Approximating hemodynamics of cerebral aneurysms with steady flow simulations

Approximating hemodynamics of cerebral aneurysms withsteady flow simulations

A.J. Geers a,b,n, I. Larrabide a,b,c, H.G. Morales a,b, A.F. Frangi b,da Center for Computational Imaging and Simulation Technologies in Biomedicine (CISTIB), Universitat Pompeu Fabra, Barcelona, Spainb Networking Biomedical Research Center on Bioengineering, Biomaterials and Nanomedicine, Barcelona, Spainc PLADEMA-CONICET and Universidad Nacional del Centro, Tandil, Argentinad Department of Mechanical Engineering, The University of Sheffield, Sheffield, UK

a r t i c l e i n f o

Article history:Accepted 17 September 2013

Keywords:Cerebral aneurysmHemodynamicsComputational fluid dynamicsSteadyPulsatile

a b s t r a c t

Computational fluid dynamics (CFD) simulations can be employed to gain a better understanding ofhemodynamics in cerebral aneurysms and improve diagnosis and treatment. However, introduction ofCFD techniques into clinical practice would require faster simulation times. The aim of this study was toevaluate the use of computationally inexpensive steady flow simulations to approximate the aneurysm'swall shear stress (WSS) field. Two experiments were conducted. Experiment 1 compared for two casesthe time-averaged (TA), peak systole (PS) and end diastole (ED) WSS field between steady and pulsatileflow simulations. The flow rate waveform imposed at the inlet was varied to account for variationsin heart rate, pulsatility index, and TA flow rate. Consistently across all flow rate waveforms, steadyflow simulations accurately approximated the TA, but not the PS and ED, WSS field. Following up onexperiment 1, experiment 2 tested the result for the TA WSS field in a larger population of 20 casescovering a wide range of aneurysm volumes and shapes. Steady flow simulations approximated thespace-averaged WSS with a mean error of 4.3%. WSS fields were locally compared by calculating theabsolute error per node of the surface mesh. The coefficient of variation of the root-mean-square errorover these nodes was on average 7.1%. In conclusion, steady flow simulations can accurately approximatethe TA WSS field of an aneurysm. The fast computation time of 6 min per simulation (on 64 processors)could help facilitate the introduction of CFD into clinical practice.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Growth and rupture of cerebral aneurysms have been associatedwith the intra-aneurysmal hemodynamics (Hashimoto et al., 2006).Better understanding of hemodynamics could improve diagnosisand treatment. In the last decade, computational fluid dynamics(CFD) simulations have been employed to study the relationshipbetween hemodynamics and rupture (Cebral et al., 2011b, 2011c;Xiang et al., 2011; Miura et al., 2013) and the hemodynamic effect ofendovascular treatment (Kakalis et al., 2008; Cebral et al., 2011a;Larrabide et al., 2013; Morales et al., 2013). In particular, researchhas focused on the wall shear stress (WSS), which is a key regulatorof vascular biology and pathology (Malek et al., 1999; Dolanet al., 2013).

The majority of studies use unsteady, pulsatile flow simulationsto capture the changing flow rate and inertia effects during the

cardiac cycle. The enormous amount of generated data is typicallyreduced by extracting and analyzing the time-averaged (TA)(Cebral et al., 2011c; Miura et al., 2013; Xiang et al., 2011), peaksystole (PS) (Castro et al., 2009; Shojima et al., 2004) or enddiastole (ED) flow field (Jou et al., 2008; Fukazawa et al., in press).Some researchers have argued, however, that in many cases themain flow features in the aneurysm remain relatively stablethroughout the cardiac cycle (Müller et al., 2012; Mantha et al.,2009) and that computationally much less expensive steady flowsimulations might already provide clinically relevant hemody-namic information (Cebral et al., 2011c). The shorter time requiredto create these simulations could aid in the introduction of CFDinto clinical practice. So far, comparisons between steady andpulsatile flow simulations have mostly been qualitative and basedon few cases, but before steady flow simulations can be reliedupon, it is essential to quantify for a large population the accuracywith which they can approximate aspects of the pulsatile flowfield in the aneurysm.

The aim of this study was to quantify the error with whichsteady flow simulations can approximate the aneurysm's TA, PSand ED WSS fields derived from pulsatile flow simulations. Since

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Journal of Biomechanics

0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jbiomech.2013.09.033

n Corresponding author at: Center for Computational Imaging and SimulationTechnologies in Biomedicine (CISTIB), Universitat Pompeu Fabra, c/ Tànger,122-140, 08018 Barcelona, Spain. Tel.: þ34 93 542 2237.

