effect of curvature on displacement forces acting on ... · pdf file¤experimental...

¤EXPERIMENTAL INVESTIGATION ¤

Effect of Curvature on Displacement Forces Acting onAortic Endografts: A 3-Dimensional ComputationalAnalysis

C. Alberto Figueroa, PhD1; Charles A. Taylor, PhD1,2; Victoria Yeh1; Allen J. Chiou1;and Christopher K. Zarins, MD2

Departments of 1Bioengineering and 2Surgery, Stanford University, Stanford,California, USA.

¤ ¤Purpose: To determine the effect of curvature on the magnitude and direction ofdisplacement forces acting on aortic endografts in 3-dimensional (3D) computationalmodels.Method: A 3D computer model was constructed based on magnetic resonanceangiography data from a patient with an infrarenal aortic aneurysm. Computational fluiddynamics tools were used to simulate realistic flow and pressure conditions of the patient.An aortic endograft was deployed in the model, and the displacement forces acting on theendograft were calculated and expressed in Newtons (N). Additional models were createdto determine the effects of reducing endograft curvature, neck angulation, and iliacangulation on displacement forces.Results: The aortic endograft had a curved configuration as a result of the patient’sanatomy, with curvature in the anterolateral direction. Total displacement force acting onthe endograft was 5.0 N, with 28% of the force in a downward (caudal) direction and 72% ofthe force in a sideways (anterolateral) direction. Elimination of endograft curvature (planargraft configuration) reduced total displacement force to 0.8 N, with the largest componentof force (70%) acting in the sideways direction. Straightening the aortic neck in the curvedendograft configuration reduced the total force acting on the endograft to 4.2 N, with areduction of the sideways component to 55% of the total force. Straightening the iliac limbsof the endograft reduced the total force acting on the endograft to 2.1 N but increased thesideways component to 91% of the total force.Conclusion: The largest component of the force acting on the aortic endograft is in thesideways direction, with respect to the blood flow, rather than in the downward (caudal)direction as is commonly assumed. Increased curvature of the aortic endograft increases themagnitude of the sideways displacement force. The degree of angulation of the proximal anddistal ends of the endograft influence the magnitude and direction of displacement force. Thesefactors may have a significant influence on the propensity of endografts to migrate in vivo.

J Endovasc Ther. 2009;16:284–294

Key words: displacement forces, aortic endografts, endograft migration, abdominal aorticaneurysm, angulation, curvature, computational fluid dynamics

¤ ¤

This study was supported by the National Institutes of Health/NHLBI grant 2R01 HL64327-05A1. In accordance with the NIHPublic Access Policy, this article is available for open access at PubMed Central.

Presented at International Congress XXI Endovascular Interventions held in Scottsdale, Arizona, USA, on February 10-14,2008.

The authors have no commercial, proprietary, or financial interest in any products or companies described in this article.

Address for correspondence and reprints: Dr. Christopher K. Zarins, Department of Surgery, Clark Center E350A, 318Campus Drive West, Stanford, CA 94305-5431 USA. E-mail: [email protected]

284 J ENDOVASC THER2009;16:284–294

� 2009 by the INTERNATIONAL SOCIETY OF ENDOVASCULAR SPECIALISTS Available at www.jevt.org

Pulsatile aortic blood flow and pressure exerttime-varying loads on all intra-aortic devic-es.1–3 Long-term durability of these devicesrequires continuous mechanical fixation ofthe vascular prosthesis to prevent displace-ment.4–6 Failure of fixation can lead to devicemigration and adverse clinical events, such assecondary interventions, endoleaks, ruptureand death.7

