**4. Morphological Biodeterminants, MBDs**

After its formation, the aneurysm trends to increase in size and change its shape as consequence of the arterial wall destructive remodeling. This phenomenon, which occurs along many years in asymptomatic way, characterizes the AAA morphology and morphometry. Aneurysm geometric characteristics have been reported to be a significant predictors of the tendency for expansion or subsequent risk of rupture [41, 42] and can be the deciding factors in the clinical management of the disease. The correlation of the rupture risk with the aneurysm geometry has been clearly depicted in cases of intracranial aneurysms, where various shape indices were proven to discriminate sufficiently between rupture and unrupture aneurysms.

For AAAs, a pioneer work to assess the rupture risk based using the biomechanical concept was recently presented [43]. The authors combined geometrical and structural factors to obtain a dimensionless severity parameter, from which, they could estimate the potential risk of a specific aneurysm in any stage of development. Later, this concept it was modified for only considering the main geometric parameters of the aneurysm which can be easily determined by computed axial tomography (CT) or magnetic resonance imaging (MRI) obtained during periodic check-up [44]. The basic idea of the method was to correlate the main simple geometric parameters of the aneurysm in order to obtain the morphologic biomechanical determinants, MBDs. This idea is supported by the hypothesis that the aneurysm shape is strongly related with its rupture potential. Here, it is important to take into consideration that this method is a baseline for the determination of a rupture risk predictor and that such a treatment decision must be made within a reasonable turnaround time. Therefore, the precision of the method should be smaller than the clinical scale of evolution of the pathology and justifies the utilization of the aneurysm morphology based on simple geometric parameters as a rupture risk predictor.

86 Aneurysm

*Vitamin E deficiency* 

concentrations of plasma vitamin E carriers.

**4. Morphological Biodeterminants, MBDs** 

wall and its influence in AAA rupture.

rupture and unrupture aneurysms.

induce local growth factor expression as well as endothelial activation, both of which can promote the progression of atherosclerosis. Since traditional atherogenic risk factors increase the likelihood of aortic CMV manifestation, CMV may play a crucial role in mediating the progression of atherosclerosis. The persistent expression of CMV-gene in the vessel wall plays a role in the vascular cellular response, including progression of atherosclerosis or vasculitis in vivo. Kilicet al [39], performed PCR analysis to demonstrate the relationship between CMV and atheromathosis at the aortic wall. CMV DNA was found

Studies have pointed to an inverse relationship between vitamin E (a-tocopherol) levels and the incidence of arterial disease. Vitamin E is an important lipid-soluble antioxidant that localizes to the hydrophobic area of biologic membranes [40]. In terms of AAA, it is hypothesized that activated polymorphonuclear cells (PMNs) release proteinases which degrade the aortic wall matrix. These same PMNs would also release oxidative enzymes, generating toxic oxygen species such as hydrogen peroxide which would lead to lipid peroxidation. Vitamin E is considered a specific, though indirect, index of in vivo peroxidation. They also showed that a small group of AAA patients had decreased vitamin E levels but not decreased vitamin E/total lipid ratios compared with controls (coronary artery disease and normal patients). Accordingly, the AAA patients may be under increased oxidative stress (e.g., increased inflammation or PMN activation) but do not have decreased

This analysis reveals how the biological information associated with AAA pathogenesis constitute the foundation on which can be defined the destructive remodeling of the aortic

After its formation, the aneurysm trends to increase in size and change its shape as consequence of the arterial wall destructive remodeling. This phenomenon, which occurs along many years in asymptomatic way, characterizes the AAA morphology and morphometry. Aneurysm geometric characteristics have been reported to be a significant predictors of the tendency for expansion or subsequent risk of rupture [41, 42] and can be the deciding factors in the clinical management of the disease. The correlation of the rupture risk with the aneurysm geometry has been clearly depicted in cases of intracranial aneurysms, where various shape indices were proven to discriminate sufficiently between

For AAAs, a pioneer work to assess the rupture risk based using the biomechanical concept was recently presented [43]. The authors combined geometrical and structural factors to obtain a dimensionless severity parameter, from which, they could estimate the potential risk of a specific aneurysm in any stage of development. Later, this concept it was modified for only considering the main geometric parameters of the aneurysm which can be easily

in 37.9% atherosclerotic and 32.7% non-atherosclerotic vascular wall specimens.

Figure 1 shows an AAA schematic representation where the simple geometric parameters involved in this method are defined. *D* is the diameter at the plane of maximum diameter, *DL* is the lumen diameter, *L* is the aneurysm length which is measured from proximal neck to distal neck, *LA* is the anterior length measured from point of intersection O to anterior wall and *LP* is the posterior length measured from point of intersection O to posterior wall. During the follow up treatment the current clinical practice establishes that only three parameters are controlled: sagittal and coronal maximum diameter and length.

**Figure 1.** AAA schematic representation with the main geometric parameters.

After careful analysis, these simple parameters have adequately been combined to define the proposed geometric biomechanical factors. Some considerations about them are listed below:

1. Deformation Rate, Characterizes the actual deformation of the aorta. It is defined as a ratio between the maximum transverse diameter *D* and infra-renal aorta diameter, *d.*

This concept considers that the aorta diameters range between 1.5 and 2.5 cm for any patient. The value that defines a low rupture risk is taken as the lower deformation condition of the artery (lower values D and higher d), and for the most critical condition, as the higher deformation (higher values *D* and lower *d*).


Once these factors were defined, it was necessary to evaluate their weight in the rupture phenomenon by means of the definition of the weighted coefficient *i* and of the weighted level risk *WLRi*.

