgorithm was used to produce level-set images that were then thresholded to extract lumen boundaries at several crosssections. The solid models were obtained via NURBS interpolation. In the rightmost column, two illustrative reconstructedgeometries are shown overlapped on volume renderings of the original medical images. (Online version in colour.)

and moderate exercise conditions. Specifically, the inlet flow waveform reported by Marshall et al.[10] was scaled according to mean flow rates collected by Sato et al. [11] on 10 healthy patientsunder rest and moderate exercise (40% of peak oxygen uptake) conditions. The Womersleysolution for pulsatile flow in rigid wall cylinders [12] was then used to prescribe the inlet profiles[13,14]. To mitigate uncertainty on the assumed heart rates, again due to the lack of patient-specific data, we constructed a sequence of four cardiac cycles whose durations varied by 0.1 sabout the mean values reported by Sato et al. [11]. The sequences of four variable consecutivecycles (0.9–1.0–0.9–0.8 s and 0.6–0.7–0.6–0.5 s for rest and stress conditions, respectively) wasiterated twice, thus a total of eight cardiac cycles were simulated for each geometry at rest andmoderate exercise. As a comparison, simulations were also performed over eight cycles at aconstant duration of 0.9 and 0.6 s, respectively.

At the outlets, we applied a three-element Windkessel model to replicate resistance (R) andcapacitance (C) effects of the distal vasculature that were not included directly in the three-dimensional domain [15]. The total resistance was first determined to match the mean arterialpressures reported by Sato et al. [11] for rest and exercise (87 and 103 mm Hg respectively), andthen distributed among internal and external carotid arteries to reproduce the flow splits reportedin the same source (QECA/QICA = 0.54 and 0.58 at rest and exercise, respectively). Similarly, thevalue for total capacitance was taken from the literature [16] and then distributed between theECA and ICA using the inverse of the coefficient used to split the resistance (table 2).

(b) Model construction: healthy infrarenal aortas and abdominal aortic aneurysmsThe luminal geometries of the abdominal aorta and its main branches were extracted from threeimage datasets of healthy patients and three abdominal CT angiography scans of patients withAAAs partially covered by a thin layer (less than 3 mm) of ILT. The procedure was similar

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Table 2. Inlet and outlet boundary conditions. Superscripts ‘r’ and ‘e’ indicate values used for rest andmild exercise simulations,respectively. Outlet conditions in healthy IAAs and AAAs were assumed to be the same for both conditions (hence, nosuperscript).

inlet flow rate Rp C Rdmodel (mm3 s−1) outlet (g mm−4 s−1) (g−1 mm4 s2) (g mm−4 s−1)

carotids 6050r 8330e int. carotid 0.295r 0.310e 0.662r 0.645e 2.66r 2.79e

ext. carotid 0.547r 0.533e 0.358r 0.375e 4.92r 4.80e. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IAAs and AAAs 52 800r 109 000e celiac 0.113 0.100 1.91. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

splenic 0.265 0.128 4.46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

left gastric 0.530 0.060 8.92. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

sup. mesenteric 0.083 0.657 1.39. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

left renal 0.365 0.600 1.46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

right renal 0.307 0.712 1.23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

left iliac 0.826 0.029 13.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

left ex. iliac 0.084 0.287 1.42. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

right iliac 0.879 0.027 14.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

right Ex. Iliac 0.084 0.281 1.42. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

to the segmentation of the carotid bifurcations (figure 1). Briefly, level-set images obtainedthrough three-dimensional segmentation in VMTK were imported into SimVascular to build aNURBS computational model. The final models included part of the suprarenal aorta, healthy oraneurysmal infrarenal aorta, superior mesenteric artery, celiac trunk, and renal and iliac arteries.The computational domains were discretized into tetrahedral elements of 1.0 mm maximum edgelength. We used three boundary layers to resolve the potentially sharp gradients in velocity inregions next to the wall, which was again modelled as rigid. Curvature-adaptive meshing wasused to reduce the size of the elements in smaller vessels and at bifurcations. The average numberof elements for these models was 5 949 585 ± 346 503.

The boundary condition at the supraceliac inlet was prescribed by imposing the averagedvolumetric flow rate and waveform obtained by Les et al. [17] from PC-MRI measurements on36 patients affected by AAA. Stress conditions were simulated using the waveform and flow ratereported by Suh et al. [18] for mild exercise (table 2). Owing to the lack of data on flow splitsin the abdominal region for the degree of exercise considered here, the Windkessel parametersat rest were kept unchanged for exercise conditions. Similar to the analyses for the carotids,simulations were performed for eight cardiac cycles of varying duration. The average cycle atrest and exercise was 0.9 and 0.6 s corresponding to heart rates of approximately 67 and 100beats per minute, respectively. A Womersley velocity profile was prescribed at the inlet [17–19]and Windkessel RCR boundary conditions were imposed at all outlets using values reported inthe full-body three-dimensional model of Xiao et al. [16] (table 2). To avoid instabilities in thenumerical solution in situations of back flow, we weakly constrained the shape of the velocityprofiles to be parabolic at the outlets of the iliac and superior mesenteric arteries using theaugmented-Lagrangian technique developed by Kim et al. [20].

(c) Finite-element modelsThe time-dependent velocity and pressure fields were obtained by solving the incompressibleNavier–Stokes equations with the stabilized finite-element method implemented in SimVascular

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[21,22]. Blood was modelled as a Newtonian fluid with constant viscosity (0.004 Pa s) and density(1060 kg m−3). All arterial walls were assumed to be rigid, a simplifying assumption dictated bythe scarcity of information on regionally varying material properties of the carotid (common,internal and external carotids) and different abdominal arteries; if our proposed metric, TFP,were to be adopted clinically, such an assumption would invariably be made as well due to thegeneral lack of patient-specific information on regional wall properties. As we discuss below,the rigid wall assumption actually leads to a conservative estimate of the TFP, which is alsodesirable clinically. Each cardiac cycle was subdivided in time steps of 0.0001 s. Snapshots of thecomputational results were saved every 50 time steps (i.e. every 0.005 s). To eliminate the effect ofinitial transients, only results collected during the last four cycles were used for post-processingand particle tracking. All the simulations were run on the Omega supercomputer of the Yale HighPerformance Computing facility, where we employed on average 512 cores.

