# Influence of morphometric and mechanical factors in thoracic aorta finite element modeling

R. ZHANG¹, M. SIGOVAN¹, and P. CLARYSSE¹

Univ Lyon, CNRS UMR 5220, Inserm U1294, INSA Lyon, Université Claude Bernard Lyon 1, Creatis, France

Ruifen.zhang@creatis.insa-lyon.fr

**Abstract.** Evaluation of mechanical properties from thoracic aorta finite element (FE) modeling can help to better stratify patients in need of intervention. This paper assesses the influence of various factors in the configuration of a FE model focused on the ascending aorta. Essential factors including morphological descriptors, material elasticity, boundary conditions (BCs), and load are investigated on a synthetic aorta model. Based on representative values retrieved from literature, the impact of these factors is studied using statistical hypothesis testing (one-way ANOVA).

Our results show that wall thickness, blood pressure, ascending aorta diameter, and material elasticity have the most dominant impacts onto the aortic stress distributions. Such an information is useful to properly configure patient-specific thoracic aorta FE modeling.

**Keywords:** Finite element modeling · ascending thoracic aorta · sensitivity analysis.

## 1 Introduction

In the cardiovascular disease landscape, the ascending thoracic aorta aneurysm (ATAA) is one of the major pathologies of the aorta. ATTA is associated with a high morbidity and high mortality rate in case of rupture [1]. The vascular diameter, measured radio-graphically, remains the conventional criterion for surgical repair and primary determinant for risk stratification [2]. However, prior studies have shown that a significant number of patients developed adverse consequences below the standard diameter criteria of 5.0 cm, whereas some patients presented mean diameters > 5.5 cm without dissection or rupture. Clearly, additional markers or measurements are needed for the diagnosis and monitoring of ATAA in view of preventing complications [3].

Promoted from advances in medical imaging, computational patient-specific Finite Element (FE) modeling of thoracic aorta mechanics can provide unique insights into vessel integrity in both healthy and diseased states. It is believed that aortic dissection or rupture may occur when the stress in the aorta wall is high enough to damage the tissue itself [4]. Consequently, wall stress distribution from a FE analysis may present a useful information for risk assessment.
However, to our knowledge, there is no comprehensive survey on the factors influencing aorta FE modeling and the resulting wall stress distributions.

In this paper, the influence of several geometrical and mechanical factors on the distribution of wall stress is investigated in an idealized but realistic FE aorta model.

## 2 Methodology

The study of the impact of factors is based on the analysis of the response of synthetic aorta models to various morphometric and mechanical factors. Statistical analysis of the response considers mean, mean minus or plus standard deviation values of the factors.

### 2.1 Constructions of synthetic aorta geometric models

<figure><img src="image_1.png" /><figcaption>**Fig. 1.** Configuration of the synthetic thoracic aorta FE modeling. (a) Schematic view of the synthetic aorta geometry with lumen centerline in blue. (b) Synthetic aorta geometry constructed in FreeCAD. The configuration of BCs in aorta FE modeling comprises: stretch in the longitudinal direction and twist at the ascending aortic root section $\partial\Omega_{asc}$; fixed displacement restriction on the descending section $\partial\Omega_{des}$ and branch sections $\partial\Omega_b$ in all degrees of freedom; support by spine modeled with a spring on the spine contact region (green surfacic elements). The systolic blood pressure is applied to the internal lumen surface $\partial\Omega_{int}$. (c) Centerlines A, B, and C sketched in the 2D-plane. (BCA: branchiocephalic; LCCA: left common carotid; LSCA: left subclavian artery.)</figcaption></figure>

