European Journal of Vascular & Endovascular Surgery
Volume 28, Issue 2 , Pages 168-176, August 2004

A Comparative Study of Aortic Wall Stress Using Finite Element Analysis for Ruptured and Non-ruptured Abdominal Aortic Aneurysms☆

Academic Vascular Unit, Vascular Laboratory, Alderson house, Hull Royal Infirmary, Hull, East Yorkshire HU3 2JZ, UK

Accepted 30 March 2004.

Article Outline

Abstract 

Background. The decision to repair an asymptomatic abdominal aortic aneurysm (AAA) is currently based on diameter (≥5.5 cm) alone. However, aneurysms less than 5.5 cm do rupture while some reach greater than 5.5 cm without rupturing. Hence the need to predict the risk of rupture on an individual patient basis is important. This study aims to calculate and compare wall stress in ruptured and non-ruptured AAA.

Methods. The 3D geometries of AAA were derived from CT scans of 27 patients (12 ruptured and 15 non-ruptured). AAA geometry, systolic blood pressure and literature derived material properties, were utilised to calculate wall stress for individual AAA using finite element analysis.

Results. Peak wall stress was significantly higher in the ruptured AAA (mean 1.02 MPa) than the non-ruptured AAA (mean 0.62 MPa). In patients with an identifiable site of rupture on CT scan, the area of peak wall stress correlated with rupture site.

Conclusions. Peak wall stress can be calculated from routinely performed CT scans and may be a better predictor of risk of rupture than AAA diameter on an individual patient basis.

Keywords:  Finite element analysis, Wall stress, Rupture-risk, Abdominal aortic aneurysm

 

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1. Introduction 

Abdominal aortic aneurysm (AAA) is a common cause of preventable death in men over the age of 65 years.1 The mortality following elective AAA repair has significantly improved over the recent years to 3–6%. However following AAA rupture, 50% of patients die before reaching hospital and emergency repair is associated with 40–50% mortality. Decision-making in regard to elective AAA repair therefore requires careful assessment of rupture risk, operative mortality and life expectancy.2

Patients are currently advised repair if the diameter of the AAA is ≥5.5cm3 or if they are symptomatic. However, although aneurysm size is an important predictor of rupture,4 not all large aneurysms rupture, while 10–24% of small aneurysms (<5.5 cm) may rupture.5, 6 Currently, no reliable criterion exists to predict risk of rupture on an individual patient basis, and a decision to operate based on AAA diameter alone may subject a significant proportion of patients to unnecessary surgery with significant mortality and morbidity. Patients with a stable aneurysm are more likely to die of other causes.7 Hence, the arbitrary setting of a single threshold diameter for elective repair to all patients may be inappropriate.2

AAA rupture occurs when the stress developed in the aneurysm wall exceeds the yield strength of the material. The stress (force per unit area) arises because the blood pressure expands the wall outwards, and this expansion is resisted and balanced by forces (stresses) in the wall. The aneurysm acts as a thin-walled pressure vessel and develops hoop and longitudinal stresses. Classical engineering stress analysis of simple axisymmetric shapes shows that the stress in both directions is directly proportional to the diameter, and inversely proportional to the thickness of the aneurysm wall. Thus, the stresses increase as the diameter increases, but decrease as the thickness increases.

Hence it is logical to relate risk of rupture to aneurysm size. However, because AAA have complicated asymmetric shapes, the relationship is more complex, and the stress in an AAA will depend on the entire geometry. It is for this reason that recent improvements in the knowledge of AAA geometry, coupled with the advances in imaging technique have focused on relating the rupture risk to AAA geometry and resultant wall stress rather than diameter alone. Thus a simplified biomechanical and clinical study has indicated that wall stress is probably a better predictor of risk of rupture than aneurysm diameter alone,8 and recently it has been shown that finite element analysis (FEA) can give a prediction of stress values in individual aneurysms using geometry derived from CT scan data.9

The purpose of this present study was to calculate and compare wall stress in a population of ruptured and non-ruptured AAA.