E-mail address: [email protected] (A.J. Geers).

Journal of Biomechanics 47 (2014) 178–185

this error might depend on the flow rate waveform (FRW)imposed at the inlet of the vascular model, we systematicallyvaried the heart rate, pulsatility index, and TA flow rate. Thestudy's dataset included both terminal and lateral aneurysms,covering a wide range of aneurysm volumes and shapes.

2. Methods

2.1. Blood flow modeling

Aneurysm models included in this study were drawn from a large databasecreated within the EU project @neurIST (Villa-Uriol et al., 2011). Patient-specificvascular models, represented by triangular surface meshes, were constructed bysegmenting three-dimensional rotational angiography (3DRA) images using ageodesic active regions approach (Bogunović et al., 2011). These models weresmoothed using a geometry-preserving smoothing algorithm (Nealen et al., 2006)to reduce small surface perturbations without displacing vertices by more than0.02 mm. Touching vessels were removed. All inlet and outlet branches wereclipped perpendicular to their axes and were, respectively, 12 and 4 diameters inlength. To define the aneurysm region, the aneurysm neck was manually delineatedand a neck surface was automatically created from this curve. This approach hasbeen demonstrated to have a low interobserver variability for volume and surfacearea measurements of the aneurysm (Larrabide et al., 2011). All mesh editingoperations were performed in @neuFuse (B3C, Bologna, Italy) (Villa-Uriol et al.,2011), a software application developed within @neurIST.

Unstructured volumetric meshes were created using the octree approach withICEM CFD 13.0 (ANSYS, Canonsburg, PA, USA). Meshes were composed of tetra-hedral elements of 0.24 mm, and three prism layers with a total height of 0.08 mmand a side length of 0.12 mm. The total number of elements ranged from 1.2 to3.8 million, the density from 1420 to 2950 elements per mm3, depending on thesurface-area-to-volume ratio of the computational domain. This mesh setup wasselected following mesh dependency tests performed on both cases of experiment 1

(see Section 2.3). These tests demonstrated the mesh independency of the WSS fieldfor steady flow simulations under the highest inflow rate conditions consi-dered in this study.

CFD simulations were created with the commercial finite volume solver CFX13.0 (ANSYS), using a second order advection scheme, a second order backwardEuler transient scheme for unsteady simulations, and CFX’ automatic time scalecontrol for steady-state simulations. Solutions converged until the normalizedresidual of the WSS everywhere in the computational domain was o10"5. Bloodwas modeled as an incompressible Newtonian fluid with density ρ¼ 1060 kg=m3

and viscosity μ¼ 0:004 Pa s. Vessel walls were assumed rigid with a no-slipboundary condition. Flow rate values (steady) or waveforms (pulsatile) wereimposed at the inlet and zero-pressure boundary conditions at all outlets. Forpulsatile flow simulations, the cardiac cycle was discretized in 200 uniformlydistributed time steps and, to reduce the effect of initial transients, the second oftwo simulated cardiac cycles was analyzed. Tests performed on both cases ofexperiment 1 demonstrated WSS differences of o1% with respect to simulationswith either 400 uniformly distributed time steps per cardiac cycle or from whichthe third cycle was analyzed.

2.2. Flow rate waveform transformation

FRWs describe how flow rate Q(t) at the inlet of the vascular models changesover time t during the cardiac cycle. In this study, we used a physiological FRWQ0ðtÞ that was derived from phase-contrast magnetic resonance images of theinternal carotid artery (ICA) of a healthy volunteer (Cebral et al., 2003). This FRWwas then linearly transformed to obtain a FRW with specified heart rate (HR), TAflow rate (QTA), and pulsatility index (PI) given by PI¼ ðQPS"QEDÞ=QTA. These threevariables will be referred to as FRW descriptors. The transformation is given by

Q ðtÞ ¼ aQ0ðctÞþb ð1Þ

where

a¼QTA

Q0TA

PIPI0

; b¼QTA 1"PIPI0

! "; c¼

HRHR0

Physiological values for the FRW descriptors (see Table 1) were derived fromthe literature. For each descriptor, we defined a baseline, an upper and a lowervalue. Values for HR and PI were obtained using the mean and standard deviation(SD) of these descriptors reported by Ford et al. (2005) and Hoi et al. (2010):baseline¼mean, lower¼mean " 2SD, and upper¼mean þ 2SD. Values for QTA

were obtained using the relationship QA ¼ 48:21 A1:84 where QA is in ml/s and A isthe inlet's cross-sectional area in cm2 (Cebral et al., 2008). This relationship wasdetermined by fitting a power-law function through measurements of Q and A ofthe ICAs and vertebral arteries of 11 normal subjects. Values for QTA were given by:baseline¼QA , lower¼ 0:73QA , and upper¼ 1:27QA . The 27% variation is theaverage relative error between prediction and measurement, derived from Cebralet al. (2008).

