**3. Regional variations in wall thickness and material properties**

8 Will-be-set-by-IN-TECH

Among the variety of constitutive equations reported in literature, the most significant difference in structural formulation were included by Holzapfel group [62] and by Baek and co-authors [4]. For a more detailed analysis on the effect of this assumption and a comparison between the two constitutive model, interested reader can refer to [17]. It is worth to stress out that, as observed by Zeinali-Davarani and co-authors [87], in parameter estimation, the larger number of parameters for a model provides more flexibility and generally gives better fitting, i.e., decreases the residual error. A common assumption in all previous models is to assume the same fibers distribution and mechanical response throughout the thickness. More recently Schriefl et al. [66] has observed that in the case of the intima layer, due to the higher fibers dispersion, the number of fiber families varying from two to four. However, not all intimas investigated had more than two fiber families while two prominent fibers families were always visible. The number of fiber families equal to two was previously reported by

Finally, it is worth to notice that it is fundamental to define a constitutive model and its material constants over some specific range, from experiments that replicate conditions (physiological or pathological), [17], in order to provide more accurate response. In fact all constitutive formulation are based on specific assumptions and hypotheses. The complexities of the artery wall poses several new conceptual and methodological challenges in the cardiovascular biomechanics. There exist several recent frameworks, in fact, to develop theories of arterial growth and remodeling (G&R) of soft tissues. Interested reader can refer to a more complete and detailed review by Humphrey and Rajagopal [38, 39] and in [41]. However, in this study, we restrict our attention to structural based formulations to emphasize

By using Finite Element analyses, Fillinger et al. [23] showed that peak wall stress is a more reliable parameter than maximum transverse diameter in predicting potential rupture event. These findings appear to be supported by the results obtained by Venkatasubramaniam et al. [79], who indicated that the location of the maximum wall stress correlates well with the site of rupture and, additionally, by the observation that AAA formation is accompanied by an increase in wall stress [55, 83], and a decrease in wall strength [84]. Simulation on 3D patient-specific models are aimed to analyze the distribution of the wall stress to estimate the rupture risk during the evolution of the pathology [23], the effect of the thrombus [29, 85] or calcification [42, 45, 68] on the peak stress. Integration of geometry data with solid modelling is used for estimation of vessel wall distension, strain and stress patterns. Studies, to date, have typically used 3D geometries usually obtained from computer tomography (CT) [52] or MRI [7] scans or have used simplified morphologies [17, 62]. Figure 5 reports as example the phases from a CT reconstruction. However, both approaches present some limitations. In particular, it is worth pointing out that 3D simulations are not fully patient-specific models but only based on 3D patient-specific geometries while the material properties are assumed as mean population values due to the difficulty of assessing *in-vivo* material properties. Consequently, to date, no fully patient-specific model has been performed. Additionally, due to the complexity of the structure and the high computational cost required by patient-specific models, sensitivity analyses have not been performed on 3D real geometries, and only univariate investigations have been performed on idealized shapes, to estimate the influence

Haskett et al. [31] by analyzing 207 aortic samples.

of a single parameter on the whole stress map [63].

their particular effects.

**2.4. Geometrical model**

As reported in previous section, starting from the observation that most AAAs are characterized by a complex not axisymmetric geometries a growing amount of literature has been published on the influence of the geometrical features. However one limitation in all the studies published so far is a constant wall thickness and homogeneous material assumed in the FE models.

*Wall thickness.* While the segmentation of the arterial lumen is a well established technique and has been performed with different modalities in living subjects, the segmentation of the wall and its connective components is not a feasible process due to the low contrast between the wall and the surrounding tissues. The conventional imaging techniques, in fact, do not provide sufficient spatial resolution to assess the wall thickness measurement and variant *in-vivo*.

