Journal of the Mechanical Behavior of Biomedical Materials

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journal of the mechanical behavior of biomedical materials 115 (2021) 104284

Available online 18 December 2020

Prediction of local strength of ascending thoracic aortic aneurysms

Xuehuan He

^a

, Stephane Avril

^b

, Jia Lu

^a^,^*

aDepartment of Mechanical Engineering, The University of Iowa, Iowa City, IA, 52242, USA

bMines Saint-Etienne, Universit´e de Lyon, INSERM, U 1059 SAINBIOSE, F - 42023 Saint-Etienne, France

A R T I C L E I N F O Keywords:

ATAA Strength Rupture risk Machine learning

A B S T R A C T

Knowledges of both local stress and strength are needed for a reliable evaluation of the rupture risk for ascending thoracic aortic aneurysm (ATAA). In this study, machine learning is applied to predict the local strength of ATAA tissues based on tension-strain data collected through in vitro inflation tests on tissue samples. Inputs to machine learning models are tension, strain, slope, and curvature values at two points on the low strain region of the tension-strain curve. The models are trained using data from locations where the tissue ruptured, and subsequently applied to data from intact sites to predict the local rupture strength. The predicted strengths are compared with the known strength at rupture sites as well as the highest tension the tissues experienced at the intact sites. A local rupture index, which is the ratio of the end tension to the predicted rupture strength, is computed. The ‘hot spots’ of the rupture index are found to match the rupture sites better than those of the peak tension. The study suggests that the strength of ATAA tissue could be reliably predicted from early phase response features defined in this work.

1. Introduction

Ascending thoracic aortic aneurysm, a local dilatation in the aorta, tends to expand gradually and sometimes results in a sudden rupture.

When an aortic aneurysm ruptures, the mortality rate is very high (Elefteriades 2008; Glower et al., 1991). Currently, the guideline for determining whether the intervention should be performed is given by the maximum-diameter criterion (55 mm) (Coady et al., 1999; Coady et al., 1997). However, it is known that aneurysms with diameter smaller than this threshold have ruptured while some larger aneurysms have remained intact (Coady et al., 1999), underlining the need for other indicators to complement the diameter criterion in assessing the rupture risk. Lately, biomechanics-based assessment has attracted considerable attention. In a study by Martin et al., the rupture potential of ATAAs was assessed in 50 patients based on wall stress and population mean rupture strength. It was found that the mean predicted ATAA diameter at rupture is around 56 mm, and that the wall compliance correlates significantly with the rupture risk (Martin et al., 2013). Tra- belsi et al. presented a method for obtaining the retrospective rupture risk taking into account both the stress distribution and the mean failure stress for 5 patients and reported that the risk index (the ratio of peak stress to rupture strength) was only weakly correlated with diameter (Trabelsi et al., 2015). Some studies also proposed stretch-based rupture

criteria using the maximum extensibility or distensibility. Duprey et al.

found the correlation between a stretch-based risk indicator and the tangent elastic modulus in ATAA, based on data collected in 31 patients undergoing in vitro bulge inflation tests (Duprey et al., 2016). Farzaneh et al. presented an inverse method to identify in vivo local extensional stiffness of ATAA walls based on 11 patients who underwent preopera- tive gated CT scans (Farzaneh et al., 2019a) and found a good correlation between the stretch-based rupture risk criterion and the local extensional stiffness (Farzaneh et al., 2019b). Together, these contri- butions highlighted the importance of biomechanical factors in rupture assessment. These efforts mirrored the developments in the related field of abdominal aortic aneurysm (AAA), where biomechanical criteria including peak wall stress (Fillinger et al., 2003; Fillinger et al., 2002;

Gasser et al., 2010; Heng et al., 2008; Raut et al., 2013; Truijers et al., 2007; Venkatasubramaniam et al., 2004), peak risk index (Erhart et al., 2015; Fillinger et al., 2003; Gasser et al., 2010; Maier et al., 2010), probabilistic rupture risk index (Polzer and Gasser 2015), and morphological indexes (Lee et al., 2013; Raut et al., 2013; Shum et al., 2009) have been extensively explored.

A challenge facing stress-based rupture assessment is the heterogeneity of ATAA tissue. It has been reported that ATAAs not only vary greatly among individuals but also exhibit significant layer-, region- and direction-specific heterogeneities in their wall properties (Genovese

* Corresponding author.

E-mail addresses: avril@emse.fr (S. Avril), jia-lu@uiowa.edu (J. Lu).

Contents lists available at ScienceDirect

Journal of the Mechanical Behavior of Biomedical Materials

journal homepage: http://www.elsevier.com/locate/jmbbm

https://doi.org/10.1016/j.jmbbm.2020.104284

Received 12 November 2019; Received in revised form 8 December 2020; Accepted 14 December 2020

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et al., 2013; Iliopoulos et al., 2009a; Iliopoulos et al., 2009b; Matsumoto et al., 2009; Okamoto et al., 2002; Pham et al., 2013; Sokolis et al., 2012a; Sokolis et al., 2012b; Yamada et al., 2015). The authors’ groups have been investigating pointwise properties of the ATAA tissue. They employed the inverse elastostatic analysis to predict the wall tension distribution (Davis et al., 2015; Davis et al., 2016; Luo et al., 2016; Romo et al., 2014). The method leverages the property of static determinacy in membrane-like structures to determine the wall tension without invoking realistic elastic properties (Lu et al., 2013; Lu and Luo 2016; Lu et al., 2008; Zhao et al., 2011a; Zhao et al., 2011b; Zhou et al., 2010).

The pointwise tension data made available by the inverse analysis, together with the surface strain derived from Digital Image Correlation (DIC), enabled the characterization of heterogeneous properties to a spatial resolution prescribed by the finite element mesh. It was found that the ATAAs are heterogeneous even at the spatial resolution of millimeters (Davis et al., 2016). Regarding the ATAA tissue strength, substantial inter- and intra-subject variations have been reported. The strength is found to vary regionally and orientationally (Duprey et al., 2016; Ferrara et al., 2016; Ferrara et al., 2018; Iliopoulos et al., 2009a;

Iliopoulos et al., 2009b; Okamoto et al., 2002). It can be affected by age (Duprey et al., 2016; Ferrara et al., 2016; Ferrara et al., 2018; Okamoto et al., 2002; Pham et al., 2013), gender (Ferrara et al., 2016; Ferrara et al., 2018; Sokolis and Iliopoulos 2014), aortic valve phenotype (Deveja et al., 2018; Ferrara et al., 2016; Ferrara et al., 2018; Forsell et al., 2014; Pichamuthu et al., 2013), and other clinical-pathological factors (Ferrara et al., 2018; Jarrahi et al., 2016; Pham et al., 2013).

