Applicability of simplified computational models in prediction of peak wall stress in abdominal aortic aneurysms

Abstract

In the paper impact of different material models on the calculated peak wall stress (PWS) and peak wall rupture risk (PWRR) in abdominal aortic aneurysms (AAAs) is assessed. Computational finite element models of 70 patient-specific AAAs were created using two different material models – a realistic one based on mean population results of uniaxial tests of AAA wall considered as reference, and a 100 times stiffer artificial model. The calculated results of PWS and PWRR were tested to evaluate statistical significance of differences caused by the non-realistic material model. It was shown that for majority of AAAs the differences are insignificant but for some 10% of them their relative differences exceed 20% which may lead to incorrect decisions on their surgical treatment. This percentage of failures favours application of realistic material models in clinical practise although they are much more time-consuming.

Keywords

Abdominal aortic aneurysm peak wall stress finite element model

1. Introduction

Abdominal aortic aneurysm (AAA) is a permanent dilatation of abdominal aorta which threatens the patient mainly by its possible rupture associated with mortality rate as high as 50% [1]. On the other hand non ruptured AAAs do seldom demonstrate themselves (are mostly asymptomatic), thus hitherto many AAAs are detected accidentally although their risk factors are well known (age above 55, male gender, smoking, hypertension, and other) [2]. However, once an AAA is detected a dilemma arises whether, in what manner (endovascular or open surgical treatment) and when it should be treated. Currently the clinical practise uses the criterion of maximal diameter combined with growth speed and some gender adjustment [3]. This means the AAAs with maximal diameter larger than 55 mm (50 mm for women) or AAAs growing faster than 5 mm in 6 months are indicated to treatment. Unfortunately this criterion is not very robust since it has been shown that the annual risk of rupture of the AAAs smaller than 70 mm is only some 10% [4, 5] which means that surgery of 90% could be postponed if we were able to identify the risky AAAs reliably. Therefore new criteria are currently under development with the aim to improve the reliability in identification of the risky AAAs.

The criteria based on analysis of wall stress in the AAA appear to be most promising among them [6, 7, 8]. The review by Khosla et al. [9] and references therein show that peak wall stress (PWS) and peak wall rupture risk (PWRR – max ratio of local stress and local strength) have a better capability in discrimination between risky and safe AAAs than maximal diameter and suggest these criteria for assessment of the AAA rupture risk. Of course different authors use very different computational models and many assumptions used in them are still not fully justified. One of the questions is the level of the material model necessary for a reliable computation of wall stresses in the AAA.

While some authors underline the importance of a proper material model in wall stress analyses of AAAs [10, 11, 12, 13] some others [14, 15, 16] state that a highly simplified model can give a comparable accuracy of PWS calculations if the magnitude of deformations is reduced by a much stiffer material model. However, these statements are supported with analyses of idealized geometries or a few patient-specific geometries only. This paper enhances the analysis of validity of this assumption on much larger cohort of 70 patients which more than doubles the number of cases in comparison with the largest study published till now on the effect of material model [17].

2. Materials and methods

A cohort of 70 patients who underwent the elective surgical repair of their AAA were included in this study. All the needed computed tomography – angiography (CT-A) scans were collected at St. Anne’s University Hospital in Brno and General University Hospital in Prague within the framework of another study [18] where also details on imaging parameters can be found. All of the AAAs were fusiform, unruptured, asymptomatic and the CT-A scans were recorded with spacing not higher than 3 mm. Clinical data for the analyzed group are summarized in Table 1.

Table 1
Clinical characteristics of the analyzed cohort of patients

	All		Men		Women
Patient info
n	70		54		16
Age [-]	74 $\pm$ 7		73 $\pm$ 7		75 $\pm$ 7
Pressure[mmHg]. Systolic( $\pm$ SD)/diastolic( $\pm$ SD)	142 $\pm$ 16/82 $\pm$ 9		142 $\pm$ 14/83 $\pm$ 9		143 $\pm$ 20/82 $\pm$ 8
AAA history [Yes/No/Unknown]	0/61/9		0/45/9		0/16/0
Smoking [Yes/No/Unknown]	37/23/10		49/11/10		4/12/0
COPD [Yes/No/Unknown]	5/58/7		4/59/7		1/15/10
AAA morphology
orthoDmax (min, max) [mm]	58	(38, 110)	58	(38, 110)	59	(44, 90)
axialDmax (min, max) [mm]	62	(40, 115)	61	(41, 115)	63	(46, 110)
Da-o (min, max) [mm]	4	( $-$ 2, 22)	4	( $-$ 2, 22)	4	( $-$ 2, 16)
AAA Volume (min, max) [cm ${}^{3}$ ]	210	(127, 347)	202	(127, 347)	236	(172, 340)