A number of factors hypothesized to affectdevice migration have been clinically investi-gated, including aortic neck diameter, length,and angulation; neck calcification and throm-bus; neck enlargement; inadequate proximaland distal fixation length; and neck enlarge-ment.8–10 In-vitro11 and in-vivo experimentalstudies,12 as well as theoretical13 and compu-tational studies,14–19 have been conducted toinvestigate the magnitude of the loads actingon endografts. These loads have been referredto as migration forces, displacement forces, ordrag forces. While some of these studies haveanalyzed several factors affecting migrationforces, such as device size, hypertensive state,and different geometrical configurations, in allcases the reported forces have been assumedto act primarily in the downward (caudal)direction of blood flow. Experimental studiesexerted linear downward displacement force

on devices and computational studies consid-ered devices in a planar and therefore ideal-ized configuration. However, in humans, aor-toiliac geometry is usually curved, withanterior angulation of the aneurysmal aortaand posterior curvature of the iliac arteries inthe pelvis. This results in significant centerlinelumen curvature of the aortoiliac segment andsignificant post-implantation curvature of im-planted aortic endografts. In cases of severeaneurysm and aortic neck angulation, thiscurvature may be quite pronounced and maybecome more accentuated over time due toendograft migration. In this study, we investi-gated the impact of curvature on the displace-ment forces experienced by aortic endograftsusing a 3-dimensional (3D) computationaltechnique.

METHOD

A 3D computer model of an abdominal aorticaneurysm (AAA) was constructed using a de-identified patient phase-contrast magneticresonance angiography (MRA) dataset. Thecomputer model included data from thesupraceliac aorta to the iliac arteries, includ-ing the celiac, superior mesenteric, and renalarteries (Fig. 1).20 Using the computer model,

Figure 1¤A 3D computer model was built based on the MRA dataset shown on the left. Thepreoperative CFD analysis is shown along with the post-endograft implantation CFD analysis.

J ENDOVASC THER2009;16:284–294

ENDOGRAFT CURVATURE AND DISPLACEMENT FORCES 285Figueroa et al.

a computational fluid dynamics (CFD) analysiswas performed to simulate blood flow andblood pressure.21,22 In this analysis, additionalpatient-specific flow data [cine phase-contrastmagnetic resonance imaging (PC-MRI)] at thesupraceliac and infrarenal levels and pressuredata (brachial pressure cuff measurementsacquired immediately after the scan) wereconsidered in order to model the baseline flowand pressure conditions in this patient asaccurately as possible.23 An aortic endograftwas then virtually implanted in the aneurysmmodel and another CFD analysis was per-formed (Fig. 1). These flow and pressurebaseline conditions remained unchanged inboth the pre and post endograft simulations.The magnitude and direction of time-varyingloads exerted by blood flow on the device werecalculated by integrating the distribution oftractions (pressure and the shearing stresses)acting on the surface of the endograft (Fig. 2).The implanted endograft model was thenmodified to (a) eliminate endograft curvature,(b) straighten aortic neck angulation, and (c)straighten iliac angulation. For each endograftconfiguration, the magnitude and direction ofdisplacement forces were calculated.

RESULTS

Preoperative Analysis

The results of the preoperative computationalmodel are shown in Figure 3. Aortic inflow

pressure matched the systolic (168 mmHg)and diastolic (92 mmHg) pressures measuredin the patient. Individual branch flow wave-forms varied depending on outflow boundaryconditions of each vessel. The differencesbetween the renal artery and iliac artery flowwaveforms are shown with forward flowin the renal artery throughout the cardiaccycle and reversal of flow in the iliac arteryduring diastole, which represent normalphysiological variations. The computed infra-renal aortic flow (1.4 L/min) matches thepatient’s measured aortic flow by PC-MRI atthat location.

Post-Implantation Analysis:Curved Endograft

In this scenario, the dimensions of theendograft were chosen to reflect the sizes ofthe infrarenal aortic neck and the iliac arteries.The course of the endograft closely followsthe preoperative aortic geometry and reflectsthe expected position of the endograft fol-lowing deployment. The curvature of theendograft is described by the formula C1 5

1/R1, where R1 is the radius that approximate-ly fits a circle through the path of the graft;thus, the smaller the radius R1 the larger thecurvature of the endograft. The magnitudeand vector of displacement force acting onthe endograft in the AP (anteroposterior),lateral, and axial projections are shown by

Figure 2¤Wall shear (left) and pressure (right) stresses representing the actions of theblood on the endograft. These stresses are integrated over the surface of the endograft tocalculate the total 3D force exerted by the pulsatile flow. Note that the pressure is severalorders of magnitude larger than the shear stress. This fact, together with the main curvatureof the graft, dictates the magnitude and orientation of the displacement force.

