Keywords: geometric modeling, abdominal aortic aneurysm, biomechanics, finite element modeling, nonlinear regression

Patient Selection.

Anonymized abdominal computed tomography angiography (CTA) scans were obtained retrospectively from Allegheny General Hospital, in Pittsburgh and Tan Tock Seng Hospital, in Singapore, for the Caucasian AAA group and Asian AAA group, respectively. Following approval of a human subjects research protocol by the Institutional Review Boards at both institutions, the existing patient medical records were obtained for 38 AAA, with 19 subjects at each institution. The choice of datasets was made using a maximum diameter matching approach, i.e., each Asian AAA had a corresponding Caucasian AAA of approximately the sameD_max. Our study population consisted of 38 subjects (19 Asian and 19 Caucasian AAA patients) who underwent elective repair within 6 months of the last preoperative CT scan follow-up. The abdominal CTA digital imaging and communications in medicine (DICOM) images had the imaging parameters indicated in Table1. The use of CTA images enabled us to make an accurate quantification of the geometric indices with our in-house segmentation code, AAAVasc [36].

Three-Dimensional Image Reconstruction and Volume Meshing.

The CT images were acquired with the DICOM format and processed with an in-housematlab (Mathworks Inc., Natick, MA) suite of scripts named AAAVasc (v1.0.3, The University of Texas at San Antonio, San Antonio, TX). AAAVasc segments the images identifying the lumen, inner wall, and outer wall contours in a semiautomated manner while generating a binary mask from each segmented image [36], as illustrated in Fig.1. Volume meshes using quadratic hexahedral elements were generated for each AAA from the binary masks using the in-housematlab script named AAAMesh. These nonuniform wall thickness meshes were created with input of the patient-specific wall thickness distributions obtained during image segmentation. All meshes were exported as NASTRAN formatted files and ranged in size from 30,000 to 90,000 quadratic hexahedral elements, resulting from surface tessellation densities with two-element layers across the wall and an average element edge length of 0.8 mm. They followed a previous mesh sensitivity analysis performed for quasi-static AAA finite element modeling [37]. Similar to previous studies [20,28,38,39], the patient-specific FEA models did not include intraluminal thrombus (ILT).

Finite Element Analysis.

Finite element analysis simulations were conducted for each AAA to obtain the wall stress metrics. The NASTRAN mesh files were imported into the finite element solver Adina (ADINA R&D, Inc., Watertown, MA) to perform quasi-static stress analysis. An intraluminal static pressure of 120 mmHg (or 16 kPa) was applied to all inner wall nodes incrementally in 24 time-steps. A hyperelastic Mooney-Rivlin constitutive model, as shown in Eq.(1), was used to characterize the AAA wall material properties with the following mean model parameters, which were derived from uniaxial tensile testing of AAA tissue specimens ex vivoex vivo [27]:$c_1\hspace{0.17em}$= 17.4 N/cm² and$c_2$ = 188.1 N/cm²

where$W$ is the strain energy distribution function and$I_1$ is the first variant of the left Cauchy-Green tensor. The global wall stress metrics were obtained after convergence of the FEA simulations using the first principal stress distributions calculated with Ansys Ensight (Ansys, Inc., Canonsburg, PA). Three metrics were computed, namely the peak wall stress (PWS), the spatially AWS, and the 99th percentile wall stress (99th WS) [20], for each of the 38 AAA. Figure2 illustrates the modeling workflow from clinical image to AAA computer model.

Geometry Quantification.

Fifty-one geometric indices were calculated for all AAA following a previously described protocol [20,21,23,25,40,41]. These geometric indices were calculated for the domain spanning the location immediately below the left renal artery until the iliac bifurcation. These indices consist of eleven 1D size indices, nine 2D shape indices, five thrombus-related indices, four 3D size indices, two 3D shape indices, thirteen wall thickness-related indices, and ten surface curvature-based indices. The complete mathematical formulation of the geometry metrics is included in Appendix A of theSupplementary Material on the ASME Digital Collection.

