**1. Introduction**

Abdominal aortic aneurysm (AAA) disease is the 18th most common cause of death in all individuals and the 15th most common in individuals aged over 65 [48]. Clinical treatment for this disorder can consist of open surgical repair or, more recently, of minimally invasive endovascular repair procedures [71]. However, both clinical treatments present significant risks and, consequently, require specific patient selection. Given the risks of current repair techniques, during the course of an aneurysm it is important to determine when the risk of rupture justifies the risk of repair. In this scenario, how to determine the rupture risk of an aneurysm is still an open question. Currently, the trend in determining the severity of an AAA is to use the maximum diameter criterion. Unfortunately, this criterion is only a general rule and not a reliable indicator since small aneurysms can also rupture, as reported in autopsy studies, while many aneurysms can become very large without rupturing [16]. The maximum diameter criterion, in fact, is based on the law of Laplace that establish a linear relationship between diameter and wall stress. However, the law of Laplace is simply based on cylindrical geometries, where only one radius of curvature is involved, whereas aneurysms are complex structures, and therefore the law fails to predict realistic wall stresses. From a biomechanical point of view, rupture events occur when acting wall stresses exceed the tensile strength of the degenerated aortic abdominal (AA) wall. Biomechanics relates the function of a physiological system to its structure and its objective is to deduce the function of a system from its geometry, material properties and boundary conditions based on the balance laws of mechanics (e.g. conservation of mass, momentum and energy). Consequently, from a more general and extensive perspective, the stress state in a body is determined by several factors such as geometry, material properties, load and boundary conditions. In order to understand the capability to estimate the potential rupture risk, it is fundamental to capture the mechanical response of the aortic tissue and its changes during aneurysmal formation. In fact, while, to date, the precise pathogenesis of AAA is poorly understood, it is well known that this change significantly impact on the structure of the aortic wall and on its mechanical behavior.

©2012 Celi and Berti, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 4 Aneurysm

This chapter will review the state of literature on the mechanical properties and modelling of AAA tissue and will present advanced computational models. The first part (Sec. 2) includes a description of the mechanical test currently used (2.1), the aortic mechanical properties (2.2) and a review of the literature on material constitutive equations (2.3) and geometrical models (2.4). To stress out the morphological complexity of the aortic segment, in Sec 3 the regional variations of material properties and wall thickness reported in literature form experimental investigations are reported. The second part (Sec. 4) describes our original contribution with a description of our Finite Element (FE) models and our probabilistic approach implemented into FE simulations to perform sensitivity analysis (Sec. 4.1). The main results are reported in Sec. 4.2 and discussed in detail in Sec. 5.

#### **2. Review**

In order to understand the biomechanical issues in the etiology and treatment of abdominal aortic aneurysms, it is important to understand the structures of the aortic wall and how they affect the mechanical response. Biological tissues are subject to the same balance laws of conservation of mass, momentum and energy of the classical engineering material. What distinguish biological tissues from materials of the field of classical engineering mechanics is their unique structure. Soft biological tissues, in fact, have a very complex structure that can be regarded as either *active* and *passive*. The active components arise from the activation of the smooth muscle cells while the passive response is governed primarily by the elastin and collagen fibres [15]. The distribution and the arrangement of the collagen fibres, in particular, have a significant influence on the mechanical properties due to they attribute anisotropic properties [49] to the tissue. Different studies have shown that this structural arrangement is very complex and varies according to the aortic segment (thoracic or abdominal) [20]. As well as being anisotropic, the material response of soft biological tissue is also highly non-linear.

#### **2.1. Experimental test**

To determine mechanical properties of AAA, studies have used both *in-vivo* "tests" and *ex-vivo*/*in-vitro* testing. As reported by Raghavan and da Silva [53], both of them offer advantages and disadvantages. In particular, in the first case the main difficult is to accurately determine the true force and the displacement distribution ascertaining stress-free configuration of the biological entity. On the other side, isolating samples may introduce as yet unknown changes to their behavior affecting the results of such tests. In vivo measurement are often performed by using imaging modality. By using ultrasound phase-locked echo-tracking, Lanne et al. [43] reported that the pressure-strain elastic modulus (*Ep*), Eq. 1, was higher on average and more widely dispersed in aneurysmal abdominal aorta compared to the non-aneurysmal aorta group. The *Ep* modulus was calculated based on the diameter (*Ds*,*Dd*) and pressure (*Ps*, *Pd*) at the systolic and diastolic values as follow:

$$E\_p = D\_p \frac{P\_s - P\_d}{D\_s - D\_d} \tag{1}$$

Using similar consideration, MacSweeney et al. [44] founded that *Ep* was higher in aneurysmal abdominal aorta compared to controls.