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2. Methods 

2.1. Patient population 

Twenty-seven patients with infra-renal AAA who had a CT scan as part of their elective or emergency evaluation were included in the study. (None of the study population had a CT scan outside the normal protocol). The data was collected retrospectively and prospectively, and the patients were divided into ruptured12 and non-ruptured15 AAA.

To calculate wall stress in AAA, three main pieces of information are required: (i) the geometry of the AAA, (ii) the material properties of the aortic wall, and (iii) the forces and constraints acting on the wall.

2.2. Geometry of AAA 

Abdominal CT scans were performed using spiral CT scanners (Mx 8000 quad slice and Mx 8000 dual slice, Philips). The standard AAA protocol included bolus tracking (scan initiation at the peak of contrast uptake), with a nominal slice thickness of 3.2 mm with 50% overlap with a helical pitch of 0.875. The scanners were set to 120kv and 250mAs with a standard algorithm. One hundred millilitres of Ultravist 300 (Schering AG, Germany, product license 0053/0174) a non-ionic contrast was administered with bolus pro software. Abdomen and pelvis were imaged to visualise the infra-diaphragmatic aorta down to the common iliac artery. The CT scans were imported into image processing software (Scion Image v4.0, Scion corporation, Maryland, USA), and each cross section of the AAA opened as a separate image file and the wall of the AAA marked manually (Fig. 1) to give its (x–y) profile in 2D which was output as a text file. The process was repeated for each slice of the AAA, and the z-coordinate of each slice added subsequently using the slice thickness information. The cloud of data points (Fig. 2) thus created was then imported into a 3D image rendering software Rhinoceros (v2, Seattle, USA) to create a 3D surface representation of the aneurysm (Fig. 3, Fig. 4, Fig. 5). The resultant surface data was then exported in a (RAW-triangle) format suitable for subsequent conversion into a finite element (FE) model. The level of detail that could be included in this exported file could be varied, effectively smoothing the surface, so the effect of this smoothing process on the predicted stresses was examined.

In this first study, the aorta wall was assumed to have a uniform thickness of 2.0 mm.10 Again the sensitivity of the predicted stresses to this value was investigated.

2.3. Material property model 

We used a previously validated mathematical model of AAA wall material properties. Raghavan et al. derived the mechanical properties of a typical AAA calculated from mean population values obtained from 69 human AAA tissue.11 A longitudinal and circumferential strip of AAA wall tissue, from the anterior surface was obtained during open repair, and subjected to uniaxial extension to derive the material parameters. They subsequently undertook a finite element analysis of a given AAA geometry while varying the material properties within the 95% confidence interval of the specimen population. Results showed that wall stress changed by 4% or less by varying the material parameters within the 95% confidence interval, implying wall stress was insensitive to variations in the values of mechanical properties within a reasonable domain, avoiding the need for patient specificity. A similar sensitivity investigation was also conducted as part of this current study.

2.4. Physiological forces acting on AAA wall 

Systolic blood pressure was taken as the maximum load acting on the aneurysm wall at any given time. A value recorded near the time of the CT scan was taken for all elective AAAs, whereas the highest systolic blood pressure recorded at any earlier admission/visit was used for those patients who presented as ruptured AAA with low blood pressure/shock. The shear stress induced by blood flow in AAA has insignificant impact on the results of stress analysis,12 hence it was ignored in the present study. In vivo, the renal arteries and the iliac arteries constrain an infra-renal AAA from deforming at the proximal and distal ends. To account for this and the effect of the lumbar vertebrae, we constrained our AAA finite element model in the longitudinal direction at the proximal and distal ends.