Unlike the heterogeneous properties, which could be estimated in vivo using aforementioned analysis if the wall deformation can be captured to a sufficient accuracy, there is no direct method to obtain the tissue strength noninvasively. Early studies have suggested that there exist correlations between the aneurysm strength and response characteristics prior to rupture. Sugita et al. reported that the strength of thoracic aortic aneurysms (TAAs) correlates moderately with a ‘yielding parameter’, which was the stress when stress-tangent elastic modulus curve reached a plateau (Sugita et al., 2011). In a further study (Sugita and Matsumoto 2013), such a correlation was also observed for normal thoracic aortas and it was pointed out that the location associated with this point may reflect a high degree of collagen fiber alignment. More- over, a moderate correlation was found between the normal thoracic aorta strength and a ‘collagen recruitment stress’, which was the stress when collagen fibers are recruited into load bearing. These in- vestigations suggested that collagen recruitment could be an important determinant of ATAA strength. Such correlations are not unreasonable, since ATAAs are collagen-rich structures (Choudhury et al., 2009;

Dobrin 1989; Dobrin et al., 1984; Fonck et al., 2007; Iliopoulos et al., 2009b; Martyn and Greenwald 1997; Sokolis et al., 2012a), and therefore both the strength and elastic properties are likely determined by the collagen content and microstructure. Degradation of elastin facilitates early collagen recruitment and rapid stiffening of the aortic wall (Fonck et al., 2007; Iliopoulos et al., 2009b), and possibly an early failure. In most published studies which characterized ATAA properties using either uniaxial (Garcia-Herrera et al., 2012; Iliopoulos et al., 2009a;

Sokolis et al., 2012a; Sokolis et al., 2012b) or biaxial (Duprey et al., 2016) tensile tests, the mechanical strength and stiffness along the axial direction of the aortic tissue were found lower than the circumferential direction. Sokolis et al. (Sokolis et al., 2012a) pointed out that this is because collagen fibers in the media tend to be aligned in the circumferential direction. In line with this finding, a recent study by the authors’ group reported that ATAA tissues consistently cracked in the direction of the maximum stiffness (Luo et al., 2016), again suggesting the influence of fiber orientation on rupture. Correlations between the strength and elastic properties, if existing, open a pathway to non-invasive estimation of the local strength from characteristics of the response curves.

The goal of our study is to evaluate the feasibility of predicting the local strength of the ATAA tissue from response features. Machine

learning techniques have shown success in exploring hidden relationships in high-dimensional space in the field of aneurysm mechanics (Canchi et al., 2017; Cilla et al., 2018; Erickson et al., 2017; Fan et al., 2018; Lee et al., 2018; Liu et al., 2019). The authors’ group used machine learning to classify tension-strain curves garnered from inflation tests on ATAA samples. It was found that the curves can be reasonably classified into rupture and nonrupture groups based on features of low strain response (Luo et al., 2018). In this study, we submit a machine learning regression approach to predict the ATAA strength from features of tension-strain curves. A total of 28,620 tension-strain curves harvested from 12 patients are used. The machine learning models are trained using data from locations where the tissue ruptured. The models are subsequently applied to data from intact sites to predict the local strength. The predictions are evaluated by comparing the pointwise strength with the highest tension the tissue experienced, and further by comparing the high-risk spots derived from the prediction with the actual rupture sites.

2. Methods

2.1. Tension-strain data and pre-rupture curve features

Tension-strain data used in the present study were obtained through in vitro inflation tests over ATAA samples collected from 12 patients who underwent surgical intervention at the University Hospital Center of St.

Etienne. (Davis et al., 2015; Davis et al., 2016; Luo et al., 2016). The study was approved by the local Internal Review Board. Tissue samples of approximated 3 × 3 cm²were subjected to inflation tests. The deformed surfaces were recorded using DIC and deforming NURBS meshes were constructed from the DIC images. An inverse stress analysis without invoking realistic material parameters was performed to compute the wall tensions at Gauss points (Davis et al., 2015; Davis et al., 2016; Luo et al., 2016). Some of these points were at or near the sites where the tissue finally ruptured while others were at locations where the tissue remained intact. A rupture site was defined as a local patch of 3-by-3 elements surrounding the point deemed to be the point of rupture initiation. For each patient, a total of 2385 tension-strain curves including 2304 nonrupture and 81 rupture were gathered.

A typical tension-strain curve as shown in Fig. 1 has a J-shape with a compliant linear region at the low strain range, an elbow transition region in the middle, and a stiffer region at the high strain end. It is believed that elastin fibers contribute mostly to the initial compliant phase, and progressive recruitment of collagen fibers is responsible for a transition phase that follows, while the much stiffer portion beyond the

Fig. 1. Typical ATAA tension strain curve.

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‘elbow’ is produced by the extension of collagen fibers (Garcia-Herrera et al., 2012; Roach and Burton 1957; Romo et al., 2014). As alluded in the introduction, it is believed that the recruitment process is correlated to strength. Therefore, eight features associated with the transitional response are extracted: the tension, strain, slope, and curvature at the maximum curvature point (MC) and a transition point (TP) (Fig. 1). The curvature at the maximum curvature point (MC curvature) may reflect how fast the collagen recruitment is, the strain (MC strain) at this point may suggest how early the recruitment occurs, and the slope (MC slope) could be a measurement of the early stiffness. The transition point is identified based on the slope change. As shown in Fig. 1, the slope of the tension-strain curve changes significantly in the transition period while becoming stable when it comes to the high strain phase. We define the transition point as the onset of decreasing of second derivative (notwithstanding the slow increase or decrease in the tailing part of the curve). This point may signify the completion of the transitional phase.

The slope (TP slope) may represent the stiffness of the tissue when most of collagen fibers are uncrimped.

The tension and strain data were fitted to third-degree NURBS curves. The third-degree was selected since it could produce continuous second derivative and curvature. The NURBS curves were set to contain 3 knot intervals (namely three segments), to mirror the three-phase structure of the curves. Using too many knot intervals may result in overfitting in the case of inhomogeneous or discontinuous data (Tjah- jowidodo et al., 2015).