286 ENDOGRAFT CURVATURE AND DISPLACEMENT FORCESFigueroa et al.


the size and direction of the arrows inFigure 4. The displacement force is primarilydirected anteriorly and laterally and is largelyin a sideways direction with respect to theaxis of blood flow in the artery. The forcevector appears to be largely determined bythe curvature of the endograft.

Quantitative assessment of displacementforces acting on the endograft using CFDanalysis is shown in Table 1. The 3D vectorrepresenting the temporal average of the totalmigration force reveals that the total displace-ment force acting on the endograft is 5.01 N;the largest components of force are in theanterior and lateral directions. The downwardcomponent of the force in the direction ofblood flow, Fz, was relatively small comparedto the sideways component, accounting foronly 28% of the total force. Thus, 72% of thetotal displacement force was directed laterallyor sideways with respect to the direction ofblood flow.

Reduced Curvature (Planar) Endograft

The post-implantation computer modelwas altered to reduce endograft curvatureby modifying the anatomy of the aorta in theregion of the aneurysm to accommodate theendograft in an almost flat or planar config-uration. The endograft dimensions and theaortic and branch flow and pressure condi-tions were unchanged. The curvature of theendograft was much reduced (Fig 5; C2 5

1/R2, C2 % C1). CFD analysis of the reducedcurvature endograft revealed a significant(more than 5-fold) reduction in the totaldisplacement force acting on the endograftto 0.8 N. The orientation of the displacementforce vector was similar to that found in thecurved endograft model (see Table 1), with30% of the total force directed downward and70% directed sideways. Thus, even in thereduced curvature model, the primary direc-tion of displacement force was sideways withrespect to the direction of blood flow.

Figure 3¤Preoperative CFD analysis of patient-specific computer model. Aortic inflowpressure and flow reflect the measured values in the patient at the time of PC-MRImeasurements. Aortic, left renal artery, and right common iliac artery pressure and flowwaveforms during 1 cardiac cycle are shown. Note: the time in the cardiac cycle shown in thevelocity magnitude plot is indicated in the flow and pressure waves with a dot.



Comparison of Curved toPlanar Endograft

Comparing the pulsatile forces acting onthe normally positioned curved endograft andthe modified planar endograft, the flow andpressure conditions in both models werevirtually identical (Fig. 6). As can be seen inthe figure, the changes in endograft curvaturedid not alter the pressure field in the aortasignificantly. However, there were markedchanges in the total displacement force actingon the endograft. The temporal average totaldisplacement force in the planar endograftwas reduced by 5-fold to 0.8 N compared to

the normally positioned curved endograft,which experienced a total force of 5 N. Inboth cases, the largest component of theforce was Fy (i.e., anterior direction), whichcan be explained by the slight angulation ofthe neck of the abdominal aorta, directing theblood stream to impinge the endograft on theanterior face.

Aortic Neck and Iliac Angulation

The curved endograft model was modifiedto (a) reduce aortic neck angulation and (b)reduce iliac artery angulation. The magnitudeand vector of total displacement force actingon the endograft are shown by the size anddirection of the arrows in Figure 7. Therewere significant differences in both the mag-nitude and the direction of the displacementforce for the 3 endograft configurations. Thenumerical values of the components of theforce vector are shown in Table 2. Thereduced neck angulation model showed a17% reduction in displacement force com-pared to the normal curved model. Thereduced iliac angulation model showed a58% reduction in total displacement forcecompared to the normal curved model. The

Figure 4¤Anterior, lateral, and axial views of the aneurysm model with the aortic endograftin place. Note the anterolateral angulation of the endograft, which follows the curvedanatomy of the aortoiliac segment. The vector of the displacement force is shown in each ofthe views.