COPD denotes chronicle obstructive pulmonary disease, orthoDmax refers to maximal diameter in a plane perpendicular to AAA centerline while axialDmax denotes maximal AAA diameter in axial plane. Da-o given as Da-o $=$ Dmax axial – Dmax ortho [18] is a measure of AAA angulation. AAA morphologies are described by mean and by their minimum and maximum values in bracket.

For calculation of PWS and PWRR in the AAA wall, computational finite element (FE) models were created and solved using A4clinics ${}^{\text{TM}}$ software (Vascops GmbH, Austria, Graz). Detailed information regarding reconstruction of AAA geometry from CT-A images can be found elsewhere [19]. Briefly, the geometry of lumen and external wall is created by expanding the virtual balloon up to the boundaries defined by an intensity threshold. Then a local wall thickness is defined in each node on the basis of intraluminal thrombus (ILT) thickness [20]. Finally, the hex-dominant FE mesh of linear elements is created with only one element across the wall. This ensures the computed stresses will be more or less constant across the wall which aims at mimicking the effect of residual stresses which are known to be present in arterial wall and act towards removal of stress gradients across the wall [21]. Size of the elements is chosen automatically and it is the same as used in previous studies [7, 22, 23, 24]. The interface between ILT and wall was considered as bonded with no slipping allowed. Boundary conditions were defined by patient-specific mean arterial pressure (1/3 systolic $+$ 2/3 diastolic blood pressure) acting on the inner surface of the wall. This mimics the fact the ILT is porous and the blood pressure is transmitted towards the wall [25, 26]. Similarly to absolute majority of computational models, both ends of AAA (mostly in the vicinity of renal arteries and below the aortic bifurcation) were fixed in all directions [7, 8, 10, 13]. Although these boundary conditions are not fully realistic as the aorta is fixed to the surrounding tissues by compliant distributed supports (connective tissue), in both real and model cases the problem is statically undetermined which results intrinsically in dependence of stresses on the material model. The analyzed models are specified below.

3. Aneurysmal wall

A hyperelastic incompressible isotropic Yeoh material model was used for the AAA wall with the strain energy density function (SEDF) in the following form

$\displaystyle\Psi=c_{10}(I_{1}-3)+c_{20}(1_{1}-3)^{2}$ (1)

where $I_{1}$ represents the first invariant of deformation tensor and $c_{10}$ and $c_{20}$ are stress-like material parameters. Their values were set to $c_{10}=$ 174 kPa and $c_{20}=$ 1888 kPa on the basis of uniaxial tensile tests of aneurysm wall [27] to reflect its mean population non-linear mechanical response [28]. This model is considered as reference below since it aims to represent real mechanical behaviour of aneurysm wall. For comparison we have used another material model with the same SEDF but being much stiffer due to its 100x higher constants ( $c_{10}=$ 17 400 kPa and $c_{20}=$ 188 800 kPa).

4. Intraluminal thrombus

If the AAA contained ILT its material behaviour was modelled by one-parameter Ogden-like SEDF with almost linear stress-strain response as follows:

$\displaystyle\Psi=c\sum\limits_{i=1}^{3}(\lambda_{i}^{4}-1),$ (2)

where $\lambda_{i}$ denotes the i-th principal stretch and $c$ is a stress-like material parameter. This parameter was set to $c=$ 2.62 kPa at the luminal side of the ILT and decreased linearly to $c=$ 1.73 kPa for the ILT being 30 mm away from the lumen. The ILT stiffness was kept at this level also for ILT thicknesses above 30 mm which corresponds to results of its ex vivo testing [29].