2.3. Experiments

Two experiments were conducted in this study. Fig. 1 schematically representsthe workflows of both experiments.

Table 1Flow rate waveform descriptors.

Descriptor Physiological variation

Lower Baseline Upper

Heart rate [bpm]a 52 68 84Pulsatility index [–]a 0.58 0.92 1.26TA flow rate [ml/s]b 0:73QA QA 1:27QA

a baseline¼mean, lower¼mean " 2 SD, and upper¼mean þ 2 SD where themean and standard deviation (SD) are taken from Ford et al. (2005) and Hoi et al.(2010).

b QA ¼ 48:21A1:84 where A is the inlet area in cm2 (Cebral et al., 2008).

Fig. 1. Workflows of experiments 1 and 2. Sets of FRW descriptors HR, PI, and QTA were created by keeping two descriptors at baseline and varying the third from lower tobaseline to upper value (see Table 1). For each set of FRW descriptors, Q0ðtÞ was linearly transformed to obtain Q(t) fromwhich then QPS and QED were derived. One pulsatileflow simulation was created with Q(t) imposed at the inlet of the vascular model and three corresponding steady flow simulations were created with flow rates QTA, QPS andQED imposed at the inlet. In the data analysis, WSS field S1 was compared to P1, S2 to P2, etc.

A.J. Geers et al. / Journal of Biomechanics 47 (2014) 178–185 179

Experiment 1 was a detailed analysis of two cases to compare between steadyand pulsatile flow simulations the TA, PS, and ED WSS fields, and to assess howthe results depend on FRW descriptors. One aneurysm was located at the internalcarotid bifurcation (terminal case) and one near the ophthalmic artery (lateralcase). Per case, seven pulsatile flow simulations were created, each with a differentFRW imposed at the inlet: one FRW with all descriptors at baseline, three FRWswith two descriptors at baseline and the third at its lower value, and three FRWswith two descriptors at baseline and the third at its upper value. Per pulsatileflow simulation, three steady flow simulations were created with correspondingflow rates imposed at the inlet: one with TA flow rate, one with PS (i.e. maximum)flow rate, and one with ED (i.e. minimum) flow rate. Because some flow rate valueswere repeated, a total of 13 steady flow simulations were created for both cases.Reynolds numbers (Re) ranged from 58 to 432 and Womersley numbers (α) from2.6 to 4.2.

Experiment 2 was a follow-up of experiment 1, in which we found that the TA,but not the PS and ED, WSS field could be accurately approximated with steadyflow simulations. To quantify in a larger population the error with which the TAWSS field could be approximated, we selected 10 terminal and 10 lateral cases,covering a wide range of aneurysm volumes and shapes. Aneurysm shape wasquantified with the nonsphericity index (NSI) given by NSI¼ 1"ð18πÞ1=3V2=3S"1

where V is the volume and S is the surface area of the aneurysm (Dhar et al., 2008).Volumes ranged from 6 to 1315 mm3 and NSIs from 0.03 to 0.32. Per case, wecreated one pulsatile flow simulation with the baseline FRW imposed at the inletand one steady flow simulation with the corresponding TA flow rate imposedat the inlet. Reynolds numbers ranged from 44 to 301 and Womersley numbersfrom 2.1 to 4.3.

2.4. Data analysis

In experiment 1, two regions of interest (ROIs) were manually defined (Fig. 2): aparent vessel segment of one diameter length just upstream from the carotidsiphon and the aneurysm. The parent vessel segment was included in the analysisto elucidate the difference with the aneurysm. From the pulsatile flow simulation,the TA, PS, and ED WSS fields of both ROIs were extracted, where PS and ED weredefined as the timesteps at which the space-averaged WSS (WSS) on the ROI wasmaximum and minimum, respectively. From the steady flow simulations, thecorresponding WSS fields were extracted. In experiment 2, data analysis focusedonly on the TA WSS field of the aneurysm and the corresponding WSS field of thesteady flow simulation.