2.5. Finite element analysis (FEA) 

Finite element analysis is used regularly in many engineering industries, especially the aerospace and motor industries, to accurately determine the stress distribution in complex components. Briefly, the geometry of the problem under consideration is divided into a finite number of small regions, called elements, which are connected together at their corners, called nodes. The behaviour of these individual elements is expressed mathematically and combined to give the behaviour of the whole geometry. The resultant set of equations is modified to take account of the applied loading and the constraint conditions, and then solved. Usually finite element models contain tens of thousands and possibly hundreds of thousands of elements, resulting in hundreds of thousands of (simultaneous) equations. Solution of these equations gives the displacements of the nodes, from which the stress distribution throughout the geometry can be determined. If a sufficient number of elements is used to model the geometry and the physical problem is represented accurately (i.e. the geometry, loads and material properties are all representative of the physical reality), then finite element analysis will give very accurate answers.23 Inevitably, however, in most biomedical engineering problems, there will be some uncertainty in one or more of the physical properties, hence it is usual to examine the sensitivity of the results to that property.

The RAW-triangle representation of the surface of the AAA was converted into a finite element model (Fig. 6) by converting each triangle into a 6-node triangular shell element (quadratic interpolation function with linked bending and membrane capabilities). This was achieved through an in-house program, which created a complete input file for the finite element program, adding in other model details such as the material properties, load and constraints. The program used for the analysis and subsequent post-processing was ANSYS 6.1 (ASN Systems Ltd, Cannonsburg, USA). The stress value used to evaluate the state of the aneurysm was the von-Mises stress. This provides a single value of stress at any point calculated from the full 3D-stress tensor, and is widely used in engineering to evaluate the state of stress of complex 3D problems.

2.6. Statistical analysis 

Data were calculated for each group as mean±standard deviation. Statistical significance was calculated by independent sample t-test. The data was computed with SPSS (v11.5) software. P≤0.05 was considered significant.

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3. Results 

There were six female (four non-rupture group, two rupture) and 21 male (11 non-ruptured and 10 ruptured) patients in this study. Further details on the patient demographics are included in Table 1. There was no significant difference in the mean diameter between the two groups (ruptured group 7.6 cm, non-ruptured 6.8 cm, P>0.1). There were two aneurysms, which ruptured at a relatively small diameter (5.0 cm and 5.7 cm).

Table 1. Comparison of patient demographics between ruptured and non-ruptured AAA
Non-ruptured AAA (n=15)Ruptured AAA (n=12)P (χ2)
Age- median years (range)75 (66–90)75 (71–84)NS
Sex- Male:female5:15:1
Hypertension (%)6725NS
Ischaemic heart disease (%)3350NS
Smoking (%)3357NS
COPD (%)5671NS

COPD, chronic obstructive pulmonary disease. NS, not significant.

The most useful output from the finite element analysis is a 3D contour plot of wall stress over the surface of each AAA studied (Fig. 7). The wall stress values are colour coded (red representing areas of high stress and blue for low stress), with the value and location of the maximum stress in each AAA identified. AAAs that ruptured, or eventually went on to rupture, had a significantly higher peak stress (mean 1.02 MPa) than the non-ruptured AAA (mean 0.62 MPa). Systolic blood pressure was also noted to be significantly higher in ruptured AAAs. The site of peak wall stress correlated with the site of rupture in all cases where the site of rupture was easily identifiable in CT or was recorded by the surgeon (as illustrated in Fig. 8) (Table 2).

  • View full-size image.
  • Fig. 7. 

    3D contour plot of wall stress over the surface of an AAA. The wall stress values are colour coded, red representing areas of high stress and blue for low stress.

  • View full-size image.
  • Fig. 8. 

    (a) Point of rupture in CT (b) correlated with area of high stress in finite element analysis. Sight of rupture and area of high stress are indicated by arrows.

Table 2. Comparison of peak wall stress in ruptured and non-ruptured AAA

SBP, systolic blood pressure, values are presented as mean, with standard deviation in parentheses.

∗ Independent sample t-test.