For each CT image slice in the axial plane, a range of wall thicknesses was obtained at 72 points along the wall circumference, from which the respective means and standard deviations were calculated. For each AAA, wall thickness distribution at each cross section is obtained from the proximal to distal axial positions. These individual wall thicknesses were then utilized to obtain the minimum, maximum, mean, mode, and median of the wall thickness variance to represent global wall thickness measures for each AAA (as indicated in theSupplementary Material on the ASME Digital Collection). Further, the biquintic Hermite finite element method (BQFE) was used to evaluate the curvature-based indices, as described in Ref. [41]. This method utilizes a high-order interpolation scheme to estimate the global curvature indices derived from local curvature distributions. BQFE discretizes the aneurysm outer wall into 12 elements and surfaces are fit to each of the elements, from which the local principal curvatures ($k_1$ and$k_2$) are computed using Eqs.(2) and(3),

where$a$,$b$, and$c$ are best-fit constants determined for each surface node. The ten global curvature indices, as described in Appendix A, (of theSupplementary Material on the ASME Digital Collection), are calculated for each AAA model as the summation of the Gaussian and mean curvatures (KG,KM), the area-averaged Gaussian, mean, first principal, and second principal curvatures (GAA, MAA, K1AA, K2AA), and the L2-norm of the aforementioned curvatures (GLN, MLN, K1LN, K2LN).

Statistical Analysis.

Biomechanical and geometric data were reported as means ± standard deviation (n = 19 for each group). As a preliminary step, we performed a multivariatet test (i.e., the HotellingT² test [42]) to assess whether the distribution of the geometric features differs between the two groups. If they do, we would carry out individual t tests to identify the features responsible for the difference. Since multiple hypotheses need to be evaluated for detecting any groupwise differences in the quantified parameters, it is important to avoid the probability of false statistical inferences resulting from multiple comparisons, which typically occurs when more than one hypothesis is tested simultaneously. Let us denote by${\mu }_{A_g}$ and${\mu }_{C_g}$ the means of the$g\text{th}$ variable (with$g=1,\hspace{0.17em}\dots ,\hspace{0.17em}54$), where the subscripts$A$ and$C$ refer to the Asian and Caucasian populations, respectively. The test hypotheses can be stated according to Eq.(4),

where${\mathit{\mu }}_{\mathit{A}}={\left({\mu }_{A_1},\dots ,{\mu }_{A_{54}}\right)}^{\prime }$ is the vector of the means of the geometric features for the Asian population and${\mathit{\mu }}_{\mathit{C}}={\left({\mu }_{C_1},\dots ,{\mu }_{C_{54}}\right)}^{\prime }$ is the vector of the means of the geometric features for the Caucasian population. To test the multivariate hypotheses, the HotellingT² test was performed across the two groups for all the variables using thefdahotelling package [42,43] in R Studio [44]. Since the number of variables is greater than the sample size, the implementation offdahotelling uses an alternative representation of theT² test that does not require computing the sample covariance matrix [42]. If$H_0^{\left(a\right)}$ is accepted, then there is not enough evidence to conclude that the population means are different. If$H_0^{\left(a\right)}$ is rejected, then the alternative hypothesis$H_1^{\left(a\right)}$ is accepted and we can conclude that the population means are not equal and there are differences across the groups.

Abdominal Aortic Aneurysm Wall Stress.

The spatial distributions of first principal stress for the Asian and Caucasian AAA groups are illustrated in Fig.3 for an exemplary pair of aneurysms. For the Asian and Caucasian groups, the mean 99th WS was 41.7 ± 15.4 N/cm² and 44.0 ± 11.6 N/cm², respectively, which were not significantly different (p = 0.987), following thet-test analysis (with Hochberg adjustment). Similarly, as shown in Tables2 and3, PWS (95.9 ± 31.9 N/cm² versus 106.6 ± 50.5 N/cm²;p = 0.987) and AWS (21.6 ± 7.9 N/cm² versus 24.1 ± 8.4 N/cm²;p = 0.987) for the Asian and Caucasian AAA, respectively, were similar across the groups.

Geometry Quantification.