In usual NURBS fitting, a knot vector is pre-defined while the control points are identified by minimizing a least squared error between the data and the fitted function (Foley et al., 1994; Piegl and Tiller 1995).

Typically, the knots are placed uniformly in space. However, this fixed-knot approach may numerically alternate the positions of MC and TP. For example, if the first NURBS segment contains the first phase and part of the second phase and if the MC lies in the second phase, the position of MC will be affected numerically by the data in the first phase while in reality, it should not. To circumvent this issue, we implemented a variable knot approach which included the knot positions as optimization variables. A grid-searching optimization procedure was devel- oped to find the optimal knot vector and a set of control points that yield the largest R²value while satisfying a shape requirement. We required

the slope of the fitted curve be positive and non-decreasing everywhere.

The computation process is illustrated in Fig. 2(a). Briefly, given n groups of possible values within 0 and 1 for knot parameters a and b, positions of the NURBS control points and the slopes of the control polygon edges were calculated and stored. The optimized parameters a and b were determined to produce a NURBS curve which yields the maximum R²value while meeting the shape requirement. Before running the optimization procedure, the parameter N which determines the grid resolution, should be determined. Typically, optimized parameters a and b stabilize when number N reaches a certain level. Fig. 2 (b) shows the tuning result for 8 randomly selected tension strain curves.

It is noticed that optimized knot parameters a and b stabilized when parameter N approaches 70. To improve the optimization accuracy while maintaining reasonable computation cost, N was selected as 100.

Note that values of R²for all cases were greater than 0.99.

If the optimal knot algorithm is effective, the resulting features should be insensitive to the tailing part of the curve. To check this, a truncation test was performed. The test consisted of four steps: 1) Ho- mogenizing the data so that each set contains 30 points. The missing points were generated by linear interpolation. 2) Removing 1–10 tailing points from each set. It was observed that, after removing 10 data points, the ends of the curves still lied outside the transition phase. 3) Fitting the truncated data using both uniform and optimal knot methods. 4) Comparing the feature values obtained from the whole curve and truncated curves.

2.2. Machine learning-predictions of local strength and rupture sites Four regression algorithms (Alpaydin 2010), namely, linear regression (LR), multi-layer perceptron regression (MLP), radial kernel support vector regression (SVR) and random forest regression (RF) were tested and it was determined that MLP gave the best overall performance. In the Results section, the outcomes of MLP will be reported.

Some comparative results from other algorithms are provided in the Discussion and Conclusion section.

MLP is a feed-forward artificial neural network paradigm that uses a backpropagation technique for training the network (Alpaydin 2010;

Bishop 1995; Bishop 2006; Erickson et al., 2017). It is a supervised

Fig. 2. Optimization procedure.

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learning algorithm that consists of computing nodes called neurons. The neurons are ordered in layers, starting with an input layer, followed by a number of hidden layers and ending with an output layer. The triggering of neurons is controlled by an activation function. Regarding a typical structure of a three-layer MLP model with m input neurons, h hidden neurons, and n output neurons. The MLP model transforms m inputs into n outputs through a non-linear mapping f: R^m→Rⁿ, where m is the size of input vector x and n is the size of the output vector f(x), such that, in matrix notation:

f(x) =A⁽²⁾(b⁽²⁾+W⁽²⁾(A⁽¹⁾(b⁽¹⁾+W⁽¹⁾x))) (1) With bias vectors b⁽¹⁾, b⁽²⁾; weight matrices W⁽¹⁾,W⁽²⁾and activation functions A⁽¹⁾and A⁽²⁾for the hidden and output layer, respectively. In the present work, a machine learning library, scikit-learn (Pedregosa et al., 2011), was used to implement the regression algorithms.

2.2.1. Regression over rupture data (training)

As a first step, the model was trained using data in the rupture group, which consisted of 972 rupture curves from 12 patients. The regressions were performed separately over 3 groups of features. The first was the four features at MC, the second was the four features at TP, and the third group was the union of these two. A 10-fold cross-validation was applied. All data were randomly shuffled and partitioned into ten sub- sets; nine of which were used to train the model while the remaining one was used to test the model. The process was repeated 10 times to cover all data. The average root mean-square error (RMSE) and average determination coefficient (R²) were used to evaluate the performance of the models.

2.2.2. Prediction of strength at intact points

During the second stage, the trained model was applied to curves in the nonrupture group to predict the strengths. A strength prediction

greater than the end tension was deemed as a positive prediction (PP), otherwise a negative prediction (NP). The percentages of PP and NP over all samples were computed.

2.2.3. Prediction of rupture sites

A rupture index was computed by dividing the end tension by the predicted strength at each point. The ‘hot spots’ of the rupture index were compared to the rupture site. Had the strength predictions from the previous step been directly used, there would be a danger of false accuracy because rupture data of each sample were already used in training. To circumvent this problem, a new 12-fold cross-validation method was used. Specifically, among all 12 samples, the model was trained over 891 rupture curves from 11 samples and applied to predict the strength at all points in the remaining sample. The process was repeated 12 times to cover all samples in the cohort. As the strength at rupture sites was predicted, the accuracy of the prediction was assessed.

3. Results

3.1. Comparison of optimal knot and fixed knot results

Features extracted from curves obtained by the optimal knot method were found to be much more stable than those from the fixed knot method. The x-y plots in Fig. 3 compare the feature values from the whole curve and the truncated curve for the case of truncating 8 points using the two methods. Clearly, the optimal knot method produced a much closer match than the fixed knot method. Although not reported, it was observed that for truncations less than 5 point, the results from the optimal knot method were more sharply concentrated along the x=y line. Evolutions of average linear fitting R²value with the number of truncated data points in both fitting methods are plotted in Fig. 4. It can be observed that the optimized knot method is much less sensitive to the truncation.

Fig. 3. Curve features from truncated and whole curves. Upper row: fixed knot method; Lower row: optimal knot method.

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3.2. Data distribution

The distributions of the eight features and the end tension in both rupture and nonrupture groups are shown in Fig. 5. For each feature, the data are normalized against the range of that of the rupture group, so that the normalized rupture data are always between 0 and 1. From the violin plot, it is clear that the end tensions in both groups have the same range. For all eight features, the high density portions in both groups fall in similar ranges, but there are small fractions of nonrupture data that lie outside the rupture data range.