¤ ¤TABLE 1

Total and 3D Components of Displacement Force(F) for the Curved and Reduced

Curvature Endografts

CurvedEndograft

Reduced CurvatureEndograft

Fx (lateral), N 22.22 20.29Fy (anterior), N 4.26 0.72Fz (axial), N 21.42 20.24Total force, N 5.01 0.81% Downward 28.35 29.54¤ ¤



axial, or downward, displacement componentwas greatest for the straight aortic neckmodel (45% of the force of 4.2 N) and leastfor the reduced iliac angulation (9% of totalforce of 2.1 N). By comparison, the curvedendograft model had a downward displace-ment component of 28%, with a total force of5.01 N. In contrast, the sideways displace-ment force was greatest in the reduced iliacangulation model (91% of total of 2.1 N) and

lowest in the reduced aortic neck angulationmodel (55% of total of 4.2 N).

DISCUSSION

Previous experimental and computationalstudies of aortic endograft fixation and mi-gration have been based on the assumptionthat endograft displacement forces are direct-ed downward, in the caudal direction of blood

Figure 5¤Anterior, lateral, and axial views of the reduced curvature endograft model in aplanar configuration that removes the anteroposterior curvature of the endograft. Theendograft dimensions are identical to the endograft shown in Figure 3. Note the reduction inmagnitude of displacement force acting on the endograft as reflected by the size of thearrows. The direction of displacement force is similar to the curved configuration in Figure 3.

Figure 6¤Comparison of normal curved endograft to reduced curvature (planar) endograftunder the same pressure and flow conditions. The curved endograft has a 5-fold greatertemporal average force (5.01 N versus 0.81 N) than the planar endograft model.



flow. Current endografts are designed with avariety of fixation strategies to resist thisdownward displacement force. Nonetheless,all currently available endografts have expe-rienced device migration. To our knowledge,no other 3D analysis of displacement forcesacting on an aortic endograft has been donein a patient-specific aneurysm model thatincludes both the endograft geometry and

the visceral branches (celiac, superior mes-enteric, and renal arteries). Our findings inthis anatomically realistic model show thatthe primary vector of displacement forceacting on an aortic endograft is not down-ward in the direction of blood flow, but ratherin a sideways direction or lateral to the axis ofblood flow. Both the magnitude and vector ofdisplacement force are greatly influenced bythe curvature of the endograft. The greaterthe curvature of the endograft the greater thetotal displacement force acting on the endo-graft and the larger the sideways componentof displacement. In addition to curvature ofthe endograft, angulation of the proximalaortic fixation zone and distal iliac fixationzone are important in determining endograftdisplacement force.

Almost all endografts have significant cur-vature due to the normal anatomy of theabdominal aorta, which is determined by theanterior angulation of the supraceliac aorta

Figure 7 ¤ Effect of reducing aortic neck angulation and iliac angulation on totaldisplacement force (B). The magnitude and direction of the displacement force vector areshown with arrows in the (A) anterior, (C) lateral, and (D) axial view.

¤ ¤TABLE 2

Total and 3D Components of Displacement Force(F) for the Curved Endograft, Straight Neck, and

Straight Iliac Arteries Simulations

CurvedEndograft

StraightNeck

StraightIliacs

Fx (lateral), N 22.22 21.48 0.28Fy (anterior), N 4.26 3.43 2.09Fz (axial), N 21.42 21.87 20.19Total force, N 5.01 4.18 2.12% downward 28.35 44.76 8.97¤ ¤



and posterior curvature of the iliac arteriesdeep in the iliac fossa. This may not beapparent on the usual AP imaging of theaorta and is best appreciated on lateral oroblique image projections. Many aneurysmshave great degrees of angulation and tortu-osity, and the degree of endograft curvatureis often much greater than the endograftmodeled in this study. Greater degrees ofendograft curvature will result in significantlylarger forces applied to the endograft.