The weighted coefficient takes into consideration the weight of a specific factor on the frequency of occurrence of the AAA rupture. The initial values of the coefficients *i* have been obtained from the opinion of a group of surgeons about the importance of each factor. Furthermore, the weighted level risk considers the impact of a factor in the probability of AAA rupture and was sorted in four intervals: low impact, middle, high and dangerous. The *WLRi* have been obtained from considerations made in open literature when the importance of a factor's value is given according to the level of risk.

Table 1 shows the threshold values assigned to each geometric biomechanical factor and their related weighted coefficient and level risk.


**Table 1.** Geometric biomechanical factors characterization.

88 Aneurysm

2. Asymmetry,

risk if it is more symmetric (

value of *D* (typical for elective repair).

feature selection algorithm based on the

potential, by means of the parameter ILT/AAA area ratio.

phenomenon by means of the definition of the weighted coefficient

which means that

3. Saccular Index,

4. Relative Thickness,

5. ILT/AAA area ratio,

6. Growth rate,

level risk *WLRi*.

This concept considers that the aorta diameters range between 1.5 and 2.5 cm for any patient. The value that defines a low rupture risk is taken as the lower deformation condition of the artery (lower values D and higher d), and for the most critical

attributed to the non-symmetry expansion of the aneurysm sac as a result of the expansionconstraints introduced by the proximity to the spinal column. Due to this, AAA geometry exhibits a high surface complexity and a significant tortuosity of the inflow conduit and the segments of the iliac arteries. An aneurysm has lower rupture

affected by the formation and further development of the aneurysm. This means that long aneurysms have more rupture possibilities than a short one. Typical values of *L* are ranged from 90 to 140 mm (some works have reported values of *L*, higher). The calculation condition of the upper threshold value is the higher value of *L* and the peak

of a variable wall thickness; both between the anterior and posterior walls and between the aneurysmatic sac and the regions close to the distal and proximal ends.� Initial studies have used uniform wall thickness in their attempt to characterize aneurysm shape. Although wall thickness was not one of the highest ranked features chosen with the

ignore [45]. Typical values of wall thickness (*t*) in aneurysmatic arteries are ranged from 0.5 to 1.5 mm [46]. This general range may vary from 0.23 mm to 4.26 mm at a calcified site [47]. The danger of aneurysm rupture will be greater when the thickness is low in the

consensus about its real influence in the AAA rupture phenomenon. Some investigators state that ILT may reduce the stress in the AAA wall, improving its compliance and significantly preventing AAA rupture. Other declared that ILT could accelerate AAA rupture. Hence, it is very important to consider the effects of ILT in the rupture

expansion rate of 0.5-1.0 cm/year is often associated with a high risk of rupture, and an elective repair should be considered even if the maximum diameter is lower than 5 cm. The value indicating that an aneurysm is in rupture risk has been determined regarding to the worst situation (the lowest value inside the range of high growth rate (0.5cm/year), the peak diameter D and the time *T* between periodic check-up (0.5 year).

The low rupture risk limits were determined for aneurysm formation conditions.

Once these factors were defined, it was necessary to evaluate their weight in the rupture

peak diameter region. This trend falls with the increase of the wall thickness.

A characteristic feature of an aneurysm is its asymmetry, which can be

. This factor assesses the portion of the aorta, with length (*L*), which is

. The aneurysm geometric characterization determines the existence

. Although 70% of AAA includes thrombus [48], there isnot

It is considered as an important indicator for AAA rupture. A high

test, its effect on aneurysm rupture cannot be

*i* and of the weighted

=1) and the risk increases as *LP* tends to be lower than *LA,* 

condition, as the higher deformation (higher values *D* and lower *d*).

trend to 0.

Hence, rupture risk quantitative indicator defined in term of AAA morphology, can be expressed as the sum of each weighted coefficient *i* multiplied by the corresponding *WLRi*:

$$RI(t) = \sum\_{1}^{6} \omega\_{l} W L R\_{1} \tag{1}$$

Regarding the results of *RI*(*t*), it is possible to advise several actions and suggestions to physicians. This is shown in Table 2.

As above indicated, the proposed method is based on six geometric biomechanical factors. But, it is possible that, for any reason, the information about some parameters is not available. In this case, the method fits its algorithm to calculate only the factors associated with the existing geometric parameters and it is able to weights the final result according to the amount of parameters taken into account.

An initial limitation of the method is associated with indirect errors in obtaining the MBDs, due to the difficulty in extracting exact values from the geometric parameters needed in determining these MBDs. The measurements of the simple geometric parameters is, usually, carried out by a radiologist, a human being with its professional customs and resources,


**Table 2.** *RI*(*t*) intervals and actions and suggestions offered by method to physicians.

with best and/or worst days, with/without personal and labor problems. Therefore, it is important to assess the influence of all these (and others) conditions on the precision of the results.

The ANSI-ASME PTC 85, ISSO 5167 standard was used to determine the indirect errors in the calculation of GBDs due to the direct measurements of the simple geometric parameters. The methodology was applied to data-base which was used for validation tests. The results that are shown in Table 3 correspond to higher values for the errors obtained. The bias limit in measuring of the geometric parameters for all parameters was considered 0.001m. The main conclusion that can be drawn from Table 3 is that the errors in determining the MBDs, are not significant.


**Table 3.** Indirect errors obtained in determining the GBDs.This standard allows defining the experimental uncertainty, *U* in determining a variable *Z*, as:

This initial set of values was validated by using one clinical case and three cases from literature.