More recently, van't Veer et al. [75] estimated the compliance and distensibility of the AAA by means of simultaneous instantaneous pressure and volume measurements obtained with the magnetic resonance imaging (MRI). By using time resolved ECG-gated CT imaging data from 67 patients, Ganten et al. [27] found that the compliance of AAA did not differ between small and large lesions. In 2011 Molacek and co-authors [47] did not find any correlation between aneurysm diameter and distensibility of AAA wall and of normal aorta. However it is worth to notice that all these studies do not provide intrinsic mechanical properties of the tissue but more general extrinsic AAA behavior. As far as the *ex-vivo* testing, there are several types of mechanical tests that can be carried out on materials to obtain information on their mechanical behavior. Such tests include simple tension test, biaxial tension, conducted on thin samples of material, and extension/inflation test of thin-walled tubes.

2 Will-be-set-by-IN-TECH

This chapter will review the state of literature on the mechanical properties and modelling of AAA tissue and will present advanced computational models. The first part (Sec. 2) includes a description of the mechanical test currently used (2.1), the aortic mechanical properties (2.2) and a review of the literature on material constitutive equations (2.3) and geometrical models (2.4). To stress out the morphological complexity of the aortic segment, in Sec 3 the regional variations of material properties and wall thickness reported in literature form experimental investigations are reported. The second part (Sec. 4) describes our original contribution with a description of our Finite Element (FE) models and our probabilistic approach implemented into FE simulations to perform sensitivity analysis (Sec. 4.1). The main results are reported in

In order to understand the biomechanical issues in the etiology and treatment of abdominal aortic aneurysms, it is important to understand the structures of the aortic wall and how they affect the mechanical response. Biological tissues are subject to the same balance laws of conservation of mass, momentum and energy of the classical engineering material. What distinguish biological tissues from materials of the field of classical engineering mechanics is their unique structure. Soft biological tissues, in fact, have a very complex structure that can be regarded as either *active* and *passive*. The active components arise from the activation of the smooth muscle cells while the passive response is governed primarily by the elastin and collagen fibres [15]. The distribution and the arrangement of the collagen fibres, in particular, have a significant influence on the mechanical properties due to they attribute anisotropic properties [49] to the tissue. Different studies have shown that this structural arrangement is very complex and varies according to the aortic segment (thoracic or abdominal) [20]. As well as being anisotropic, the material response of soft biological tissue is also highly non-linear.

To determine mechanical properties of AAA, studies have used both *in-vivo* "tests" and *ex-vivo*/*in-vitro* testing. As reported by Raghavan and da Silva [53], both of them offer advantages and disadvantages. In particular, in the first case the main difficult is to accurately determine the true force and the displacement distribution ascertaining stress-free configuration of the biological entity. On the other side, isolating samples may introduce as yet unknown changes to their behavior affecting the results of such tests. In vivo measurement are often performed by using imaging modality. By using ultrasound phase-locked echo-tracking, Lanne et al. [43] reported that the pressure-strain elastic modulus (*Ep*), Eq. 1, was higher on average and more widely dispersed in aneurysmal abdominal aorta compared to the non-aneurysmal aorta group. The *Ep* modulus was calculated based on the diameter (*Ds*,*Dd*)

and pressure (*Ps*, *Pd*) at the systolic and diastolic values as follow:

aneurysmal abdominal aorta compared to controls.

*Ep* = *Dp*

Using similar consideration, MacSweeney et al. [44] founded that *Ep* was higher in

More recently, van't Veer et al. [75] estimated the compliance and distensibility of the AAA by means of simultaneous instantaneous pressure and volume measurements obtained with the

*Ps* − *Pd Ds* − *Dd*

(1)

Sec. 4.2 and discussed in detail in Sec. 5.