Based on the general schematic thoracic aortic geometry in Figure 1(a), the synthetic aorta models were constructed in FreeCAD software by sweeping the sino-tubular junction (STJ) cross-section along a curved centerline down to the
diaphragm level (Figure 1(b)). To integrate shape variations, three centerlines were sketched in a 2D plane that correspond to typical ascending aorta shapes from normal to less or more curved arch (Figure 1(c)). Line A presents highest curvature on the ascending part, line B has moderate curvature (healthy case), and line C is the most opened aorta with smaller curvature. The lengths for ascending, aortic arch, and descending parts are 89, 64 and 202 mm, respectively (obtained from one patient in our dataset). Mean and standard deviation (SD) of the ascending thoracic aorta diameter (ATD), descending thoracic aorta diameter (DTD) and aortic wall thickness (T) were set from population data available in the scientific literature (Table. 2) according to

$$ \bar{X} = \frac{\sum N_i \bar{X}_i}{\sum N_i}, \quad s(X) = \sqrt{\frac{\sum N_i ((\bar{X}_i - \bar{X})^2 + s(X_i)^2)}{\sum N_i}} \quad (1) $$

where $\bar{X}_i$, $s_i(X)$, $N_i$ are the mean, standard deviation and number of subjects in each data set indexed with $i$. Due to the limitations in spatial resolution hampering estimation of wall thickness in medical imaging, the measurements of diameters and wall thickness from in-vitro samples were included. The statistics of the parameters are: ATD= 32.8±4.4 mm, DTD= 22.3±2.6 mm, and $T$ = 2.2±0.5 mm. Supra-aortic branches of branchiocephalic (BCA), left common carotid (LCCA), and left subclavian artery(LSCA) originating from the top of the aortic arch were attributed the same diameter of 10 mm. The generated geometries correspond to a geometry at rest, often named zero-pressure geometry.

<figure><img src="image_2.png" /><figcaption>**Fig. 2.** The non-dilated control idealized mesh in yellow and the dilated pathological idealized mesh in grey after RBF morphing. One control point (red) in the middle of the ascending major curvature part controls the morphing of the aneurysm. Other control points in blue impose restriction onto the mesh.</figcaption></figure>

In order to introduce pathological aorta shapes into the study, virtual ascending aortic aneurysms were created by a localized dilation of the healthy FE mesh using a radial basis functions (RBF) based algorithm with a Gaussian kernel [5, 13]. The major curvature part smoothly protrudes outward by 20% and 40% of
its normal diameter representing moderate to more severe aneurysms with sectional radius of 40 and 46 mm, respectively (Fig. 2, right). One control point in the middle of the ascending major curvature domain was selected to pilot the aneurysm deformation in the outward direction, as well as several control points around the arch and the descending parts to constrain the mesh. The weights and the polynomial coefficients for RBF morphing function were calculated from coordinates of the control points in the reference and deformed frames.

## 2.2 Material law for aorta wall

The constitutive law (i.e. strain energy function SEF $\Psi$) proposed by Gasser et al. [14] is adopted for aorta FE modeling; it is based on the histological information in aortic wall, referred as GOH law (Eq.(2)) in the sequel. The SEF is formulated as follows

$$ \begin{aligned} \Psi &= \Psi_{\text{vol}}(J) + \bar{\Psi}_{g} + \sum_{i=1,2} \bar{\Psi}_{fi} \\ &= \Psi_{\text{vol}}(J) + \frac{1}{2} c_{10} (\bar{I}_1 - 3) + \sum_{i=1,2} \frac{k_1}{2k_2} [\exp(k_2 \bar{E}_i^2) - 1], \end{aligned} \quad (2) $$

where $\Psi_{\text{vol}}(J)$ denotes a Lagrange contribution, which enforces the associated volume incompressibility kinematical constraint; $\bar{\Psi}_g$ a ground-matrix term and $\bar{\Psi}_{fi}$ a collagen fiber reinforcement term for the purely isochoric contribution; $\bar{\Psi}_g$ is modeled by an isotropic Neo-Hookean law with coefficient $c_{10}$ (Pa); $\bar{\Psi}_{fi}$ is modeled by an exponential function weighted by parameter $k_1$ (Pa) and dimensionless parameter $k_2$; $\bar{E}_i = \kappa \bar{I}_1 + (1 - 3\kappa) \bar{I}_{4i} - 1$, where $\bar{I}_1$, $\bar{I}_{4i}$ are the first and the fourth invariants of the right Cauchy-Green strain tensor; the angle $\gamma$ in a generalized structure tensor $\mathbf{H} = \kappa \mathbf{I} + (1 - 3\kappa) \mathbf{a}_0 \otimes \mathbf{a}_0$ represents the mean orientations of two family fibers, which are symmetric about the circumferential direction and $\mathbf{a}_0$ is the orientation direction; $0 < \kappa < 1/3$ accounts for the fiber orientation dispersion. With $\kappa = 0$, the collagen fibers are aligned to the mean direction $\gamma$ (no dispersion). On the contrary, $\kappa = 1/3$ characterizes an isotropic distribution of the orientations of the collagen fibers around $\gamma$.