The means and standard deviations of all 51 geometric indices are included in Appendix B of theSupplementary Material on the ASME Digital Collection. The detailed results of thet-test analysis (with Hochberg adjustment) of the geometric indices across the two AAA groups are described as follows:

• Wall Thickness Indices—Of 13 wall thickness indices, two were statistically significant, as shown in Figs.4(a) and4(b). Maximum wall thickness${(t}_{w,\max }$) (5.47 ± 1.83 mm versus 3.53 ± 1.23 mm;p = 0.0244) and the maximum variance of wall thickness$\left(t_{w,\mathrm{maxVar}}\right)$ (1.07 ± 0.78 mm versus 0.34 ± 0.39;p = 0.0339) were higher for Asian AAA patients in contrast to their Caucasian counterparts.

• Second Order Curvature Based Indices—Eight of ten curvature indices were found to be statistically significant across the groups and with higher magnitudes for the Asian AAA versus Caucasian AAA: (i) L2-norm of the Gaussian curvature (GLN) (20.12 ± 10.34 versus 6.00 ± 2.03;p < 0.0001), (ii) L2-norm of the mean curvature (MLN) (2.41 ± 1.57 versus 0.64 ± 0.18;p = 0.0012), (iii) area-averaged major principal curvature (K1AA) (0.15 ± 0.06 mm⁻¹ versus 0.07± 0.01 mm⁻¹;p < 0.0001), (iv) area-averaged minor principal curvature (K2AA) (−0.1 ± 0.05 mm⁻¹ versus −0.02± 0.01 mm⁻¹;p < 0.0001), (v) L2-norm of the major principal curvature (K1LN) (531.79 ± 445.44 versus 155.74 ± 69.45;p = 0.0421), (vi) L2-norm of the minor principal curvature (K2LN) (423.57 ± 306.29 versus 113.06 ± 50.64;p = 0.0054), (vii) square root sum of the Gaussian curvature (KG) (3.12 ± 1.66 mm⁻¹ versus 0.69 ± 0.25 mm⁻¹;p < 0.0001), and (viii) square root sum of the mean curvature (KM) (43.31 ± 24.44 mm⁻² versus 9.63 ± 2.19 mm⁻²;p < 0.0001). This comparison is illustrated in Figs.5(a)–5(f).

Association of Biomechanical Parameters and Geometric Indices.

Of the three biomechanical parameters, only 99th WS showed significant association with the geometric indices; the correlation coefficients and respectivep-values are listed in Table4 (Asian AAA group) and Table5 (Caucasian AAA group). For the Asian AAA group, 9 of the 51 geometric indices were found to be highly correlated with 99th WS, whereas seven significant predictors of 99th WS were found for the Caucasian AAA group. GAA andDDr were the common indices for the two groups, with correlation coefficients of 0.6 and 0.544 (Asian AAA), and 0.523 and 0.646 (Caucasian AAA), respectively. Further, GAA had the highest positive correlation coefficient (r = 0.6) with the Asian 99th WS, whereast_t,minLoc had the highest negative correlation coefficient (r = −0.579) in this group. For the Caucasian AAA,DDr was found to have a high positive correlation with 99th WS (r = 0.646), while botht_w,minVar andt_w,modeVar exhibited similar negative correlation coefficients (r = −0.562).

Multivariate Data Analysis.

The coefficient of determination of the multivariate regression model can explain the ability of the statistically significant geometric indices to predict wall stress. These indices were utilized as independent variables for the multivariate regression model to predict 99th WS for each group, which resulted in the following predictive models:

• (i)
  For the Asian AAA group (R² = 0.77, Fig.6(a)),

  $$99\text{th}\hspace{0.17em}\text{WS}=65.0-28.32*T+0.08*V_{\text{ILT}}+10465*\text{GAA}$$

  (7)

• (i)
  For the Caucasian AAA group (R² = 0.87, Fig.7(a)),

  $$\begin{array}{c}99\text{th}\hspace{0.17em}\text{WS}=24.13+0.67\hspace{0.17em}*\hspace{0.17em}{D}_{\text{max}}+0.31*D_{\text{neck},d}-47.0*\mathit{DHr}\\ -79.95*t_{w,\text{minVar}}+55626*\text{GAA}\end{array}$$

  (8)

Noteworthy is that the coefficient of determination of 99th WS using maximum diameter (D_max) as the only geometric predictor was lower than that for the aforementioned multivariate models [Eqs.(7) and(8)] by more than 50%: 99th WS versusD_max for Asian AAA (R² = 0.24, Fig.6(b)) and Caucasian AAA (R² = 0.28, Fig.7(b)).