5. Model comparison

All the models were solved using firstly the reference model, and the PWS and PWRR have been evaluated; more details can be found in [7]. It is noted our setup for the reference material is the same as used for the most complex model in [7] and also the same as used in the study where its ability to discriminate between ruptured and intact AAAs was confirmed [22]. Secondly, the same approach was repeated for the stiffer model and the obtained results were compared statistically. A non-parametric simple sign test was used to verify the null hypothesis that there was no difference between both material models. The alternative hypothesis states the reference model gives higher PWS and PWRR than the stiffer model. Finally, absolute values of differences in both PWS and PWRR for all patient-specific geometries were turned into histograms for which a 3 parameter Weibull distribution was used with the following probability density function:

$\displaystyle\sigma(x)=\frac{\beta}{\eta}\left({\frac{x-\gamma}{\eta}}\right)^% {\beta-1}\cdot e^{-\left({\frac{x-\gamma}{\eta}}\right)^{\beta}}$ (3)

where $\eta$ is a scale parameter, $\beta$ refers to shape parameter and $\gamma$ denotes location parameter.

Figure 1.

Rupture risk index evaluated for a chosen case for both reference (left) and stiffer (right) model. It is noted the location of PWS is shifted to the neck area of the AAA for the stiffer model and overall rupture risk index is increased by some 25% compared to the reference material. It is underlined both figures use different scales.

6. Results

Values of PWS and PWRR were calculated for all the models with exclusion of bifurcation areas as well as areas close to the supports. The effect of material model for a particular case can be observed in Fig. 1 while all the results are summarized in Table 2. Comparison of the individual data is shown in Figs 2 and 3. It is evident that although in many cases the differences are insignificant, there are AAAs for which the stiffer model gives significantly different results. Statistical testing confirmed that medians of PWS and PWRR for the stiffer model compared to the reference one are lower by 11 kPa ( $p=$ 0.0003) and 0.017[-] ( $p=$ 0.0009), respectively. Absolute values of the relative differences between the stiffer and reference model were then summarized into histograms and fitted by 3 parameter Weibull distribution. The obtained values were $\eta=$ 10.94, $\beta=$ 1.408, $\gamma=-$ 0.2404 for PWS and $\eta=$ 9.796, $\beta=$ 1.266, $\gamma=-$ 0.075 for PWRR. Using these distributions it can be assessed that, for instance, for 10% of cases the difference in PWS between both models will be larger than 20% as shown in Fig. 4 left. Another example shows that the difference in PWRR will be above 25% for 5% of cases when the stiffer model is used as shown in Fig. 4 right.

Table 2
Statistical descriptors obtained for PWS and PWRR for both the reference and stiffer models

	PWS [kPa]		PWRR [-]
	Reference model	Stiffer model	Reference model	Stiffer model
Mean $\pm$ Standard deviation	226 $\pm$ 59	215 $\pm$ 63	0.594 $\pm$ 0.296	0.577 $\pm$ 0.318
Median (Q1; Q3)	210 (174; 262)	198 (165; 243)	0.531 (0.422; 0.681)	0.502 (0.384; 0.663)

Figure 2.

PWS for reference and stiffer materials for all cases ranked with respect to PWS for the reference material.

Figure 3.

PWRR for reference and stiffer materials for all cases ranked with respect to PWS for the reference material.

Figure 4.

Histograms and fitted distributions of absolute values of differences between the reference and stiffer models for PWS; (left) and PWRR (right). Chosen quantiles are marked as well.

7. Discussion

In this study we have assessed the effect of material model on computed values of PWS and PWRR. The reference model was based on experimental testing while the stiffer model with initial Young’s modulus of some 100 MPa was artificial. To our best knowledge this study is the largest one where the effect of material model is analyzed. We believe a sufficient number of real cases is necessary to include most possible shapes of AAAs.