<table><caption>**Table 1.** GOH material parameters for three levels of material elasticity.</caption><thead><tr><th>Elasticity</th><th>c<sub>10</sub> [MPa]</th><th>k<sub>1</sub> [MPa]</th><th>k<sub>2</sub> [-]</th><th>κ</th><th>γ°</th><th>Refs</th></tr></thead><tbody><tr><td>Stiff</td><td>0.06</td><td>1</td><td>5</td><td>0</td><td>35</td><td>[18]</td></tr><tr><td>Mid</td><td>0.0095</td><td>5.15</td><td>8.64</td><td>0.24</td><td>38.8</td><td>[19]</td></tr><tr><td>Compliant</td><td>0.014</td><td>0.81</td><td>12.42</td><td>0.18</td><td>39.1</td><td>[19]</td></tr></tbody></table>

Three levels of material elasticity were identified from literature based on GOH biaxial stretch-stress experiments (Table. 1).
## 2.3 Physiological conditions for the synthetic aorta wall

Applied loads and BCs arise primarily from three sources: the hemodynamic load that acts on the luminal surface, the spine constraint on the aorta, and an inherent pre-stretch of the aorta tissue. It is generally observed that the ascending aorta moves up-and-down in the longitudinal direction following the systolic-diastolic motion of the heart. Aortic root stretch of 11.6 ± 2.9 mm [15] and twists of 0°, 6°, 12° were applied to mimic this behavior at the aortic root section ∂Ωasc (Figure 1(b)). A spring with stiffness k = 10⁸ N/mm, a dashpot of damping coefficient c = 10⁵ N/(mm/s) [16] were imposed to the spine contact region. The aortic descending ∂Ωdes and branch ∂Ωb sections were kept fixed (Figure 1(b)). Aortic lumen ∂Ωint was submitted to blood pressure to deform the aorta from the zero-pressure geometry to the end-systolic (ES) state.

## 2.4 Statistical analysis of FE modeling factors

<table><caption>**Table 2.** Three levels of each biomechanical factor in synthetic aortic FE modeling.</caption><thead><tr><th>Factors</th><th>Metrics</th><th>-SD</th><th>Mean</th><th>+SD</th><th>#Subjects</th><th>References</th></tr></thead><tbody><tr><td rowspan="5">Aortic geometry</td><td>ATD [mm]</td><td>28.55</td><td>32.83</td><td>37.11</td><td>1520</td><td>[6–8, 12]</td></tr><tr><td>DTD [mm]</td><td>19.7</td><td>22.3</td><td>24.6</td><td>38</td><td>[7, 9]</td></tr><tr><td>Thickness T [mm]</td><td>1.7</td><td>2.2</td><td>2.7</td><td>427</td><td>[9–12]</td></tr><tr><td>Curvature [-]</td><td>A</td><td>B</td><td>C</td><td>-</td><td></td></tr><tr><td>Aneurysm [%]</td><td>0</td><td>20</td><td>40</td><td>-</td><td></td></tr><tr><td>Load</td><td>Systolic BP [mmHg]</td><td>100</td><td>120</td><td>140</td><td>-</td><td></td></tr><tr><td rowspan="2">BCs</td><td>Stretch [mm]</td><td>8.7</td><td>11.6</td><td>14.5</td><td>73</td><td>[15]</td></tr><tr><td>Twist [degrees]</td><td>0</td><td>6</td><td>12</td><td>-</td><td></td></tr><tr><td>Elasticity</td><td>GOH law</td><td>stiff</td><td>mid</td><td>compliant</td><td>-</td><td>[18, 19]</td></tr></tbody></table>