**2. Review**

**2.1. Experimental test**

*Uniaxial test.* Uniaxial extension testing is the simplest and most common of *ex-vivo* testing methods. Here, a rectangular planar sample is subjected to extension along its length at a constant displacement (or load) rate while the force (or the displacement) is recorded during extension. Under the assumption of incompressibility (zero changes of volume during the tensile test, Eq. 2) the recorded force-extension data are converted to stress/strain:

$$A\_0 L\_0 = AL \tag{2}$$

where *A*<sup>0</sup> and *L*<sup>0</sup> are the initial cross sectional area and the initial length while *A* and *L* are the values in the current configuration. Interested reader can refer to Di Puccio et al. [19] for a recent review on the incompressibility assumption on soft biological tissue.

*Biaxial test.* Due to the presence of the collagen fibers, the uniaxial testing is not sufficient for highlighting the aorta tissue and the stress distribution does not fully conform to physiological conditions. Therefore, biaxial tension tests should be performed. During biaxial test, an initial square thin sheet of material is stress normally to both edges. Even if, theoretically, the biaxial test are not sufficient to fully characterized anisotropic materials, [40, 50] they are able to capture additional information regarding the mechanical behavior of the specimens with respect to uniaxial one. By contrast, biaxial tests provides a complete characterization of the material properties for isotropic material. To some extent soft biological tissues can be considered as isotropic within certain limitation, however, in their general formulation, they respond anisotropically under loads. Figure 1 depicts as example the mechanical test and response of a soft tissue under uniaxial (a) and biaxial (b) test.

As we can observe, a distinctive mechanical characteristic of soft tissue in tension tests is its initial flat response and relatively large extensions followed by an increased stiffening at higher extension. As it is well known, this behavior is the result of collagen fibres recruitment as proposed by Roach and Burton [61]. The non-linear stress strain curve arises from the phenomenon of the fibres recruitment. As the material is stretched, the fibres gradually become uncrimped and become more aligned with the direction of applied load.

The results of uniaxial and biaxial tests are used to characterize the mechanical behavior of soft tissue under investigation. Due to the large deformation that characterizes this type of tissue, from a mathematical point of view, a Strain Energy Function (SEF) denoted by *W* is introduced. The Cauchy stress tensor (σ) is calculated as:

$$\boldsymbol{\sigma} = \boldsymbol{J}^{-1} \mathbf{F} \frac{\partial \mathcal{W}}{\partial \mathbf{F}} \tag{3}$$

**Figure 1.** Schematic of uniaxial (a) and biaxial (b) test and curves. The *t* value is a rappresentative tension value and *e* a typical extension dimension.

where **F** is the deformation gradient tensor, defined as **F** = *∂***x**/*∂***X**, i.e. the derivative of the current position **x** as regards to the initial position **X** during a deformation process and *J* is the determinant of **F**. Under the assumption of incompressibility (*J*=1), the SEF is split in a volumetric (W*vol*) and isochoric (W*isoch*) component. In case of uniaxial test we have:

$$
\sigma\_{11} = \lambda\_{11} \frac{\partial W\_{\text{isoch}}}{\partial \lambda\_{11}} \tag{4}
$$

where *λ*<sup>11</sup> is the stretch in the 1-1 direction (see Fig. 1 (a)). For biaxial test both components can be calculated:

$$
\sigma\_{\theta\theta} = \lambda\_{\theta\theta} \frac{\partial W\_{\text{isoch}}}{\partial \lambda\_{\theta\theta}} \tag{5}
$$

$$
\sigma\_{zz} = \lambda\_{zz} \frac{\partial W\_{\text{isoch}}}{\partial \lambda\_{zz}} \tag{6}
$$

Equations 5-6 represent the stress components used in the follow sections. With respect to Fig. 1 direction 1-1 and 2-2 are now defined as the circumferential *σθθ* and the axial *σzz* ones, respectively.

#### **2.2. Mechanical properties of healthy and pathological aortic tissue**

AAA development is multifactorial phenomenon. A mechanism postulated for AAA formation focuses on inflammatory processes where macrophages recruitment leads to MMP production and elastase release. The biomechanical change associated with enzymatic degradation of structural proteins suggests that AAA expansion is primarily related to elastolysis [21]: a decreasing quadratic relationship was found between elastin concentration and diameter for normal aortas and for pathological increasing diameter [65]. Despite universal recognition of the importance of wall mechanics in the natural history of AAAs [2, 38, 81], there are few detailed studies of the mechanical properties.