While clinical endograft migration is usuallydescribed as longitudinal or caudal migra-tion,3,24,25 image analysis often reveals signif-icant lateral movement of the endograft, asdepicted in Figure 8. The 5.5-cm aneurysmwas treated with an endograft, and post-

implantation computed tomography (CT)showed good position of the endograft withcomplete aneurysm exclusion and no endo-leak. Lateral projection of the 3D CT imageshows anterior curvature of the endograft,which is similar to the curved computationalendograft model in this study. At 1 year, therewas 1.6-cm anterior displacement of theendograft (Fig. 8); after 3 years, there wascontinued anterior endograft displacement,with 2.3 cm lateral movement of the endograftrelative to the spine. The aneurysm remainedunchanged in size, and there was no endoleak;however, there was significant movement ofthe endograft in the same direction as thesideways displacement force demonstrated inthe computational model in this study.

Figure 8¤ Lateral and cross-sectional CT images of an aortic endograft used to treat apatient with an abdominal aortic aneurysm. Note the anterior curvature of the endograft postimplantation. There is anterior displacement of the aorta and endograft at 1 year withincreased anterior displacement at 3 years. The movement is in the same direction of thesideways vector of displacement force demonstrated in the computational analysis. Note:these datasets correspond to a different patient than the one considered for thecomputational analysis.



A recent clinical study has shown that lateralmovement of endografts within the aneurysmsac is an indicator of stent-graft instability. In astudy of 60 high-risk patients treated with bothsuprarenal and infrarenal endografts, Rafii etal.26 showed that lateral endograft movementwithin the aneurysm sac at 1 year wasassociated with an increased risk of lateadverse events, including development of typeI endoleaks, stent-graft migration, and theneed for secondary procedures. Other reportsof endograft migration have shown evidenceof lateral endograft movement in publishedimages,27,28 which are consistent with thedisplacement force analysis findings in ourstudy. The results presented here indicate thatthe total force experienced by AAA endograftsin vivo greatly depends on the level ofcurvature to which these devices are exposed.This curvature is determined to some extentby the neck and iliac angulations, but it is alsoa function of the level of tortuosity of theaneurysm sac. Therefore, graft curvature invivo might be an important predictor ofmigration, since we have demonstrated thatgeometries exposed to nearly identical levelsof pressure experience very different migra-tion forces. Our results show that the larger thecurvature, the larger is the migration force forthe same level of internal pressure.

Many patients treated with aortic endograftshave severe degrees of aortic neck and iliacangulation and aneurysm tortuosity. Thesepatients are likely to be at an increased risk ofmigration based on the migration forcesacting on the endograft. In general, in tortuousgeometries with important curvatures, thelargest component of the migration force willbe contained in the plane of the highestcurvature. Currently available endografts havenot been designed to deal with such degreesof tortuosity. Future endograft device designshould take into consideration the orientationof the displacement force vector in order toensure long-term stability of the device.

Limitations

Our simulations have been based on theassumption that the baseline supraceliac flowwaveform and outflow boundary conditionsfor the untreated and stented AAA models

remain unchanged. This is a reasonable firstapproximation for the resting conditions mod-eled since it is unlikely that repair of AAA willhave much of an effect on the already lowresistance of the infrarenal aortic segmentunder this low-flow state. As a result, sincestent-graft implantation would be unlikely toalter the pressure drop across the infrarenalregion (as the first graph of Figure 6 indicates),the effect on supraceliac flow waveform or theoutflow boundary conditions would be expect-ed to be minimal under resting conditions. Inany event, further refinement of the inflow andoutflow boundary conditions under restingconditions or the extension to higher flowstates would require flow and pressure mea-surements before and after stent-graft implan-tation on a patient-specific basis.