During the AAA formation the artery wall is subjected to the remodelling process [77] and, as a consequence, the ratio between AA and AAA wall thickness changes. Di Martino et al. [18] noted a significant difference in wall thickness between ruptured and elective AAAs (3.6±0.3 mm vs 2.5±0.1 mm, respectively). By comparing the wall thickness between healthy and pathological samples, Vande Geest et al. [77] reported that the mean measured thickness values were 1.49±0.11 and 1.32±0.08 mm for the AA and AAA specimens, respectively. In all these studies, samples were measured only in the anterior area and consequently no information regarding regional variation between ventral and dorsal was reported. Thubrikar et al. [70] obtained five whole unruptured AAA specimens during surgical resection. Raghavan et al. [54] performed similar measures on three unruptured and one ruptured AAA, harvested as a whole during necropsy. More recently Celi et al. [10] performed measurements on 12 harvested unrupture ascending segments. In Table 1, the main results of these experimental measures are reported for both anterior and posterior region (mean±sd).

It is worth to notice that the thickness distribution seems to be opposite of that in the normal abdominal aorta where the wall is thicker than the posterior wall in 64% of cases [74].

From the computational point of view, in literature only few authors have investigated the effect on wall thickness reduction. Scotti et al. [67] used a non uniform wall thickness in an

#### 10 Will-be-set-by-IN-TECH 12 Aneurysm


**Table 1.** Wall thickness measurements, reported in literature, categorized by circumferential location as anterior and posterior

idealized isotropic model to performed FSI simulations. Their results show that the models with a non uniform wall thickness have a maximum wall stress nearly four times that of a uniform one. Starting from experimental measurement on 12 human harvested ascending aortic samples, Celi et al. [10] developed structural 3D models of ascending AAA by including wall thickness regional variation between dorsal and ventral areas.

*Material properties.* As far as the material properties, to date, different behavior has funded between healthy (HAA) and pathological samples. However, due to the lack of sufficient biaxial data, a full characterization in regional variations are not provided (in circumferential direction in particular), and mechanical tests have been performed mainly in the ventral area where the bulge was formed. As well as the material properties change during the AAA progression, also the wall strength value changes. This aspect plays a fundamental role in the rupture phenomenon. In fact, the concept is that AAA rupture follows the basic principles of material failure; i.e., an aneurysm ruptures when the mural stresses or deformation meets an appropriate failure criterion. In the filed of the classical mechanics, this concept is defined by means of the potential rupture risk (RPI) parameter and quantify as the ratio of local wall stress to local wall strength:

$$RPI = \frac{local\,\,stress}{local\,\,strength} \tag{15}$$

In the same manner the safety factor (SF) can be used as the inverse of the RPI.

Thubrikar et al. [70] performed uniaxial tensile tests in both longitudinal and circumferential direction, on samples from five aneurysms. To study the regional variation they obtained samples from anterior, lateral (without distinction between left and right side) and posterior regions. In this study, however, authors did not perform tests until failure and they recorded the yield stress to define the initial point of a permanent damage. Thubrikar et al. observed that in both directions, the yield stress was greater in the lateral region with respect to the anterior and the posterior region, Figure 6(a). Experimental values regarding ultimate stress were reported by Raghavan et al. [54], Figure 6(b). They cut multiple longitudinally oriented rectangular specimen strips at various locations from three unruptured AAA and one ruptured AAA for a total of 48 strips. Samples were tested uniaxially until failure. They observed that the failure tension (ultimate) of specimen strips varied regionally from 55 kPa (near the rupture site) to 423 kPa at the undilated neck. However they report that there was no perceptible pattern in failure properties along the circumference.

Using multiple linear regression, Vande Geest et al. [78] proposed a mathematical model to estimate the wall strength by including several mixed parameters such as the gender, the presence of the intraluminal thrombus (ILT) and the family history. The final statistical model for local Cauchy wall strength (Eq. 16, dimension in kPa) was given by:

$$
\sigma\_{\mu} = 719 - 379 \left( \text{ILT}^{\frac{1}{2}} - 0.81 \right) - 156 \left( D\_{\text{NORM}} - 2.46 \right) - 213 \,\text{HIST} + 193 \,\text{SEX} \tag{16}
$$

**Figure 6.** Yield stress in circumferential (upper side) and longitudinal (lower side) direction from [70] (a). Failure stress by circumferential location as anterior (0◦), left (90◦), posterior (180◦) and right (270◦) regions from [54] (b).

where *ILT* <sup>1</sup> <sup>2</sup> is the square root of the ILT thickness whose units, *DNORM* is a dimensionless parameter for local normalized diameter, HIST is a dimensionless binary variable (1/2 for positive family history, -1/2 for no family history), and SEX is a dimensionless binary variable for gender (1/2 for males, -1/2 for females). Table 2 depicts some examples of the effect of the coefficients of Eq. 16 by varing the gender and the ILT thickness.