To globally compare WSS fields, we calculated the WSS of the ROI and thecorresponding relative error. To locally compare the fields, we visualized theabsolute error of the WSS field and calculated the root-mean-square error (RMSE)over all nodes i¼ 1;2;…;M of the ROI:

RMSE¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1M

∑M

i ¼ 1ðWSSpulsatile;i"WSSsteady;iÞ2

s

ð2Þ

Additionally, the coefficient of variation (CV) of the RMSE was calculated to putthe error measurement in context of WSS:

CVRMSE¼ RMSE=WSSpulsatile ð3Þ

3. Results

3.1. Experiment 1

Fig. 3 shows WSS on the parent vessel segment as a function ofFRW descriptors. For both cases, steady flow simulations accu-rately approximated the TA WSS with errors ranging from 0.0 to1.5%. PS WSS was underestimated with errors ranging from 2.2 to34.6% and ED WSS was overestimated with errors ranging from1.7 to 23.3%. The observed trends were consistent across all FRWs,but the size of the errors varied with changing FRW descriptors.

Fig. 4 shows WSS on the aneurysm as a function of FRWdescriptors. The same as for the parent vessel segment, theaneurysm's TA WSS field was accurately approximated by steadyflow simulations (error range: 1.9–7.8%). Most notable differencewith respect to the parent vessel segment was that steady flowsimulations overestimated PS WSS (8.0–48.8%) and underesti-mated ED WSS (8.5–30.0%). Again, the observed trends wereconsistent across all FRWs, but the size of the errors varied withchanging FRW descriptors. Highlighting this variation for TA WSS:errors remained approximately constant with increasing heartrate, increased with increasing pulsatility index, and decreasedwith increasing TA flow rate.

To visually compare the pulsatile and steady flow simulations,Fig. 5 shows contour plots of the TA, PS, and ED WSS fields and thenode-by-node absolute error. These plots correspond to the flowsimulations with all three FRW descriptors at baseline. Steadyflow simulations locally accurately approximated the TA WSS fieldon the aneurysm with a CVRMSE of 6.3% for the terminal caseand 5.2% for the lateral case. Besides an offset in magnitude, thedistribution of the PS and EDWSS fields was mostly similar. However,relatively large differences in distribution could be found in someregions.

3.2. Experiment 2

Figs. 6 and 7 show contour plots of the TA WSS field and theabsolute error of the TA WSS field, respectively. Colormaps werenormalized to place the error in context of the aneurysm's WSS.Relatively large local errors were found to coincide with regionsof high WSS, such as the impingement region, and with regionsof fluctuating WSS, which was assessed from animations of thechanging WSS field during the cardiac cycle (see supplementarymaterial).

Table 2 shows the relative error in WSS and CVRMSE. Moredetails on the geometric and hemodynamic characteristics of eachaneurysm can be found in the supplementary material. Steadyflow simulations accurately approximated the WSS on the aneur-ysmwith a relative error of 4.371.7% (mean7standard error) anda CVRMSE of 7.173.1%. For all cases, WSS was slightly under-estimated. To quantify the dependency of CVRMSE on the geo-metric characteristics of the aneurysm, we split the cases threetimes into two groups of 10: (1) by aneurysm type, (2) by volume,and (3) by NSI. On average, higher values of CVRMSE were foundfor terminal (8.4%) vs. lateral (5.9%) aneurysms, for large (7.9%) vs.small (6.4%) aneurysms, and for aneurysms with large (8.1%) vs.small (6.2%) NSI. The Wilcoxon rank-sum test revealed no signifi-cant differences (Po0:05).

3.3. Computational load

All simulations were run on a cluster with 64 processors(Intel Xeon X5355 2.66 GHz Quad Core) distributed over eightnodes, sharing 16 GB per node. In experiment 2, steady flowsimulations were created on average 49 times faster than pulsatileflow simulations. In terms of wall-clock time, pulsatile flow

Fig. 2. Experiment 1. Vascular models with the aneurysms in light gray and theparent vessel segments in dark gray.

A.J. Geers et al. / Journal of Biomechanics 47 (2014) 178–185180

PARENT VESSEL SEGMENT OF TERMINAL CASE

PARENT VESSEL SEGMENT OF L ATERAL CASE

Fig. 3. Experiment 1. TA, PS, and ED WSS of the parent vessel segment as a function of three FRW descriptors.

ANEURYSM OF TERMINAL CASE

ANEURYSM OF L ATERAL CASE

Fig. 4. Experiment 1. TA, PS, and ED WSS of the aneurysm as a function of three FRW descriptors.

A.J. Geers et al. / Journal of Biomechanics 47 (2014) 178–185 181

simulations took on average 5 h (range: 3–10 h) and steady flowsimulations took on average 6 min (range: 3–14 min).