3.1. Effect of number of elements 

The number of elements in a finite element model is critical to the accuracy of the results, as discussed previously, and for the aneurysm models the number of elements was controlled by the smoothing applied to the 3D surface representation obtained from the CT scans. To determine the element resolution required, a typical aneurysm was solved with an increasing number of elements. The results showed that increasing the number of elements from 9,000 to nearly 20,000 made only a 1.2% difference in the maximum stress value predicted. Thus increasing the number of elements beyond this level would provide only a minimal increase in accuracy. As a result, the aneurysm models in this study typically contained 20–30,000 elements depending on the complexity of the shape being modelled.

3.2. Effect of blood pressure 

Blood pressure (BP) is a significant parameter in wall stress calculation, since the stress is directly proportional to the applied pressure. In our study, patients whose AAA ruptured had significantly higher systolic BP than the non- ruptured group. To examine the association of AAA geometry to risk of rupture, wall stress was calculated in both groups by standardising the blood pressure at 120 mmHg. With blood pressure as a constant, patients in the ruptured group still had a significantly higher wall stress suggesting the significance of association of AAA geometry to risk of rupture.

3.3. Effect of wall thickness 

AAA wall thickness has been quoted to be around 2 mm in previous studies,10 but is difficult to measure accurately. Hence, in our current study we assumed a uniform AAA wall thickness of 2.0 mm, but we also studied the effect of varying the wall thickness on AAA geometry in a typical aneurysm geometry. Wall stress was found to be inversely proportional to thickness (Fig. 9), as expected from basic engineering principles governing the behaviour of thin walled pressure vessels. Thus, increasing or decreasing the wall thickness by 25% led approximately to a 20% decrease or increase in maximum stress, respectively.

3.4. Effect of varying the material properties 

For simplicity we used the material properties defined by Raghavan et al. in our analysis.11 We also studied the effect of varying the material properties over the complete range reported by Raghavan, and found that wall stress changed by less than 4%.

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4. Discussion 

Over the recent decades there has been little change in the overall mortality associated with ruptured AAA. The high perioperative mortality associated with AAA repair highlights the need to predict risk of rupture on an individual patient basis. There have been several attempts in the past to identify risk factors associated with rupture, and several studies have analysed the factors that influence the risk of rupture. Increasing size, AAA expansion rate, hypertension, chronic obstructive pulmonary disorders, smoking, family history and a large relative AAA size compared with the individual body size are some of the variables known to increase the risk of rupture.4 However, only recently have studies focused on relating the risk of rupture on an individual patient basis. Studies by Fillinger et al. have related risk of rupture to AAA geometry.13 We have adopted similar principles to calculate wall stress and our work validates the earlier work in a different population. This study demonstrates that wall stress can be calculated from routinely performed CT scans.

Wall stress distribution is found to be inhomogeneous due to the difference in geometry at various points within an aneurysm. Even subtle changes in geometry can affect the stresses, however, we have examined the effect of geometry smoothing and found only small differences in the predicted peak stresses. Thus while modest rippling of the surface will have some affect, it is more likely to be due to discretization errors (i.e. division of the surface into finite elements) rather than localised stress effects. We have attempted to overcome any such problems by using a very large number of elements in our analyses, and undertaken convergence studies to demonstrate that we have a sufficient number to ensure accuracy of the predictions. It is the overall shape and asymmetry of the aneurysm, including the anterior and superior limits that are the determining factor. Presumably this is why an AAA of 45 mm diameter can have the same stress as that of an AAA of 65 mm, and why the natural anterior–posterior asymmetry of AAA frequently leads to higher posterior stresses. Significant difference in wall stress between ruptured and non-ruptured AAA (with similar diameter) may indicate that wall stress is a better predictor of risk of rupture than diameter alone.

There are limitations in our study. We have assumed the material properties defined by Raghavan et al. in our study population,11 but examined the sensitivity of the results to those values. We found that wall stress values changed by 4% or less over the relatively wide range considered. Hence, these calculations will hold true if the material properties for our study population are found to be within the Raghavan's sample population. Currently, to the best of our knowledge, there are no studies to measure the material properties of AAA wall in UK population. We plan to undertake this in the near future.