In literature artificial models are suggested to replace more realistic models because their use can speed up the whole computational process significantly [14, 15]. In our analyses a typical solution time for the reference model was about 20 minutes on a standard computer while we were able to cut it down to some 5 minutes when the stiffer model was used. The difference in computational time would be even much higher if the stiffer model was compared with some more advanced model using AAA wall material with a more pronounced strain stiffening, respecting the zero-pressure configuration and including residual stresses [8]. Solution of one case using such a model takes typically some 24 hours thus use of a much stiffer model without these features could result is some 300-fold speed up. Therefore the stiffer (and simplified) model is suitable for computationally demanding analyses which require multiple computing of one case. Typical application would be stochastic analyses using Monte Carlo simulations. For clinical use on the other hand, a typical patient is scheduled for surgery some two weeks after the AAA detection, so there is no need to speed up the patient-specific AAA stress analyses at the expense of their accuracy.

The accuracy or reliability of stresses computed using a stiffer material model represents the most critical issue in their practical application. Our results show that for majority of cases the differences in both PWS and PWRR are less than 10% (see Fig. 4) which explains why some other studies [14, 15] analyzing only few cases have not found any significant effect of the material model on wall stress. Another study [10] found average 18% stress increase (for 99-percentile) when using the same reference material as in this study compared to a material with a 2.6 times higher initial stiffness. Our results have also shown that globally the PWS decreases with a stiffer material. We observed this was mostly true for smooth convex balloon-like AAA shapes where local curvatures are indeed small thus the assumptions of Laplace law are valid. Therefore a stiffer material results in lower stress gradients across the wall which was observed here. However, our cohort contained also about 10% of cases where differences in PWS and PWRR were higher than 20% (extremes were 32% difference in PWS and 38% in PWRR) which could cause wrong conclusions regarding riskiness of these cases. It is worth to mention that such error induces a much higher error in the prediction of probability of rupture and it may occur not only in AAAs with extremely high PWRR where a 20% change does not affect the overall conclusion defining the analyzed AAA as risky (left end of curves in Fig. 3). It occurs occasionally also in the range of PWRR about 0.5 which corresponds to the AAA diameter close to 55 mm [30]. Use of a stiffer material in such cases would result in a dangerous error in conclusion transferring a risky AAA into the group of safe cases. Therefore we conclude that the observed threat of such errors is a strong argument against clinical use of artificially stiff material models for assessment of susceptibility of AAAs to rupture.

All results must be assessed with respect to their limitations. We used the reference material which was based on uniaxial tensile test data and it is known that these models differ significantly from those based on biaxial tensile tests [31] and produce also significantly different wall stresses [11]. Nevertheless, the superiority of any of these models was not yet conclusively shown. Another limitation lays in the fact that residual stresses are mimicked by using one element across the thickness. Naturally, such computational model cannot reliably simulate wall response to an elevated blood pressure as it can be done by using more advanced (but also much more computationally demanding) models [8]. Therefore all conclusions made here are related to average wall stress only.

Finally, there is another limitation of the applied models, namely the reconstructed geometry is considered as unloaded. The used Vascops A4Clinics software is not capable to reconstruct the real unloaded (zero-pressure) geometry and the large cohort of patients analyzed in this study was feasible by virtue of its automated procedures. Taking the unloaded geometry into consideration would certainly change the magnitude of wall stresses [10, 11, 17] but it was shown that neglecting the difference between the CT-A recorded and unloaded geometries does not deteriorate the capability of the models to discriminate between intact and ruptured AAAs [7, 22]. It can be expected that this simplification as well as all the others mentioned above would increase the differences in PWS and PWRR against the reference (i.e. most sophisticated) model and consequently also the impact of oversimplifications in the stiffer model. Therefore they cannot change our main conclusion that artificially stiff materials lead occasionally to unrealistic wall stresses; on the contrary, they could even enhance its relevance. These large differences in stresses may occur especially in AAAs where the local curvature is very high (see Fig. 1) [32] and consequently the assumptions of membrane stress state (independent of the material model) are violated.

8. Conclusion

It was confirmed that use of an artificially stiff material ensured a much faster convergence and resulted overall in values of PWS and PWRR similar to the realistic (reference) material model. Therefore a stiffer material model is applicable for investigation of the effect of other AAA features especially when manifold repeated FE solutions are needed. On the other hand, it was shown that for some 10% of the analyzed patient-specific cases the differences in both PWS and PWRR were higher than 20% which may lead to wrong conclusions regarding assessment of riskiness of the analyzed AAA; thus non-realistic material models should not be used for clinical assessment of AAAs.