Early studies focused on simple uniaxial tests. He and Roach [32] obtained rectangular specimen strips during surgical resection of eight AAA patients and subjected them to uniaxial extension tests up to a pre-defined maximum load rather than until failure. They showed that the stress-strain behavior of AAA tissue was non-linear. Later, in two reports, Vorp et al. [82] and Raghavan et al. [57] reported on uniaxial extension testing of strips harvested from the anterior midsection of 69 AAA. The specimens were extended until failure. In most cases, the rectangular specimens' length was in the axial orientation, but in a small population, they were oriented circumferentially. Results have found that aneurysmal tissue is substantially weaker and stiffer than normal aorta [18, 57, 70].

To date, the most complete data on both the biaxial mechanical behavior of aorta and AAAs comes from Vande Geest et al. [76, 77]. The source of these specimens becomes from AAA ventral tissue available during the open surgical repair of unruptured lesions. They reported biaxial mechanical data for AAA (26 samples) and normal human AA as a function of age: less than 30, between 30 and 60 and over 60 years of age. In particular Vande Geest and co-workers confirmed that the aortic tissue becomes less compliant with age and that AAA tissue is significantly stiffer than normal abdominal aortic tissue, Figure 2.

**Figure 2.** Stress-stretch plot comparing the equibiaxial response for AAA and HAA for four patient groups, for circumferential direction (a) and axial direction (b). Modified from [22].

The specimen was subjected to force-controlled testing with varying prescribed forces between the two orthogonal directions. A CCD camera was used to track the displacement of markers forming a 5x5 mm square placed on the specimen. It is worth to stress out that the use of optical extensometer (markers tracking with CCD camera) is fundamental to measure the deformation during test avoiding the potential tissue slippage from the clamps. Figure 3 depicts a representative biaxial stress-stretch data for healthy (a-b) and pathological (c-d) samples considering three different tension ratios (*T<sup>θ</sup>* : *Tz*) equal to 1:1 , 0.75:1 and 1:0.75.

#### **2.3. Material models**

4 Will-be-set-by-IN-TECH

**Figure 1.** Schematic of uniaxial (a) and biaxial (b) test and curves. The *t* value is a rappresentative

volumetric (W*vol*) and isochoric (W*isoch*) component. In case of uniaxial test we have:

*σ*<sup>11</sup> = *λ*<sup>11</sup>

*σθθ* = *λθθ*

*σzz* = *λzz*

**2.2. Mechanical properties of healthy and pathological aortic tissue**

[2, 38, 81], there are few detailed studies of the mechanical properties.

where **F** is the deformation gradient tensor, defined as **F** = *∂***x**/*∂***X**, i.e. the derivative of the current position **x** as regards to the initial position **X** during a deformation process and *J* is the determinant of **F**. Under the assumption of incompressibility (*J*=1), the SEF is split in a

where *λ*<sup>11</sup> is the stretch in the 1-1 direction (see Fig. 1 (a)). For biaxial test both components

Equations 5-6 represent the stress components used in the follow sections. With respect to Fig. 1 direction 1-1 and 2-2 are now defined as the circumferential *σθθ* and the axial *σzz* ones,

AAA development is multifactorial phenomenon. A mechanism postulated for AAA formation focuses on inflammatory processes where macrophages recruitment leads to MMP production and elastase release. The biomechanical change associated with enzymatic degradation of structural proteins suggests that AAA expansion is primarily related to elastolysis [21]: a decreasing quadratic relationship was found between elastin concentration and diameter for normal aortas and for pathological increasing diameter [65]. Despite universal recognition of the importance of wall mechanics in the natural history of AAAs

Early studies focused on simple uniaxial tests. He and Roach [32] obtained rectangular specimen strips during surgical resection of eight AAA patients and subjected them to uniaxial extension tests up to a pre-defined maximum load rather than until failure. They showed that the stress-strain behavior of AAA tissue was non-linear. Later, in two reports,

*∂Wisoch ∂λ*11

*∂Wisoch ∂λθθ*

*∂Wisoch ∂λzz*

(4)

(5)

(6)

tension value and *e* a typical extension dimension.

can be calculated:

respectively.

Equations that characterize a material and its response to applied loads are called constitutive relations since they describe the gross behavior resulting from the internal constitution of a material. Constitutive modelling of vascular tissue is an active field of research and numerous descriptions have been reported. Constitutive models for biological tissues can be established following a so-called phenomenological or structural approach. The first type of formulation [14, 26, 36, 73] does not take into account any histological constituents and attempt to describe the global mechanical behavior of the tissue without referring to its underlying microstructure. The phenomenological approach is commonly used but has led to a number of difficulties in describing the mechanical behavior of tissues. Among phenomenological

**Figure 3.** Experimental biaxial data for both healthy (a-b) and pathological (c-d) samples with different tension ratio. Open diamonds, 1 : 1; open squares, 0.75 : 1 and open circles, 1 : 0.75. Modified from [9].

