Madhavan L. Raghavan

Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: identification of a finite strain constitutive model and evaluation of its applicability

Knowledge of the wall stresses in an abdominal aortic aneurysm (AAA) may be helpful in evaluating the need for surgical intervention to avoid rupture. This must be preceded by the development of a more suitable finite strain constitutive model for AAA, as none currently exists. Additionally, reliable stress analysis of in vivo AAA for the purposes of clinical diagnostics requires patient-specific values of the material parameters, which are difficult to determine noninvasively. The purpose of this work, therefore, was three-fold: (1) to develop a finite strain constitutive model for AAA; (2) to estimate the variation of model parameters within a sample population; and (3) to evaluate the sensitivity of computed stress distribution in AAA due to this biologic variation. We propose here a two parameter, hyperelastic, isotropic, incompressible material model and utilize experimental data from 69 freshly excised AAA specimens to both develop the functional form of the model and estimate its material parameters. Parametric analyses were performed via repeated finite element computations to determine the effect of varying each of the two model parameters on the stress distribution in a three-dimensional AAA model. The agreement between experimental data and the proposed functional form of the constitutive law was very good (R2 > 0.9). Our finite element simulations showed that the computed AAA wall stresses changed by only 4% or less when both the parameters were varied within the 95% confidence intervals for the patient population studied. This observation indicates that in lieu of the patient-specific material parameters, which are difficult to determine the use of population mean values is sufficiently accurate for the model to be reasonably employed in a clinical setting. We believe that this is an important advancement toward the development of a computational tool for the estimation of rupture potential for individual AAA, for which there is great clinical need.

Mechanical wall stress in abdominal aortic aneurysm: Influence of diameter and asymmetry

PURPOSE Risk for rupture of an abdominal aortic aneurysm is widely believed to be related to its maximum diameter. From a biomechanical standpoint, however, risk is probably more precisely related to mechanical wall stress. Many abdominal aortic aneurysms are asymmetric (for example because of anterior bulging with posterior expansion limited by the vertebral column). The purpose of this work was to investigate the effect of maximum diameter and asymmetric bulge on wall stress. METHODS Three-dimensional computer models of abdominal aortic aneurysms were generated. In one protocol, maximum diameter was held constant while bulge shape factor was varied. The shape factor took into account the asymmetric shape of the bulge. In a second protocol, the shape of the aneurysmal wall was held constant while maximum diameter was varied. Wall stress was computed in each instance with a commercial software package and assumption of physiologic intraluminal pressure. RESULTS Both maximum diameter and the shape factor were found to have substantial influence on the distribution of wall stress within the aneurysm. In some instances the maximum stress occurred at the midsection, and in others it occurred elsewhere. The magnitude of peak stress acting on the aneurysm increased nonlinearly with increasing maximum diameter or increasing asymmetry. CONCLUSIONS Our computer models showed that the stress within the wall of an abdominal aortic aneurysm and possibly the potential for rupture are as dependent on aneurysm shape as they are on maximum diameter. This information may be important in determining severity of individual abdominal aortic aneurysms and in improving understanding of the natural history of the disease.

Ex vivo biomechanical behavior of abdominal aortic aneurysm: Assessment using a new mathematical model

Knowledge of the biomechanical behavior of abdominal aortic aneurysm (AAA) as compared to nonaneurysmal aorta may provide information on the natural history of this disease. We have performed uniaxial tensile testing of excised human aneurysmal and nonaneurysmal abdominal aortic specimens. A new mathematical model that conforms to the fibrous structure of the vascular tissue was used to quantify the measured elastic response. We determined for each specimen the yield σy and ultimate σu strengths, the separate contribution to total tissue stiffness by elastin (EE) and collagen (EC) fibers, and a collagen recruitment parameter (A), which is a measure of the tortuosity of the collagen fibers. There was no significant difference in any of these mechanical properties between longitudinal and circumferential AAA specimens, nor inEE andEC between longitudinally oriented aneurysmal and normal specimens.A, σy, and σu were all significantly higher for the normal than for the aneurysmal group:A=0.223±0.046versus A=0.091±0.009 (mean ± SEM;p<0.0005), σyversus σy (p<0.05), and σuversus σu (p<0.0005), respectively. Our findings suggest that the AAA tissue is isotropic with respect to these mechanical properties. The observed difference inA between aneurysmal and normal aorta may be due to the complete recruitment and loading of collagen fibers at lower extensions in the former. Our data indicate that AAA rupture may be related to a reduction in tensile strength and that the biomechanical properties of AAA should be considered in assessing the severity of an individual aneurysm.