(d) Haemodynamic wall parameters and endothelial cell activationConsiderable attention has been directed over the past few decades towards characterizing effectsthat different haemodynamic stimuli may have on the endothelial cell gene expression thatleads to either pro-atherosclerotic or pro-thrombogenic phenotypes. For example, several indicesbased on wall shear stress (WSS) and its temporal and spatial variations have been proposedto capture these mechanobiological effects with different degrees of success [23]. We evaluatedmultiple indices (e.g. WSS, relative residence times, particle residence times near the wall) andcombinations thereof, but ultimately combined two of the most commonly employed ones tobetter localize endothelial regions exposed to both low and oscillatory shear flows, conditionsthat promote a pro-thrombogenic phenotype [24,25]. The time-averaged magnitude of the WSS(TAWSS) is defined as

TAWSS = 1T

∫T

0|τw| dt, (2.1)

where T is the total duration of the sequence of the last four simulated cardiac cycles (e.g. 3.6and 2.4 s, respectively, for rest and stress conditions), and |τw| is the Euclidean norm of the WSS(actually in-plane components of the traction vector) on the luminal surface. To facilitate thecomparison between the different simulations, and to reduce the effects of imposing the sameboundary conditions for all models, nodal values of TAWSS (having units of Pa) were normalizedby the average value of the most proximal segment of the domain (i.e. the common carotid arteryin carotid simulations and the supraceliac portion of the aorta in infrarenal abdominal aorta (IAA)and AAA simulations), then denoted as TAWSS.

The oscillatory shear index (OSI) has been proved successful in identifying atheroproneregions of the vasculature [26,27]. It is defined as follows:

OSI = 12

⎛⎝1 −

∣∣∣∫T0 τwdt

∣∣∣∫T

0 |τw|dt

⎞⎠ . (2.2)

This dimensionless scalar index assumes its maximum value (0.5) within regions where theWSS field changes directions due to complex flow patterns. For example, a purely oscillatoryflow with equal forward and backward contributions will produce an OSI of 0.5; conversely, inunidirectional flows in regular geometries, the OSI will be identically zero.

In an attempt to localize regions of the wall exposed to both high OSI and low TAWSS,we propose using the ratio of these two indices to characterize the degree of ‘thrombogenicsusceptibility’ of the vessel wall. We refer to this metric as the endothelial cell activation potential(ECAP), namely

ECAP = OSI

TAWSS. (2.3)

Higher values of the ECAP index will thereby correspond to situations of large OSI and smallTAWSS, that is, of endothelial susceptibility.

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(e) Particle tracking and platelet activationComputational fluid dynamics (CFD) studies typically produce values of velocity and pressure atevery node of a fixed Eulerian computational domain. Such techniques do not provide, however,any information on the loading history that a fluid particle experiences along its path. Becausethe mechanisms that trigger platelet activation can involve prolonged mechanical stimulation[28–30], a Lagrangian interpretation of the flow field can help discern which endothelial regionsare more likely to receive flow rich in activated platelets [31]. Hence, we implemented a particletracking procedure to enrich our CFD results with information on the shear history of all particlesthroughout the flow domain. Following a strategy similar to the investigations of Lagrangiancoherent structures in the vasculature [19,32], we first re-meshed the computational domain toenhance the resolution of the particle tracking analysis (nominal edge size of 0.15 and 0.5 mmfor the carotid and the abdominal aorta simulations, respectively). The nodal coordinates ofthe newly refined meshes were used as initial particle injection sites in an in-house parallelparticle tracking code built on top of VTK and MPI open source libraries. Specifically, the codeintegrated the Lagrangian trajectories of fluid elements from the CFD velocity field using a fourth-order Runge–Kutta scheme. VTK classes (e.g. vtkCellLocator), endowed with methods for linearbasis interpolation on finite elements, were then used to extract the velocity vectors at arbitrarylocations within the domain. Owing to the discrete size of the integration time step, a smallpercentage of particles could erroneously be lost through the wall boundaries. Following theprocedure reported in Duvernois et al. [33], we added a small inward velocity component ofmagnitude 1 cm s−1 to the otherwise zero velocity vector at the wall nodes to limit the leakageof particles without significantly affecting the tracking.

For each fluid element and at each integration time step, we collected the values of thevelocity gradient and thus flow-induced shear encountered along a particle’s path. Suchinformation was used to compute a PLatelet Activation Potential (PLAP) recently proposedby Shadden & Hendabadi [31] to investigate the flow through an arterial stenosis. Briefly, thePLAP is a non-dimensional scalar index that represents the magnitude of shear rates that a fluidparticle accumulates while travelling throughout the fluid domain. In our analyses, we trackedparticles by looping twice over the four cardiac cycle sequence prescribed at the inlet. PLAP isdefined as