Stress, by definition is force per unit area, hence it is not a surprising observation that wall stress was found to vary very little with change in material property. Several studies have supported the enzymatic theory of aneurysm pathogenesis, and the concept of focally increased levels of activated matrix metalloproteinases (MMP9 and MMP2) is generally accepted. Our study does not contradict the above in anyway. While the wall stress is not sensitive to material properties, it seems likely that the strength of the tissue is altered by increased MMP9 and MMP2 activity. (In 1996 Raghavan et al., reported a 50% difference in the failure stress between healthy and aneurysmal aorta). It is also worth noting that while wall stress is relatively insensitive to material properties, wall deflection is not. So even though stress will remain comparatively constant for decreasing wall stiffness, distension will increase. (It is actually this change in geometry that brings about the change in stress). We are examining this effect further.

A further limitation in our study is the assumption of a uniform AAA wall thickness of 2 mm. We have shown that the wall stress is significantly affected by wall thickness. With refinement in imaging techniques we will be able to measure wall thickness more accurately, and further work may be needed to compute wall stress using accurate wall thickness measurements in different parts of each AAA.

In our study, we have ignored the effect of thrombus on wall stress. There have been conflicting views of the effect of thrombus on wall stress and risk of rupture14, 15, 16, 17, 18, 19, 20, 21, 22 We aim to determine the material properties of thrombus and include them in future studies.

Although there are limitations in our study, the early results are promising and make the case for a more detailed examination of the role of wall stress in assessing the risk of rupture on an individual patient basis. With the AAA population being generally elderly, the need to assess the rupture risks and offer early surgery in some, may alter the AAA related mortality significantly. Moreover wall stress may identify patients with a low risk of rupture, who may avoid an unnecessary procedure, as not all patients with AAA die of rupture. Currently, wall stress calculations are undertaken manually which is time consuming, however, further research is expected to culminate in a fully automated process which should significantly assist clinicians and patients in their decision making.

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Acknowledgments 

The authors thank the Department of Vascular Radiology, Hull Royal Infirmary and the Radiologists Dr A. Early, Dr T. Nicholson, Dr D. Ettles and Dr G. Robinson for their contribution in data collection. We would like to thank Mrs R. Waddington for her invaluable assistance in statistical analysis.

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References 

  1. Reid DB, Welch GH, Pollock JG. Abdominal aortic aneurysm: a preventable cause of death?. J R Coll Surg Edinb. 1990;35(5):284–288
  2. Brewster DC, Cronenwett JL, Hallett JW, Johnston KW, Krupski WC, Matsumura JS. Guidelines for the treatment of abdominal aortic aneurysms. Report of a subcommittee of the Joint Council of the American Association for Vascular Surgery and Society for Vascular Surgery. J Vasc Surg. 2003;37(5):1106–1117
  3. Mortality results for randomised controlled trial of early elective surgery or ultrasonographic surveillance for small abdominal aortic aneurysms. The UK Small Aneurysm Trial Participants. Lancet 1998;352(9141):1649–55.
  4. Sonesson B, Sandgren T, Lanne T. Abdominal aortic aneurysm wall mechanics and their relation to risk of rupture. Eur J Vasc Endovasc Surg. 1999;18(6):487–493
  5. Darling RC, Messina CR, Brewster DC, Ottinger LW. Autopsy study of unoperated abdominal aortic aneurysms. The case for early resection. Circulation. 1977;56(Suppl. 3):161–164
  6. Nicholls SC, Gardner JB, Meissner MH, Johansen HK. Rupture in small abdominal aortic aneurysms. J Vasc Surg. 1998;28(5):884–888
  7. Ashton HA, Buxton MJ, Day NE, Kim LG, Marteau TM, Scott RA, et al.  The Multicentre Aneurysm Screening Study (MASS) into the effect of abdominal aortic aneurysm screening on mortality in men: a randomised controlled trial. Lancet. 2002;360(9345):1531–1539
  8. Hall AJ, Busse EF, McCarville DJ, Burgess JJ. Aortic wall tension as a predictive factor for abdominal aortic aneurysm rupture: improving the selection of patients for abdominal aortic aneurysm repair. Ann Vasc Surg. 2000;14(2):152–157
  9. Raghavan ML, Vorp DA, Federle MP, Makaroun MS, Webster MW. Wall stress distribution on three-dimensionally reconstructed models of human abdominal aortic aneurysm. J Vasc Surg. 2000;31(4):760–769
  10. Sacks MS, Vorp DA, Raghavan ML, Federle MP, Webster MW. In vivo three-dimensional surface geometry of abdominal aortic aneurysms. Ann Biomed Engng. 1999;27(4):469–479
  11. Raghavan ML, Vorp DA. Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: identification of a finite strain constitutive model and evaluation of its applicability. J Biomech. 2000;33(4):475–482
  12. Peattie RA, Asbury CL, Bluth EI, Riehle TJ. Steady flow in models of abdominal aortic aneurysms. Part II: Wall stresses and their implication for in vivo thrombosis and rupture. J Ultrasound Med. 1996;15(10):689–696
  13. Fillinger MF, Marra SP, Raghavan ML, Kennedy FE. Prediction of rupture risk in abdominal aortic aneurysm during observation: wall stress versus diameter. J Vasc Surg. 2003;37(4):724–732
  14. Wolf YG, Thomas WS, Brennan FJ, Goff WG, Sise MJ, Bernstein EF. Computed tomography scanning findings associated with rapid expansion of abdominal aortic aneurysms. J Vasc Surg. 1994;20(4):529–535