In shortly. In the clinical case, the state of a 74 year-old male patient with an aneurysm was assessed. The geometrical characterization shows that the peak diameter is lower than the threshold value (50 mm), therefore under current medical practice; the patient should be kept under observation. But, on the other hand, the values of the deformation rate and the asymmetry index fall into the high risk level interval. It must be noticed that by means of statistical analysis these geometric biomechanical factors are considered as the most influential factors on the aneurysm potential rupture.Other two MBDs are also sorted as high risk level, although their weight on the rupture phenomenon is lower. Therefore, the value of the patient-specific quantitative predictor calculated by equation (1) is *RI*(t)=0.64, which indicates that the elective repair should be considered. This result was confirmed because, during the period of check-up examination, the patient underwent an emergency surgical procedure for aneurysm rupture in the posterior wall.

90 Aneurysm

results.

are not significant.

literature.

Deformation Rate,

Asymmetry,

Saccular Index,

ILT/AAA ratio,

Relative Thickness,

Growth rate,

experimental uncertainty, *U* in determining a variable *Z*, as:

RI(t) Actions/Suggestions

< 0.2 Rupture risk is very low. No action is suggested. 0.2 ÷ 0.45 Rupture risk is low. A close observation is required.

÷ 0.7 Elective repair should be considered. Other symptoms such a back and abdominal pain, syncope or vomiting, should be observed.

with best and/or worst days, with/without personal and labor problems. Therefore, it is important to assess the influence of all these (and others) conditions on the precision of the

The ANSI-ASME PTC 85, ISSO 5167 standard was used to determine the indirect errors in the calculation of GBDs due to the direct measurements of the simple geometric parameters. The methodology was applied to data-base which was used for validation tests. The results that are shown in Table 3 correspond to higher values for the errors obtained. The bias limit in measuring of the geometric parameters for all parameters was considered 0.001m. The main conclusion that can be drawn from Table 3 is that the errors in determining the MBDs,

MDBs Uncertainty, *Uz* Relative uncertainty,

1.81E-01 0.0464

2.55E-02 0.075

1.23E-02 0.022

1.81E-03 3.13E-03

1.18E-02 1.8

1.67E-02 0.027

**Table 3.** Indirect errors obtained in determining the GBDs.This standard allows defining the

This initial set of values was validated by using one clinical case and three cases from

In shortly. In the clinical case, the state of a 74 year-old male patient with an aneurysm was assessed. The geometrical characterization shows that the peak diameter is lower than the threshold value (50 mm), therefore under current medical practice; the patient should be kept under observation. But, on the other hand, the values of the deformation rate and the asymmetry index fall into the high risk level interval. It must be noticed that by means of statistical analysis these geometric biomechanical factors are considered as the most influential factors on the aneurysm potential rupture.Other two MBDs are also sorted as high risk level, although their weight on the rupture phenomenon is lower. Therefore, the value of the patient-specific quantitative predictor calculated by equation (1) is *RI*(t)=0.64, which indicates that the elective repair should be considered. This result was confirmed

*U* (%)

> 0.7 Rupture risk is very high. Surgical intervention must be necessary.

**Table 2.** *RI*(*t*) intervals and actions and suggestions offered by method to physicians.

In another test, a triple validation was performed comparing the results documented in the original papers [49], [50] and [51], the results presented by [43] and the results obtained with the proposed set of values [52]. The geometries of the different analyzed AAAs are very different, however the value of RI(t) is able to sort patients correctly. In the model presented in [49], it is noticed that the aneurysm affects a significant region of the aorta and has a high rate of growth, which has a high relative importance in the value of *RI*(*t*). In the model [50], the two biomechanical factors that have more influence in the deterioration of the aneurysm increase in comparison with the previous one, but they stay in the range of elective repair, although it was expected that the indicator value would be higher.

Analyzing the model [51], it is noticed that there is a worsening of most of the geometric parameters; the most important are a high growth rate, a maximum diameter 20% greater than the threshold value and an aneurysm affecting a significant region of the artery. This behavior justifies that the value of the rupture risk indicator falls into the category of possible rupture.

These results encouraged the implementation of another validation test: a broader control study with a population of two hundred and one patients at the Clinic Hospital of Valladolid-Spain, who were submitted to Endovascular Aneurysm Repair (EVAR) treatment.Previously, a new the set of values for the weighted coefficient was defined by using a statistical tool to contrast the hypothesis that certain events have a probability of occurring. In this case, the event is associated to the AAA rupture due to a specific MBD.

According to this statistical tool, the new set of values resulting for *i* is: Deformation Rate= 0.35, Asymmetry=0.07, Saccular Index=0.1, Relative Thickness=0.07, ILT/AAA area ratio=0.07 and Growth rate=0.34.

For this new test, the population of the sample was divided in three groups: Group I (*n*=174) - patients without later consequences after EVAR treatment; Group II (*n*=5) - patients who died from causes associated with the AAA pathology; Group III (*n*=22) - patients whose AAA ruptures. As all these patients were submitted to EVAR treatment, the main objective of this test is to verify if some of the surgical procedures in patients whose aneurysm has a maximum diameter higher than threshold value could have been avoided, and/or if the method can predict the rupture of aneurysm with a diameter less than the threshold value.

The results showed that in 88% of the patients who belongs to group I is justified the surgical procedure, because the *RI(t)* values fall into dangerous and high level rupture risk. In the group II, the results suggest that the five patients should be submitted to surgical procedure because their rupture risk index is dangerous + high risk condition. All these patients died either during repair treatment or during recovering of it. The state of health of all these patients was not good, because they presented other diseases like renal chronic insufficiency, atheromatic plaque, previous complications related with cardiovascular diseases, digestive hemorrhages.