PLAP(x, t) =∫ t

t−2T|D(x(τ ), τ )| dτ , (2.4)

where |D(x(τ ), τ )| is the Frobenius norm of the symmetric part of the spatial gradient of velocitytensor, t is the time of injection of the particle and 2T indicates how long the particle hasbeen tracked (i.e. eight cardiac cycles in our case). Collecting particle information for multiplecardiac cycles allows one to capture flow stagnation events that might be of importance inthrombogenesis. If a particle were to exit the domain before completing the eight cycles,integration of its trajectory was interrupted and its PLAP congealed at the value computed atthe last time step before exiting the domain. Although the PLAP was calculated for all particles inthe fluid domain, our primary interest was in the trajectories of those particles that at some timecame close to the endothelial layer. Hence, we tracked all particles backwards in time and PLAPwas thereby computed on fluid elements whose positions at the end of the CFD simulations wereknown (and thus used as initial locations for particle tracking). After averaging results obtainedfor different injection times, we assigned final PLAP values to their corresponding nodes forfurther post-processing and plotting. Given that we were primarily interested in investigating thepresentation of activated platelets to the endothelium, we probed the PLAP field at each boundarynode at 20 equispaced points lying on a segment starting at the wall node and pointing towardsthe inward normal direction. Each vessel wall node was then assigned the averaged segmentalvalue of the PLAP. To take into account potential effects of the probing distance from the wall,we computed PLAP for segments of length equal to 5% and 10% of the local vessel radius. Lastly,similar to what done for TAWSS, PLAP values were also normalized individually for each CFD

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>5.00

4.00

3.00

2.00

1.00

0

ECAP

C1L C2L C3L C4L C5L

C1R C2R C3R C4R C5R

Figure 2. Colour-coded spatial distributions of the ECAP for all 10 carotid arteries: C1L and C1R denote the left and right carotids,respectively, of patient C1 and so forth. Within each box, the left plot shows simulations of rest conditions and the right plotshows simulations of mild exercise conditions. 99th percentile values are listed in table 3. (Online version in colour.)

model by a nominal value obtained by averaging PLAP on the most proximal segment of thecomputational domain (common carotid artery in carotid bifurcation analyses, and supraceliacaorta in aortic simulations).

(f) Thrombus formation potentialOf the numerous combinations of metrics considered for WSS and flow-induced plateletactivation, a simple multiplicative index, the TFP, was finally defined to combine on the onehand the WSS-based ECAP obtained via standard CFD computations and on the other handthe fluid shear history-based PLAP obtained by particle tracking. The main purpose of this newindex was to identify, if present, local regions of the wall that at the same time were exposed topro-thrombotic WSS stimuli and a flow rich in activated platelets. We defined TFP as

TFP = ECAP · PLAP = OSI · PLAP

TAWSS. (2.5)

3. Results

(a) Non-thrombogenic carotid arteriesThe ECAP highlights those regions where WSS is characterized by both a low intensity andpronounced changes in direction during a cardiac cycle. As expected, we observed high values ofECAP near the carotid bifurcation, where the flow was disturbed further by the sinus (i.e. a localdilatation). Conversely, the proximal segment of the common carotid and the distal segmentsof the internal and external carotids exhibited values of ECAP close to zero, confirming thatsuch regions are not exposed to disturbed flows. Figure 2 shows distributions of ECAP in all10 carotid arteries, both at rest and under mild exercise conditions. In all vessels save one, regionsof high ECAP were on the outer wall of the bifurcation; the highest values of ECAP were on theinner wall in C2R (patient C2, right carotid), which corresponded to a site of calcification in thatparticular artery.

In all cases, prescribing exercise conditions did not cause evident changes in terms ofthe magnitude or distribution of ECAP. Despite an increase in WSS under exercise, indexnormalization with respect to the proximal value of TAWSS was sufficient to maintain peakvalues of ECAP similar. Yet, exercise conditions led to a slight broadening of the region ofhigh ECAP towards the internal carotid and along the direction of flow in the bifurcation ofC5L. Interestingly, this vessel was characterized by the largest bifurcation angle (81◦), consistent

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Table 3. 99th percentile values of ECAP, PLAP and TFP. Values for healthy IAAs and AAAs were computed considering a clippeddomain that included the infrarenal and iliac arteries up to their first bifurcation. Italic values reveal the overall maxima of eachcolumn for each group.

index (99th percentile)

PLAP TFP

model geometry conditions ECAP 5% 10% 5% 10%

radius angle

carotid bifurcations (mm) (deg)

C1L 3.01 69 rest 1.45 2.01 2.06 0.92 1.31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 1.34 1.98 1.91 1.10 1.52. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C1R 2.98 54 rest 2.70 2.33 1.81 1.58 1.78. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 2.47 2.10 1.63 1.33 1.95. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C2L 3.11 76 rest 0.95 2.16 2.03 0.62 0.81. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.98 2.01 2.03 0.66 0.90. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C2R 3.12 57 rest 1.79 2.45 2.18 0.91 1.76. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 2.24 2.29 2.12 1.31 2.15. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C3L 3.16 56 rest 0.90 2.02 2.08 0.39 0.85. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.96 1.95 2.18 0.39 0.82. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C3R 3.01 65 rest 0.46 2.08 2.21 0.28 0.50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.68 2.02 2.25 0.44 0.78. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C4L 2.76 60 rest 0.72 1.96 2.29 0.50 0.70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.79 1.75 2.29 0.70 0.94. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C4R 2.98 43 rest 1.37 2.04 1.86 0.83 1.31. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 1.30 1.92 1.79 0.82 1.40. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C5L 3.35 81 rest 1.16 2.17 2.10 0.32 0.48. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 1.29 2.10 2.09 0.40 0.57. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

C5R 3.45 67 rest 1.27 2.09 1.95 0.72 1.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 1.62 2.09 1.94 1.29 1.89. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

carotid average rest 1.28 2.13 2.06 0.71 1.06. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 1.37 2.02 2.02 0.84 1.29. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

infrarenal aortas IAA radius (mm)

IAA1 7.70 rest 0.66 1.01 1.42 0.22 0.43. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.58 1.13 1.54 0.19 0.36. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IAA2 8.12 rest 0.53 1.58 2.31 0.70 0.97. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.41 1.66 2.39 0.52 0.76. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IAA3 7.65 rest 0.50 1.87 2.94 0.80 1.21. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.42 2.07 2.76 0.69 1.07. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

infrarenal aorta average rest 0.56 1.49 2.23 0.58. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.47 1.62 2.23 0.47. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

(Continued.)