Asian Abdominal Aortic Aneurysm Are Anatomically Different Than Caucasian Abdominal Aortic Aneurysm.

The majority of EVG manufacturers are based in U.S. or Europe [47], and their designs are based on the anatomical features of the local patient population (i.e., Caucasian). While AAA are less common in Asian population, the numbers have been steadily rising in several countries [3,6,8–10]. The reduced feasibility of currently available EVG for Asian AAA patients is a growing concern [12,14–17] and clinicians have to modify or customize these grafts prior to EVAR. As a tortuous aorta is one of the risk factors for AAA [48], which limits EVAR access to the aneurysm sac [49], Lee and colleagues [50] suggested lengthening the main body or limb of the EVG as a potential solution for severely tortuous aneurysms. Other than patient or clinician preferences, AAA surgeries are performed due to complexity in anatomical structures such as aortic neck anatomy, iliac diameter, presence of iliac aneurysm, and iliac occlusive disease [17,49,51]. When comparing morphological differences across two patient populations (30 Asian AAA patients and 30 Caucasian AAA patients), Mladenovic et al. [12] found that the biggest difference in anatomy was length and volume of the common iliac artery, and transverse diameter of the aneurysm (D_max). Cheng and colleagues reported thatD_max was larger and neck lengths were shorter in Asian AAA patients compared to Caucasian AAA patients from Europe and U.S. [14]. However, Banzic and et al. reported no differences inD_max when comparing AAA patients from Europe and China [13]. In addition, they found that Asian patients had longer infrarenal abdominal aortas and longer AAA. We are unable to compare the measures ofD_max from our study with those in Refs. [12–14] as our patient groups were diameter-matched. Furthermore, we did not find any differences in neck length, AAA length, or neck anatomy across the two patient groups (Appendix B of theSupplementary Material on the ASME Digital Collection). Conversely, we found that aneurysm wall thickness measures and surface curvature indices were significantly different across the Asian and Caucasian AAA populations.

The importance of wall thickness in predicting aneurysm growth has been stated previously [52,53] and postulated as a rupture risk predictor [54]. From a biomechanical viewpoint, the highest stress to strength ratio is experienced prior to rupture and local wall thickness strongly influences it [55]. To this end, Raghavan et al. [52] reported that the wall near the rupture site is thinner compared to other regions in the dilated abdominal aorta. We found two wall thickness-based indices to be statistically different across the two groups while the maximum wall thickness for Asian AAA was 55% greater than for Caucasian AAA. Remarkably, the maximum wall thickness in the Caucasian group was similar to the ruptured AAA group (also Caucasian) reported by Di Martino et al. [56]

Changes in AAA curvature are primarily accounted for due to their bulging and tortuous anatomy. The tortuosity of the aneurysmal infrarenal aorta yields irregular surfaces and kinking [57] that lead to changes in normal hemodynamics [18,19] and increased arterial wall stress [18,19,58], which are problematic during EVAR [49]. Interestingly, mildly tortuous aneurysms had 3.3 times greater risk of rupture in the Caucasian AAA population, following a multivariate gender-controlled analysis [48]. There is a greater inclination angle (i.e., less tortuosity) in Caucasian AAA (n = 182) compared to Asian AAA (n = 113) [13]. Boonruangsri et al. [57] performed a study on 85 Thai cadavers (58 males and 27 females) and while only four had AAA, these did not exhibit tortuosity or kinking. In our study, we found no statistically significant differences in aortic tortuosity across the groups. This is likely due to the fact that all patients from both groups had an elective aneurysm repair within 6 months of their last CTA follow-up and were at similar stages of AAA severity. Hence, they did not exhibit the dissimilar tortuosity values typical of emergently repaired AAA.

Association of Aneurysmal Wall Stress With Geometry Measures.

