An approach to quantitative assessment of hemodynamic differences between unruptured and ruptured ophthalmic artery aneurysms

Background and purpose: Hemodynamic parameters are important in the pathogenesis, evolution and rupture of intracranial aneurysm. Energy loss (EL) has been applied for the rupture risk prediction of artery aneurysms recently. We proposed a new EL and further investigate its effects on the rupture of aneurysms. Materials and methods: Sixty-four patient-specific ophthalmic aneurysm datasets were divided into ruptured and unruptured groups based on their clinical history. Based on patient-specific 3D-DSA data, realistic models were retrospectively reconstructed and then analyzed by using computational fluid dynamic method. Results: The flow field feature EL in ruptured cases was significantly higher than that in unruptured cases. The average wall shear stress (WSS) and the maximum WSS in ruptured cases were higher than those in unruptured cases. Modified pressure loss coefficient (PLC_M) in ruptured cases was slight higher than that in unruptured cases but the difference has no statistical significance. Multivariate logistic regression analysis demonstrated flow field feature EL (p < 0.05) and the maximum WSS (p < 0.05) were the only independently significant variables to predict rupture of ophthalmic aneurysm. There were no differences in PLC_M, the maximum oscillatory shear index (OSI), the average OSI and AR between the two groups. Conclusion: Flow field feature EL may be a reliable factor to predict the rupture risk of aneurysms.     

Biomechanical wall properties of human intracranial aneurysms resected following surgical clipping (IRRAs Project)

Background and purposeIndividual rupture risk assessment of intracranial aneurysms is a major issue in the clinical management of asymptomatic aneurysms. Aneurysm rupture occurs when wall tension exceeds the strength limit of the wall tissue. At present, aneurysmal wall mechanics are poorly understood and thus, risk assessment involving mechanical properties is inexistent. Aneurysm computational hemodynamics studies make the assumption of rigid walls, an arguable simplification. We therefore aim to assess mechanical properties of ruptured and unruptured intracranial aneurysms in order to provide the foundation for future patient-specific aneurysmal risk assessment. This work also challenges some of the currently held hypotheses in computational flow hemodynamics research.MethodsA specific conservation protocol was applied to aneurysmal tissues following clipping and resection in order to preserve their mechanical properties. Sixteen intracranial aneurysms (11 female, 5 male) underwent mechanical uniaxial stress tests under physiological conditions, temperature, and saline isotonic solution. These represented 11 unruptured and 5 ruptured aneurysms. Stress/strain curves were then obtained for each sample, and a fitting algorithm was applied following a 3-parameter (C₁₀, C₀₁, C₁₁) Mooney–Rivlin hyperelastic model. Each aneurysm was classified according to its biomechanical properties and (un)rupture status.ResultsTissue testing demonstrated three main tissue classes: Soft, Rigid, and Intermediate. All unruptured aneurysms presented a more Rigid tissue than ruptured or pre-ruptured aneurysms within each gender subgroup. Wall thickness was not correlated to aneurysmal status (ruptured/unruptured). An Intermediate subgroup of unruptured aneurysms with softer tissue characteristic was identified and correlated with multiple documented risk factors of rupture.ConclusionThere is a significant modification in biomechanical properties between ruptured aneurysm, presenting a soft tissue and unruptured aneurysms, presenting a rigid material. This finding strongly supports the idea that a biomechanical risk factor based assessment should be utilized in the to improve the therapeutic decision making.     

Morphological and Hemodynamic Discriminators for Rupture Status in Posterior Communicating Artery Aneurysms

Background and Purpose

The conflicting findings of previous morphological and hemodynamic studies on intracranial aneurysm rupture may be caused by the relatively small sample sizes and the variation in location of the patient-specific aneurysm models. We aimed to determine the discriminators for aneurysm rupture status by focusing on only posterior communicating artery (PCoA) aneurysms.

Materials and Methods

In 129 PCoA aneurysms (85 ruptured, 44 unruptured), clinical, morphological and hemodynamic characteristics were compared between the ruptured and unruptured cases. Multivariate logistic regression analysis was performed to determine the discriminators for rupture status of PCoA aneurysms.