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Table 3. (Continued.)

index (99th percentile)

PLAP TFP

model geometry conditions ECAP 5% 10% 5% 10%

aortic aneurysms IAA rad. (mm) max diam. (mm)

AAA1 12.3 43.5 rest 4.13 2.09 2.62 6.27 7.98. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 1.87 2.32 3.15 3.54 4.74. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AAA2 11.0 51.5 rest 1.16 3.19 2.90 3.22 3.02. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.88 3.14 3.25 2.25 2.48. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AAA3 12.1 51.8 rest 0.89 2.92 3.22 1.46 1.72. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 0.79 3.39 3.81 1.40 1.75. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

aneurysm average rest 2.06 2.73 2.91 3.65 4.24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

exercise 1.18 2.95 3.40 2.40 2.99. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

with the importance that geometric features have in determining the peculiar haemodynamics incarotids [34]. In particular, large bifurcation angles shield some regions of the proximal internalcarotid from high shear flows, which could give rise to complex haemodynamics in exerciseconditions as well. Table 3 shows the 99th percentile values of the ECAP index for all 10 carotidsimulations. We believe that analysing upper percentiles might be more convenient and robustthan using peak nodal values in discerning those cases of major interest where ECAP (or anyother index) maintains high values at least in a small non-negligible region rather than only atisolated computational nodes. Among the carotids, the largest 99th percentiles were observed inC1R (upper ECAP percentile = 2.70 and 2.47 under rest and exercise conditions, respectively) andC2R (upper ECAP percentile = 1.79 and 2.24 under rest and exercise conditions, respectively).Carotid C1R also presented the second largest nodal value (max ECAP = 9.15 at rest), while thelargest peak was observed in carotid C4L (max ECAP = 11.4 at rest).

Tracking particle trajectories backwards, rather than forwards, in time provided severaladvantages in terms of analysis and visualization. By plotting results for PLAP at the initial(injection) position of the particles, one can easily locate regions of the luminal surface that wereexposed to either high- or low-shear fluid elements. Moreover, the procedure to obtain a nearwall PLAP was greatly simplified because we could use the same particle tracking simulationto probe and average the PLAP field at different distances from the wall boundaries. Figure 3shows distributions of PLAP in all 10 carotid models averaged over a near wall distance equal to5% of the local radius. Of note, general patterns could be observed in all cases: PLAP was highthroughout the straight part of the common carotid artery (particularly on the external side), butlower in the vicinity of the bifurcation. Gradually higher values were also found when movingdistally along the internal carotid artery, whereas the portion of the external carotid included inthe computational domain was probably too short to observe a similar behaviour. These resultssuggest that large arterial segments (especially regions with relatively simple haemodynamicslike the common carotid artery) might naturally be in contact with fluid elements that haveencountered high shear flows along their trajectory, and thus may potentially be rich in activatedplatelets. Wall regions subjected to more complex haemodynamics, like the proximal bifurcationand the sinus, instead seem to be relatively spared and could present a lower PLAP. Particlesinjected in the vicinity or downstream of these locations were, in fact, advected back throughthe complex flow that characterized the bifurcation; hence, their distribution of PLAP values waslikely dictated by other haemodynamic features (e.g. presence of vortices, recirculation eventsand stagnation).

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>3.0

2.4

1.8

1.2

0.6

0

PLAP

C1L C2L C3L C4L C5L

C1R C2R C3R C4R C5R

Figure 3. Colour-coded spatial distributions of the PLAP for all 10 carotid arteries. The colour-coded maps refer to PLAP valuesaveraged over a distance from the wall equal to 5% of the local radius. 99th percentile values averaged over a distance of both5 and 10% of the wall are listed in table 3. (Online version in colour.)

Simulations of mild exercise did not significantly affect the magnitudes or distributions ofPLAP. The higher velocity gradients (and thus higher shear-induced activation potentials) thatcould be expected in exercise due to increased flow rates were counterbalanced by the lowertimes that particles resided within the domain because of the higher heart rates and overall flowvelocities. In general, exercise conditions flattened the PLAP distributions, with a slight reductionin the peak values and localized increases in PLAP in some of the low-index regions, as, forexample, in C2R (on the interior side) and C4L (on the outer wall) proximal to the bifurcation.Index scaling was effective in removing a weak correlation that we initially observed betweenpeak values of PLAP and common carotid radii. Prior to normalization, the maximum value ofPLAP was observed in C5L (radius = 3.35 mm, second largest of the 10 common carotids), whilethe minimum value was observed in C4L (radius = 2.76 mm, the smallest of the 10). As shownin table 3, scaled 99th percentile values of PLAP were instead very similar for all simulations(average 99th percentile of PLAP = 2.13 ± 0.15 and 2.02 ± 0.14 for a 5% probing distance underrest and exercise conditions, respectively).

Table 3 also shows effects of the probing distance on the upper percentile values of PLAP. Whiledifferences due to probing over a larger distance could be considered to be negligible on average(3% reduction and 1% increase under rest and exercise conditions, respectively), the effects weremore apparent on a case to case basis. The most pronounced differences were observed in thesmallest carotids, C1R (22% decrease from a 5% to 10% probing distance both at rest and exercise)and C2R (16% and 30% increase under rest and exercise conditions, respectively). The averagingdistance did not change the regional distribution of PLAP, however.