4. Discussion

This study has demonstrated that steady flow simulations canaccurately approximate the TA WSS field of an aneurysm. Errorsfor the aneurysm's PS and ED WSS fields were considered toolarge, although the WSS distributions were similar. The similaritiesbetween steady and pulsatile flow simulations have been pre-viously reported, but have never before been quantified for a largepopulation. In summary, the main contributions of this study are:(1) WSS fields were compared node-by-node and both global andlocal errors were quantified, (2) the influence of the FRW imposedat the inlet was assessed by systematically varying the heart rate,pulsatility index, and TA flow rate, and (3) a large population of 22cases was studied, including both terminal and lateral aneurysms,covering a large range of aneurysm volumes and NSIs.

Despite the complexity of pulsatile flow in blood vessels andaneurysms, relatively small errors were found for the TAWSS field.The parent vessel segment's WSS waveform was shifted in phaseahead of the FRW, and steady flow simulations underestimated PSWSS and overestimated ED WSS. In contrast, the aneurysm's WSSwaveform lagged behind the FRW, and steady flow simulationsoverestimated PS WSS and underestimated ED WSS. This phaselag can be attributed to the inertia of the blood; it takes time forblood in the aneurysm to accelerate and decelerate in response toflow rate changes in the parent vessel. As a result, the PS and EDWSS fell short of reaching the WSS of the corresponding steadyflow simulation. Interestingly, TA WSS was slightly underesti-mated for all cases in this study. A hint to the explanation of thisobservation was offered by the series of 13 steady flow simulationsof experiment 1. For both cases, we found a quadratic relationshipbetween the flow rate into the aneurysm, which itself is linearlydependent on the flow rate at the inlet, and the aneurysm's WSS(second-order polynomial, coefficient of determination R240:999).Although this analysis does not take into account the inertia effects,

the quadratic relationship could explain the consistent underesti-mation of TA WSS.

The approximation of the TA WSS field was for some aneurysmsmore accurate than for others. To gain more insight into when steadyflow simulations can provide a better approximation, we related theerror measurements to the aneurysm's geometry. As might beexpected for this statistically small sample, Wilcoxon rank-sum testrevealed no significant differences for the investigated geometricvariables. However, we did observe a trend toward more accurateapproximations for aneurysms that were lateral, small and/orspherical. Perhaps those aneurysms typically have simpler, morestable flow patterns, which are likely associated with smaller errors,whereas terminal, large and/or multilobular aneurysms may havemore complex, unstable flow patterns.

To put in perspective the 4.3% error we found for TA WSS inexperiment 2, we will compare it with the sensitivity of CFDsimulations to two key parameters. The first parameter is thevascular geometry. Failure to properly model the parent vessel hasbeen reported to significantly influence the aneurysm's WSS field(Castro et al., 2006) and even small differences in the vasculargeometry can sometimes cause substantial differences in flowstructure (Cebral et al., 2005). Applying different levels of smooth-ing to the vascular model can lead to 18.0% difference in WSS(Gambaruto et al., 2011) and reconstructing the vascular modelfrom either 3DRA or computed tomographic angiography (CTA)has been shown to give a mean difference in TA WSS of 44.2%(Geers et al., 2011), which is one order of magnitude above theerror reported in this work. The second parameter is the inflowboundary condition. In experiment 1, the TA flow rate of the FRWwas systematically varied 27% above and below the baseline value.With respect to the baseline simulations, we found variations in TAWSS of up to 70%, which confirms results from previous studiesby Marzo et al. (2011) and Karmonik et al. (2010). Even changeswithin the same patient from rest to exercise increases the TA WSSfrom 3 to 34% (Bowker et al., 2010). In light of these uncertainties,the error with which steady flow simulations can approximate theTA WSS field is relatively small.

As mentioned in the introduction, the majority of studies usepulsatile flow simulations to investigate the intra-aneurysmal

Fig. 5. Experiment 1. TA, PS, and EDWSS fields of the pulsatile (row 1) and steady (row 2) flow simulations, and the absolute error of the WSS field calculated at each node ofthe surface mesh (row 3). All three FRW descriptors were at baseline. View points were selected to best visualize the aneurysm, so images are not necessarily at the samescale. Part of the terminal case's parent vessel was clipped to prevent it from obstructing the view of the aneurysm. The percentages in parentheses indicate the error inspace-averaged WSS. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