In this study, we investigated a set of noninvasive explanatory geometric measures to predict in vivo wall stress across Asian and Caucasian AAA groups. This is the first report on the association of geometric indices with biomechanical stresses in Asian AAA. The use of wall stress, as opposed to maximum AAA diameter, has been strongly advocated as an efficient rupture risk predictor [19,20,22,27–31,33–35,59]. The calculation of wall stress is not straightforward and cannot be calculated directly from routine clinical imaging such as CT or MRI. Hence, we could rely on geometric surrogates for estimating AAA wall stress. We employed 99th WS for estimating this stress, which is advantageous as it eliminates the highest 1% of wall stresses in the patient-specific models (usually due to irregular mesh elements and thus are more erroneous [32]. 99th WS exhibited significant correlation with nine and seven geometric indices characteristic of Asian and Caucasian AAA, respectively. Interestingly, Caucasian AAA wall stress had more associations with diameter-based and thickness-based indices compared to Asian AAA wall stress, which was equally associated with shape, diameter, thrombus, and surface curvature indices.

In an experimental study by Mower et al. [55], wall thickness was found to be more associated with biomechanical stress than maximum AAA diameter. In addition, more wall thickness-based indices were associated with Caucasian AAA than Asian AAA, a finding that concurs with Chauhan et al. [20] for emergently repaired Caucasian AAA.

Thrombus-based indices showed that wall stress is associated positively with ILT volume and negatively with the location of minimum thrombus thickness in the Asian patient population. A negative correlation of wall stress with the location of minimum thrombus thickness indicates that if the thrombus thickness is closer to the left renal artery the wall stresses are greater. Clinical studies have associated a thick ILT layer and an increase in ILT volume with strong rupture risk predictors [60,61]. ILT volume has shown a strong association with maximum wall shear stress and peak wall rupture index in a study of 23 AAA patients [62]. From multivariate analyses, we correspondingly found that 99th WS was positively associated with ILT volume for the Asian AAA population, but not the Caucasian AAA population. Overall, thrombus-based indices were stronger predictors of wall stress for Asian AAA, thereby indicating a significant role of ILT in their overall biomechanical measures.

The increase in surface curvature exhibited by an abdominal aorta as the AAA grows is potentially due to a change in proximal neck angle [54] or bending of the infrarenal aorta [55]. This can lead to abnormal blood flow patterns, which in turn results in increased wall stresses [54] that further exacerbate the pathological condition. Therefore, regionally varying surface curvatures are likely to have a high predictive power for elevated AAA wall stress [55]. Previously, we reported on the quantification of geometric indices for predicting wall stress in 75 emergently repaired AAA, which led to wall thickness-based, surface curvature-based, and size-based indices being strong predictors of spatially averaged wall stress [20]. The relative importance of surface curvatures in discriminatory geometric analysis was corroborated by Lee et al. [41] with 205 electively and emergently repaired AAA. They calculated ten curvature-based indices using BQFE for the spatial discretization and machine learning algorithms to obtain the indices with the highest aneurysm classification accuracy, which were GLN and MLN. We infer that our nine and seven significant indices are sufficient geometric surrogates of wall stress in Asian and Caucasian AAA, respectively, and can potentially be used to make an informed assessment of their rupture risk.

D_max Is Not an Accurate Predictor of Abdominal Aortic Aneurysm Wall Stress.

For decades,D_max has been hailed as the clinical standard for AAA rupture risk assessment. However, several studies provide support for biomechanical stress as a more accurate rupture risk predictor, which is not accurately associated with increasing aneurysm diameter [20–22,28,33,56]. Previously, we reported on the geometrical attributes of 312 Asian abdominal aortas comprising unruptured AAA, ruptured AAA, and normal infrarenal aorta without aneurysm [23]. Using various classification models, we found that a multivariate model built with four geometry measures and one clinical variable serves as a rupture risk predictor with 95.2% accuracy. Using a similar principle, we utilized stepwise regression with wall stress to generate reduced attribute models consisting of 3–5 significant geometric indices that adequately predict 99th WS.