3.3. Performance of MLP

Before testing the regression model, optimal algorithmic parameters were determined. Specifically, in MLP, the accuracy of the model im- proves with the number of neurons. Nevertheless, high number of neurons may induce overfitting (Fan et al., 2018). As shown in Fig. 6, the training accuracy saturated at around 100 neurons. Therefore, the number of neurons was selected as 100.

3.4. Training results

Fig. 7 shows the predicted rupture tension vs the experimental rupture tension for the training dataset using the three feature sets. Note that the union produced better results compared to MC and TP alone.

The average cross-validation R²from the union is approximately 0.98

while the root mean squared error is of the order of 10⁻².

3.5. Prediction of strength at intact points

Fig. 8 shows the strength prediction results at intact points. The positive predictions account for 92% of all points.

3.6. Prediction of rupture sites

The strengths at rupture sites were predicted in the 12-fold test. The model yielded an average R2 of 0.84 and an average RMSE of 0.175.

Although the accuracy is lower than that of training, it is still reasonably good.

The predicted rupture index distributions are depicted in Fig. 9(a) and (b), along with the rupture image and the tension distribution in the last pressurized state for each sample. Note that the first 1–6 samples and the last 7–12 samples are shown in Fig. 9(a) and (b), respectively. A purple square is used to mark the site of rupture initiation (a window of 3 ×3 elements) in both the rupture index and tension plots. It can be observed that, among 12 samples, the predicted hot spots and the actual rupture sites match generally well in 10 samples. For sample 11, there are two predicted ‘hot spots’ and one of them matches with the actual site. For sample 12, the hot spot does not match the rupture site. The Fig. 4. R²values vs the number of truncated points.

Fig. 5. Data distributions in rupture and nonrupture groups.

Fig. 6.Training accuracy of MLP versus number of neurons.

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RMSE for this sample was 0.30, which was the main contributor to the elevated average RMSE in the 12-fold test.

Comparing the rupture index with the tension contours, for 5 samples (2, 3, 6, 9 and 10), the high-risk spots match the high-tension regions and they both match the rupture sites. However, for the remaining 7 samples, they are different, and the high-tension spots do not coincide with the rupture sites. Clearly, the rupture index provides a better prediction of the rupture sites.

4. Discussion and conclusion

Machine learning was employed to establish a relationship between

pre-rupture response features and the strength of ATAAs. The regression models were trained over data from rupture sites, and subsequently applied to predict the strength at intact points. A three-stage approach was employed to progressively examine the regression results: 1) perform the standard 10-fold cross validation over the rupture group (all 12 samples) during training; 2) predict the strength in nonrupture group (12 samples) and collect the positive and negative predictions; 3) predict the rupture sites using the 12-fold cross validation. The strengths at rupture sites were also predicted and compared with the known values.

Results of these steps suggest that the local rupture strength can be reliably estimated from the pre-rupture features defined in this work. To our best knowledge, it is the first time that machine learning is used to Fig. 7. Training performance from three feature sets. Upper left: MC; Upper Right: TP; Lower: MC +TP.

Fig. 8.Strength prediction at intact point.

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Fig. 9a.Rupture index (first column), actual rupture image (second column) and tension distribution (third column) for samples 1 to 6. The rupture images are not drawn to scale.

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Fig. 9b.Rupture index (first column), actual rupture image (second column) and tension distribution (third column) for samples 7 to 12.

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noninvasively predict the local strength of ATAA tissue and rupture sites based on pre-rupture response features.

An underlying premise of this work is that the ATAA strength correlates with response features, in particular, those related with the progressive collagen recruitment manifested in the transition phase of response. As alluded earlier, Sugita et al. (Sugita and Matsumoto 2013;

Sugita et al., 2011) have reported the existence of the correlations between the TAAs strength and the stress value at which tangent elastic modulus reaches 63% of the plateau level. This position was found to strongly correlate with the stage when most of collagen fibers are uncrimped. However, as reported in (Sugita and Matsumoto 2013;

Sugita et al., 2011), using this stress alone yielded a correlation coefficient of only 0.49, which is insufficient for making prediction. Other studies (Luo et al., 2016; Romo et al., 2014) have suggested that, when assessing rupture propensity at tissue scale, both stress and strain should be concurrently considered. In the present work, up to eight response features associated with the transition phase of the response were employed. It is noted that using eight features produced superior outcome compared with using two subgroups of four features each. This finding indicates that, in the machine learning approach, providing sufficient information that reasonably characterizes the transition phase is imperative.

To check whether the results are insensitive to machine learning algorithms, we also tested three other regression methods, namely, linear regression (LR), support vector regression (SVR), and random forest regression (RF). Before testing, the optimal parameters associated with each model were also determined. As a result, a lasso regularization method with parameter α =10⁻³under second order polynomial features expansion was selected for LR, a radial basis kernel function with parameters C =10 and γ =0.1 were picked for SVR, and the optimal number of trees was determined as 50 for RF. The regression results in the training stage are summarized in Table 1. In overall, the perfor- mances were very similar to that of MLP. RF and SVR slightly out- performed MLP and LP in the training stage, however, the opposite was observed in the latter two steps. Table 2 reports the strength predictions for intact points in stages 2 and 3. Both SVR and RF produced markedly lower positive predictions compared to MLP and LR. This could be attributed to the fact that SVR and RF are poor at extrapolation when the testing data is outside the training range (Quinlan 1986; Riedel and Stulp, 2019). As seen in Fig. 5, there are small fractions of nonrupture data that fall outside the corresponding ranges of the rupture data.

It is worth-noting that the LR method, although slightly under- performing MLP, has the potential of facilitating the development of closed-form statistical models for predicting the strength. For abdominal aortic aneurysm (AAA), Vande Geest et al. (Vande Geest et al., 2006) proposed a statistical model to noninvasively estimate the local AAA strength from clinical factors including patient’s age, gender, smoking history, and local parameters such as wall diameter and thrombus thickness. This model has been incorporated in several studies to evaluate the rupture risk of AAAs (Biehler et al., 2016; Conlisk et al., 2016;

Jarrahi et al., 2016; Polzer and Gasser 2015). In our work, the regu- larized linear regression model with second order polynomial features yielded a R²value of 0.95 for training and 87% positive prediction. The model seems promising for deriving a regression formula for estimating

ATAA strength. This line of work will be explored in the future.