    see also discussion 535–8

  15. Vorp DA, Wang DH, Webster MW, Federspiel WJ. Effect of intraluminal thrombus thickness and bulge diameter on the oxygen diffusion in abdominal aortic aneurysm. J Biomech Engng. 1998;120(5):579–583
  16. Pillari G, Chang JB, Zito J, Cohen JR, Gersten K, Rizzo A, et al.  Computed tomography of abdominal aortic aneurysm. An in vivo pathological report with a note on dynamic predictors. Arch Surg. 1988;123(6):727–732
  17. Satta J, Laara E, Juvonen T. Intraluminal thrombus predicts rupture of an abdominal aortic aneurysm. J Vasc Surg. 1996;23(4):737–739
  18. Schurink GW, van Baalen JM, Visser MJ, van Bockel JH. Thrombus within an aortic aneurysm does not reduce pressure on the aneurysmal wall. J Vasc Surg. 2000;31(3):501–506
  19. Mower WR, Quinones WJ, Gambhir SS. Effect of intraluminal thrombus on abdominal aortic aneurysm wall stress. J Vasc Surg. 1997;26(4):602–608
  20. Di Martino E, Mantero S, Inzoli F, Melissano G, Astore D, Chiesa R, et al.  Biomechanics of abdominal aortic aneurysm in the presence of endoluminal thrombus: experimental characterisation and structural static computational analysis. Eur J Vasc Endovasc Surg. 1998;15(4):290–299
  21. Thubrikar MJ, Robicsek F, Labrosse M, Chervenkoff V, Fowler BL. Effect of thrombus on abdominal aortic aneurysm wall dilation and stress. J Cardiovasc Surg (Torino). 2003;44(1):67–77
  22. Wang DH, Makaroun MS, Webster MW, Vorp DA. Effect of intraluminal thrombus on wall stress in patient-specific models of abdominal aortic aneurysm. J Vasc Surg. 2002;36(3):598–604
  23. Fagan MJ. Finite Element Analysis—Theory and Practice. London: Longmans; 1992;

☆ Abstract presented at: European Society for Vascular Surgery XVII Annual Meeting, September 2003, Dublin, Ireland.

PII: S1078-5884(04)00178-9

doi:10.1016/j.ejvs.2004.03.029

European Journal of Vascular & Endovascular Surgery
Volume 28, Issue 2 , Pages 168-176, August 2004

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