Footnotes

Acknowledgments

This work is an output of project NETME CENTRE PLUS (LO1202), created with financial support from the Ministry of Education, Youth and Sports under the National Sustainability Programme I. We also gratefully acknowledge the collaboration with surgeons and radiologists from St. Anne’s University Hospital and Faculty of Medicine, Masaryk University, Brno and First Faculty of Medicine, Charles University in Prague and General University Hospital in Prague who collected the CT-A data used in this study. Finally, we would like to acknowledge prof. T. Christian Gasser from KTH Stockholm for providing us support in use of the Vascops A4Clinics software.

Conflict of interest

None to report.

References

Hoornweg

Storm-Versloot

Ubbink

, et al. Meta Analysis on Mortality of Ruptured Abdominal Aortic Aneurysms. Eur J Vasc Endovasc Surg 2008; 35: 558–570.

Kent

Zwolak

Egorova

, et al. Analysis of risk factors for abdominal aortic aneurysm in a cohort of more than 3 million individuals. J Vasc Surg 2010; 52: 539–548.

Moll

Powell

Fraedrich

, et al. Management of abdominal aortic aneurysms clinical practice guidelines of the European society for vascular surgery. Eur J Vasc Endovasc Surg; 41. Epub ahead of print 2011. DOI: 10.1016/j.ejvs.2010.09.011.

Lederle

Johnson

Wilson

, et al. Rupture rate of large abdominal aortic aneurysms in patients refusing or unfit for elective repair. JAMA 2002; 287: 2968–2972.

Brewster

Cronenwett

Hallett

, et al. 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: 1106–1117.

McGloughlin

Doyle

. New approaches to abdominal aortic aneurysm rupture risk assessment: Engineering insights with clinical gain. Arterioscler Thromb Vasc Biol 2010; 30: 1687–1694.

Gasser

Auer

Labruto

, et al. Biomechanical rupture risk assessment of abdominal aortic aneurysms: Model complexity versus predictability of finite element simulations. Eur J Vasc Endovasc Surg 2010; 40: 176–185.

Polzer

Gasser

. Biomechanical rupture risk assessment of abdominal aortic aneurysms based on a novel probabilistic rupture risk index. J R Soc Interface; 12. Epub ahead of print 2015. DOI: 10.1098/rsif.2015.0852.

Khosla

Morris

Moxon

, et al. Meta-analysis of peak wall stress in ruptured, symptomatic and intact abdominal aortic aneurysms. Br J Surg 2014; 101: 1350–1357.

10.

Speelman

Bosboom

EMH

Schurink

GWH

, et al. Initial stress and nonlinear material behavior in patient-specific AAA wall stress analysis. J Biomech 2009; 42: 1713–1719.

11.

Polzer

Christian Gasser

Bursa

, et al. Importance of material model in wall stress prediction in abdominal aortic aneurysms. Med Eng Phys; 35. Epub ahead of print 2013. DOI: 10.1016/j.medengphy.2013.01.008.

12.

Vande Geest

Schmidt

Sacks

, et al. The effects of anisotropy on the stress analyses of patient-specific abdominal aortic aneurysms. Annals of Biomedical Engineering 2009; 36: 921–932.

13.

Rodríguez

Martufi

Doblaré

, et al. The effect of material model formulation in the stress analysis of abdominal aortic aneurysms. Ann Biomed Eng 2009; 37: 2218–2221.

14.

Zelaya

Goenezen

Dargon

, et al. Improving the Efficiency of Abdominal Aortic Aneurysm Wall Stress Computations. PLoS One 2014; 9: e101353.

15.

Joldes

Miller

Wittek

, et al. A simple, effective and clinically applicable method to compute abdominal aortic aneurysm wall stress. J Mech Behav Biomed Mater 2016; 58: 139–148.

16.

Drewe

Parker

Kelsey

, et al. Haemodynamics and stresses in abdominal aortic aneurysms: A fluid-structure interaction study into the effect of proximal neck and iliac bifurcation angle. J Biomech. Epub ahead of print June 2017. DOI: 10.1016/j.jbiomech.2017.06.029.