The existence of geometric indices other than the standard maximum diameter, which are highly correlated with wall stress, suggests that they could be used as effective surrogates in the assessment of AAA rupture risk, in lieu of an FEA-based approach for wall stress quantification. Previously, Pappu et al. [63] and Speelman [59] demonstrated that the degree of aneurysm tortuosity is associated with increased wall stress and thereby with a high probability of aortic wall rupture. Evidently, tortuosity was one of the predictors of 99th WS for the Asian AAA group, albeit negatively associated with wall stress (Eq.(7)). In addition, 99th WS was positively associated withV_ILT and GAA for this group. Overall, the multivariate model for the Asian AAA group was able to predict 99th WS with an accuracy of approximately 77% in contrast toD_max alone, which predicted 99th WS with an accuracy of 24%. For the Caucasian AAA group, the geometric surrogate model includedD_max and four additional surrogates (D_neck,d,DHr,t_w,minVar, and GAA) (Eq.(8)), which resulted in an improved accuracy of 87% compared toD_max alone (28%). The second-order surface curvature index GAA was the key geometric measure in predicting wall stress for both AAA groups, which underscores the importance of aneurysm shape (rather than size) as a surrogate for wall stress.

Limitations and Recommendations for Future Work.

The outcome of this study should be interpreted within the context of important limitations. The role of gender was not investigated in this work, although it is known that AAA in female patients is more aggressive, expands rapidly, and females have a higher mortality rate compared to their male counterparts [64]. In addition, we did not consider the geometry of the iliac arteries and their influence on wall stress, while common iliac arteries are known to be different in shape between the two patient populations [12,14]. The image segmentation algorithms are not fully automated, which could lead to uncertainty in the calculation of the wall thickness related indices and the generation of the volume meshes with patient-specific wall thickness. Nevertheless, such uncertainty was not quantified, as no ground truth exists for comparison of individual AAA wall thickness. Moreover, the variability in the CTA imaging parameters at the clinical centers, as described in Table1, may introduce systematic bias in the prediction of some wall thickness related measures. Such bias was not quantified and is an important limitation of this work. It is acknowledged that the application of standard uniform pressure (120 mmHg) on the inner surface of the aneurysmal wall, in lieu of patient-specific blood pressure measurements, is a limitation of the FEA modeling approach. However, this is a common practice in patient-specific AAA computational studies [20,22,27,29] as these measurements are usually unknown at the time of image acquisition and due to the high variability of blood pressure with relative physical activity. Moreover, the volume meshes were not subject to a zero-pressure or unstressed geometry derivation protocol prior to FEA. Using a zero-pressure AAA configuration would yield a finite element mesh representative of a geometry different than the computational model utilized for the geometric calculations. Consequently, the geometry for the biomechanical analyses would be smaller than the one utilized for image-based geometric calculations. It is unclear how such discrepancy would affect the underlying correlations between descriptors of AAA size and shape, and wall stress. ILT has been reported to act as a “stress-shield” barrier between blood flow and the AAA wall [65–72]. Since ILT was excluded from the patient-specific computational analyses, it is suspected that 99th WS could be overestimated for both groups. However, patient-specific computational studies without ILT have been reported in the literature previously [20,28,38,39]. Finally, this work can be improved upon with a larger patient cohort to determine if the same geometric surrogate models can be derived using a larger sample size of Asian and Caucasian AAA.

The contribution of this work to AAA biomechanics and geometric analysis can be summarized as follows. First, we established that there are significant geometric differences (wall thickness based and curvature based) between Asian and Caucasian AAA. This provides support to the notion that a patient's ethnic origin and anatomical characteristics may be important considerations for assessing the disease severity of AAA patients. Second, we infer that geometric indices can be more efficient predictors of AAA wall stress compared to maximum aneurysm diameter. The use of geometric surrogates of wall stress provides a clear advantage over the clinical gold standard (D_max), and their calculation can be potentially translated to the clinic for improved diagnosis and interventional planning. Our data support the fact that, within the limited sample sizes used for the current work, there is an evident difference in aneurysmal anatomy between Asian and Caucasian populations. Further studies are necessary to delineate the intricate differences in pathology across these populations.

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