Figure 4 shows computed distributions of TFP (recall equation (2.5)) for all 10 carotidbifurcations at rest and exercise conditions. Consistent with the above, the 99th percentile valuesof TFP (cf. table 3) were in all cases smaller than at least one of the corresponding values of theECAP and PLAP indices, and in most cases (e.g. for all measurements taken with a 5% probingdistance of the local radius) smaller than both despite the upper percentiles of PLAP alwaysbeing larger than 1.75. The same observation remains valid when considering peak values atcomputational nodes. On average, the peak nodal values of TFP were 4.73 ± 1.80 and 5.05 ± 2.07for a probing distance equal to 5% of the local radius under rest and exercise conditions,respectively, whereas the average peak nodal values of ECAP were 6.64 ± 2.41 and 5.96 ± 1.58under rest and exercise conditions, respectively. Comparison of figures 2 and 3 highlights howregions of high ECAP consistently coincided with regions of low PLAP and vice versa, whichthereby attenuated the values of TFP (figure 4) in the healthy carotid arteries. This finding mightprovide some new insight into the mechanisms that inherently protect regions of the normal

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>7.0

5.6

4.2

2.8

1.4

0

TFP

C1L C2L C3L C4L C5L

C1R C2R C3R C4R C5R

Figure 4. Colour-coded spatial distributions of the TFP. These distributions show values of TFP computed using PLAP averagedover a distance equal to 5% of the local radius (cf. figures 2 and 3). 99th percentile values are listed in table 3. (Online versionin colour.)

>6.0

4.8

100

95

90

100

95

90

3.6

2.4

1.2

0

>3.5

2.8

2.1

1.4

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>2.5

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1.0

0.5

0

>7.0

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0

>3.0

2.4

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4.2

2.8

1.4

0

percentiles

C1R C3R

C1R

C3R

ECAP

ECAP PLAP

PLAP TFP

ECAP PLAP TFP

Figure 5. Detailed summary of results for the ECAP, PLAP and TFP for carotid arteries C1R and C3R (cf. figures 2–4 as well astable 3), which revealed the overall maximum and minimum 99th percentile values, respectively. On the left, areas of highECAP and PLAP were evident on the external surface of the carotids. Darker colour tones were assigned to higher percentileswith respect to the total surface areas. On the right, colour-coded distributions of ECAP, PLAP and TFP are shown in detail nearthe bifurcation and sinus, where the TFP values were higher. (Online version in colour.)

vasculature that are subjected to complex haemodynamic stimuli, such as the carotid bifurcation,from de novo thrombus formation despite their otherwise high vulnerability for atherosclerosis.

Indeed, it was this observation that motivated our definition of the TFP index. ECAP andPLAP are distinct in simple geometries: if a high shear flow rich in activated platelets comes closeenough to the wall (high PLAP), it increases the magnitude of the wall shear (i.e. traction vector)and hinders its directional oscillation (low ECAP). High ECAP and PLAP regions thus tend tonot overlap in the normal vasculature. Figure 5 illustrates this concept further for the two carotid

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arteries that revealed, respectively, the minimum and maximum values of peak TFP (C3R andC5L). Locations of high ECAP and PLAP, where ‘high’ is defined as the upper 10th percentile onthe local surface areas, were marked on the external surfaces but with little overlap between thetwo high index regions. The area of intersection between the two high index regions was less than1% of the total area in both cases (0.7% and 0.2% for C1R and C3R, respectively). This figure alsoshows a detailed view of the carotid sinus, where values of TFP were the highest. Notably, regionsof high TFP tended to coincide with regions where ECAP was high, mainly because ECAP wasclose to zero in most regions of the wall subjected to relatively simple haemodynamics (followingthe well-known behaviour of OSI, on which the definition of ECAP is based). Thus, the PLAP didnot contribute to changing the boundaries of non-zero TFP regions (which are dictated primarilyby ECAP); rather, it confined the locations of the areas of peak TFP. Note, also, that in both casesthe peak value of TFP was outside the overlap area of high ECAP and PLAP regions.

In summary, despite complex geometries and blood flows due primarily to the bifurcationand the sinus, the human carotid artery tends not to harbour intraluminal thrombosis in theabsence of an eroded or ruptured atherosclerotic plaque [25,35]. Such was the case for the 10non-atherosclerotic carotid arteries studied herein. The present simulations suggest further that,despite the complex haemodynamics, regions within the carotid artery that may be presentedwith platelets that have recently experienced a high shear history are also regions that experiencehigher values of non-oscillatory WSS. Increased WSS tends to upregulate endothelial productionof nitric oxide and prostacyclin, both of which are anti-thrombotic, hence suggesting that regionsof high (i.e. normal, not pathologically high) WSS should be protected. The postulated TFP wasfound to be similar under rest and exercise conditions, with the maximum 99th percentile value of1.58 (for C1R at rest, max TFP = 8.06) for an averaging distance of 5% of the local radius and 2.15(for C2R under exercise conditions, max TFP = 7.86) for an averaging distance of 10% of the localradius. Although higher values may be found for other normal human carotids, the present set of10 results suggest that upper percentile threshold values for TFP should be above approximately2.0 or 2.5, depending on the computational scheme. With this in mind, let us now consider threemodels of healthy abdominal aortas free of thrombus and three aortic aneurysms that harbouredonly a localized, thin ILT.

(b) Healthy abdominal aortasThe fluid dynamic analyses conducted on three healthy infrarenal aortas suggested that ECAPvalues observed in this region are, in general, lower than in the carotid sinus (figure 6). Despiteobserving high OSI values in large portions of the infrarenal aorta, as expected due to thebackflow induced by the renal flow during diastole, TAWSS was also comparatively higher inthe same region. Average 99th percentile values of ECAP were then approximately 56% andapproximately 66% less than their corresponding values in carotids under rest and exerciseconditions, respectively (table 3). Such differences were partially reflected in TFP percentilevalues, which were on average approximately 18% (under rest) and approximately 44% (underexercise) larger in carotids than in the infrarenal aortas, regardless of the probing distanceemployed. Lastly, exercise conditions (figure 6) seemed to have a moderate effect in reducing TFPin infrarenal aortas (approx. 20% reduction in average upper percentile TFP value under exerciseconditions). These findings substantiated our original idea to study the carotid bifurcation asa region of highly disturbed flow that is yet normally thrombo-resistant. Such regions provideimportant insight into possible threshold values of candidate metrics.

