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TECHNICAL PAPERS: Soft Tissue

# Nonlinear Anisotropic Stress Analysis of Anatomically Realistic Cerebral Aneurysms

[+] Author and Article Information
Baoshun Ma1

Department of Biomedical Engineering, 1402 SC, University of Iowa, Iowa City, IA 52242

Jia Lu

Department of Mechanical and Industrial Engineering, University of Iowa, Iowa City, IA 52242jia-lu@uiowa.edu

Robert E Harbaugh

Departments of Neurosurgery, and Engineering Science and Mechanics, Penn State University, Hershey, PA 17033rharbaugh@psu.edu

Department of Biomedical Engineering, 1402 SC, University of Iowa, Iowa City, IA 52242ml-raghavan@uiowa.edu

1

Present address: Department of Biomedical Engineering, Boston University, Boston, MA 02215.

2

Corresponding author.

J Biomech Eng 129(1), 88-96 (Jul 21, 2006) (9 pages) doi:10.1115/1.2401187 History: Received December 11, 2005; Revised July 21, 2006

## Abstract

Background. Static deformation analysis and estimation of wall stress distribution of patient-specific cerebral aneurysms can provide useful insights into the disease process and rupture. Method of Approach. The three-dimensional geometry of saccular cerebral aneurysms from 27 patients (18 unruptured and nine ruptured) was reconstructed based on computer tomography angiography images. The aneurysm wall tissue was modeled using a nonlinear, anisotropic, hyperelastic material model (Fung-type) which was incorporated in a user subroutine in ABAQUS . Effective material fiber orientations were assumed to align with principal surface curvatures. Static deformation of the aneurysm models were simulated assuming uniform wall thickness and internal pressure load of $100mmHg$. Results. The numerical analysis technique was validated by quantitative comparisons to results in the literature. For the patient-specific models, in-plane stresses in the aneurysm wall along both the stiff and weak fiber directions showed significant regional variations with the former being higher. The spatial maximum of stress ranged from as low as $0.30MPa$ in a small aneurysm to as high as $1.06MPa$ in a giant aneurysm. The patterns of distribution of stress, strain, and surface curvature were found to be similar. Sensitivity analyses showed that the computed stress is mesh independent and not very sensitive to reasonable perturbations in model parameters, and the curvature-based criteria for fiber orientations tend to minimize the total elastic strain energy in the aneurysms wall. Within this small study population, there were no statistically significant differences in the spatial means and maximums of stress and strain values between the ruptured and unruptured groups. However, the ratios between the stress components in the stiff and weak fiber directions were significantly higher in the ruptured group than those in the unruptured group. Conclusions. A methodology for nonlinear, anisotropic static deformation analysis of geometrically realistic aneurysms was developed, which can be used for a more accurate estimation of the stresses and strains than previous methods and to facilitate prospective studies on the role of stress in aneurysm rupture.

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## Figures

Figure 1

A basilar tip cerebral aneurysm (CA1) and the contiguous vasculature. The aneurysm surface was isolated from the vasculature by a cutting plane. This isolated aneurysm is used as a representative case and shown in its top view for illustration of results. The boundary of the isolated aneurysm was fixed in space in static deformation analysis.

Figure 2

A half ellipsoid aneurysm model and the computed stiff fiber directions. The circumferential (local 1) direction is the stiff fiber direction and the meridional (local 2) direction is the weak fiber direction according to the criteria proposed in this study.

Figure 3

Computed surface curvature, fiber direction, stress, and strain for a realistic aneurysm model (CA1, top view). The deformed aneurysm surface was used in the plots. The surface curvature was computed based on the deformed surface geometry. The range of contour levels of k1 have been adjusted to better review the patterns of curvature distribution. The results based on linear elasticity theory are also shown (f).

Figure 4

Spatial mean and maximum values of σ11 and σ22 in the study population categorized based on their presentation in the clinic as either unruptured (UR) or ruptured (R). σ11 and σ22 were stress values in the local 1 and 2 directions, respectively. The boxes denote the 25th, 50th (median), and 75th percentiles.

Figure 5

Spatial mean and maximum values of ε11 and ε22 in the study population categorized based on their presentation in the clinic as either unruptured (UR) or ruptured (R). The boxes denote the 25th, 50th (median), and 75th percentiles.

Figure 6

Spatial mean and maximum values of σ11∕σ22 in the study population categorized based on their presentation in the clinic as either unruptured (UR) or ruptured (R). The ratio was different with statistical significance between the two groups for the spatial mean values (p=0.004), but not for spatial maximum values (p=0.142). The boxes denote the 25th, 50th (median), and 75th percentiles.

Figure 7

Stresses and deformed shapes for half ellipsoid aneurysm model by using the material model in Ref. 21. σ11 and σ22 are stress values in the circumferential and meridional directions, and p80 and p160 refer to internal pressure of 80 and 160mmHg, respectively. The material property varies from isotropic at the dome (c1=c2) to three times circumferentially stiffer at the neck (c1=3c2) as a linear function of normalized arc length.

Figure 8

Stresses for the half ellipsoid aneurysm model by using averaged aneurysm material properties. The results for shell and axisymmetric membrane elements, and isotropic and anisotropic analyses were compared. σ11 and σ22 were stress values in the circumferential and meridional directions, respectively. For σ22 the results for both shell and membrane elements and isotropic and anisotropic analysis were almost indistinguishable.

Figure 9

Sensitivity to material model parameters for a representative aneurysm (CA1)

Figure 10

Sensitivity of stresses to mesh density (total number of nodes in mesh) for a realistic aneurysm model (CA1). The total number of nodes for this aneurysm is 2695 and ranges from 1270 to 5444 for the database of 27 aneurysms in this study in all other computations.

Figure 11

Total elastic strain energy in a realistic aneurysm model when the fiber directions were perturbed from the optimum curvature-determined directions. Note that the strain energy is minimized when the proposed criteria is used.

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