It has been argued in an early study that the tension values at the maximum curvature points mostly lied in the physiological tension range (Luo et al., 2018). A typical aorta of 5 cm diameter under physiological blood pressure ranging between 11 and 16 kPa would bear a wall tension of 0.28–0.40 N/mm according to Laplace law (Stamler et al., 1993). We checked randomly selected rupture and nonrupture curves and found that most of tension values at the TP point were also within this range. For members of the rupture group, we also checked the ratio of the tension values at the TP point to the strength. Fig. 10 presents the histogram of this ratio. Note that most of them are less than 0.4, which is in good agreement with the study by Duprey et al. (Duprey et al., 2016) where the ratio of the in vivo stress to the maximum stress for 31 ATAAs varies from 0.04 to 0.45. Therefore, it is reasonable to conclude that both MC and TP are within the physiological response range. This is desirable because, if the present work is eventually applied to in vivo application, only the physiological range of response will be available.

There are several limitations in this study. First, the patient cohort was relatively small. Although ~28,000 curves were harvested, they were obtained from 12 samples out of 12 patients. The tension and strain ranges in the rupture group were not large enough to cover the data in the non-rupture group, and this caused SVM and RF to perform poorly in prediction. As reported in the literature (Genovese et al., 2013; Ilio- poulos et al., 2009a; Iliopoulos et al., 2009b; Matsumoto et al., 2009;

Okamoto et al., 2002; Pham et al., 2013; Sokolis et al., 2012a; Sokolis et al., 2012b; Yamada et al., 2015), many clinical and pathological factors such as age, gender, and disease conditions as the ones investi- gated in Ferrara et al. (Ferrara et al., 2016; Ferrara et al., 2018) could contribute to inter-subject heterogeneities in material properties.

Therefore, more samples covering a wider range of patient types are expected to improve the prediction. Second, the identification of the rupture site can be improved. In this study, the method proposed in (Luo et al., 2016) was employed to define the rupture site. The location of crack initiation was assumed at the geometric center of cracks identified from post-rupture image. A local window containing 3 ×3 elements centered on the crack was designated as the rupture site. This method has not been fully verified. For orifices with regular shapes and located

Table 1

Regression results for training datasets.

Feature set Accuracy Metrics MLP LR SVR RF

MC R² 0.882 0.61 0.948 0.977

RMSE 0.156 0.247 0.106 0.070

TP R² 0.939 0.912 0.965 0.983

RMSE 0.120 0.137 0.089 0.061

Union R² 0.978 0.957 0.979 0.991

RMSE 0.072 0.097 0.068 0.043

Table 2

PP outcomes in the nonrupture group.

Evaluation Stage Accuracy Metrics MLP LR SVR RF

Stage 2 PP 92.5 88.1 77.2 87.7

Stage 3 PP 90.1 87.5 70.9 77.4

Fig. 10.Tension ratio (TP (tension)/Strength) in rupture group.

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in the center regions of the specimens, this identification can be effective. For orifices that are irregular and located near the boundary, it could be unclear as to where exactly the rupture initiated. More information, for example footages of high-speed camera, is expected to better locate the initial crack position. Recall that in sample 12 the predicted high-risk region does not match the rupture location. This could be attributed to the boundary effect. In this sample the crack was situated in the proximity of the mesh boundary or the physical boundary. It is known that the inverse method of stress analysis could suffer the loss of accuracy in the boundary region (Lu and Zhao 2009; Zhao et al., 2009).

It is also possible that rupture was affected by the fixture. The crescent orifice curved along the clamped boundary, suggesting the influence of the fixture. Thirdly, since the local tissue thickness was not recorded, the tension instead of the stress was used in this study. A stress-based regression could permit scaling the tension values with the local thickness. This does not entail changing the algorithm, however, since such a test was not performed. we refrain from making speculation about possible outcome. In addition, the layer-specific heterogeneity of the aortic wall was also not considered. Sokolis et al. (Sokolis et al., 2012b) reported differences in the uniaxial stress–strain and failure responses for the three layers of the ATAA wall. Incorporating layer-specific information could be important. Lastly, the tension strain data were generated from inflation test for which one does not have control over the local loading condition (in the present study, most points underwent approximately equal-biaxial loading). Ideally, experimental data from different loading protocols should be included in the machine learning.

In conclusion, while some limitations remain, the work suggests that the tissue strength can be related to features of the early phase of the tension-strain response. Eventually, this constitutes a step forward towards non-invasive prediction of the local strength and rupture site in ATAAs.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

JL acknowledges a research grant from ANSYS Inc. SA is grateful to the European Research Council (ERC grant Biolochanics, grant number 647067) for financial support.

References

Alpaydin, E., 2010. Introduction to Machine Learning. Introduction to Machine Learning {Adaptive Computation and Machine Learning. Place of publication not identified MIT Press. Place of publication not identified.

Biehler, J., et al., 2016. Probabilistic noninvasive prediction of wall properties of abdominal aortic aneurysms using Bayesian regression. Biomech. Model.

Mechanobiol. 16 https://doi.org/10.1007/s10237-016-0801-6.

Bishop, C.M., 1995. Neural Networks for Pattern Recognition. Oxford University Press, Inc.

Bishop, C.M., 2006. Pattern Recognition and Machine Learning (Information Science and Statistics). Springer-Verlag.

Canchi, T., Patnaik, S.S., Ng, E.Y., Srinivasan, D.K., Narayanan, S., Finol, E.A., 2017. On the relative effectiveness of machine learning and statistical methods in predicting abdominal aortic aneurysm rupture in the Asian population. Arterioscl Throm Vas Choudhury, N., 2009. Local mechanical and structural properties of healthy and diseased 37.

human ascending aorta tissue. Cardiovasc. Pathol. 18, 83–91. https://doi.org/

10.1016/j.carpath.2008.01.001.

Cilla, M., Perez-Rey, I., Martinez, M., Pena, E., Martinez, J., 2018. On the use of machine learning techniques for the mechanical characterization of soft biological tissues. Int J Numer Method Biomed Eng 34. https://doi.org/10.1002/cnm.3121.

Coady, M.A., Rizzo, J.A., Hammond, G.L., Kopf, G.S., Elefteriades, J.A., 1999. Surgical intervention criteria for thoracic aortic aneurysms: a study of growth rates and complications. Ann. Thorac. Surg. 67, 1922–1926. https://doi.org/10.1016/S0003- 4975(99)00431-2.

Coady, M.A., Rizzo, J.A., Hammond, G.L., Mandapati, D., Darr, U., Kopf, G.S., Elefteriades, J.A., 1997. What is the appropriate size criterion for resection of

thoracic aortic aneurysms? J. Thorac. Cardiovasc. Surg. 113, 476–489. https://doi.

org/10.1016/S0022-5223(97)70360-X.