A statistical hypothesis testing was designed to answer two questions: (i) Do different values of factors have significant impact onto the stress distribution? (ii) Which factors have the most impact onto FE modeling? In this test, the independent variables are the 9 factors in Table 2 where we consider three value levels: mean, mean minus and plus SD. The dependent variable is the von Mises stress value distribution on the ascending part of the synthetic aorta model. One-way analysis of variance (one-way ANOVA) was selected for this multi-group testing, as the independent variables are categorical and the dependent variable is numerical. The null hypothesis $H_0$ of this one-way ANOVA is that the mean values of von Mises stress for each level are same $x_1 = x_2 = x_3$. If the F-statistic p-value is lower than critical value $\alpha = 0.05$, then at least one of the levels is statistically significantly different from the others, and the null hypothesis is rejected. On the contrary, subgroup means are fairly close to the overall mean and/or distributions overlap and are hard to distinguish. Thus the first question was addressed through the p values. The importance of the 9 factors
was assessed based on the ranking of the effect size $\eta^2$. $\eta^2$ is the percentage of variance in the dependent variable accounting for independent variables. The higher the values, the more influential the factors.

## 3 Results

<figure><img src="image_3.png" /><figcaption>**Fig. 3.** QQ-plot for the thickness parameter to validate the normal distribution assumption for one-way ANOVA testing.</figcaption></figure>

Value of von Mises stress at the hexahedra element first Gaussian integration points distributed over the ascending aorta were collected. With a meshing seed size of 5 mm, there were around 630 elements and therefore about 630x3 (levels) data per case. As a precondition to one-way ANOVA, the normality assumption of sample distributions was validated by the correlation value $R^2$ of the ANOVA model's residual probability with Gaussian distribution (e.g. quantile-quantile, QQ-plot). For example, normal assumption for the wall thickness was confirmed with residuals closely distributed along the 45 degree line with $R^2 = 0.98$, as shown in Fig. 3. In addition, the homogeneity assumption of variances was checked with Bartlett's test.

The results of the ANOVA in Table. 3 shows that except for twist and DTD, all the other investigated factors show significant differences between their three levels with p-values <0.05. In addition to this p-value which tells if at least one level is significantly different from the others, 'post-hoc' Tukey's honestly significantly difference (HSD) tests were conducted to identify the level that is significantly different from the others. For example in Fig. 4, for twist, large p-values indicate no difference between the three level pairs. In contrast, significant differences exists between all the three pairs for thickness (p-values<0.01).
<table><caption>**Table 3.** Summary table of one-way ANOVA tests, including residuals (Res.), sum of square (SS), degrees of freedom (df), mean square (MS), F-statistic, *p* value and effect size η².</caption><thead><tr><th>Factors</th><th>SS</th><th>df</th><th>MS</th><th>F-statistic</th><th>p-value</th><th>η<sup>2</sup></th></tr></thead><tbody><tr><td>ATD</td><td>0.428</td><td>2.0</td><td>0.214</td><td>489.48</td><td>&lt;0.001</td><td>0.335</td></tr><tr><td>Res.</td><td>0.847</td><td>1939</td><td>4.37e-4</td><td></td><td></td><td></td></tr><tr><td>Pressure</td><td>1.053</td><td>2.0</td><td>0.526</td><td>1310.48</td><td>0.0</td><td>0.576</td></tr><tr><td>Res.</td><td>0.775</td><td>1929</td><td>4.02e-4</td><td></td><td></td><td></td></tr><tr><td>Material</td><td>0.348</td><td>2.0</td><td>0.174</td><td>6732.14</td><td>&lt;0.001</td><td>0.294</td></tr><tr><td>Res.</td><td>0.839</td><td>1929</td><td>4.35e-4</td><td></td><td></td><td></td></tr><tr><td>Stretch</td><td>0.004</td><td>2.0</td><td>0.00208</td><td>5.19</td><td>0.0056</td><td>0.0054</td></tr><tr><td>Res.</td><td>0.773</td><td>1929</td><td>4.01e-4</td><td></td><td></td><td></td></tr><tr><td>Twist</td><td>1.13e-29</td><td>2.0</td><td>5.65e-30</td><td>1.44e-26</td><td>1.0</td><td>1.496e-29</td></tr><tr><td>Res.</td><td>7.54e-01</td><td>1929</td><td>3.91e-4</td><td></td><td></td><td></td></tr><tr><td>Thickness</td><td>2.242</td><td>2.0</td><td>1.12</td><td>2586</td><td>0.0</td><td>0.723</td></tr><tr><td>Res.</td><td>0.858</td><td>1979</td><td>4.34e-4</td><td></td><td></td><td></td></tr><tr><td>DTD</td><td>4.19e-4</td><td>2.0</td><td>2.09e-4</td><td>0.53</td><td>0.589</td><td>5.88e-4</td></tr><tr><td>Res.</td><td>0.711</td><td>1796</td><td>3.96e-4</td><td></td><td></td><td></td></tr><tr><td>Curvature</td><td>0.0187</td><td>2.0</td><td>9.37e-3</td><td>11.69</td><td>&lt;0.001</td><td>0.0107</td></tr><tr><td>Res.</td><td>1.735</td><td>2164</td><td>8.02e-4</td><td></td><td></td><td></td></tr><tr><td>Aneurysm</td><td>0.038</td><td>2.0</td><td>0.0192</td><td>29.9</td><td>&lt;0.001</td><td>0.0301</td></tr><tr><td>Res.</td><td>1.241</td><td>1929</td><td>6.43e-4</td><td></td><td></td><td></td></tr></tbody></table>