17.

Merkx

MAG

van ’t Veer

Speelman

, et al. Importance of initial stress for abdominal aortic aneurysm wall motion: Dynamic MRI validated finite element analysis. J Biomech 2009; 42: 2369–2373.

18.

Novak

Polzer

Krivka

, et al. Correlation between transversal and orthogonal maximal diameters of abdominal aortic aneurysms and alternative rupture risk predictors. Comput Biol Med 2017; 83: 151–156.

19.

Holzapfel

Sommer

Auer

, et al. Layer-Specific 3D Residual Deformations of Human Aortas with Non-Atherosclerotic Intimal Thickening. Ann Biomed Eng 2007; 35: 530–545.

20.

Kazi

Thyberg

Religa

, et al. Influence of intraluminal thrombus on structural and cellular composition of abdominal aortic aneurysm wall. J Vasc Surg 2003; 38: 1283–1292.

21.

Fung

. What are the residual stresses doing in our blood vessels? Ann Biomed Eng 1991; 19: 237–249.

22.

Erhart

Hyhlik-Dürr

Geisbüsch

, et al. Finite element analysis in asymptomatic, symptomatic, and ruptured abdominal aortic aneurysms: In search of new rupture risk predictors. Eur J Vasc Endovasc Surg 2015; 49: 239–245.

23.

Siika

Liljeqvist

Hultgren

, et al. AAA Rupture Often Occurs Outside the Maximal Diameter Region and is Preceded by Rapid Local Growth and an Increased Biomechanical Rupture Risk Index. Epub ahead of print 2015. DOI: 10.1016/j.ejvs.2015.06.028.

24.

Hyhlik-Dürr

Krieger

Geisbüsch

, et al. Reproducibility of deriving parameters of AAA rupture risk from patient-specific 3D finite element models. J Endovasc Ther 2011; 18: 289–298.

25.

Schurink

GWH

van Bockel

Visser

MJT

. Thrombus within an aortic aneurysm does not reduce pressure on the aneurysmal wall. J Vasc Surg 2000; 31: 501–506.

26.

Polzer

Gasser

Markert

, et al. Impact of poroelasticity of intraluminal thrombus on wall stress of abdominal aortic aneurysms. Biomed Eng Online; 11. Epub ahead of print 2012. DOI: 10.1186/1475-925X-11-62.

27.

Raghavan

Webster

Vorp

. Ex vivo biomechanical behavior of abdominal aortic aneurysm: assessment using a new mathematical model. Ann Biomed Eng 1996; 24: 573–582.

28.

Raghavan

Vorp

. 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: 475–482.

29.

Gasser

Görgülü

Folkesson

, et al. Failure properties of intraluminal thrombus in abdominal aortic aneurysm under static and pulsating mechanical loads. J Vasc Surg 2008; 48: 179–188.

30.

Gasser

Nchimi

Swedenborg

, et al. A novel strategy to translate the biomechanical rupture risk of abdominal aortic aneurysms to their equivalent diameter risk: Method and retrospective validation. Eur J Vasc Endovasc Surg 2014; 47: 288–295.

31.

Geest

Sacks

Vorp

. The effects of aneurysm on the biaxial mechanical behavior of human abdominal aorta. J Biomech 2006; 39: 1324–1334.

32.

Sacks

Vorp

Raghavan

, et al. In Vivo Three-Dimensional Surface Geometry of Abdominal Aortic Aneurysms. Ann Biomed Eng 1999; 27: 469–479.

Applicability of simplified computational models in prediction of peak wall stress in abdominal aortic aneurysms

Abstract

Keywords

1. Introduction

2. Materials and methods

Table 1 Clinical characteristics of the analyzed cohort of patients

Table 2 Statistical descriptors obtained for PWS and PWRR for both the reference and stiffer models

8. Conclusion

Footnotes

Acknowledgments

Conflict of interest

References

Table 1
Clinical characteristics of the analyzed cohort of patients

Table 2
Statistical descriptors obtained for PWS and PWRR for both the reference and stiffer models