Conlisk, N., Geers, A., McBride, O., Newby, D., Hoskins, P., 2016. Patient-specific modelling of abdominal aortic aneurysms: the influence of wall thickness on predicted clinical outcomes. Med. Eng. Phys. 38 https://doi.org/10.1016/j.

medengphy.2016.03.003.

Davis, F.M., Luo, Y.M., Avril, S., Duprey, A., Lu, J., 2015. Pointwise characterization of the elastic properties of planar soft tissues: application to ascending thoracic aneurysms. Biomech. Model. Mechanobiol. 14, 967–978. https://doi.org/10.1007/

s10237-014-0646-9.

Davis, F.M., Luo, Y.M., Avril, S., Duprey, A., Lu, J., 2016. Local mechanical properties of human ascending thoracic aneurysms. J Mech Behav Biomed Mater 61, 235–249.

https://doi.org/10.1016/j.jmbbm.2016.03.025.

Deveja, R.P., Iliopoulos, D.C., Kritharis, E.P., Angouras, D.C., Sfyris, D., Papadodima, S.

A., Sokolis, D.P., 2018. Effect of aneurysm and bicuspid aortic valve on layer-specific ascending aorta mechanics. Ann. Thorac. Surg. 106, 1692–1701. https://doi.org/

10.1016/j.athoracsur.2018.05.071.

Dobrin, P.B., 1989. Patho-physiology and pathogenesis of aortic-aneurysms - current concepts. Surg. Clin. 69, 687–703.

Dobrin, P.B., Baker, W.H., Gley, W.C., 1984. Elastolytic and collagenolytic studies of arteries - implications for the mechanical-properties of aneurysms. Arch. Surg. 119, 405–409. https://doi.org/10.1001/archsurg.1984.01390160041009.

Duprey, A., Trabelsi, O., Vola, M., Favre, J.P., Avril, S., 2016. Biaxial rupture properties of ascending thoracic aortic aneurysms. Acta Biomater. 42, 273–285. https://doi.

org/10.1016/j.actbio.2016.06.028.

Elefteriades, J.A., 2008. Thoracic aortic aneurysm: reading the enemy’s playbook. Curr.

Probl. Cardiol. 33, 203–277. https://doi.org/10.1016/j.cpcardiol.2008.01.004.

Erhart, P., et al., 2015. Finite element analysis in asymptomatic, symptomatic, and ruptured abdominal aortic aneurysms: in search of new rupture risk predictors. Eur.

J. Vasc. Endovasc. Surg. 49, 239–245. https://doi.org/10.1016/j.ejvs.2014.11.010.

Erickson, B.J., Korfiatis, P., Akkus, Z., Kline, T.L., 2017. Machine learning for medical imaging(1). Radiographics 37, 505–515. https://doi.org/10.1148/rg.2017160130.

Fan, F.L., Cong, W.X., Wang, G., 2018. A new type of neurons for machine learning. Int J Numer Method Biomed Eng 34. https://doi.org/10.1002/cnm.2920.

Farzaneh, S., Trabelsi, O., Avril, S., 2019a. Inverse identification of local stiffness across ascending thoracic aortic aneurysms. Biomech. Model. Mechanobiol. 18, 137–153.

https://doi.org/10.1007/s10237-018-1073-0.

Farzaneh, S., Trabelsi, O., Chavent, B., Avril, S., 2019b. Identifying local arterial stiffness to assess the risk of rupture of ascending thoracic aortic aneurysms. Ann. Biomed.

Eng. 47, 1038–1050. https://doi.org/10.1007/s10439-019-02204-5.

Ferrara, A., Morganti, S., Totaro, P., Mazzola, A., Auricchio, F., 2016. Human dilated ascending aorta: mechanical characterization via uniaxial tensile tests. J Mech Behav Biomed Mater 53, 257–271. https://doi.org/10.1016/j.jmbbm.2015.08.021.

Ferrara, A., Totaro, P., Morganti, S., Auricchio, F., 2018. Effects of clinico-pathological risk factors on in-vitro mechanical properties of human dilated ascending aorta.

J Mech Behav Biomed Mater 77, 1–11. https://doi.org/10.1016/j.

jmbbm.2017.08.032.

Fillinger, M.F., Marra, S.P., Raghavan, M.L., Kennedy, F.E., 2003. Prediction of rupture risk in abdominal aortic aneurysm during observation: wall stress versus diameter.

J. Vasc. Surg. 37, 724–732. https://doi.org/10.1067/mva.2003.213.

Fillinger, M.F., Raghavan, M.L., Marra, S.P., Cronenwett, J.L., Kennedy, F.E., 2002. In vivo analysis of mechanical wall stress and abdominal aortic aneurysm rupture risk.

J. Vasc. Surg. 36, 589–597. https://doi.org/10.1067/mva.2002.125478.

Foley, J.D., Phillips, R.L., Hughes, J.F., Dam, Av, Feiner, S.K., 1994. Introduction to Computer Graphics. Addison-Wesley Longman Publishing Co., Inc.

Fonck, E., Prod’hom, G., Roy, S., Augsburger, L., Rufenacht, D.A., Stergiopulos, N., 2007.

Effect of elastin degradation on carotid wall mechanics as assessed by a constituent- based biomechanical model. Am. J. Physiol. Heart Circ. Physiol. 292, H2754–H2763.

https://doi.org/10.1152/ajpheart.01108.2006.

Forsell, C., Bjorck, H.M., Eriksson, P., Franco-Cereceda, A., Gasser, T.C., 2014.

Biomechanical properties of the thoracic aneurysmal wall: differences between bicuspid aortic valve and tricuspid aortic valve patients. Ann. Thorac. Surg. 98, 65–71. https://doi.org/10.1016/j.athoracsur.2014.04.042.

Garcia-Herrera, C.M., et al., 2012. Mechanical behaviour and rupture of normal and pathological human ascending aortic wall. Med. Biol. Eng. Comput. 50, 559–566.

https://doi.org/10.1007/s11517-012-0876-x.

Gasser, T.C., Auer, M., Labruto, F., Swedenborg, J., Roy, J., 2010. Biomechanical rupture risk assessment of abdominal aortic aneurysms: model complexity versus predictability of finite element simulations. Eur. J. Vasc. Endovasc. Surg. 40, 176–185. https://doi.org/10.1016/j.ejvs.2010.04.003.