Very interesting results are obtained in the analysis of the group III. The values of *RI(t)* indicate that95.4% of the patients, present levels of rupture risk sorted as dangerous and high and the surgical procedure could have been considered before rupture. All these patients had aneurysms whose maximum diameter was less than the threshold value for surgical treatment and a systematic (time between two consecutive revisions lower than 1 year) follow-up check are suggested to diminishing the risks associated to emergency surgery by ruptures.

The fact that one patient presented a middle rupture index was somewhat unexpected and it is probably attributable to a combination of other factors not considered here, associated to factors of biological and/or structural nature. It was verified that the geometric parameters are lower than the threshold values.

The obtained outcomes are promising and have motivated further actions. Recent studies [53] have identified other MBDs based on the lumen centerline geometry. According to [54], the resulting centerline is a piecewise linear line dened on the Voronoi diagram, whose vertices lie on Voronoi polygon boundaries [55]. Values of Voronoi sphere radius *R*(**x**) are therefore dened on centerlines, so that centerline points are associated with maximal inscribed spheres. Since centerlines were constructed to lie on local maxima of distance from the boundary, there is a tight connection between maximal sphere radius and minimum projection diameter used in clinical evaluation. In fact, classic angiographic vessel diameter evaluation is performed considering the minimum diameter obtained by measurements on different projections. The availability of a robust method for centerline computation and diameter measurement allows to characterize blood vessel geometry in a synthetic way, therefore giving the opportunity of performing a study on a population of models. Since it has been shown that planarity, tortuosity and branching angles have a major inuence on complex blood ow patterns, such a study may reveal if particular vessel congurations are involved in vascular pathology.

Three MBDs have been defined using this approach: tortuosity, curvature and torsion centerline. Today, VMTK software havebeen developed to 3D reconstruction of the lumen centerline geometry. Figure 2 shows the visual representation of these determinants. Tortuosity, an absolute number, expresses the fractional increase in length of a tortuous vessel in relation to the imaginary straight line and has been described in [55]. Torsion is measured in 1/cm2 and curvature is measured in 1/cm.

**Figure 2.** Schematic visualization of tortuosity, curvature and torsion [53].

Recently, it has been postulated that aneurysm peak wall stress (PWS) may be superior to diameter as predictor of the rupture risk. This statement has its theoretical foundation in the physical principle of the aneurysm rupture. Complex AAA geometry contributes to equivalent complex wall stress distribution over the entire AAA, with the higher stresses associated with regions of high curvature [56].

92 Aneurysm

surgery by ruptures.

are lower than the threshold values.

involved in vascular pathology.

measured in 1/cm2 and curvature is measured in 1/cm.

**Figure 2.** Schematic visualization of tortuosity, curvature and torsion [53].

Very interesting results are obtained in the analysis of the group III. The values of *RI(t)* indicate that95.4% of the patients, present levels of rupture risk sorted as dangerous and high and the surgical procedure could have been considered before rupture. All these patients had aneurysms whose maximum diameter was less than the threshold value for surgical treatment and a systematic (time between two consecutive revisions lower than 1 year) follow-up check are suggested to diminishing the risks associated to emergency

The fact that one patient presented a middle rupture index was somewhat unexpected and it is probably attributable to a combination of other factors not considered here, associated to factors of biological and/or structural nature. It was verified that the geometric parameters

The obtained outcomes are promising and have motivated further actions. Recent studies [53] have identified other MBDs based on the lumen centerline geometry. According to [54], the resulting centerline is a piecewise linear line dened on the Voronoi diagram, whose vertices lie on Voronoi polygon boundaries [55]. Values of Voronoi sphere radius *R*(**x**) are therefore dened on centerlines, so that centerline points are associated with maximal inscribed spheres. Since centerlines were constructed to lie on local maxima of distance from the boundary, there is a tight connection between maximal sphere radius and minimum projection diameter used in clinical evaluation. In fact, classic angiographic vessel diameter evaluation is performed considering the minimum diameter obtained by measurements on different projections. The availability of a robust method for centerline computation and diameter measurement allows to characterize blood vessel geometry in a synthetic way, therefore giving the opportunity of performing a study on a population of models. Since it has been shown that planarity, tortuosity and branching angles have a major inuence on complex blood ow patterns, such a study may reveal if particular vessel congurations are

Three MBDs have been defined using this approach: tortuosity, curvature and torsion centerline. Today, VMTK software havebeen developed to 3D reconstruction of the lumen centerline geometry. Figure 2 shows the visual representation of these determinants. Tortuosity, an absolute number, expresses the fractional increase in length of a tortuous vessel in relation to the imaginary straight line and has been described in [55]. Torsion is The role of these geometric biodeterminants in the prediction of AAA it has been assessed taking into consideration the presence of intra-luminal thrombus (ILT) [53]. In the study were included nineteen patients whose-which AAA maximum diameters ranged from 5 to 12 cm. Statistical analysis confirmed that the maximum diameter significantly influenced PWS and the tortuosity may also affect PWS values in models with ILT in the same direction.

On the other hand, it has been demonstrated [57] that PWS is strongly correlated with the maximum diameter as well as the centreline asymmetry. It is notable, however, that in 73% of the analyzed models in this work a significant correlation was found between asymmetry and maximum diameter. Therefore, if diameter strongly correlated with peak stress then asymmetry would also score high.

Perhaps one of the most ambiguous issues in the assessment of rupture risk is the existence as well as the develop of the ILT. Despite ILT´s impact on aneurysm disease, little is known about its development, and it is unclear whether it increases or decreases the risk of aneurysm rupture. That is, the ILT reinforces proteolytic activity [58], which weakens the wall [59], or buffers against wall stress [50]. It has been hypothesized that ILT develops either from rupture of vulnerable plaques or as a more continuous process characterized by blood-flow induced activation of platelets and their deposition at non-endothelialized sites of the wall exposed to low (sub-physiological) wall shear stress [60].