Results

While univariate analyses showed that the size of aneurysm dome, aspect ratio (AR), size ratio (SR), dome-to-neck ratio (DN), inflow angle (IA), normalized wall shear stress (NWSS) and percentage of low wall shear stress area (LSA) were significantly associated with PCoA aneurysm rupture status. With multivariate analyses, significance was only retained for higher IA (OR = 1.539, p < 0.001) and LSA (OR = 1.393, p = 0.041).

Conclusions

Hemodynamics and morphology were related to rupture status of intracranial aneurysms. Higher IA and LSA were identified as discriminators for rupture status of PCoA aneurysms.     

Flow instability detected in ruptured versus unruptured cerebral aneurysms at the internal carotid artery

Flow instability has emerged as a new hemodynamic metric hypothesized to have potential value in assessing the rupture risk of cerebral aneurysms. However, diverse findings have been reported in the literature. In the present study, high-resolution hemodynamic simulations were performed retrospectively on 35 aneurysms (10 ruptured & 25 unruptured) located at the internal carotid artery (ICA). Simulated hemodynamic parameters were statistically compared between the ruptured and unruptured aneurysms, with emphasis on examining the correlation of flow instability with the status of aneurysm rupture. Pronounced flow instability was detected in 20% (2 out of 10) of the ruptured aneurysms, whereas in 44% (11 out of 25) of the unruptured aneurysms. Statistically, the flow instability metric (quantified by the temporally and spatially averaged fluctuating kinetic energy over the aneurysm sac) did not differ significantly between the ruptured and unruptured aneurysms. In contrast, low wall shear stress area (LSA) and pressure loss coefficient (PLC) exhibited significant correlations with the status of aneurysm rupture. In conclusion, the present study suggests that the presence of flow instability may not correlate closely with the status of aneurysm rupture, at least for ICA aneurysms. On the other hand, the retrospective nature of the study and the small sample size may have to some extent compromised the reliability of the conclusion, and therefore large-scale prospective studies would be needed to further address the issue.     

Improving the Efficiency of Abdominal Aortic Aneurysm Wall Stress Computations

An abdominal aortic aneurysm is a pathological dilation of the abdominal aorta, which carries a high mortality rate if ruptured. The most commonly used surrogate marker of rupture risk is the maximal transverse diameter of the aneurysm. More recent studies suggest that wall stress from models of patient-specific aneurysm geometries extracted, for instance, from computed tomography images may be a more accurate predictor of rupture risk and an important factor in AAA size progression. However, quantification of wall stress is typically computationally intensive and time-consuming, mainly due to the nonlinear mechanical behavior of the abdominal aortic aneurysm walls. These difficulties have limited the potential of computational models in clinical practice. To facilitate computation of wall stresses, we propose to use a linear approach that ensures equilibrium of wall stresses in the aneurysms. This proposed linear model approach is easy to implement and eliminates the burden of nonlinear computations. To assess the accuracy of our proposed approach to compute wall stresses, results from idealized and patient-specific model simulations were compared to those obtained using conventional approaches and to those of a hypothetical, reference abdominal aortic aneurysm model. For the reference model, wall mechanical properties and the initial unloaded and unstressed configuration were assumed to be known, and the resulting wall stresses were used as reference for comparison. Our proposed linear approach accurately approximates wall stresses for varying model geometries and wall material properties. Our findings suggest that the proposed linear approach could be used as an effective, efficient, easy-to-use clinical tool to estimate patient-specific wall stresses.     

Automated methodology for determination of stress distribution in human abdominal aortic aneurysm

Knowledge of impending abdominal aortic aneurysm (AAA) rupture can help in surgical planning. Typically, aneurysm diameter is used as the indicator of rupture, but recent studies have hypothesized that pressure-induced biomechanical stress may be a better predictor Verification of this hypothesis on a large study population with ruptured and unruptured AAA is vital if stress is to be reliably used as a clinical prognosticator for AAA rupture risk. We have developed an automated algorithm to calculate the peak stress in patient-specific AAA models. The algorithm contains a mesh refinement module, finite element analysis module, and a postprocessing visualization module. Several aspects of the methodology used are an improvement over past reported approaches. The entire analysis may be run from a single command and is completed in less than 1 h with the peak wall stress recorded for statistical analysis. We have used our algorithm for stress analysis of numerous ruptured and unruptured AAA models and report some of our results here. By current estimates, peak stress in the aortic wall appears to be a better predictor of rupture than AAA diameter. Further use of our algorithm is ongoing on larger study populations to convincingly verify these findings.     