Genovese, K., Lee, Y.U., Lee, A.Y., Humphrey, J.D., 2013. An improved panoramic digital image correlation method for vascular strain analysis and material characterization.

J Mech Behav Biomed Mater 27, 132–142. https://doi.org/10.1016/j.

jmbbm.2012.11.015.

Glower, D.D., Speier, R.H., White, W.D., Smith, L.R., Rankin, J.S., Wolfe, W.G., 1991.

Management and long-term outcome of aortic dissection. Ann. Surg. 214, 31–41.

Heng, M.S., Fagan, M.J., Collier, J.W., Desai, G., McCollum, P.T., Chetter, I.C., 2008.

Peak wall stress measurement in elective and acute abdominal aortic aneurysms.

J. Vasc. Surg. 47, 17–22. https://doi.org/10.1016/j.jvs.2007.09.002.

Iliopoulos, D.C., et al., 2009a. Regional and directional variations in the mechanical properties of ascending thoracic aortic aneurysms. Med. Eng. Phys. 31, 1–9. https://

doi.org/10.1016/j.medengphy.2008.03.002.

Iliopoulos, D.C., Kritharis, E.P., Giagini, A.T., Papadodima, S.A., Sokolis, D.P., 2009b.

Ascending thoracic aortic aneurysms are associated with compositional remodeling and vessel stiffening but not weakening in age-matched subjects. J. Thorac.

Cardiovasc. Surg. 137, 101–109. https://doi.org/10.1016/j.jtcvs.2008.07.023.

(11)

Jarrahi, A., Karimi, A., Navidbakhsh, M., Ahmadi, H., 2016. Experimental/numerical study to assess mechanical properties of healthy and Marfan syndrome ascending thoracic aorta under axial and circumferential loading. Mater. Technol. 31, 247–254.

https://doi.org/10.1179/1753555715y.0000000049.

Lee, K., et al., 2013. Surface curvature as a classifier of abdominal aortic aneurysms: a comparative analysis. Ann. Biomed. Eng. 41, 562–576. https://doi.org/10.1007/

s10439-012-0691-4.

Lee, R., et al., 2018. Applied machine learning for the prediction of growth of abdominal aortic aneurysm in humans. EJVES Short Rep 39, 24–28. https://doi.org/10.1016/j.

ejvssr.2018.03.004.

Liu, M.L., Liang, L., Sun, W., 2019. Estimation of in vivo constitutive parameters of the aortic wall using a machine learning approach. Comput. Methods Appl. Mech. Eng.

347, 201–217. https://doi.org/10.1016/j.cma.2018.12.030.

Lu, J., Hu, S.H., Raghavan, M.L., 2013. A shell-based inverse approach of stress analysis in intracranial aneurysms. Ann. Biomed. Eng. 41, 1505–1515. https://doi.org/

10.1007/s10439-013-0751-4.

Lu, J., Luo, Y.M., 2016. Solving membrane stress on deformed configuration using inverse elastostatic and forward penalty methods. Comput. Methods Appl. Math.

308, 134–150. https://doi.org/10.1016/j.cma.2016.05.017.

Lu, J., Zhao, X., 2009. Pointwise identification of elastic properties in nonlinear hyperelastic membranes—Part I: theoretical and computational developments.

J. Appl. Mech. 76 https://doi.org/10.1115/1.3130805.

Lu, J., Zhou, X.L., Raghavan, M.L., 2008. Inverse method of stress analysis for cerebral aneurysms. Biomech. Model. Mechanobiol. 7, 477–486. https://doi.org/10.1007/

s10237-007-0110-1.

Luo, Y.M., Duprey, A., Avril, S., Lu, J., 2016. Characteristics of thoracic aortic aneurysm rupture in vitro. Acta Biomater. 42, 286–295. https://doi.org/10.1016/j.

actbio.2016.06.036.

Luo, Y.M., Fan, Z.W., Baek, S., Lu, J., 2018. Machine learning-aided exploration of relationship between strength and elastic properties in ascending thoracic aneurysm.

Int J Numer Method Biomed Eng 34. https://doi.org/10.1002/cnm.2977.

Maier, A., Gee, M.W., Reeps, C., Pongratz, J., Eckstein, H.H., Wall, W.A., 2010.

A comparison of diameter, wall stress, and rupture potential index for abdominal aortic aneurysm rupture risk prediction. Ann. Biomed. Eng. 38, 3124–3134. https://

doi.org/10.1007/s10439-010-0067-6.

Martin, C., Sun, W., Pham, T., Elefteriades, J., 2013. Predictive biomechanical analysis of ascending aortic aneurysm rupture potential. Acta Biomater. 9, 9392–9400. https://

doi.org/10.1016/j.actbio.2013.07.044.

Martyn, C.N., Greenwald, S.E., 1997. Impaired synthesis of elastin in walls of aorta and large conduit arteries during early development as an initiating event in pathogenesis of systemic hypertension. Lancet 350, 953–955. https://doi.org/

10.1016/S0140-6736(96)10508-0.

Matsumoto, T., et al., 2009. Biaxial tensile properties of thoracic aortic aneurysm tissues.

J. Biomech. Sci. Eng. 4, 518–529. https://doi.org/10.1299/jbse.4.518.

Okamoto, R.J., Wagenseil, J.E., DeLong, W.R., Peterson, S.J., Kouchoukos, N.T., Sundt, T.M., 2002. Mechanical properties of dilated human ascending aorta. Ann.

Biomed. Eng. 30, 624–635. https://doi.org/10.1114/1.1484220.

Pedregosa, F., et al., 2011. Scikit-learn: machine learning in Python. J. Mach. Learn. Res.

12, 2825–2830.

Pham, T., Martin, C., Elefteriades, J., Sun, W., 2013. Biomechanical characterization of ascending aortic aneurysm with concomitant bicuspid aortic valve and bovine aortic arch. Acta Biomater. 9, 7927–7936. https://doi.org/10.1016/j.actbio.2013.04.021.

Pichamuthu, J.E., Phillippi, J.A., Cleary, D.A., Chew, D.W., Hempel, J., Vorp, D.A., Gleason, T.G., 2013. Differential tensile strength and collagen composition in ascending aortic aneurysms by aortic valve phenotype. Ann. Thorac. Surg. 96, 2147–2154. https://doi.org/10.1016/j.athoracsur.2013.07.001.

Piegl, L., Tiller, W., 1995. The NURBS Book.