SEFs, Vande Geest and co-authors [77] found that a constitutive functional form used earlier by Choi and Vito [12], Equation 7, would best suit their experimental data:

$$\mathcal{W}\_{\text{isoch}} = b\_0 \left( e^{\frac{1}{2}b\_1 E\_{\theta\theta}^2} + e^{\frac{1}{2}b\_2 E\_{zz}^2} + e^{\frac{1}{2}b\_3 E\_{\theta\theta} E\_{zz}} - 3 \right) \tag{7}$$

where *b*0, *b*1, *b*<sup>2</sup> and *b*<sup>3</sup> are the material parameters and *Eθθ* and *Ezz* are the components of the Green-strain tensor (Eq. 8) defined as follows:

$$\mathbf{E} = \frac{1}{2} \left( \mathbf{C} - \mathbf{I} \right) = \frac{1}{2} \left( \mathbf{F} \mathbf{F}^T - \mathbf{I} \right) \tag{8}$$

where **I** is the identity matrix and **C** is the right Cauchy-Green strain tensor.

Alternatively, structural constitutive descriptions [5, 28, 34, 35] overcome this limitation and integrate histological and mechanical information of the arterial wall. In particular, the contributions of constitutive cells, fibers and networks of elements are added together to depict the whole tissue behavior. The structural-based approach has become common with the advent of microstructural imaging methods [64, 80]. In fact, soft biological tissues have a very complex microstructure, consisting of many different components and including elastin fibres, collagen fibres, smooth muscle cells and extracellular matrix.

The same experimental data obtained by vande Geest et al. [77] were then fitted by using an invariant based constitutive equation with two fibre families (2FF) by Basciano et al. [5], Eq. 9, and Rodriguez et al. [62, 63], Eq. 10.

6 Will-be-set-by-IN-TECH

**Figure 3.** Experimental biaxial data for both healthy (a-b) and pathological (c-d) samples with different tension ratio. Open diamonds, 1 : 1; open squares, 0.75 : 1 and open circles, 1 : 0.75. Modified from [9].

SEFs, Vande Geest and co-authors [77] found that a constitutive functional form used earlier

where *b*0, *b*1, *b*<sup>2</sup> and *b*<sup>3</sup> are the material parameters and *Eθθ* and *Ezz* are the components of the

Alternatively, structural constitutive descriptions [5, 28, 34, 35] overcome this limitation and integrate histological and mechanical information of the arterial wall. In particular, the contributions of constitutive cells, fibers and networks of elements are added together to depict the whole tissue behavior. The structural-based approach has become common with the advent of microstructural imaging methods [64, 80]. In fact, soft biological tissues have a very complex microstructure, consisting of many different components and including elastin

The same experimental data obtained by vande Geest et al. [77] were then fitted by using an invariant based constitutive equation with two fibre families (2FF) by Basciano et al. [5], Eq.

2 **FF***<sup>T</sup>* <sup>−</sup> **<sup>I</sup>** 

<sup>2</sup> (**<sup>C</sup>** <sup>−</sup> **<sup>I</sup>**) <sup>=</sup> <sup>1</sup>

<sup>2</sup> *<sup>b</sup>*3*EθθEzz* <sup>−</sup> <sup>3</sup>

(7)

(8)

by Choi and Vito [12], Equation 7, would best suit their experimental data:

 *e* 1 <sup>2</sup> *<sup>b</sup>*1*E*<sup>2</sup> *θθ* + *e* 1 <sup>2</sup> *<sup>b</sup>*2*E*<sup>2</sup> *zz* + *e* 1

**<sup>E</sup>** <sup>=</sup> <sup>1</sup>

fibres, collagen fibres, smooth muscle cells and extracellular matrix.

where **I** is the identity matrix and **C** is the right Cauchy-Green strain tensor.