In addition, we have assumed smoothlumen transitions of the endograft in boththe neck and iliac regions. This is often notthe case, particularly in cases of severeangulation, and this may affect the forceanalysis results presented here. We have alsoassumed that pressure in the aortic aneurysmsac after endograft deployment is zero. This isnot the case in patients who have endoleaksafter endovascular aneurysm repair.

Furthermore, these results have been ob-tained under the assumption that both thearterial and graft walls are rigid. However,both the aortic wall and endograft fabric mayhave varying degrees of movement. Futurework, which includes the compliance of thevessel walls, is needed to evaluate how muchchanges in compliance might affect the forcesexperienced by endografts.29

Lastly, the analysis presented here providesjust an estimate of the total force acting on anendograft. Actual endograft migration willdepend not only on this force, but also on anumber of other factors that are not consid-ered in this analysis, such as the frictiondeveloped at the fixation points between thegraft and the wall, fixation length, longitudi-nal columnar force, and the influence ofpenetrating hooks and barbs.

Conclusion

The primary orientation of the displacementforce vector acting on an endograft implanted



in an AAA is sideways rather than downwardwith respect to the direction of blood flow. Themagnitude of the displacement force is greatlyinfluenced by the curvature of the endograft;greater degrees of endograft curvature resultin increased total and increased sideways-directed displacement force. Three-dimen-sional computational analysis may provide apowerful tool to evaluate the risk of endograftmigration in vivo.

Acknowledgments: The authors gratefully acknowledgeAndrea Les for the patient-specific image data, flow andpressure measurements, and computer model utilized forthe preoperative analysis utilized in this work.

REFERENCES

1. Fogarty TJ, Arko FR, Zarins CK. Endografttechnology: highlights of the past 10 years.J Endovasc Ther. 2004;11(Suppl II):II-192–II-199.

2. Zarins CK, Bloch DA, Crabtree T, et al. Stentgraft migration after endovascular aneurysmrepair: importance of proximal fixation. J VascSurg. 2003;38:1264–1272.

3. Zarins CK. Stent-graft migration: how do weknow when we have it and what is itssignificance? [commentary] J Endovasc Ther.2004;11:364–365.

4. Arko FR, Heikkinen MA, Lee ES, et al. Iliacfixation length and resistance to in-vivo stentgraft displacement. J Vasc Surg. 2005;41:664–671.

5. Benharash P, Lee JT, Abilez OJ, et al. Iliacfixation inhibits migration of both suprarenaland infrarenal aortic endografts. J Vasc Surg.2007;45:250–257.

6. Heikkinen MA, Alsac JM, Arko FR, et al. Theimportance of iliac fixation in prevention ofstent graft migration. J Vasc Surg. 2006;43:1130–1137.

7. Sampram ES, Karafa MT, Mascha EJ, et al.Nature, frequency, and predictors of secondaryprocedures after endovascular repair of ab-dominal aortic aneurysm. J Vasc Surg.2003;37:930–937.

8. Hobo R, Kievit J, Leurs LJ, et al. Influence ofsevere infrarenal aortic neck angulation oncomplications at the proximal neck followingendovascular AAA repair: a EUROSTAR study.J Endovasc Ther. 2007;14:1–11.

9. Fulton JJ, Farber MA, Sanchez LA, et al. Effectof challenging neck anatomy on mid-termmigration rates in AneuRx endografts. J VascSurg. 2006;44:932–937.

10. Zarins CK, AneuRx Clinical Investigators. TheUS AneuRx Clinical Trial: 6-year clinical update2002. J Vasc Surg. 2003;37:904–908.

11. Resch T, Malina M, Lindblad B, et al. The impactof stent design on proximal stent-graft fixationin the abdominal aorta: an experimental study.Eur J Vasc Endovasc Surg. 2000;20:190–195.

12. Murphy EH, Johnson ED, Arko FR. Device-specific resistance to in vivo displacement ofstent-grafts implanted with maximum iliacfixation. J Endovasc Ther. 2007;14:585–592.