Inflation of an artery leading to aneurysm formation and rupture

Formation and rupture of aneurysms due to the inflation of an artery with collagen fibers distributed in two preferred directions, subjected to internal pressure and axial stretch are examined within the framework of nonlinear elasticity. A two layer tube model with a fiber-reinforced composite based incompressible anisotropic hyperelastic constitutive material is employed to model the stress-strain behavior of the artery wall with distributed collagen fibers. The artery wall takes up a uniform inflation deformation, and there are no aneurysms in the artery under the normal condition. But an aneurysm may be formed in arteries when the stiffness of the fibers is decreased to a certain value or the direction of the fibers is changed to a certain degree towards the circumferential direction. The aneurysm may expand to much large extent and become complex in shape. One portion of the aneurysm becomes highly distended as a bubble while the rest remains lightly inflated. The rupture of the aneurysm is discussed along with the distribution of stresses. Critical pressures and the rupture pressures are given for different collagen fiber orientations or stiffness. Furthermore, the stability of the solutions is discussed to explain the formation of aneurysm.     

A fluid–structure interaction study using patient-specific ruptured and unruptured aneurysm: The effect of aneurysm morphology,hypertension and elasticity

Fluid–structure interaction (FSI) simulations using five patient-specific aneurysm geometries are carried out to investigate the difference between ruptured and unruptured aneurysms.     

Posterior Cerebral Artery Angle and the Rupture of Basilar Tip Aneurysms

Since the initial publication of the International Study of Unruptured Intracranial Aneurysms (ISUIA), management of unruptured intracranial aneurysms has been mainly based on the size of the aneurysm. The contribution of morphological characteristics to treatment decisions of unruptured aneurysms has not been well studied in a systematic and location specific manner. We present a large sample of basilar artery tip aneurysms (BTA) that were assessed using a diverse array of morphological variables to determine the parameters associated with ruptured aneurysms. Demographic and clinical risk factors of aneurysm rupture were obtained from chart review. CT angiograms (CTA) were evaluated with Slicer, an open source visualization and image analysis software, to generate 3-D models of the aneurysms and surrounding vascular architecture. Morphological parameters examined in each model included aneurysm volume, aspect ratio, size ratio, aneurysm angle, basilar vessel angle, basilar flow angle, and vessel to vessel angles. Univariate and multivariate analyses were performed to determine statistical significance. From 2008–2013, 54 patients with BTA aneurysms were evaluated in a single institution, and CTAs from 33 patients (15 ruptured, 18 unruptured) were available and analyzed. Aneurysms that underwent reoperation, that were associated with arteriovenous malformations, or that lacked preoperative CTA were excluded. Multivariate logistic regression revealed that a larger angle between the posterior cerebral arteries (P1-P1 angle, p = 0.037) was most strongly associated with aneurysm rupture after adjusting for other morphological variables. In this location specific study of BTA aneurysms, the larger the angle formed between posterior cerebral arteries was found to be a new morphological parameter significantly associated with ruptured BTA aneurysms. This is a physically intuitive parameter that can be measured easily and readily applied in the clinical setting.     

Morphometry and hemodynamics of posterior communicating artery aneurysms: Ruptured versus unruptured

Posterior communicating artery (PCoA) aneurysms frequently rupture in small size (<7 mm). The aim of the study is to demonstrate morphometric and hemodynamic analyses in ruptured and unruptured PCoA aneurysms to improve predictive accuracy for rupture. Geometrical models were reconstructed from rotational DSA images of 57 ruptured and 22 unruptured side-wall PCoA aneurysms, which were classified into four two-dimensional (2D) groups by a combination of H/D and H/S ratios (H: dome height, D: dome diameter, and S: semi-axis height). Surface area ratio (SAR) of low time-averaged wall shear stress (TAWSS, ≤4 dynes cm⁻²) and high oscillatory shear index (OSI, ≥0.15) were computed in aneurysms. We hypothesized that a two-step analysis method, i.e., one-dimensionally morphometric and hemodynamic analyses in each 2D group, can enhance accuracy of PCoA aneurysm rupture evaluation. There was the highest incidence of H/D > 1 and H/S ≤ 2 with the largest surface area and SAR-TAWSS, but the lowest incidence of H/D ≤ 1 and H/S > 2 with the smallest surface area and SAR-TAWSS in ruptured PCoA aneurysms. PCoA aneurysms of H/D > 1 and H/S ≤ 2 with surface area > 70 mm², H/D ≤ 1 and H/S > 2 with neck diameter > 2.3 mm, H/D ≤ 1 and H/S ≤ 2 with aneurysmal height/parent diameter ratio > 1.0, and H/D > 1 and H/S > 2 with aneurysmal angle > 115° need special attention for clinical diagnosis and treatment. The study highlighted the importance of the two-step analysis method for clinical evaluation of PCoA aneurysm rupture.     