(c) Abdominal aortic aneurysms with thin thrombiFigure 7 shows the geometric model used to simulate the haemodynamics associated with anapproximate 4.4 cm diameter abdominal aortic aneurysm (AAA1), including major arteries of theabdominal vasculature. The lesion dilatation was more pronounced on the anterior side, with theproximal neck located approximately 3.5 cm below the renal arteries and shifted forward with

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>2.0

1.6

1.2

0.8

0.4

0

>2.0

1.6

1.2

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0

>2.0

1.6

1.2

0.8

0.4

0

>2.0

1.6

1.2

0.8

0.4

0

TFP

Figure 6. Colour-coded spatial distribution of TFP for the three healthy IAA simulations for both rest (left of each pair) andmild exercise (right of each pair) conditions. 99th percentile values are listed in table 3. Note that the maximum values of TFPare lower than those observed in the healthy carotids, as expected due to the more complex geometry within the latter (i.e.combination of a proximal bifurcation and sinus in the carotid). (Online version in colour.)

respect to the path of the healthy aorta. Shown, too, are full-field distributions of ECAP, PLAP andTFP together with a detailed lateral view of the lesion, with the actual thin ILT visualized via atransparent overlay on the central surface. Not surprisingly, our computational analyses on AAA1predicted haemodynamic conditions that are typically expected in AAAs. After detaching fromthe wall immediately below the neck, the flow promptly decelerated giving rise to recirculationand mixing. Accordingly, the ECAP index (a measure of low and oscillatory WSS) was highthroughout the upper and central portion of the anterior portion of the lesion, with an upperpercentile value higher than that found in carotid bifurcations C1R and C2R (upper ECAP

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>543210

>543210

>3.02.41.81.20.60

>7.05.64.22.81.40

>2.52.01.51.00.50

>7.05.64.22.81.40

100TFP

percentiles

95

90

lumen

thrombus

TFP/thrombus colocalization

ECAP PLAP TFP

Figure 7. Colour-coded spatial distributions of ECAP, PLAP and TFP in an abdominal aortic aneurysm (AAA1) having a thinthrombus on the central anterior surface. Upper row, front view of the full geometry. Lower row, lateral view zoomed at thethrombotic region of the lesion. The two plots presented for each index refer to simulations of rest (left) and mild exercise(right) conditions. Rightmost column, simulation results overlapped on a CT slice showing colocalization of high TFP regionsand ILT. The geometry of the thin ILT is visualized as a partially transparent white surface. (Online version in colour.)

percentile = 4.13 in AAA1 versus upper ECAP percentile = 2.70 and 1.79 under rest conditionsin C1R and C2R, respectively). Conversely, the peak nodal values were significantly smaller(max ECAP = 5.54 in AAA1 versus max ECAP = 9.15 and 11.4 under rest conditions in C1R andC2R, respectively).

The PLAP index was also higher in the anterior and central portions of the lesion, in contrastwith that which was observed for the carotid arteries wherein regions of high PLAP weresystematically characterized by low values of ECAP. This finding for the aneurysm is likely due tothe more complex haemodynamics, where secondary vortices and prolonged recirculation mightfavour the mixing of (and thus transfer of particles between) low and high shear flows. Althoughpeak nodal values of both ECAP and PLAP were higher in some of the carotid simulations, themaximum TFP was higher in AAA1 (max TFP = 8.65 in AAA1 versus max TFP = 8.06 in C1R atrest and for a probing distance equal to 5% of the local radius). Differences were more pronouncedin terms of 99th percentiles, with the aneurysmal value being more than three times larger than thelargest carotid value (upper TFP percentile = 6.27 in AAA1 versus upper TFP percentile = 1.58 inC1R under rest conditions and for a probing distance equal to 5% of the local radius). The regioncovered by the actual ILT encompassed areas of high TFP, including the region wherein PLAPwas found to reach its peak value.

Figure 8 shows spatial distributions of TFP in two other aneurysms (AAA2 and AAA3)of approximately 5.1 cm and approximately 5.2 cm diameter, respectively. Despite their largersize, close to the clinically accepted criterion for intervention (5–5.5 cm), these lesions harbouredvery thin (and in the case of AAA3 even fragmented) ILT. Interestingly, both AAAs presentedcalcified proximal necks with peculiar morphologies that proved to affect significantly theirhaemodynamics. In AAA2, a slight stenosis at the inlet of the aneurysm caused a promptacceleration of the entering blood flow. The high shear jet was directed towards the anterior wallof the lesion, locally increasing the WSS and thus explaining the very low TFP values observed

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5.6

100

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>7.0

4.22.81.40

5.6>7.0

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4.22.81.40

AAA2

AAA3

TFP

TFP/thrombus colocalization

TFP

percentiles

100

95

90

TFP

percentiles

Figure 8. On the left, full-field distributions of TFP in AAA2 (left) and AAA3 (right). Upper row, frontal view of the lesions.Lower row, posterior view zoomed in at the thrombus-covered locations. For each lesion, the two plots refer to simulations ofrest (left) and exercise (right) conditions. On the right, colocalization of thrombus with regions of high TFP. Darker colour tonesare assigned to higher percentiles with respect to the total surface areas. (Online version in colour.)

throughout the anterior region. It was only after prolonged recirculation that the newly injectedflow reached the posterior wall, gradually losing intensity from the bottom towards the top.Two high TFP islands were visible in the upper posterior wall, one of which coincided with theregion covered by the actual ILT. The peak value of TFP was found within the other region ofhigh TFP; it was approximately 70% greater than the maximum value observed in the carotids(max TFP = 13.54 in AAA2 versus max TFP = 8.06 in C1R for a probing distance equal to 5%of the radius and rest conditions) and similarly the 99th percentile was about twofold higher(99th upper TFP percentile = 3.22 in AAA1 versus 99th upper TFP percentile = 1.58 in C1R for aprobing distance equal to 5% of the radius and rest conditions). Nevertheless, differences in bothpercentile and peak values of ECAP or PLAP were not evident between this aneurysm and thecarotid arteries, thus suggesting that it was critical to combine the information provided by theECAP and PLAP to discriminate between regions protected from or prone to the formation of ILT.