These two illustrated cases are representative observations for non-prominent and prominent factors.

The effect size η² ranking shows that wall thickness, blood pressure, ascending diameter (ATD), and material elasticity have dominant impacts onto the aortic stress distributions, with respectively η² = 0.723, 0.576, 0.335, 0.294.

## 4 Discussion

This synthetic study gives attention points for the proper configuration of patient specific FE based aorta modeling. From the conclusions of the above analysis, FE modeling was performed on one patient (43 years old, male) with a bicuspid aortic valve (BAV) and congenital disease represented by a narrowing of the aorta (i.e. Co-arctation). Approval was obtained from the local ethics committee (ID-RCB: 2018-A01742-53). Contrast-enhanced CT imaging with 0.58*0.58*0.45 mm resolution was performed on a dual-energy CT scanner (IQon, Philips, Best, The Netherlands). Aorta lumen was segmented from CT images at end-diastole (ED) with in-house software CreaTools, and meshed with a uniform wall thickness of 1.5 mm. FE simulation was performed within Abaqus/Standard, fixing the descending and branches sections and imposing ED to ES displacements at the aortic root section (estimated from 3D image registration), and ES blood pressure of 120 mmHg. In this simulation, mechanical GOH parameters were estimated from an inverse modeling simulation (c₁₀ = 0.445 MPa, k₁ = 0.206
<figure><img src="image_4.png" /><figcaption>**Fig. 4.** Raincloud plots of sample's histograms together with boxplot. P-values are from Tukey's pair comparison for one-way ANOVA 'post-hoc' testing.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 5.** Spatial distribution of von Mises stress in MPa from a patient-specific aorta FE modeling.</figcaption></figure>
MPa, $k_2 = 5$, $\kappa = 0.091$, $\gamma = 25.1^{\circ}$). Several parameters therefore remain not subject based. The obtained spatial distribution for von Mises stress is shown in Fig. 5. Apart unrealistic stress values at the aortic root section, and focusing on the ascending part, high stress values located on the minor curvature region and range of stress values were comparable to other works [20, 21].

The importance of wall thickness in aorta FE modeling has been recognized, notably in studies on abdominal aorta [22, 23]. Usually, and as in this work, a uniform thickness is assumed, which can be unrealistic, in particular in presence of a pathology. As demonstrated recently by our group [17], the black blood dual CT technique produces new contrast allowing to identify the vascular wall, therefore allowing the integration of patient-specific wall thickness. This is one aspect we would like to investigate. Blood pressure appears also as an essential factor. Relative aortic blood pressure may be assessed from 4D flow Magnetic Resonance Imaging [24], another imaging techniques that can also be used for patient specific FE aorta modeling. The elasticity of the aortic wall material that we studied above is to treat the five GOH parameters as a single set, the global sensitivity study for the influence of each individual parameter on the aortic FE model will be published later in our inverse model. Material parameters reveal also as influential but remain difficult to assess for individuals. This is the object of inverse modeling approaches that aim at estimating patient specific biomechanical wall properties. Several works have shown the challenge to uniquely determine material law parameters from available data. This study can help to set up such modeling and even extended to analyze the sensitivity to all material law parameters.