Recently, the investigationsare addressed to the integration of ILT in the computational models and, consequently, its effects in patient-specific on PWS values and distribution. A significant difference in PWS when including the ILT in 3D AAA computational model it has been reported [61]. Wang et al [50] showed that computational integration of ILT in 3D models could actually modify not only the value but also the distribution of PWS, thus playing a protective role against rupture but this conclusion was not supported in [62]. On the other hand, there is still some concern regarding the protective role of ILT, since many authors who evaluate the influence of ILT on hemodynamic stress transmission, reported that he presence of ILT fails to reduce the transmission of this stress on the AAA wall, consequently, leaving the AAA rupture risk equalled [63]. AAAs can experience higher stresses at regions of inflection, regardless of wall thickness variation. In such cases, the concentric or eccentric location of ILT in the AAA sac cannot be effectively reduce PWS values or changes its distribution [64]. A question of interest arises here, regarding whether such PWS values derived from computational estimation should be taken into consideration, since AAA rupture rarely takes places at these sites, reserving this possibility only for thrombosed AAAs [65]. Therefore, all these ideas reinforce the need to quantify and take into consideration the effect of the ILT.

Finally, it is important to address the topic related to the use either simple geometric parameters or biodeterminants in the AAA rupture risk assessment. To answer this question some aspects should be analyzed. The first one is related to the temporal scales of the disease progression which is higher than the results´ precision in the determination of the geometric parameters. This conclusion justify the use of morphological determinants. On the other hand, is the fact that aneurysm shape has a significant influence on flow patterns and consequently in its rupture potential. Recent findings have shown that the aneurysm geometrical shape may be related to the rupture risk. The morphological nature determinants (MBDs) are defined by appropriate relations among simple geometric parameters to characterize the influence of the aneurysm morphology on its rupture potential.

Utilizing idealized aneurysm models of the true vessel lumen surface geometry, the role of the geometric characteristics in the hemodynamic stresses prediction by using of Pearson´s rank correlation coefficients was assessed [66]. In this work, the model was modified to allow the parametrization of the main parameters assessed: maximum diameter D, length L and asymmetry, β. Figure 3, shows a schematic view of the models used in this study.

The results show that hemodynamics stresses correlate better with MBDs. For hemodynamic pressure, the relation with saccular index and deformation rate are strong and negative (r=- 0.75, p=0.000 and r=-0.7, p=0.000 respectively). The asymmetry coefficient has no-significant correlation (r=-0.25, p=0.00).

**Figure 3.** Schematic view of the parametrized aneurysm models. a) *L* and are constant and *D* varies; b) *D* and are constant and L varies.

The relation of asymmetry and deformation rate with WSS is weak with significance less than 15% and saccular index is no significant.

The main conclusions of this study are: luminal pressure is the primary mechanical load on the aneurismal wall and that MBDs are better predictors than simple parameters of the hemodynamic stresses.

On the other hand, Raghavanet al [67], showed that the deviation of the aneurysm shape from spherical configuration, the level of its surface ondulation or ellipticity and the norm of the surface mean curvature are a good predictors of rupture.

This analysis confirms that MBDs may become a useful addition to current clinical criteria, mainly maximum diameter, in the decision-making process of the aneurysm treatment. Certainly, as in the same way that other biomechanical consideration, the suggested models require further studies.

#### **5. Structural Biodeterminants, SBDs**

94 Aneurysm

potential.

b) *D* and

hemodynamic stresses.

correlation (r=-0.25, p=0.00).

Finally, it is important to address the topic related to the use either simple geometric parameters or biodeterminants in the AAA rupture risk assessment. To answer this question some aspects should be analyzed. The first one is related to the temporal scales of the disease progression which is higher than the results´ precision in the determination of the geometric parameters. This conclusion justify the use of morphological determinants. On the other hand, is the fact that aneurysm shape has a significant influence on flow patterns and consequently in its rupture potential. Recent findings have shown that the aneurysm geometrical shape may be related to the rupture risk. The morphological nature determinants (MBDs) are defined by appropriate relations among simple geometric parameters to characterize the influence of the aneurysm morphology on its rupture

Utilizing idealized aneurysm models of the true vessel lumen surface geometry, the role of the geometric characteristics in the hemodynamic stresses prediction by using of Pearson´s rank correlation coefficients was assessed [66]. In this work, the model was modified to allow the parametrization of the main parameters assessed: maximum diameter D, length L and asymmetry, β. Figure 3, shows a schematic view of the models used in this study.

The results show that hemodynamics stresses correlate better with MBDs. For hemodynamic pressure, the relation with saccular index and deformation rate are strong and negative (r=- 0.75, p=0.000 and r=-0.7, p=0.000 respectively). The asymmetry coefficient has no-significant

The relation of asymmetry and deformation rate with WSS is weak with significance less

The main conclusions of this study are: luminal pressure is the primary mechanical load on the aneurismal wall and that MBDs are better predictors than simple parameters of the

On the other hand, Raghavanet al [67], showed that the deviation of the aneurysm shape from spherical configuration, the level of its surface ondulation or ellipticity and the norm of

are constant and *D* varies;

**Figure 3.** Schematic view of the parametrized aneurysm models. a) *L* and

the surface mean curvature are a good predictors of rupture.

are constant and L varies.

than 15% and saccular index is no significant.

In addition to morphological factors, numerically predicted wall stress, finite element analysis rupture index, rupture potential index and severity parameters have been proposed as alternative approaches to assessing rupture risk [68].