**Table 2.** Effect of the gender (case 1 vs. case 2) and of ILT thickness (case 2 vs. case 3) on the wall strength by using Eq. 16.

As we can notice the presence of the ILT decreases significantly the *σ<sup>u</sup>* of about 63%. However, it is worth to notice that Eq. 16 describes local variation of the wall strength only in terms of normalized diameter and ILT thickness. Indeed Fillinger et al. [24] report that aneurysms likely rupture at stresses of 450 kPa or lower.

#### **4. Finite element analyses**

10 Will-be-set-by-IN-TECH

N. of samples District *Thkanterior* (mm) *Thkposterior* (mm) Ref AAA 2.09±0.51 2.73±0.46 [70] AAA 2.25±0.37 2.34±0.48 [54] aTAA 1.63±0.48 2.18±0.35 [10]

**Table 1.** Wall thickness measurements, reported in literature, categorized by circumferential location as

idealized isotropic model to performed FSI simulations. Their results show that the models with a non uniform wall thickness have a maximum wall stress nearly four times that of a uniform one. Starting from experimental measurement on 12 human harvested ascending aortic samples, Celi et al. [10] developed structural 3D models of ascending AAA by including

*Material properties.* As far as the material properties, to date, different behavior has funded between healthy (HAA) and pathological samples. However, due to the lack of sufficient biaxial data, a full characterization in regional variations are not provided (in circumferential direction in particular), and mechanical tests have been performed mainly in the ventral area where the bulge was formed. As well as the material properties change during the AAA progression, also the wall strength value changes. This aspect plays a fundamental role in the rupture phenomenon. In fact, the concept is that AAA rupture follows the basic principles of material failure; i.e., an aneurysm ruptures when the mural stresses or deformation meets an appropriate failure criterion. In the filed of the classical mechanics, this concept is defined by means of the potential rupture risk (RPI) parameter and quantify as the ratio of local wall

*RPI* <sup>=</sup> *local stress*

Thubrikar et al. [70] performed uniaxial tensile tests in both longitudinal and circumferential direction, on samples from five aneurysms. To study the regional variation they obtained samples from anterior, lateral (without distinction between left and right side) and posterior regions. In this study, however, authors did not perform tests until failure and they recorded the yield stress to define the initial point of a permanent damage. Thubrikar et al. observed that in both directions, the yield stress was greater in the lateral region with respect to the anterior and the posterior region, Figure 6(a). Experimental values regarding ultimate stress were reported by Raghavan et al. [54], Figure 6(b). They cut multiple longitudinally oriented rectangular specimen strips at various locations from three unruptured AAA and one ruptured AAA for a total of 48 strips. Samples were tested uniaxially until failure. They observed that the failure tension (ultimate) of specimen strips varied regionally from 55 kPa (near the rupture site) to 423 kPa at the undilated neck. However they report that there was

Using multiple linear regression, Vande Geest et al. [78] proposed a mathematical model to estimate the wall strength by including several mixed parameters such as the gender, the presence of the intraluminal thrombus (ILT) and the family history. The final statistical model

In the same manner the safety factor (SF) can be used as the inverse of the RPI.

no perceptible pattern in failure properties along the circumference.

for local Cauchy wall strength (Eq. 16, dimension in kPa) was given by:

<sup>2</sup> − 0.81

*local strength* (15)

− 156 (*DNORM* − 2.46) − 213 *HIST* + 193 *SEX* (16)

wall thickness regional variation between dorsal and ventral areas.

anterior and posterior

stress to local wall strength:

*σ<sup>u</sup>* = 719 − 379

 *ILT* <sup>1</sup> In order to get some indications on how regional variation of wall thickness and material properties affect the wall stress, two different FE models were developed. The first case describes a simplified model where an isotropic SEF has been adopted [56]. The tissue was described as homogeneous and consequently no distinction between healthy and pathological tissues was modeled. The second model introduces anisotropy and material regional variation to obtain more realistic simulations. For this last model, three different regions were considered and characterized with specific anisotropic SEFs: healthy material for the necks (HAA), pathological for the anterior bulge (AAA) and pathological for the posterior (AHA). Due to no data were available for the posterior region, a simple data manipulation was applied to define the new AHA pathological dataset starting from AAA experimental data as previously described in [9]. Figure 7 depicts the anisotropic dataset for the three rappresentative materials. As we can observe, the AHA dataset is able to reproduce an intermediate mechanical behavior between full healthy and pathological material.

**Figure 7.** Example of HAA and AAA material models and virtual dataset (AHA) adopted for the transition region.

Both FE models are characterized by a wall thickness reduction in longitudinal and circumferential direction. For the necks, a wall thickness value equal to 1.8 mm was used while reduction of 30% and 50% was applied for the ventral area and of 20% for the dorsal one. Aneurysm shapes were defined as idealized 3D geometries with circular cross sections. Meridian lines describing the interior surface were based on a SZ-shaped function reported in Equation 17:

$$z(r) = \begin{cases} \begin{array}{cc} \mathcal{R}\_0 & 0 \le z \le a \\ 2(\mathcal{R}\_{AAA} - \mathcal{R}\_0) \left(\frac{z-a}{b-a}\right)^2 + \mathcal{R}\_0 & a < z \le \frac{a+b}{2} \\ \left(\mathcal{R}\_{AAA} - \mathcal{R}\_0\right) - 2\left(\frac{z-b}{b-a}\right)^2 + \mathcal{R}\_0 & \frac{a+b}{2} < z \le b \\ \mathcal{R}\_{AAA} & b < z \le \frac{L}{2} \end{array} \tag{17}$$

where the parameter *a* and *b* locate the extremes of the slope portion of the curve. Due to symmetry only one-half of the profile is reported. Geometrical profiles are reported in Fig. 8.

**Figure 8.** Lateral (a) and frontal (b) view of an asymmetrical aneurysmatic shape.

where where *Ra* is the radius of the healthy artery, *RAAA*|*max* is the maximum radius of the aneurysm in the ventral region, *RAAA*|*min* is the maximum radius of the aneurysm in the dorsal site. *L* (equal to 80 mm) defines the length of the abdominal vessel and *LAAA* is the length of the aneurysmatic area. Figure 9(a) depicts an example of meshed asymmetric aneurysm with indication of the three anisotropic materials (in accordance with Fig. 7, while in Figure 9(b)) transversal cross section at the maximum diameter is reported with indication of circumferential wall thickness reduction.

**Figure 9.** Example of asymmetric aneurysm and assignment of local material properties (a) and transversal cross section (b) with a wall thickness reduction of 50% and 20% in the ventral and dorsal region respectively.

For the constitutive equations for both healthy and pathological tissue, an invariant-based anisotropic polynomial SEF was chosen, as reported in Eq. 18. The material coefficients were calculated by using a specific weighted non-linear regression procedure implemented in Matlab and based on the Levenberg-Marquardt algorithm.

$$\mathcal{W}\_{\text{isoch}} = \sum\_{i=1}^{3} a\_i \left(\overline{I}\_1 - 3\right)^i + 2 \sum\_{j=2}^{6} b\_j \left(\overline{I}\_4 - 1\right)^j \tag{18}$$

Aneurysms were inflated applying a uniform inner pressure of 16 kPa, corresponding to the nominal value of peak systolic pressure. The ends of the vessels were left free to move in the radial direction.

#### **4.1. Sensitivity analysis**

12 Will-be-set-by-IN-TECH

rappresentative materials. As we can observe, the AHA dataset is able to reproduce an

intermediate mechanical behavior between full healthy and pathological material.