# 5 Acknowledgements

This study was conducted within the framework of the LABEX PRIMES (ANR-11-LABX-0063) project of the University of Lyon, within the "Investissements d'Avenir" (ANR-11-IDEX-0007) program operated by the French National Research Agency (ANR) and the International Research Project METISLAB from the CNRS. It is supported by China Scholarship Council (CSC), and the QUANTAAS ANR (ANR-18-CE19-0025-01) project.

# References

1. Mokashi, S.A. and Svensson, L.G., 2019. Guidelines for the management of thoracic aortic disease in 2017. General thoracic and cardiovascular surgery, 67(1), pp.59-65.

2. American College of Cardiology Foundation, American Heart Association Task Force on Practice Guidelines, American Association for Thoracic Surgery, American College of Radiology, American Stroke Association, Society of Cardiovascular Anesthesiologists, Society for Cardiovascular Angiography and Interventions, Society of Interventional Radiology, Society of Thoracic Surgeons, Society for Vascular Medicine and North American Society for Cardiovascular Imaging, 2010. 2010 ACCF/AHA/AATS/ACR/ASA/SCA/SCAI/SIR/STS/SVM guidelines for the diagnosis and management of patients with thoracic aortic disease. Journal of the American College of Cardiology, 55(14), pp.e27-e129.
3. Evangelista, A., Isselbacher, E.M., Bossone, E., Gleason, T.G., Eusanio, M.D., Sechtem, U., Ehrlich, M.P., Trimarchi, S., Braverman, A.C., Myrmel, T. and Harris, K.M., 2018. Insights from the international registry of acute aortic dissection: a 20-year experience of collaborative clinical research. Circulation, 137(17), pp.1846-1860.

4. Wang, D.H., Makaroun, M.S., Webster, M.W. and Vorp, D.A., 2002. Effect of intraluminal thrombus on wall stress in patient-specific models of abdominal aortic aneurysm. Journal of vascular surgery, 36(3), pp.598-604.

5. Buhmann, M.D., 2003. Radial basis functions: theory and implementations (Vol. 12). Cambridge university press.

6. Mao, S.S., Ahmadi, N., Shah, B., Beckmann, D., Chen, A., Ngo, L., Flores, F.R., lin Gao, Y. and Budoff, M.J., 2008. Normal thoracic aorta diameter on cardiac computed tomography in healthy asymptomatic adults: impact of age and gender. Academic radiology, 15(7), pp.827-834.

7. Morrison, T.M., Choi, G., Zarins, C.K. and Taylor, C.A., 2009. Circumferential and longitudinal cyclic strain of the human thoracic aorta: age-related changes. Journal of vascular surgery, 49(4), pp.1029-1036.

8. Doyle, B.J., Norman, P.E., Hoskins, P.R., Newby, D.E. and Dweck, M.R., 2018. Wall stress and geometry of the thoracic aorta in patients with aortic valve disease. The Annals of thoracic surgery, 105(4), pp.1077-1085.

9. García-Herrera, C.M., Celentano, D.J., Cruchaga, M.A., Rojo, F.J., Atienza, J.M., Guinea, G.V. and Goicolea, J.M., 2012. Mechanical characterisation of the human thoracic descending aorta: experiments and modelling. Computer Methods in Biomechanics and Biomedical Engineering, 15(2), pp.185-193.

10. Mousavi, S.J. and Avril, S., 2017. Patient-specific stress analyses in the ascending thoracic aorta using a finite-element implementation of the constrained mixture theory. Biomechanics and modeling in mechanobiology, 16(5), pp.1765-1777.

11. Di Giuseppe, M., Alotta, G., Agnese, V., Bellavia, D., Raffa, G.M., Vetri, V., Zingales, M., Pasta, S. and Pilato, M., 2019. Identification of circumferential regional heterogeneity of ascending thoracic aneurysmal aorta by biaxial mechanical testing. Journal of molecular and cellular cardiology, 130, pp.205-215.