Morphological Parameters Associated with Ruptured Posterior Communicating Aneurysms

The rupture risk of unruptured intracranial aneurysms is known to be dependent on the size of the aneurysm. However, the association of morphological characteristics with ruptured aneurysms has not been established in a systematic and location specific manner for the most common aneurysm locations. We evaluated posterior communicating artery (PCoA) aneurysms for morphological parameters associated with aneurysm rupture in that location. CT angiograms were evaluated to generate 3-D models of the aneurysms and surrounding vasculature. Univariate and multivariate analyses were performed to evaluate morphological parameters including aneurysm volume, aspect ratio, size ratio, distance to ICA bifurcation, aneurysm angle, vessel angles, flow angles, and vessel-to-vessel angles. From 2005–2012, 148 PCoA aneurysms were treated in a single institution. Preoperative CTAs from 63 patients (40 ruptured, 23 unruptured) were available and analyzed. Multivariate logistic regression revealed that smaller volume (p = 0.011), larger aneurysm neck diameter (0.048), and shorter ICA bifurcation to aneurysm distance (p = 0.005) were the most strongly associated with aneurysm rupture after adjusting for all other clinical and morphological variables. Multivariate subgroup analysis for patients with visualized PCoA demonstrated that larger neck diameter (p = 0.018) and shorter ICA bifurcation to aneurysm distance (p = 0.011) were significantly associated with rupture. Intracerebral hemorrhage was associated with smaller volume, larger maximum height, and smaller aneurysm angle, in addition to lateral projection, male sex, and lack of hypertension. We found that shorter ICA bifurcation to aneurysm distance is significantly associated with PCoA aneurysm rupture. This is a new physically intuitive parameter that can be measured easily and therefore be readily applied in clinical practice to aid in the evaluation of patients with PCoA aneurysms.     

Correlation Between the Rupture Risk and 3D Geometric Parameters of Saccular Intracranial Aneurysms

The purpose of this study is to evaluate the association of the location and geometric parameters of intracranial aneurysm with the risk of rupture. A retrospective study consisted of 284 patients diagnosed with saccular intracranial aneurysm between January 2009 and May 2013 at Wuxi Third People’s Hospital was conducted. 3D digital subtraction angiography images from all patients (240 ruptured, 44 unruptured) were obtained and analyzed. The location of the aneurysms and the 3D geometric parameters including the aneurysm depth, the neck size, diameter of the parent artery, aneurysm angle, aspect radio, size ratio, and the neck-to-parent-artery ratio (NPR) were compared between ruptured and unruptured groups. Results: In ruptured group, anterior communicating artery, posterior communicating artery (PCoA), and the bifurcation of internal carotid artery (ICA) were the top three locations for aneurysm occurrence, accounting for 40.00, 30.42, and 12.08 % respectively. While in the unruptured group, top three locations were PCoA (36.36 %), posterior cerebral circulation (18.18 %), and the bifurcation of the ICA (15.91 %). Distribution of aneurysm location is significantly different (p < 0.05) between ruptured and unruptured aneurysms. For the 3D geometric parameters characterizing aneurysm, aneurysm depth (p < 0.05), parent artery diameter (p < 0.05), aneurysm angle (p < 0.01), aspect ratio (p < 0.01), and size ratio (p < 0.01) all showed a significant difference between ruptured and unruptured group. No difference was found in the neck size and the NPR ratio between the two groups. 3D geometric parameters such as aneurysm depth, parent artery diameter, aneurysm angle, aspect ratio, and size ratio can be helpful in evaluating the rupture risk of saccular intracranial aneurysm for a better prevention and prognosis.     