**Figure 7.** Example of HAA and AAA material models and virtual dataset (AHA) adopted for the

Both FE models are characterized by a wall thickness reduction in longitudinal and circumferential direction. For the necks, a wall thickness value equal to 1.8 mm was used while reduction of 30% and 50% was applied for the ventral area and of 20% for the dorsal one. Aneurysm shapes were defined as idealized 3D geometries with circular cross sections. Meridian lines describing the interior surface were based on a SZ-shaped function reported in

> � *z*−*a b*−*a* �2

where the parameter *a* and *b* locate the extremes of the slope portion of the curve. Due to symmetry only one-half of the profile is reported. Geometrical profiles are reported in Fig. 8.

where where *Ra* is the radius of the healthy artery, *RAAA*|*max* is the maximum radius of the aneurysm in the ventral region, *RAAA*|*min* is the maximum radius of the aneurysm in the dorsal site. *L* (equal to 80 mm) defines the length of the abdominal vessel and *LAAA* is the length of the aneurysmatic area. Figure 9(a) depicts an example of meshed asymmetric aneurysm with indication of the three anisotropic materials (in accordance with Fig. 7, while

� *<sup>z</sup>*−*<sup>b</sup> b*−*a* �2

*R*<sup>0</sup> 0 ≤ *z* ≤ *a*

*RAAA <sup>b</sup> <sup>&</sup>lt; <sup>z</sup>* <sup>≤</sup> *<sup>L</sup>*

+ *R*<sup>0</sup> *<sup>a</sup>*+*<sup>b</sup>*

<sup>+</sup> *<sup>R</sup>*<sup>0</sup> *<sup>a</sup> <sup>&</sup>lt; <sup>z</sup>* <sup>≤</sup> *<sup>a</sup>*+*<sup>b</sup>*

2

(17)

2

<sup>2</sup> *< z* ≤ *b*

transition region.

Equation 17:

*z*(*r*) =

⎧ ⎪⎪⎪⎪⎪⎨

2(*RAAA* − *R*0)

(*RAAA* − *R*0) − 2

**Figure 8.** Lateral (a) and frontal (b) view of an asymmetrical aneurysmatic shape.

⎪⎪⎪⎪⎪⎩

To evaluate the sensitivity of the maximum stress state with respect to geometrical features, sensitivity and multivariate analyzes were also carried out by means of ANSYS Probabilistic Design Toolbox. This type of investigation presents two main advantages: the spread of the response of the output variables can be found, and it is possible to define the parameters that mainly influence the response of the system, for further details see [3, 46, 58]. Correlation coefficients are used as a measure of the strength of the relationship between input parameter and output measure.

In this study, analyzes were performed using the Monte Carlo method, in which the correlations between input and output variables are defined in a completely statistical way. In order to reduce the number of samples, the Latin Hypercube technique, instead of a direct sampling, was adopted. The effectiveness of these procedures was previously tested by Celi [8] and Celi et al. [11]. In order to study the effect of the AAA geometry on the distribution of the wall stresses, we introduced three dimensionless geometrical parameters:

$$F\_{\rm R} = \frac{R\_{\rm AAA}}{R\_{\rm d}}; \quad F\_{\rm L} = \frac{L\_{\rm AAA}}{R\_{\rm AAA}}; \quad F\_{\rm sym} = \frac{R\_{\rm AAA|min} - R\_{\rm d}}{R\_{\rm AAA|max} - R\_{\rm d}}; \quad F\_{\rm tth} = \frac{t \hbar k\_0 - t \hbar k\_{\rm min}}{t \hbar k\_0} 100 \tag{19}$$

The parameter *FR* defines the ratio between the maximum AAA radius and the healthy arterial radius, *FL* defines the ratio between the length of the aneurysm and the maximum AAA radius, while *Fsym* ∈ [0,1] is a measure of the aneurysmal eccentricity. The extreme cases *Fsym*=1 and *Fsym*=0 define the symmetrical situation and the most asymmetric geometry, respectively. Table 3 summarizes the FE analyses performed in this study.