12. Cosentino, F., Agnese, V., Raffa, G.M., Gentile, G., Bellavia, D., Zingales, M., Pilato, M. and Pasta, S., 2019. On the role of material properties in ascending thoracic aortic aneurysms. Computers in biology and medicine, 109, pp.70-78.

13. Capellini, K., Vignali, E., Costa, E., Gasparotti, E., Biancolini, M.E., Landini, L., Positano, V. and Celi, S., 2018. Computational fluid dynamic study for aTAA hemodynamics: an integrated image-based and radial basis functions mesh morphing approach. Journal of biomechanical engineering, 140(11).

14. Gasser, T.C., Ogden, R.W. and Holzapfel, G.A., 2006. Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. Journal of the royal society interface, 3(6), pp.15-35.

15. Plonek, T., Berezowski, M., Kurcz, J., Podgorski, P., Sasiadek, M., Rylski, B., Mysiak, A. and Jasinski, M., 2018. The evaluation of the aortic annulus displacement during cardiac cycle using magnetic resonance imaging. BMC cardiovascular disorders, 18(1), pp.1-6.

16. Pagoulatou, S.Z., Ferraro, M., Trachet, B., Bikia, V., Rovas, G., Crowe, L.A., Vallée, J.P., Adamopoulos, D. and Stergiopulos, N., 2021. The effect of the elongation of the proximal aorta on the estimation of the aortic wall distensibility. Biomechanics and Modeling in Mechanobiology, 20(1), pp.107-119.
17. Rotzinger, D.C., Si-Mohamed, S.A., Shapira, N., Douek, P.C., Meuli, R.A. and Boussel, L., 2020. "Dark-blood" dual-energy computed tomography angiography for thoracic aortic wall imaging. European radiology, 30(1), pp.425-431.

18. Farzaneh, S., Trabelsi, O. and Avril, S., 2019. Inverse identification of local stiffness across ascending thoracic aortic aneurysms. Biomechanics and modeling in mechanobiology, 18(1), pp.137-153.

19. Weisbecker, H., Pierce, D.M., Regitnig, P. and Holzapfel, G.A., 2012. Layer-specific damage experiments and modeling of human thoracic and abdominal aortas with non-atherosclerotic intimal thickening. Journal of the mechanical behavior of biomedical materials, 12, pp.93-106.

20. Wisneski, A.D., Mookhoek, A., Chitsaz, S., Hope, M.D., Guccione, J.M., Ge, L. and Tseng, E.E., 2014. Patient-specific finite element analysis of ascending thoracic aortic aneurysm. The Journal of heart valve disease, 23(6), p.765.

21. Nathan, D.P., Xu, C., Plappert, T., Desjardins, B., Gorman III, J.H., Bavaria, J.E., Gorman, R.C., Chandran, K.B. and Jackson, B.M., 2011. Increased ascending aortic wall stress in patients with bicuspid aortic valves. The Annals of thoracic surgery, 92(4), pp.1384-1389.

22. Shang, E.K., Nathan, D.P., Woo, E.Y., Fairman, R.M., Wang, G.J., Gorman, R.C., Gorman III, J.H. and Jackson, B.M., 2015. Local wall thickness in finite element models improves prediction of abdominal aortic aneurysm growth. Journal of vascular surgery, 61(1), pp.217-223.

23. Joldes, G.R., Miller, K., Wittek, A. and Doyle, B., 2016. A simple, effective and clinically applicable method to compute abdominal aortic aneurysm wall stress. Journal of the mechanical behavior of biomedical materials, 58, pp.139-148.

24. Bouaou, K., Bargiotas, I., Dietenbeck, T., Bollache, E., Soulat, G., Craiem, D., Houriez-Gombaud-Saintonge, S., De Cesare, A., Gencer, U., Giron, A. and Redheuil, A., 2019. Analysis of aortic pressure fields from 4D flow MRI in healthy volunteers: Associations with age and left ventricular remodeling. Journal of Magnetic Resonance Imaging, 50(3), pp.982-993.