AAA3 also presented a stenotic segment preceding its inlet, within which flow gained enoughmomentum to maintain shear stresses high along the wall of the lesion despite a complexhaemodynamics characterized by multiple recirculation vortices. Thin ILT was present in 4separate regions located in the upper and posterior wall. An analysis of the distribution of theTFP over the entire lesion showed high values only in one of these locations (max TFP = 8.33 at

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6

5

4

3

2

1

0ECAP PLAP TFP

aver

age

inde

x (9

9th

perc

entil

e)

carotidsIAAsAAAs

Figure 9. Mean 99th percentile values under rest conditions of ECAP, PLAP (for a 5% probing distance) and TFP for the allthree sets of vessels considered (n= 16 total). Error bars indicate the s.e.m., with n= 10 for the carotids, n= 3 for the IAAs,and n= 3 for the AAAs. Variations were the greatest for the AAAs, as expected due to the diverse geometries presented bydifferent patients. Larger subsets will be needed for meaningful statistical comparisons, but the TFP alone delineated well thethrombo-resistant and the thrombo-prone regions.

rest for a probing distance equal to 5% of the radius, slightly above the maximum value found inthe carotid bifurcations), while the index was uniformly low throughout the rest of the wall. As aresult, the upper TFP percentile of AAA3 was the lowest among the aneurysms and comparablewith the maximum value observed in the carotid analyses (upper TFP percentile = 1.46 for AAA3versus upper TFP percentile = 1.58 for C1R at rest and for a probing distance equal to 5% ofthe local radius). The fact that three out of four thrombus pockets seemed to be in regions nothaemodynamically prone to thrombus deposition (mainly due to high WSS) would suggestthat these ILTs might not be the foci of a new formation event, but rather the remnants of apreviously deposited (and more uniform) thrombus that had been excavated by the high WSSflow characterizing the lesion. Indeed, this was found to be the case by examining a prior image.Hence, this peculiar aneurysm could be considered as a cautioning case, with TFP distributionsthat do not fully represent a novel thrombus deposition, but rather provide valuable lowerbounds for the desired threshold value of the TFP.

Figure 9 summarizes results from table 3 graphically and reveals that the TFP alone delineatedwell between all 13 normal thrombo-resistant arteries and the three pathological thrombo-proneaneurysms. These results well support our original hypothesis that one must consider boththe condition of the endothelium and the flow-induced shear history experienced by plateletspresented to the endothelium.

4. DiscussionThe primary goal of this work was to motivate a new phenomenological approach for identifyingregions of possible formation of ILT on an intact but susceptible endothelium within AAAs.That is, in contrast to prior haemodynamic studies (see reviews [4–6]) of thrombus formationin regions of arterial injury (e.g. eroded stenoses or ruptured atherosclerotic plaques) or withinmedical implants (e.g. a bioprosthetic valve or metallic device), we focused on regions whereblood would not be exposed to collagen or other highly thrombogenic material. In particular,this study was motivated by the hypothesis that two basic haemodynamic features must coincideto promote the formation of a thrombus on an intact endothelium—platelets must experience asufficiently high shear history and then be presented to a susceptible region. We submit that the

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proposed TFP—which combines three prior useful metrics of low WSS, high oscillatory WSS andhigh flow-induced particle shear history—captures these two key features (cf. figure 9).

Simulations for 10 non-diseased carotid arteries from five patients revealed that regions thatare usually susceptible to low and oscillatory shear stress (and hence atheroprone and potentiallypro-thrombotic) did not overlap with regions that are likely to be presented with platelets thathave recently experienced a high shear history. Although explored in detail for only one locationof complex flow within the human vasculature, this finding seems reasonable in hindsight—theremust be a mechanistic reason why the healthy carotid artery, with its geometric complexity, isnot susceptible to thrombus formation under physiologic conditions despite being susceptibleto atherogenesis. There is a need nonetheless to evaluate this new index and its lower boundvalue (e.g. 99th percentile of TFP ∼ 2.0 for a 5% averaging distance) for other locations, perhapsincluding the aortic arch as well as other bifurcations. The carotid artery was selected because ofthe uniqueness of its geometry, which includes both a bifurcation and a distal sinus that effectivelyrepresents a local dilatation of the wall, and the availability of prior studies on the carotid thatprovided reliable information on inlet and outlet boundary conditions [10,11]. As confirmed byour analyses, this choice proved to be particularly insightful because the endothelium within thecarotid sinus is constantly subjected to low and oscillatory WSSs and at the same time containsflows potentially rich in activated platelets. The carotid bifurcation was thus a convenient test casefor our hypothesis of combined mechanisms. Conversely, three analyses conducted on thrombus-free normal infrarenal aortas (figure 6) revealed that albeit oscillatory, aortic blood flow wasnormally not stagnant and maintained a relatively high shear stress on the vessel wall. Suchfactors led to very low values of TFP index, in agreement with the observation that healthy aortassimilarly do not normally develop thrombus, and that the carotid artery was a better comparatorfor identifying a safe (lower) bound value of the TFP.