*Wisoch* = *b*<sup>0</sup>

Green-strain tensor (Eq. 8) defined as follows:

$$\mathcal{W}\_{\text{isoch}} = \alpha \left(\overline{I}\_1 - 3\right)^2 + \beta \left(\overline{I}\_4 - 1\right)^6 + \gamma \left(\overline{I}\_6 - 1\right)^6 \tag{9}$$

$$\mathcal{W}\_{\text{isoch}} = \mathbb{C}\_1 \left( \overline{I}\_1 - \mathfrak{Z} \right) + \sum\_{i=3}^4 \frac{k\_1^i}{2k\_2^i} \left( e^{k\_2^i \left( \left( 1 - \rho \right) \left( \overline{I}\_1 - \mathfrak{Z} \right)^2 + \rho \left( \overline{I}\_4 - I\_0 \right)^2 \right)} - 1 \right) \tag{10}$$

where *I*<sup>1</sup> is the first invariant of the isochoric portion of the right Cauchy-Green stretch tensor (Eq. 11) and *I*<sup>4</sup> and *I*<sup>6</sup> are mixed invariants of the isochoric portion of the right Cauchy-Green deformation tensor (Eq. 12-13), introduced from embedded fibers [6, 33, 69].

$$
\overline{I}\_1 = tr\overline{\mathbf{C}}\tag{11}
$$

$$
\overline{I}\_4 = \mathbf{a}\_0 \cdot \overline{\mathbf{C}} \,\mathbf{a}\_0 \tag{12}
$$

$$
\overline{I}\_6 = \mathbf{b}\_0 \cdot \overline{\mathbf{C}} \,\mathbf{b}\_0 \tag{13}
$$

where **a**0, **b**<sup>0</sup> are the direction of the fibers as reported in Figure 4(a-b). In Eq. 9, *α* is the coefficients for the isotropic part while *β* and *γ* for the anisotropic component. In the same manner, in Eq. 10, *C*<sup>1</sup> is a stress-like material parameter for the purely elastin contribute, and *ki <sup>j</sup>* are material parameters corresponding to the fibers (*k*<sup>3</sup> <sup>1</sup> <sup>=</sup> *<sup>k</sup>*<sup>4</sup> <sup>1</sup> and *<sup>k</sup>*<sup>3</sup> <sup>2</sup> <sup>=</sup> *<sup>k</sup>*<sup>4</sup> <sup>2</sup>). The parameter *ρ* ∈ [0;1] is a (dimensionless) measure of anisotropy, *I*<sup>0</sup> > 1 is dimensionless parameters regarded as the initial crimping of the fibers (Fig. 4(b)).

**Figure 4.** Collagen fiber orientation (**a**0, **b**0) in a square specimen of tissue for the 2FF model (a), the 2FF model with dispersion (b) and for the 4FF model (c). Note the crimp and fibres dispersion in case (b) and the two additional fibre family (**c**0, **d**0) in (c).

A constitutive relation based on four fibres family (4FF) (Fig. 4(c)) was proposed by Baek et al. [4], including two additionally fibres family (in longitudinal and circumferential direction, [86]), Equation 14:

$$\mathcal{W}\_{\text{isoch}} = \frac{c}{2} \left( \overline{I}\_1 - 3 \right) + \sum\_{i=1}^{4} \frac{c\_1^i}{4c\_2^i} \left( e^{\zeta\_2^i \left( \overline{I}\_4^{\dagger} - 1 \right)^2} - 1 \right) \tag{14}$$

where *c*, *c<sup>i</sup>* <sup>1</sup> and *<sup>c</sup><sup>i</sup>* <sup>2</sup> are material parameters for this specific SEF. Ferruzzi et al. [22] assumed that diagonal families of collagen were regarded as mechanically equivalent, hence *c*<sup>3</sup> <sup>1</sup> <sup>=</sup> *<sup>c</sup>*<sup>4</sup> 1, *c*3 <sup>2</sup> <sup>=</sup> *<sup>c</sup>*<sup>4</sup> <sup>2</sup>. By fitting the biaxial data, the model parameter associated with the isotropic term decreased with increasing age for AA specimens and decreased markedly for AAA specimens [22, 31]. These finding are in good agreement with histopathological results of reduced elastin in ageing [30, 51] and AAAs, e.g. [32, 60].

For all models, the diagonal fibres are accounted for by **a**0=−**b**0; in the 4FF model, axial (**c**0) and circumferential (**d**0) fibres are fixed at 90◦ and 0◦, respectively.

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

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

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 their particular effects.

#### **2.4. Geometrical model**

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

**Figure 5.** Example of AAA (a), segmentation of a CT cross section (b) and 3D reconstruction of a AAA (c), from [10].