A.J. Geers et al. / Journal of Biomechanics 47 (2014) 178–185182

hemodynamics. However, some studies have argued before thatsteady flow simulations could already provide useful informationabout the intra-aneurysmal hemodynamics, including the hemo-dynamic effect of stents (Kim et al., 2008; Augsburger et al., 2009;Janiga et al., 2013), the relationship between the WSS and thebiological response of the arterial wall (Tremmel et al., 2010;Kadasi et al., 2013), and associations between hemodynamicvariables and aneurysm rupture (Cebral et al., 2011c). The latterrefers to a study of 210 aneurysms by Cebral et al., the largest onthis topic to date, in which TA hemodynamic variables werecompared between groups of ruptured and unruptured aneur-ysms. The analysis was repeated five times: twice with pulsatileflow simulations for different heart rates and three times withsteady flow simulations under different inflow rate conditions. Theauthors found that associations between hemodynamic variablesand rupture were mostly unaffected by the choice of simulationand concluded that steady flow simulations could be used toextract some hemodynamic variables related to rupture. Ourfindings suggest that also other studies linking WSS to rupture,such as the 119-aneurysm study by Xiang et al. (2011) or the 106-aneurysm study by Miura et al. (2013), could be reproduced usingsteady flow simulations.

The main motivation to use steady flow simulations is toreduce simulation time. In our study, it took on average 6 minon a cluster of 64 processor to compute the flow field for one case,nearly 50 times faster than computing the pulsatile equivalent.Even for a ruptured aneurysm that requires immediate treatmentthis might be fast enough for use during interventional treatmentplanning (Karmonik et al., 2013). Further speed-ups couldbe achieved by allowing higher convergence errors or changingmeshing strategy (Spiegel et al., 2011). Please note that thepreparation of the CFD simulations is currently a manual process,which also requires significant amounts of time. However, effortsare underway to automate image segmentation (Bogunović et al.,2011), branch labeling (Bogunović et al., 2013), aneurysm detection(Lauric et al., 2010), and aneurysm neck delineation (Larrabideet al., 2011). This will further facilitate the introduction of CFD into

clinical practice. An additional advantage of fast simulation timesis that it allows for quick sweeping over input parameters (Mülleret al., 2012), which could help estimate the propagation ofuncertainty of, for instance, in- and outflow conditions.

Although the use of steady flow simulations is an attractiveoption to significantly reduce simulation times, we must stressthat this approach may lead to loss of important information aboutthe pulsatile flow field in the aneurysm. Previous studies havedemonstrated the additional value of pulsatile flow simulations.For example, high resolution CFD simulations have revealed highfrequency fluctuations in the aneurysm's flow field (Baek et al.,2010; Ford and Piomelli, 2012; Valen-Sendstad et al., 2013), whichmight be relevant to aneurysm rupture, and associations havebeen found between temporal variations of the WSS field and theformation and rupture of aneurysms (Cebral et al., 2011b; Manthaet al., 2006; Xiang et al., 2011). In other words, steady flowsimulations can be useful to quickly evaluate the aneurysm's flowfield, but additional information can be extracted from the morerealistic pulsatile flow simulations.

5. Conclusions

Steady flow simulations can accurately approximate the TAWSS field of an aneurysm. This has been demonstrated for bothterminal and lateral aneurysms, covering a large range of aneur-ysm volumes and NSIs, and for a systematically varied FRWimposed at the inlet. In light of the sensitivity of hemodynamicsimulations to uncertainties in vascular geometry and inflowboundary conditions, this approximation error can be considerednegligible when studying associations between the TA WSS fieldand, for instance, aneurysm rupture. On a cluster of 64 processors,steady flow simulations were computed on average within 6 min,nearly 50 times faster than computing the pulsatile equivalent.Such fast computation times could help facilitate the introductionof CFD into clinical practice.

Fig. 6. Experiment 2. TA WSS field of the pulsatile (left) and steady (right) flow simulations. Colormaps were normalized by using WSS derived from the pulsatile flowsimulation as unit. View points were selected to best visualize the aneurysm, so images are not necessarily at the same scale. The percentages in parentheses indicate theerror in space-averaged WSS. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

A.J. Geers et al. / Journal of Biomechanics 47 (2014) 178–185 183

Conflict of interest statement

All authors declare that there are no conflicts of interest.

Acknowledgments

The authors would like to thank Dr. Salvatore Cito for valuablediscussions and technical advice and Dr. Catalina Tobon Gomezfor proofreading the manuscript. This work was supported bythe @neurIST project (financed by the European Commissionthrough contract no. IST-027703) and Philips Healthcare, Best,The Netherlands.

Appendix A. Supplementary material

Supplementary data associated with this article can be found in theonline version, at http://dx.doi.org/10.1016/j.jbiomech.2013.09.033.

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Fig. 7. Experiment 2. Absolute error of the TA WSS field calculated at each node of the surface mesh. Colormaps were normalized by using WSS derived from the pulsatileflow simulation as unit. View points were selected to best visualize the aneurysm, so images are not necessarily at the same scale. The percentages in parentheses indicatethe error in space-averaged WSS. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

Table 2Experiment 2. Summary of error measurements for all 20 aneurysms and foraneurysms grouped by type, volume, and NSI.