Polzer, S., Gasser, T.C., 2015. Biomechanical rupture risk assessment of abdominal aortic aneurysms based on a novel probabilistic rupture risk index. J. R. Soc. Interface 12.

ARTN 20150852, 10.1098/rsif.2015.0852.

Quinlan, J.R., 1986. Induction of decision trees %. J Mach. Learn. 1, 81–106. https://doi.

org/10.1023/a:1022643204877.

Raut, S.S., Chandra, S., Shum, J., Finol, E.A., 2013. The role of geometric and biomechanical factors in abdominal aortic aneurysm rupture risk assessment. Ann.

Biomed. Eng. 41, 1459–1477. https://doi.org/10.1007/s10439-013-0786-6.

Riedel, S., Stulp, FJapa, 2019. Comparing Semi-parametric Model Learning Algorithms for Dynamic Model Estimation in Robotics.

Roach, M.R., Burton, A.C., 1957. The reason for the shape of the distensibility curves of arteries. Can. J. Biochem. Physiol. 35, 681–690.

Romo, A., Badel, P., Duprey, A., Favre, J.P., Avril, S., 2014. In vitro analysis of localized aneurysm rupture. J. Biomech. 47, 607–616. https://doi.org/10.1016/j.

jbiomech.2013.12.012.

Shum, J., Martufi, G., DiMartino, E.S., Finol, E.A., 2009. Quantitative assessment of abdominal aortic aneurysm geometry and rupture potential. Proceedings of the Asme Summer Bioengineering Conference - 2009, Pt a and B, pp. 1303–1304.

Sokolis, D.P., Iliopoulos, D.C., 2014. Impaired mechanics and matrix metalloproteinases/

inhibitors expression in female ascending thoracic aortic aneurysms. J Mech Behav Biomed Mater 34, 154–164. https://doi.org/10.1016/j.jmbbm.2014.02.015.

Sokolis, D.P., Kritharis, E.P., Giagini, A.T., Lampropoulos, K.M., Papadodima, S.A., Iliopoulos, D.C., 2012a. Biomechanical response of ascending thoracic aortic aneurysms: association with structural remodelling. Comput. Methods Biomech.

Biomed. Eng. 15, 231–248. https://doi.org/10.1080/10255842.2010.522186.

Sokolis, D.P., Kritharis, E.P., Iliopoulos, D.C., 2012b. Effect of layer heterogeneity on the biomechanical properties of ascending thoracic aortic aneurysms. Med. Biol. Eng.

Comput. 50, 1227–1237. https://doi.org/10.1007/s11517-012-0949-x.

Stamler, J., Stamler, R., Neaton, J.D., 1993. Blood-pressure, systolic and diastolic, and cardiovascular risks - united-states population-data. Arch. Intern. Med. 153, 598–615. https://doi.org/10.1001/archinte.153.5.598.

Sugita, S., Matsumoto, T., 2013. Yielding phenomena of aortic wall and intramural collagen fiber alignment: possible link to rupture mechanism of aortic aneurysms.

J. Biomech. Sci. Eng. 8, 104–113. https://doi.org/10.1299/jbse.8.104.

Sugita, S., Matsumoto, T., Ohashi, T., Kumagai, K., Akimoto, H., Tabayashi, K., Sato, M., 2011. Evaluation of rupture properties of thoracic aortic aneurysms in a pressure- imposed test for rupture risk estimation. Cardiovasc Eng Technol 3. https://doi.org/

10.1007/s13239-011-0067-1.

Tjahjowidodo, T., Dung, V.T., Han, M.L., 2015. A Fast Non-uniform Knots Placement Method for B-Spline Fitting. In: 2015 Ieee/Asme International Conference on Advanced Intelligent Mechatronics (Aim), pp. 1490–1495.

Trabelsi, O., Davis, F.M., Rodriguez-Matas, J.F., Duprey, A., Avril, S., 2015. Patient specific stress and rupture analysis of ascending thoracic aneurysms. J. Biomech. 48, 1836–1843. https://doi.org/10.1016/j.jbiomech.2015.04.035.

Truijers, M., Pol, J.A., SchultzeKool, L.J., van Sterkenburg, S.M., Fillinger, M.F., Blankensteijn, J.D., 2007. Wall stress analysis in small asymptomatic, symptomatic and ruptured abdominal aortic aneurysms. Eur. J. Vasc. Endovasc. Surg. 33, 401–407. https://doi.org/10.1016/j.ejvs.2006.10.009.

Vande Geest, J., Wang D, H.J., R Wisniewski, S., S Makaroun, M., Vorp D, A., 2006.

Towards A noninvasive method for determination of patient-specific wall strength distribution in abdominal aortic aneurysms. Ann. Biomed. Eng. 34, 1098–1106.

https://doi.org/10.1007/s10439-006-9132-6.

Venkatasubramaniam, A.K., et al., 2004. A comparative study of aortic wall stress using finite element analysis for ruptured and non-ruptured abdominal aortic aneurysms.

Eur. J. Vasc. Endovasc. Surg. 28, 168–176. https://doi.org/10.1016/j.

ejvs.2004.03.029.

Yamada, H., Sakata, N., Wada, H., Tashiro, T., Tayama, E., 2015. Age-related distensibility and histology of the ascending aorta in elderly patients with acute aortic dissection. J. Biomech. 48, 3267–3273. https://doi.org/10.1016/j.

jbiomech.2015.06.025.

Zhao, X., Chen, X., Lu, J., 2009. Pointwise identification of elastic properties in nonlinear hyperelastic membranes—Part II: experimental validation. J. Appl. Mech. 76 https://doi.org/10.1115/1.3130810.

Zhao, X.F., Raghavan, M.L., Lu, J., 2011a. Characterizing heterogeneous properties of cerebral aneurysms with unknown stress-free geometry: a precursor to in vivo identification. J Biomech Eng-T Asme 133. Artn 051008, 10.1115/1.4003872.

Zhao, X.F., Raghavan, M.L., Lu, J., 2011b. Identifying heterogeneous anisotropic properties in cerebral aneurysms: a pointwise approach. Biomech. Model.

Mechanobiol. 10, 177–189. https://doi.org/10.1007/s10237-010-0225-7.

Zhou, X.L., Raghavan, M.L., Harbaugh, R.E., Lu, J., 2010. Patient-specific wall stress analysis in cerebral aneurysms using inverse shell model. Ann. Biomed. Eng. 38, 478–489. https://doi.org/10.1007/s10439-009-9839-2.