**Table 3.** Range of the geometric parameters defining the aneurysmal shape.


**Table 4.** Scheme of the simulations performed in this study.

#### **4.2. Results**

*Fitting procedure.* The results of the best fit procedures for the anisotropic SEFs are reported in Figure 10. For all models very good results were obtained and, as a metric of the goodness of fit, the root mean square of the fitting error were computed: *R*2=0.992, *R*2=0.992 and *R*2=0.983 from healthy to pathological tissue, respectively.

Table 5 lists the parameters values for the HAA, AHA and AAA models. The angle *θ* represents the embedded fiber orientations, as illustrated in Fig. 4(a).


**Table 5.** Coefficients for the models for the three SEFs. Vales in kPa

Thus, the new models adequately reproduce the experimental data sets for HAA, AHA and AAA tissues using only one SEF with six parameters per model. Figure 10(c-f-i) points out the changes in the anisotropic effect by increasing the pathological response of a tissue from healthy to aneurysmatic state.

*FE simulations.* Distributions of the circumferential stresses for the isotropic and anisotropic models for three different values of *Fthk* are shown in Fig. 11. It can be observed that for all models and geometries, the maximum stress is localized in the interior wall surface and in the proximity of the minor radius of curvature, due to the geometrical effect of the curvature itself, and that there exists a stress gradient through the aneurysm wall thickness. As expected, the isotropic model underestimates the peak stress value of about 40%, 38%, 42% with respect to the 2FF homogeneous anisotropic model, Fig. 11(d-e-f), of about 44%, 40%, 43% with respect to the 2FF heterogeneous model, Fig. 11(g-h-i). By focusing our attention on the anisotropic

14 Will-be-set-by-IN-TECH

The parameter *FR* defines the ratio between the maximum AAA radius and the healthy arterial radius, *FL* defines the ratio between the length of the aneurysm and the maximum AAA radius, while *Fsym* ∈ [0,1] is a measure of the aneurysmal eccentricity. The extreme cases *Fsym*=1 and *Fsym*=0 define the symmetrical situation and the most asymmetric geometry,

> Parameter Definition Distribution Range F*<sup>R</sup>* Dilatation ratio uniform 1.5-3 F*<sup>L</sup>* Shape factor uniform 1.5-4 F*sym* Symmetry factor uniform 0-1 F*thk* Thickness ratio uniform 0-50

Model Thk N. of materials Type of Material Type of Analysis Iso1 variable 1 (AAA) Iso Det./Prob. Aniso1 variable 1 (AAA) Aniso Det. Aniso3 variable 3 (AAA, AHA, HAA) Aniso Det.

*Fitting procedure.* The results of the best fit procedures for the anisotropic SEFs are reported in Figure 10. For all models very good results were obtained and, as a metric of the goodness of fit, the root mean square of the fitting error were computed: *R*2=0.992, *R*2=0.992 and *R*2=0.983

Table 5 lists the parameters values for the HAA, AHA and AAA models. The angle *θ*

Model *a*<sup>1</sup> *a*<sup>3</sup> *a*<sup>3</sup> *b*<sup>4</sup> *b*<sup>5</sup> *b*<sup>6</sup> *θ R*<sup>2</sup> HAA 2.503 1.641 896.714 3.467 1.564 102.677 45.510 0.992 AHA 12.194 40.869 2166.994 2.483 41.883 64.650 52.199 0.992 AAA 1.5 0.1 4966.781 54.381 3856.291 4997.367 45.989 0.983

Thus, the new models adequately reproduce the experimental data sets for HAA, AHA and AAA tissues using only one SEF with six parameters per model. Figure 10(c-f-i) points out the changes in the anisotropic effect by increasing the pathological response of a tissue from

*FE simulations.* Distributions of the circumferential stresses for the isotropic and anisotropic models for three different values of *Fthk* are shown in Fig. 11. It can be observed that for all models and geometries, the maximum stress is localized in the interior wall surface and in the proximity of the minor radius of curvature, due to the geometrical effect of the curvature itself, and that there exists a stress gradient through the aneurysm wall thickness. As expected, the isotropic model underestimates the peak stress value of about 40%, 38%, 42% with respect to the 2FF homogeneous anisotropic model, Fig. 11(d-e-f), of about 44%, 40%, 43% with respect to the 2FF heterogeneous model, Fig. 11(g-h-i). By focusing our attention on the anisotropic

respectively. Table 3 summarizes the FE analyses performed in this study.