Cases Error in WSS CVRMSE

Mean [%] SE [%] p-Value Mean [%] SE [%] p-Value

All 4.3 0.4 7.1 0.7Terminal 4.7 0.6 0.50 8.4 1.2 0.26Lateral 3.9 0.5 5.9 0.4Large volume 4.5 0.4 0.55 7.9 1.0 0.20Small volume 4.1 0.7 6.4 0.9Large NSI 4.4 0.5 0.82 8.1 1.2 0.33Small NSI 4.3 0.6 6.2 0.7

NSI¼nonsphericity index; WSS¼space-averaged WSS; CVRMSE¼coefficient ofvariation of the root-mean-square error; SE¼standard error; p-value was calcu-lated with Wilcoxon rank-sum test. Per category, aneurysms were split into twogroups of 10, e.g. ‘Large volume’ and ‘Small volume’ refer to subsets of 10aneurysms with largest and smallest volume.

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Appendix 1 – Table experiment 2

Note: This appendix was included in the Journal of Biomechanics publication as onlinesupplementary material.

Table 1: Experiment 2. Geometric and hemodynamic characteristics of all 20 aneurysms.Geometry WSS RMSE

Case Type V [mm3] NSI [-] Pulsatile [Pa] Steady [Pa] e [%] RMSE [Pa] CVRMSE [%]1 T 1315 0.23 0.060 0.057 5.2 0.009 14.32 T 201 0.24 0.603 0.583 3.3 0.031 5.23 T 6 0.03 0.505 0.490 3.0 0.018 3.54 T 91 0.27 0.506 0.474 6.2 0.056 11.05 T 67 0.21 0.288 0.281 2.2 0.011 3.8

6 T 88 0.24 0.533 0.515 3.4 0.027 5.17 T 197 0.30 0.107 0.099 7.2 0.013 12.18 T 114 0.17 0.178 0.164 7.5 0.020 11.39 T 25 0.15 0.195 0.186 4.5 0.014 7.3

10 T 275 0.23 0.167 0.159 4.8 0.018 10.7

11 L 101 0.16 0.491 0.480 2.2 0.017 3.612 L 1122 0.26 0.054 0.052 4.0 0.004 7.413 L 175 0.32 1.120 1.081 3.5 0.053 4.714 L 106 0.16 0.095 0.089 6.1 0.007 7.815 L 72 0.09 6.755 6.683 1.1 0.309 4.6

16 L 12 0.10 2.438 2.316 5.0 0.142 5.817 L 288 0.25 2.033 1.953 3.9 0.132 6.518 L 147 0.20 0.365 0.351 4.0 0.023 6.219 L 163 0.18 0.193 0.183 5.3 0.013 6.520 L 674 0.19 0.448 0.429 4.2 0.025 5.6

mean 4.3 7.1SE 0.4 0.7

T = terminal; L = lateral; V = volume; NSI = nonsphericity index; WSS = space-averaged WSS; e = relative error; RMSE =root-mean-square error; CVRMSE = coefficient of variation of the RMSE; SE = standard error.

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Appendix 2 – Animation of pulsatile WSS fields

Note: This appendix was included in the Journal of Biomechanics publication as onlinesupplementary material.

Click here to download a 4MB animation of the pulsatile WSS fields. All 20 casesof experiment 2 are displayed.

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Appendix 3 – Analytical study of steady and pulsatile

flow in a tube

Note: This appendix was not included in the Journal of Biomechanics publication be-cause it does not have direct relevance to steady and pulsatile flow in aneurysms. How-ever, it might provide an insight into the behavior of steady and pulsatile flow in tubes.

This appendix considers viscous, incompressible flow of a Newtonian fluid. TheNavier-Stokes equations that govern the motion of this fluid have analytical solutionsfor fully developed steady and pulsatile flow in a rigid tube with circular cross section[5]. Although the flow field in tortuous blood vessels and aneurysms is highly complexand can not be solved analytically, a deeper understanding of steady and pulsatile flowin arteries can be gained from studying these analytical solutions.