Biomechanical failure properties and microstructural content of ruptured and unruptured abdominal aortic aneurysms

PurposeTo test the hypothesis that ruptured abdominal aortic aneurysms (AAA) are globally weaker than unruptured ones.MethodsFour ruptured and seven unruptured AAA specimens were harvested whole from fresh cadavers during autopsies performed over an 18-month period. Multiple regionally distributed longitudinally oriented rectangular strips were cut from each AAA specimen for a total of 77 specimen strips. Strips were subjected to uniaxial extension until failure. Sections from approximately the strongest and weakest specimen strips were studied histologically and histochemically. From the load-extension data, failure tension, failure stress and failure strain were calculated. Rupture site characteristics such as location, arc length of rupture and orientation of rupture were also documented.ResultsThe failure tension, a measure of the tissue mechanical caliber was remarkably similar between ruptured and unruptured AAA (group mean±standard deviation of within-subject means: 11.2±2.3 versus 11.6±3.6 N/cm; p=0.866 by mixed model ANOVA). In post-hoc analysis, there was little difference between the groups in other measures of tissue mechanical caliber as well such as failure stress (95±28 versus 98±23 N/cm²; p=0.870), failure strain (0.39±0.09 versus 0.36±0.09; p=0.705), wall thickness (1.7±0.4 versus 1.5±0.4 mm; p=0.470) , and % coverage of collagen within tissue cross section (49.6±12.9% versus 60.8±9.6%; p=0.133). In the four ruptured AAA, primary rupture sites were on the lateral quadrants (two on left; one on left-posterior; one on right). Remarkably, all rupture lines had a longitudinal orientation and ranged from 1 to 6 cm in length.ConclusionThe findings are not consistent with the hypothesis that ruptured aortic aneurysms are globally weaker than unruptured ones.     

Analysis of Morphological Parameters to Differentiate Rupture Status in Anterior Communicating Artery Aneurysms

In contrast to size, the association of morphological characteristics of intracranial aneurysms with rupture has not been established in a systematic manner. We present an analysis of the morphological variables that are associated with rupture in anterior communicating artery aneurysms to determine site-specific risk variables. One hundred and twenty-four anterior communicating artery aneurysms were treated in a single institution from 2005 to 2010, and CT angiograms (CTAs) or rotational angiography from 79 patients (42 ruptured, 37 unruptured) were analyzed. Vascular imaging was evaluated with 3D Slicer© to generate models of the aneurysms and surrounding vasculature. Morphological parameters were examined using univariate and multivariate analysis and included aneurysm volume, aspect ratio, size ratio, distance to bifurcation, aneurysm angle, vessel angle, flow angle, and parent-daughter angle. Multivariate logistic regression revealed that size ratio, flow angle, and parent-daughter angle were associated with aneurysm rupture after adjustment for age, sex, smoking history, and other clinical risk factors. Simple morphological parameters such as size ratio, flow angle, and parent-daughter angle may thus aid in the evaluation of rupture risk of anterior communicating artery aneurysms.     

Regional distribution of wall thickness and failure properties of human abdominal aortic aneurysm

The regional distribution of wall thickness and failure properties in human abdominal aortic aneurysm (AAA) was explored. Three unruptured and one ruptured AAA were harvested as a whole during necropsy. Thickness was measured at about every 1.5 cm² wall surface area for an average of 100 measurement sites per AAA. Multiple longitudinally oriented rectangular specimen strips were cut at various locations from each AAA for a total of 48 strips. The strips were subjected to uniaxial extension until failure. Wall thickness varied regionally and between AAA from as low as 0.23 mm at a rupture site to 4.26 mm at a calcified site (median=1.48 mm). Wall thickness was slightly lower in the posterior and right regions. The failure tension (ultimate) of specimen strips varied regionally and between AAA from 5.5 N/cm close to a blister site in the ruptured AAA to 42.3 N/cm at the undilated neck of a 4 cm diameter unruptured AAA (median=14.8 N/cm). Failure stress (ultimate) varied from 33.6 to 235.1 N/cm² (median=126.6 N/cm²). There was no perceptible pattern in failure properties along the circumference. Failure tension of specimen strips at or close to blisters was mostly low. The rupture site in the ruptured aneurysm had the lowest recorded wall thickness of 0.23 mm with only slightly higher readings within a 1 cm radius. The failure tension of the specimen strip close to the rupture site was low (11.1 N/cm) compared to its neighborhood in the ruptured aneurysm.     