**Table 3.** Range of the geometric parameters defining the aneurysmal shape.

represents the embedded fiber orientations, as illustrated in Fig. 4(a).

**Table 5.** Coefficients for the models for the three SEFs. Vales in kPa

**Table 4.** Scheme of the simulations performed in this study.

from healthy to pathological tissue, respectively.

healthy to aneurysmatic state.

**4.2. Results**

16 Aneurysm Biomechanics and FE Modelling of Aneurysm: Review and Advances in Computational Models <sup>15</sup> Biomechanics and FE Modelling of Aneurysm: Review and Advances in Computational Models 17

**Figure 10.** Representative stress-stretch data and fitting results for an HAA (a-b), AHA (d-e) and AAA (g-h) in circumferential and axial direction, see Fig. 7. The isolines of the SEFs for all models are reported in (c), (f) and (i).

models, as we can observe, the 2FF heterogeneous models present the highest maximum stress values due to the presence of the additional two material (HAA and AHA). The effect of these materials is to increase deformation in both radial and axial directions of the ventral and dorsal regions by changing, as a consequence, the local curvature.

As far as the stress gradient, Figure 12 depicts the transmural circumferential stress for model Aniso1 and Aniso3 for the two extreme cases of constant wall thickness and of maximum reduction. In the bulge area, Fig. 12(a), models Aniso1 and Aniso3 present the same stress gradient behavior due to the use of the same material (AAA). The effect of the wall thickness reduction is an increase of about 30% in the bulge region where the maximum diameter is reached. In the dorsal region, Fig. 12(b) the wall thickness reduction increases the maximum stress of about 8% for both models, while, the combined effect of the wall thickness reduction and different material produces an increase of about 21%. As far as the multivariate analysis, under the assumption of a constant wall thickness, the peak stress, is primarily affected by *FR*, while if the wall thickness reduction in the bulge (*Fthk*) is considered, *Fthk* plays the main role.

16 Will-be-set-by-IN-TECH 18 Aneurysm

**Figure 11.** Contour plots of the circumferential stress for model Iso1 (a-c), Aniso1 (d-f) and Aniso3 (g-i) with progressive wall thickness reduction. Constant wall *thk* (a,d,g), reduction of 30% and 20% (b,e,h) and reduction of 50% and 20% (c,f,i) in ventral and dorsal area. Stress in kPa.

**Figure 12.** Stress gradient (kPa) in the ventral (a) and dorsal (b) area for the anisotropic models by considering constant wall thickness and the reduction of 50% and 20% in ventral and dorsal area.

Moreover, the same stress value is obtained both in fusiform aneurysm with critical dimension (*Fthk* 2.5) and in saccular with *Fthk* 2, see Figure 13(a). Including the wall thickness variation in the multivariate analyses points out the importance of these parameters. Figure 13(b) depicts the correlation coefficients (C.C.) for each input variable: the stresses increase as the diameter increases, but decrease as the thickness increases. Additionally, shorter models had higher wall stress.

**Figure 13.** Maximum circumferential stresses as a function of *FR* and *FL* parameters (a) and correlation coefficients (C.C.) for each geometrical parameters (b).

#### **5. Discussion and conclusion**

16 Will-be-set-by-IN-TECH

**Figure 11.** Contour plots of the circumferential stress for model Iso1 (a-c), Aniso1 (d-f) and Aniso3 (g-i) with progressive wall thickness reduction. Constant wall *thk* (a,d,g), reduction of 30% and 20% (b,e,h)

**Figure 12.** Stress gradient (kPa) in the ventral (a) and dorsal (b) area for the anisotropic models by considering constant wall thickness and the reduction of 50% and 20% in ventral and dorsal area.

and reduction of 50% and 20% (c,f,i) in ventral and dorsal area. Stress in kPa.