The criterion currently used by the medical community is that you can relate directly the risk of rupture with the maximum diameter of the aneurysm. However, as noted above, the biomechanics states that rupture occurs when wall stress exceeds its strength. This assumes a linear relationship between the maximum stress and the maximum diameter of the aneurysm. Thus, we propose an equation to describe this approach.

$$
\sigma\_{\max} = k R\_{\max} \tag{2}
$$

where *max* is the maximum stress in the aneurysm, *k* is a constant determined by experience, and *Rmax* the maximum radius of the aneurysm. The maximum diameter criterion has many limitations.

Since there is currently no method to determine the stresses in the wall in vivo, it is necessary to develop models of the mechanical behavior of the arterial wall. These models can be generated from ideal parameterized geometries created by three-dimensional design software (CATIA, SolidWorks, etc.), or can be obtained through the processing of medical images.

Once the geometry is generated, we calculate using finite element method software (ANSYS, ABAQUS, etc.) in order to determine the stress distribution in the wall of AAA.

#### *Structural biomechanical determinant of VandeGeest*

After evaluating the stresses, and using the ultimate strength of arterial tissue or an assessment of the strength of the wall, you can define a structural biomechanical factor. This factor or biodeterminant, can allow us to estimate how close an aneurysm can be of the rupture and, consequently, the appropriateness of the surgical procedure in the patient.

Thus, it is proposed [69] the following factor:

$$RI(t) = \frac{Stress\_l}{Strength\_l} \tag{3}$$

where*i*is the chosen point on aneurysm geometry.

It is noted that when the rupture index approaches the value of 1, the state of risk of aneurysm rupture increases, ie when the stress observed in the wall reaches the value of strength.

If the strength is only an estimated value for the entire aneurysm, use the maximum stress given by the simulation. But, when using the strength distribution in the whole geometry, the rupture index is evaluated at each point of the geometry of the aneurysm.

#### *Rupture criterion of Li and Kleinstreuer*

This approach [70] is based not only on statistical analysis of some cases of abdominal aortic aneurysms, but also on results of numerical simulations. To do so, tests were conducted with 10 patients whose data were known, in order to verify the accuracy criterion used to calculate max:

$$
\sigma\_{\max} = 0.006 \frac{(1 - 0.69\lambda)e^{\left(0.0123 \left(0.05 P\_{\text{slgt}} + \lambda \text{mA}\right)}\right)}{t^{0.63} \beta^{0.125}}}{} \tag{4}
$$

where *max*is the maximum stress that appears frequently in an area whose diameter is equal to two thirds of the maximum diameter of AAA, is the ratio of the areas in the plane of maximum diameter ( = *AILT*,max / *AAAA,max*), *β* is the coefficient of asymmetry, *Psis* is the systolic blood pressure (mmHg), *D* is the maximum diameter of AAA (cm) and *t* is the thickness of the wall in the plane of maximum diameter.

If the thickness of the arterial wall cannot be determined from images taken by the TAC, can be approximated by the following equation:

$$t = 3.9 \left(\frac{D}{2}\right)^{-0.2892} \tag{5}$$

According to the authors, this approach presents a very low error in the determination of the maximum stress compared to other models. Whatever the feature is used to calculate stress, the results are very similar to the stress determined by finite element method software.

Clearly, the geometry should not be too complex, which is a limitation. Furthermore, the location of the maximum stress cannot determine, although the value is known.

This approach appears to be quite accurate results, and its application is very simple. So it could be used to determine the maximum stress of the aneurysm with a very simple approach. However, we emphasize that in no other study has been applied.

#### *Rupture criterion based on remodeling history model*

These models allow determining a stress value, which is compared with the strength of the arterial wall to evaluate if the break is close or not.

The value of strength can be obtained:


#### *Criteria based on two-dimensional modeling*

It is a very simple model in two dimensions of the arterial wall;


calculate max:

maximum diameter (

where 

*Rupture criterion of Li and Kleinstreuer* 

to two thirds of the maximum diameter of AAA,

thickness of the wall in the plane of maximum diameter.

be approximated by the following equation:

*Rupture criterion based on remodeling history model* 

arterial wall to evaluate if the break is close or not.

The value of strength can be obtained:

*Criteria based on two-dimensional modeling* 

personal information.

If the strength is only an estimated value for the entire aneurysm, use the maximum stress given by the simulation. But, when using the strength distribution in the whole geometry,

This approach [70] is based not only on statistical analysis of some cases of abdominal aortic aneurysms, but also on results of numerical simulations. To do so, tests were conducted with 10 patients whose data were known, in order to verify the accuracy criterion used to

���� = 0.006 (���.���)���.������.����������.����

systolic blood pressure (mmHg), *D* is the maximum diameter of AAA (cm) and *t* is the

If the thickness of the arterial wall cannot be determined from images taken by the TAC, can

According to the authors, this approach presents a very low error in the determination of the maximum stress compared to other models. Whatever the feature is used to calculate stress,

Clearly, the geometry should not be too complex, which is a limitation. Furthermore, the

This approach appears to be quite accurate results, and its application is very simple. So it could be used to determine the maximum stress of the aneurysm with a very simple

These models allow determining a stress value, which is compared with the strength of the

By an empirical approach based on an expression that takes into account the patient's

Form literature, which are based on uni-axial tests aneurysmal tissue of patients.

� = �.� �� � � ��.����

the results are very similar to the stress determined by finite element method software.

location of the maximum stress cannot determine, although the value is known.

approach. However, we emphasize that in no other study has been applied.