13. Morris L, Delassus P, Walsh M, et al. Amathematical model to predict the in vivopulsatile drag forces acting on bifurcated stentgrafts used in endovascular treatment ofabdominal aortic aneurysms (AAA). J Bio-mech. 2004;37:1087–1095.

14. Li Z, Kleinstreuer C, Farber M. Computationalanalysis of biomechanical contributors to pos-sible endovascular graft failure. Biomech Mod-el Mechanobiol. 2005;4:221–234.

15. Li Z, Kleinstreuer C. Analysis of biomechanicalfactors affecting stent-graft migration in anabdominal aortic aneurysm model. J Biomech.2006;39:2264–2273.

16. Fung GS, Lam SK, Cheng SW, et al. On stent-graft models in thoracic aortic endovascularrepair: a computational investigation of thehemodynamic factors. Comput Biol Med.2008;38:484–489.

17. Morris L, Delassus P, Grace P, et al. Effects offlat, parabolic and realistic steady flow inletprofiles on idealised and realistic stent graft fitsthrough abdominal aortic aneurysms (AAA).Med Eng Phys. 2006;28:19–26.

18. Howell BA, Kim T, Cheer A, et al. Computa-tional fluid dynamics within bifurcated abdom-inal aortic stent-grafts. J Endovasc Ther.2007;14:138–143.

19. Molony DS, Callanan A, Morris LG, et al. Geo-metrical enhancements for abdominal aorticstent-grafts. J Endovasc Ther. 2008;15:518–529.

20. Wilson NM, Wang KC, Dutton RW, et al. Asoftware framework for creating patient specificgeometric models from medical imaging data forsimulation based medical planning of vascularsurgery. In Lecture Notes in Computer Science.vol. 2208. Berlin: Springer; 2001:449–456.

21. Taylor CA, Hughes TJR, Zarins CK. Finite ele-ment modeling of blood flow in arteries. ComputMeth Appl Mech Eng. 1998;158:155–196.

22. Vignon-Clementel IE, Figueroa CA, Jansen KE,et al. Outflow boundary conditions for three-dimensional finite element modeling of bloodflow and pressure in arteries. Comput MethAppl Mech Eng. 2006;195:3776–3796.



23. Les AS, Figueroa CA, Draney Blomme MT, et al.Quantification of hemodynamics in abdominalaortic aneurysms during rest and exercise usingmagnetic resonance imaging and computation-al fluid dynamics. Presented at: InternationalSociety for Magnetic Resonance in MedicineConference; Toronto, Canada; May 3–8, 2008.

24. Cao P, Verzini F, Zannetti S, et al. Devicemigration after endoluminal abdominal aorticaneurysm repair: analysis of 113 cases with aminimum follow-up period of 2 years. J VascSurg. 2002;35:229–235.

25. Parra JR, Ayerdi J, McLafferty R, et al. Confor-mational changes associated with proximalseal zone failure in abdominal aortic endo-grafts. J Vasc Surg. 2003;37:106–111.

26. Rafii BY, Abilez OJ, Benharash P, et al. Lateralmovement of endografts within the aneurysmsac is an indicator of stent-graft instability.J Endovasc Ther. 2008;15:335–343.

27. Kramer SC, Seifarth H, Pamler R, et al. Geometricchanges in aortic endografts over a 2-year obser-vation period. J Endovasc Ther. 2001;8:34–38.

28. Umscheid T, Stelter WJ. Time-related alter-ations of shape, position, and structure in self-expanding modular aortic stent-grafts. J En-dovasc Surg. 1999;6:17–32.

29. Figueroa CA, Vignon-Clementel IE, Jansen KE,et al. A coupled momentum method formodeling blood flow in three-dimensionaldeformable arteries. Comput Meth Appl MechEng. 2006;195:5685–5706.



effect of curvature on displacement forces acting on ... · pdf file¤experimental...

Documents