By contrast, we also sought patient-specific cases of AAAs that harboured but a nascentILT. Lesions with large thrombus would obviously not be useful in searching for conditionsunder which the ILT initiated because the overall geometry of the lumen and wall likelychange following the accumulation of thrombus. Conversely, lesions without thrombus couldprovide additional information on the lower bound value of the TFP, but would not help inidentifying how far above this bound the ILT might initiate. We were thus fortunate to findthree rare cases wherein there was a small, thin ILT within a sizable aneurysm (figures 7and 8). That the average 99th percentile of TFP in these aneurysms (approx. 3.60) was morethan five times the corresponding value found in carotids (approx. 0.70) and healthy aortas(approx. 0.58), as shown in table 3 and figure 9, and that in all cases peak values of TFP co-localized with the regions of the actual ILTs (which were removed from the lesion numericallyto perform unbiased simulations) provided additional confidence that the TFP merits furtherstudy. There is a need, therefore, to identify and analyse many more AAAs wherein there is buta nascent thrombus.

Albeit not shown (see the electronic supplementary material, tables and figures), we alsoreperformed all simulations for constant rather than variable heart rates. Although values of TFPchanged slightly in both the arteries and the aneurysms, the co-localization of regions of highTFP with the actual thin ILT in the aneurysms was even stronger. Hence, the results presentedvisually in figures 7 and 8 may be conservative, which is desirable of a potential clinical metric.We also performed additional simulations on aneurysm AAA2 after numerically removing theslight stenosis at the proximal neck (i.e. inlet). Doing so resulted in higher values of TFP in theanterior portion of the lesion, where ILT is often found in similarly shaped lesions. In this case,one could speculate that the presence of the stenosis and associated inlet ‘jet-like’ flow may havebeen somewhat protective from the perspective of ILT formation. If so, categorization of sucheffects on the haemodynamics and associated platelet activation/adhesion could become a usefulclinical aid.

There have been many prior studies of possible haemodynamic initiators of thrombuswithin AAAs. For example, Bluestein et al. [36] presented results many years ago for steadyflow simulations within idealized axisymmetric lesions. They concluded that ‘the recirculation

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zone formed inside the aneurysm cavity creates conditions that promote thrombus formation’[36, p. 280]. In particular, they suggested that ‘Once trapped in the recirculation zone, plateletswith elevated shear histories and higher incidence of activation were readily deposited to thewall in areas of low wall shear stress’ [36, p. 284]. Although these observations were extremelyinsightful, they did not suggest a simple metric to capture these multiple effects. The TFPproposed herein represents such a metric that is consistent with their seminal findings.

More recently, attention has been directed more towards the strain history experienced bythe platelets, particularly in vortical structures. For example, Biasetti et al. [37] focused on highfluid shear stresses generated by vortical structures within fusiform aneurysms, which movewithin the body of the lesion and potentially release activated platelets upon the break-up ofthe vortices. Shadden & Hendabadi [31] similarly focused on elevated fluid shear stresses andexposure times experienced by platelets, albeit for thrombus formation in stenoses. The PLAPused herein was taken directly from the latter work. Neither of these studies attempted to tietogether the activation of the platelets due to elevated shear histories and their more likelyadhesion and aggregation on intact but susceptible endothelium in regions of low WSS or highOSI. Again, the TFP is consistent with these important studies.

We emphasize again that our goal was to identify a potentially useful clinical metric, notto explore underlying biochemomechanical mechanisms of thrombus formation. The latter isyet important and necessitates multiscale, multiphysics approaches as being pursued by others(cf. [4–6]). Nevertheless, a phenomenological metric could also be very important, particularlyin treatment planning. Such a clinical metric necessarily should be based on informationusually available in the monitoring of AAAs, namely, medical images that reveal the patient-specific geometry. That is, even though large-scale fluid–solid interaction models are nowpractical [16], information on regional variations in patient-specific material properties remainwanting. We thus assumed rigid walls in all of our simulations, as would be expected ofmost computations based on current clinically available data. Albeit not shown, we performeda series of computations to determine TFP in idealized abdominal aortas with or withoutaxisymmetric aneurysms that were generated using our growth and remodelling codes [38].In this way, we could compare directly the results based on a rigid versus a compliant wall(i.e. one endowed with appropriate regional variations in biaxial material stiffness and wallthickness). Our findings revealed, consistent with the definition of ECAP, that the computedvalue of TFP was lower in the deformable wall simulations for the healthy infrarenal aortabecause the WSS maintained higher values during late systole and diastole (despite an overalllower peak value), thus resulting in generally higher TAWSS values when taking into accountwall elasticity. In other words, a TFP computed for a healthy artery based on a rigid wallassumption (cf. figure 9 for the carotids and infrarenal aorta) assumes a higher (i.e. conservative)value, which is desirable clinically. Interestingly, the computed value of TFP for idealized,deformable aneurysms was similar to that obtained via the rigid wall assumption, in part becauseof the extreme stiffening that occurs following the loss of elastin in aneurysms [39]. Hence,the rigid wall assumption adopted here was both consistent with expected future simulationsof clinical cases and computationally conservative as desired when comparing normal versuspathologic cases.

In conclusion, for complex ‘pathologic’ geometries and haemodynamics with pronouncedstagnation and recirculation (e.g. as in aneurysms), particles that have previously accumulateda high shear history could potentially be advected to the wall even by a lower shear flow thatwould result in low WSSs on the endothelium. In this case, the computed TFP may provide muchmore information than simply a WSS-based parameter like ECAP, a measure such as (near wall)residence time, or a single measure of platelet shear history [31,37,40,41]. That is, the presentfindings motivate further consideration of the TFP, or a similarly combined index, that accountsfor the potential of activated platelets being presented to a quiescent (non-thrombogenic) versusa susceptible (thrombogenic) endothelium. Being able to predict when a lesion might develop anILT could enable another level of computational sophistication (cf. [42]) and thereby contribute totreatment planning for AAAs.

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Ethics statement. This study was approved by the Institutional Review Boards of Yale University and theVeterans Affairs Connecticut Healthcare System.Funding statement. This work was supported, in part, by the staff and facilities of the Yale University Faculty ofArts and Sciences High Performance Computing Center as well as grants from the NIH (R01 HL086418, U01HL116323).

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