It is a very simple model in two dimensions of the arterial wall;

*max*is the maximum stress that appears frequently in an area whose diameter is equal

= *AILT*,max / *AAAA,max*), *β* is the coefficient of asymmetry, *Psis* is the

��.����.��� (4)

is the ratio of the areas in the plane of

(5)

the rupture index is evaluated at each point of the geometry of the aneurysm.

From these criteria leads to a simple equation that relates the pressure *P*, the wall thickness *t* and the maximum radius *Rmax* of the aneurysm:

$$
\sigma\_{\max} = P \frac{R\_{\max}}{t} \tag{6}
$$

This modeling, which leads to the stress calculation, presents the following limitations:


This approach is similar to the criterion of maximum diameter used today.

#### *Criteria based on three-dimensional modeling*

a. Modeling of material behavior: linear elasticity.

Many authors have used an elastic model of the arterial wall in their research [71, 72]

Commenting on the approach proposed in [73], the authors have attempted to determine the influence of the diameter and symmetry in the mechanical stress of the arterial wall of abdominal aortic aneurysm using an elastic behavior of the wall.

This approach has the merit of taking into account the behavior of the material used, and the authors are aware of the limits of their model, since the aim of their study was to show the influence of symmetry. However, other studies [74, 75] showed that the hyperelastic behavioral model is more suitable for simulating an aneurysm under pressure due to the large strains that can undergo aneurysmal arterial wall (20-40%).

b. Modeling of material behavior: hyperelasticity.

Given the fact that the tissue of the aneurysmal arterial wall can be deformed the order of 20-40%, the behavior can no longer be considered as elastic.

Hyperelastic materials are characterized by the existence of an energy function W, which depends on the state of deformation.

Tensions can be calculated with this energy function *W*, which depends on the material, which can be isotropic or anisotropic, which will influence in *W*.

b.1) Isotropic hyperelasticity

In 1940, Mooney and Rivlin established a behavioral model for the material like rubber, whose behavior is similar to the tissue of the arterial wall due to the incompressibility of both materials.

Heng et al. [76], used the Mooney-Rivlin equation to establish one of the simplest hyperelastic models. The problem with this model with only two parameters is that is more suited to the study of polymers. This law was made by Mooney to model the behavior of rubbers, and it seems too simple for the study of tissues, whose behavior seems much more complex because its composition is not homogeneous.

You can also use a more complex form of Mooney-Rivlin model. In [77] it is performed a study which uses this model and the results seem that calculate appropriately the real tensions of the arterial wall. This form uses 9 parameters addition to the incompressibility parameter.

In 2000, it is defined a mathematical model using a regression from experimental results [75]. This is part of the theory of finite deformations and is based on the first principle of mechanics of continuous media. The assumptions underlying this model were that the wall is non-linear, homogeneous, incompressible and isotropic.

In 2006, this model is modified using another form of the density function [78]. It is observed that for incompressible materials considered, this equation is the same as proposed in [75].

In 2008, it is proposed a model based on the concept of material failure energy Φ [79]. This energy is the maximum amount of energy that the wall can withstand before breaking, because of the deformations. This value depends on the atomic or microscopic structure of the wall of an AAA.

b.2) Anisotropic hyperelasticity

single transverse anisotropy.

In 1976, Tong and Fung [80], developed a cross-anisotropic hyperelastic model, which allows a behavioral model of the arterial wall aneurysm.

	- Rodríguez anisotropichyperelasticmodel [81];
	- Holzapfel anisotropic hyperelastic model [82]. Proposed model for biological materials with two families of collagen fibers, as they really are the arterial walls.

The anisotropic hyperelastic behavior models better approximate the actual behavior of the aneurysmal arterial wall, but according to the model used, the results can be very different. One can see that the Rodríguez hyperelastic anisotropic model is closer (at the level of stress distribution) to an isotropic hyperelastic that the Holzapfelhyperelastic anisotropic model, as shown in Figure 4.

c. Fluid-Structure Interaction

All approaches that have been presented are based on the physical principle of fault material of aortic wall. However, all these approaches use a constant pressure value (often the peak systolic pressure), whereas, in reality, not only the pressure varies, but also the blood moves. In an attempt to make models as realistic as possible, we have developed the modeling fluid-structure interaction (FSI), in which the model considers simultaneously the effect of blood flow on the arterial wall and vice versa.

Isotropic Holzapfel Rodríguez

**Figure 4.** Stress in different models.

98 Aneurysm

parameter.

proposed in [75].

the wall of an AAA.

as shown in Figure 4.

c. Fluid-Structure Interaction

b.2) Anisotropic hyperelasticity single transverse anisotropy.

Heng et al. [76], used the Mooney-Rivlin equation to establish one of the simplest hyperelastic models. The problem with this model with only two parameters is that is more suited to the study of polymers. This law was made by Mooney to model the behavior of rubbers, and it seems too simple for the study of tissues, whose behavior seems much more

You can also use a more complex form of Mooney-Rivlin model. In [77] it is performed a study which uses this model and the results seem that calculate appropriately the real tensions of the arterial wall. This form uses 9 parameters addition to the incompressibility

In 2000, it is defined a mathematical model using a regression from experimental results [75]. This is part of the theory of finite deformations and is based on the first principle of mechanics of continuous media. The assumptions underlying this model were that the wall

In 2006, this model is modified using another form of the density function [78]. It is observed that for incompressible materials considered, this equation is the same as

In 2008, it is proposed a model based on the concept of material failure energy Φ [79]. This energy is the maximum amount of energy that the wall can withstand before breaking, because of the deformations. This value depends on the atomic or microscopic structure of

In 1976, Tong and Fung [80], developed a cross-anisotropic hyperelastic model, which

 Holzapfel anisotropic hyperelastic model [82]. Proposed model for biological materials with two families of collagen fibers, as they really are the arterial walls.