Modeling rupture of growing aneurysms

Growth and rupture of aneurysms are driven by micro-structural alterations of the arterial wall yet precise mechanisms underlying the process remain to be uncovered. In the present work we examine a scenario when the aneurysm evolution is dominated by turnover of collagen fibers. In the latter case it is natural to hypothesize that rupture of individual fibers (or their bonds) causes the overall aneurysm rupture. We examine this hypothesis in computer simulations of growing aneurysms in which constitutive equations describe both collagen evolution and failure. Failure is enforced in constitutive equations by limiting strain energy that can be accumulated in a fiber. Within the proposed theoretical framework we find a range of parameters that lead to the aneurysm rupture. We conclude in a qualitative agreement with clinical observations that some aneurysms will rupture while others will not.     

On the potentially protective role of contact constraints on saccular aneurysms

Most intracranial saccular aneurysms remain asymptomatic until rupture. Yet, some unruptured lesions present with various symptoms, often related to the compression of a nerve or other intracranial tissue. An obvious question, therefore, is whether or not symptomatic unruptured lesions necessarily have a greater rupture-potential than asymptomatic ones. In this paper, we show numerically that contact constraints can have a protective effect on certain lesions. Specifically, finite element analyses of stress fields in model axisymmetric lesions, with and without the presence of a rigid contacting obstacle at the fundus, reveal that with the exception of near point loads, the constraint decreases the stresses near the fundus. Given that it is well accepted that rupture occurs when wall stress exceeds wall strength, these findings suggest that the rupture-potential will be lower in at least one sub-class of constrained versus comparable unconstrained lesions. Because of the myriad of sizes, shapes, and compositions of saccular aneurysms, however, there is a need to examine this important issue further, hopefully based on an increased awareness for clinical data on lesion-tissue interactions.     

Realistic non-Newtonian viscosity modelling highlights hemodynamic differences between intracranial aneurysms with and without surface blebs

Most computational fluid dynamic (CFD) simulations of aneurysm hemodynamics assume constant (Newtonian) viscosity, even though blood demonstrates shear-thinning (non-Newtonian) behavior. We sought to evaluate the effect of this simplifying assumption on hemodynamic forces within cerebral aneurysms, especially in regions of low wall shear stress, which are associated with rupture. CFD analysis was performed for both viscosity models using 3D rotational angiography volumes obtained for 26 sidewall aneurysms (12 with blebs, 12 ruptured), and parametric models incorporating blebs at different locations (inflow/outflow zone). Mean and lowest 5% values of time averaged wall shear stress (TAWSS) computed over the dome were compared using Wilcoxon rank-sum test. Newtonian modeling not only resulted in higher aneurysmal TAWSS, specifically in areas of low flow and blebs, but also showed no difference between aneurysms with or without blebs. In contrast, for non-Newtonian analysis, bleb-bearing aneurysms showed significantly lower 5% TAWSS compared to those without (p=0.005), despite no significant difference in mean dome TAWSS (p=0.32). Non-Newtonian modeling also accentuated the differences in dome TAWSS between ruptured and unruptured aneurysms (p<0.001). Parametric models further confirmed that realistic non-Newtonian viscosity resulted in lower bleb TAWSS and higher focal viscosity, especially when located in the outflow zone. The results show that adopting shear-thinning non-Newtonian blood viscosity in CFD simulations of intracranial aneurysms uncovered hemodynamic differences induced by bleb presence on aneurysmal surfaces, and significantly improved discriminant statistics used in risk stratification. These findings underline the possible implications of using a realistic model of blood viscosity in predictive computational hemodynamics.     

Dynamic effects on the formation and rupture of aneurysms

Dynamic analysis of an axially stretched arterial wall with collagen fibers distributed in two preferred directions under a suddenly applied constant internal pressure along with the possibility of the formation and rupture of aneurysm are examined within the framework of nonlinear dynamics. A two layer tube model with the fiber-reinforced composite-based incompressible anisotropic hyper-elastic material is employed to model the mechanical behavior of the arterial wall. The maximum amplitudes and the phase diagrams are given by numerical computation of the differential relation. It is shown that the arterial wall undergoes nonlinear periodic oscillation and no aneurysms are formed under the normal condition. However, an aneurysm may be formed under such abnormal conditions as the stiffness of the fibers is deduced or the direction of the fibers is oriented towards the axial direction. Furthermore, the possibility for the rupture of aneurysm is discussed with the distribution of stresses.