In the first part of this work a literature survey on AAA biomechanics is reported by including several aspects from experimental test to constitutive model formulations. In the second part our FE models are presented, aimed at simulating and enhancing the computational study of the aneurismatic pathology. With respect to previous works, a more realistic type of AAA, even if idealized, was considered defined by means of regional variation of wall thickness and material properties. Notwithstanding many important findings from prior finite element stress analyses, all models are limited by the assumption of material homogeneity and constant wall thickness, e.g. [17, 55, 59, 62, 63]. Starting from the principle that intramural cells seek to remodel the arterial wall in order to maintain and to restore stresses towards homeostatic values, the material and geometrical properties must vary from region to region. This concept is the base of the remodeling phenomena as suggest by Humphrey [37]. In order to include material regional variation, in this work, we have introduced a simplified form of the stored energy function (Equation 18), motivated directly by microstructural information on two collagen fiber families [66]. This constitutive form fits well (e.g. mean *R*<sup>2</sup> of about 0.9) healthy and pathological available human biaxial data without complexity. Our SEF, in fact, is able to cope the progressive decrease in the elastin contribute (associated with the isotropic contribution attributed to an elastin-dominated amorphous matrix [34]) and increase in the anisotropic effects (associated with the predominant families of collagen fibers). The decrease of the elastin from heathy to pathological state as well as the increase of anisotropy are reported in Figure 14 where the isotropic (*Wiso*) and anisotropic (*Waniso*) components of the SEF for HAA and AAA models are reported.

The present nonlinear regression focused not only on the estimating global best-fit values of model parameters suitable for performing stress analyses, but also on their effect in terms of energy behavior and changes from heathy to pathological tissue (Figure 10(c-i)). In

**Figure 14.** Isotopic (a) and anisotropic (b) component of the SEF reported in Eq. 18 for healthy (HAA) and pathological (AAA) tissue.

parallel to the marked decrease in the isotropic stored energy for AAA tissues (as describe above), we can observed an increase of stiffness capturing the observed biaxial reduction in extensibility/distensibility in particular in the circumferential direction (see Fig. 10(g)). As mention in Sec. 1, in current clinical practise, the aortic diameter is the main feature that is used to predict the risk of rupture. The more reliable quantification of the rupture risk is provided by the RPI parameter of Eq. 15 (and similar), however, which stress (principal stress, circumferential stress or von Mises stress) and strength is still matter of controversy. Gaining an understanding of the mechanical properties of the AAA tissue therefore is of clinical significance. Due to the difficult to reliably predict abdominal aortic aneurysm expansion and rupture in individuals several clinical trials have been performed [25, 72]. At the same time, from the computational point of view, literature systematically reports the evidence to support the role of patient-specific biomechanical profiles in the management of patients with AAA both from imaging and FE approach [1, 13, 81]. In order to accurately predict the risk of rupture of AAA, is necessary to predict the AAA wall strength distribution and the material properties non-invasively. With regard to our work, our specific FE simulations (both deterministic and probabilistic), reveal the importance to define a more realistic geometrical shape by including also wall thickness regional variation. Several previous studies were devoted to the definition of the geometrical parameters that mainly influence the wall stress (Vorp et al. [83] and more recently Rodríguez et al. [62] found that wall stress is substantially increased by an asymmetric bulge in AAAs, just to cite but a few), but, to the best of our knowledge, this is the first structural study in which also the wall thickness is considered as variable. Figure 15 depicts results from two deterministic simulations extracted by the multivariate analysis: aneurysm with a large diameter and constant wall thickness (a) and FE model small diameter and a wall thickness reduction of 50% in the ventral area. The stress contour plot points out how the wall thickness reduction influence the maximum stress value and its localization.

To conclude, there is, therefore, a pressing need to include patient specific regional variations to identify regions within AAAs that have the highest ratio of stress to strength. Future studies on patient-specific geometries of AAAs should consider the actual wall thickness. Moreover, the understanding the mechanical properties of the AAA wall will enhance our ability to design implants that can stay in place and/or protect the aneurysm wall from blood pressure.

**Figure 15.** Stress contour plot of aneurysm with a large diameter and constant wall thickness (a) and a FE model small diameter and a wall thickness reduction of 50% in the ventral area.