The anisotropic hyperelastic behavior models better approximate the actual behavior of the aneurysmal arterial wall, but according to the model used, the results can be very different. One can see that the Rodríguez hyperelastic anisotropic model is closer (at the level of stress distribution) to an isotropic hyperelastic that the Holzapfelhyperelastic anisotropic model,

All approaches that have been presented are based on the physical principle of fault material of aortic wall. However, all these approaches use a constant pressure value (often the peak systolic pressure), whereas, in reality, not only the pressure varies, but also the blood moves. In an attempt to make models as realistic as possible, we have developed the

complex because its composition is not homogeneous.

is non-linear, homogeneous, incompressible and isotropic.

allows a behavioral model of the arterial wall aneurysm.

Rodríguez anisotropichyperelasticmodel [81];

Anisotropy with two families of fibers

Some authors try to use a method of modeling of blood flow, to study its influence on the stresses of the aneurysm wall. These approaches are also used mechanical simulations to assess the stress in the wall of the aneurysm.

From the results obtained with FSI simulations [46], has been determined that in the simulations using the computational analysis of the static stress incurred in an underestimation of wall tension, which is shown in Figure 5. This value can reach 12.5%, as reported [83].

**Figure 5.** Stream lines which characterize the blood flow inside the aneurysm and surface distribution of stresses, obtained using a modeling FSI.

In 2006, a simulation of aneurysm under pressure [41] and blood flow was carried out in addition to demonstrating that when taking into consideration the bloodstream, the stresses change little while the time required for the simulation is three to four times greater.

The authors concluded that the fluid-structure interaction approach is interesting, but a modeling of the wall with the systolic pressure is sufficient to calculate the stresses in it.

$$Strength = 141.26 - 17.16ILT + 3.394GE - 257.3NORD - 69.5HIST \tag{7}$$

$$Strength = 72.9 - 33.5 \left( \left( lLT^{0.5} \right) - 0.79 \right) - 12.3 \left( lNRD - 2.31 \right) - 24 HIST + 15 SEX \quad \text{(8)}$$


Overall analysis by using Finite Element Method is an orderly process that will include the following steps:

100 Aneurysm

After the revision presented we can conclude that an anisotropic hyperelastic model, using systolic pressure load, and geometry with the important details of the AAA, is the best

At this point, is already known that the evaluation of the wall stress cannot be considered as an isolated indicator to assess the risk of rupture of AAAs, as an aneurysmal wall region which is subjected to high stresses, may also have a high strength, thus equalizing potential rupture. According to the remodeling history model, the strength of the wall is different from patient to patient and in the same patient at different regions and time scales. To resolve this situation, has been developed a technique for noninvasive estimation of the

where *ILT* is the thickness of the *ILT* (in cm), *AGE* is the patient age in years, *NORD* is the diameter normalized to the maximum diameter of AAA, *HIST* is ± ½ according to family history (½ if the history is positive, - ½ if no background) and *SEX* is ± ½ by sex of the

These authors have increasingly improved this criterion, being the last, expressed by

The logical process for estimating the risk of aneurysm rupture using structural

Subsequently, using the Rupture Index (*RI*) proposed in Equation 1 can be estimated if a ruptured aneurysm is close. Obviously, it will require a medical evaluation of patient state

Unable to provide a method for determining the in vivo distribution of wall stress, nowadays it is used the finite element method (FEM), which is recognized as a very precise technique, which aims to find approximate solutions of partial differential equations and integral equations. Equations are solved at the nodes of the meshes that are generated and interpolated within the element, generating a continuous solution throughout the domain.

�������� � ���� � �����(������) � ����� � ����(���� � ����) � ������ � ����� (8)

�������� � ������ � �������� � ������� � ��������� � �������� (7)

distribution of strength, defining a potential rupture index (RPI)[84], with equation:

choice for calculating the stresses in aneurysmal wall.

patient (½ if the patient is a man, - ½ for women).

*Rupture of aneurysm prediction* 

2. CT of the patient's aneurysm.

of health (PSH).

equation 8, thatbest approximates the strength of the wall.

biomechanical factors would be the one described below:

6. State estimation of risk of rupture of the aneurysm.

*Simulation using the Finite Element Method (FEM)* 

3. Geometric model of the aneurysm from medical imaging. 4. Simulation of the aneurysm using data specific to the patient.

5. Estimating the strength of the arterial wall aneurysmal of the patient.

1. Obtain the blood pressure of the patient.

*Evaluation of the arterial wall strength.* 

1. Generation of geometry. The geometry can be generated or imported. In the case of aneurysm geometry is imported directly from the patient CT using some of the commercial software or open source currently available, so it has the actual geometry of the aneurysm affecting the patient under study. Figure 6, shows the geometric model of AAA obtained by the processing of medical images using the public software MeVisLab.

**Figure 6.** Geometric model of AAA obtained from the processing of medical images.

2. Discretization of meshing domain: The structure or part is divided into elements and modeled as a finite element mesh. In this step the analyst must decide the type, number, size and order of items to be used. This decision will characterize the degree of confidence results thereafter. An example that represents the arterial wall mesh is presented in Figure 7.

**Figure 7.** Mesh representation of a geometry that represents the arterial wall of an AAA.

3. Application of the boundary conditions: apply the loads which will be under the model (in this case the blood pressure) and the restrictions of the same (in this case is assumed to be attached to the remainder of the artery limiting their movement and must be taken into account if organs or body parts that limit their movement).


**Figure 8.** Stress distribution in the arterial wall obtained by finite element simulation.