Let Q(t) be a FRW with period T and Qk its discretization in N real data points.The discrete Fourier transform gives Fourier coefficients Q̂n ⌘ 1

N ÂN�1k=0 Qk e�2p ink/N ,

such that

Q(t)⇡N�1

Ân=0

Q̂n eiwn t (1)

where wn is the angular frequency, given by

wn =

(2p n

�T if n N/2

�2p (N �n)�

T if n > N/2

For fully developed pulsatile flow in a tube, the velocity profile v(r, t) is a functionof radius r and time t alone [4], namely

v(r, t) =2Q̂0

p R2

1�

⇣ rR

⌘2�+

N�1

Ân=1

Q̂n

p R2

2

641� J0(bn

rR )

J0(bn)

1� 2J1(bn)bnJ0(bn)

3

75 eiwn t (2)

where R is the tube’s radius, J0 and J1 are Bessel functions of the first kind, andbn = i3/2 an with an = R

qwn r

�µ . Note that a1 is the Womersley number. The cor-

responding WSS tw(t) is given by

tw(t) = �µ✓

∂v∂ r

◆

r=R

=4 µ Q̂0

p R3 �N�1

Ân=1

µ bn Q̂n

p R3

2

4J1(bn)J0(bn)

1� 2J1(bn)bnJ0(bn)

3

5 eiwn t (3)

The equations describing pulsatile flow in a tube are linear in both velocity andpressure and can therefore be split in a steady and an oscillatory part [6]. The steadypart is represented by the first term in Eqs. (1)-(3) and describes a constant forward flowwith TA flow rate Q̂0 and a parabolic velocity profile. The oscillatory is represented

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Table 2: Flow rate waveform descriptors.Descriptor Physiological variation

lower baseline upperHeart rate [bpm]a 52 68 84Pulsatility index [-]a 0.58 0.92 1.26TA flow rate [ml/s]b 0.73QA QA 1.27QA

a baseline = mean, lower = mean� 2SD, and upper = mean+ 2SDwhere the mean and standard deviation (SD) are taken from (au-

thor?) [2] and (author?) [3].b QA = 48.21A1.84 where A is the inlet area in cm2 [1].

Figure 1: Analytical solution of steady and pulsatile flow in a tube. (left) FRW with allthree descriptors at baseline and corresponding WSS waveform. (right) Velocity profilescorresponding to TA, PS and ED WSS.

by the rest of the terms and describes the back and forth motion of the fluid, whichproduces zero net flow over the cardiac cycle.

The Womersley number characterizes the pulsatile flow field. For low Womers-ley numbers (<1), the oscillatory part of v(r, t) describes an approximately parabolicvelocity profile and the flow field can be considered quasi-steady. For physiologicalWomersley numbers (⇠3), however, the oscillatory part of v(r, t), and therefore v(r, t)itself, noticeably deviates from a parabolic profile. Especially around PS, during whichthe blood experiences a relatively large acceleration, inertial forces dominate viscousforces and the velocity profile is more plug-like. As a result, maximum or PS WSS inpulsatile flow is higher than the WSS in steady flow under maximum flow rate condi-tions and, similarly, minimum or ED WSS in pulsatile flow is lower than the WSS insteady flow under minimum flow rate conditions. TA WSS in pulsatile flow is equal tothe WSS in steady flow under TA flow rate conditions given that all but the first termin Eqs. (1)-(3) vanish when integrated over the cardiac cycle.

To illustrate the difference in WSS between steady and pulsatile flow, we calculatedv(r, t) and tw(t) for a tube with a radius of 2.5 mm, which is the typical radius of anICA [1], and blood with a density of 1060 kg/m3 and a viscosity of 0.004 Pa·s. Theeffect of FRW descriptors on the difference was assessed by fixing two descriptors atbaseline and varying the third from lower to upper value as specified in Table 2.

Figure 1 shows the FRW with all three descriptors at baseline and the correspondingWSS waveform. A phase shift of the WSS waveform ahead of the FRW can be clearlyobserved at PS. The figure also shows the pulsatile velocity profiles correspondingto TA, PS and ED WSS. They are shown together with parabolic velocity profiles of

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Figure 2: Analytical solution of pulsatile and steady flow in a tube. TA, PS and ED WSS asfunction of the three FRW descriptors.

steady flow under TA, PS and ED flow rate conditions.Figure 2 shows the TA, PS and ED WSS as function of each of the three FRW

descriptors. As mentioned before, TA WSS is equal, PS WSS is lower and ED WSS ishigher in steady than in pulsatile flow. In terms of the relative difference with respect tothe WSS in pulsatile flow: for increasing heart rate, the difference in PS WSS increasedfrom 29.2% to 37.6% and the difference in ED WSS increased from 10.7% to 17.3%;for increasing pulsatility index, the difference in PS WSS increased from 27.5% to38.1% and the difference in ED WSS increased from 7.3% to 24.4%; and for all TAflow rates, the difference in PS WSS was constant at 33.9% and the difference in EDWSS was constant at 14.0%.

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