Three-dimensional geometrical characterization of cerebral aneurysms.

The risk of rupture of cerebral aneurysms has been correlated with the size of the aneurysm sac. It is conceivable that geometrical shape, not just size may also be related to aneurysm rupture potential. Further, aneurysm shape may also be a factor in deciding on treatment modalities, i.e., to clip or coil. However, our ability to make use of available information on aneurysm shape remains poor. In this study, methods were developed to quantify the seemingly arbitrary three-dimensional geometry of the aneurysm sac, using differential and computational geometry techniques. From computed tomography angiography (CTA) data, the three-dimensional geometry of five unruptured human cerebral aneurysms was reconstructed. Various indices (maximum diameter, neck diameter, height, aspect ratio, bottleneck factor, bulge location, volume, surface area, Gaussian and mean curvatures, isoperimetric ratio, and convexity ratio) were utilized to characterize the geometry of these aneurysm surfaces and four size-matched hypothetical control aneurysms. The physical meanings of various indices and their possible role as prognosticators for rupture risk and presurgical planning were discussed.

In vivo three-dimensional surface geometry of abdominal aortic aneurysms.

AbstractAbdominal aortic aneurysm (AAA) is a local, progressive dilation of the distal aorta that risks rupture until treated. Using the law of Laplace, in vivo assessment of AAA surface geometry could identify regions of high wall tensions as well as provide critical dimensional and shape data for customized endoluminal stent grafts. In this study, six patients with AAA underwent spiral computed tomography imaging and the inner wall of each AAA was identified, digitized, and reconstructed. A biquadric surface patch technique was used to compute the local principal curvatures, which required no assumptions regarding axisymmetry or other shape characteristics of the AAA surface. The spatial distribution of AAA principal curvatures demonstrated substantial axial asymmetry, and included adjacent elliptical and hyperbolic regions. To determine how much the curvature spatial distributions were dependent on tortuosity versus bulging, the effects of AAA tortuosity were removed from the three-dimensional (3D) reconstructions by aligning the centroids of each digitized contour to the z axis. The spatial distribution of principal curvatures of the modified 3D reconstructions were found to be largely axisymmetric, suggesting that much of the surface geometric asymmetry is due to AAA bending. On average, AAA surface area increased by 56% and abdominal aortic length increased by 27% over those for the normal aorta. Our results indicate that AAA surface geometry is highly complex and cannot be simulated by simple axisymmetric models, and suggests an equally complex wall stress distribution.

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

PURPOSE To test the hypothesis that ruptured abdominal aortic aneurysms (AAA) are globally weaker than unruptured ones. METHODS Four 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. RESULTS The 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±3.6N/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/cm(2); 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±0.4mm; 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. CONCLUSION The findings are not consistent with the hypothesis that ruptured aortic aneurysms are globally weaker than unruptured ones.

Non-Invasive Determination of Zero-Pressure Geometry of Arterial Aneurysms

Arterial aneurysms are in a pre-deformed state in vivo under non-zero pressure. The ability to determine their zero pressure geometry may help in improving accuracy of determination of stress distribution and reverse estimation of material properties from dynamic imaging data. An approximate method to recover the zero pressure geometry of the AAA is proposed. This method is motivated by the observation that the patterns in displacement field for a given AAA are strikingly consistent in an AAA under all physiological pressures. The basic principle is to leverage this observation to iteratively identify the geometry that when subjected to the in vivo pressure, will recover the geometry reconstructed from in vivo imaging. The methodology is demonstrated and validated using patient-specific AAA models.

Three-dimensional finite element analysis of residual stress in arteries.

Calculation of residual stress in arteries, using the analytical approach has been quite valuable in our understanding of its critical role in vascular mechanics. Stresses are calculated at the central section of an infinitely long tube by imposing a constant axial stretch while deforming the artery from the stress-free state to its unloaded state. However, segments used to perform opening-angle measurements have finite lengths. Further, the stress-free artery configuration is assumed to be circular. Experiments show that they are slightly noncircular. The numerical approach to residual stress calculation can allow us to study both these issues. Using 3D cylindrical geometries and an isotropic material model, we investigated how segment length can affect residual stress calculations and identified the appropriate segment length for experiments. Further, we recorded and used the true noncircular stress-free state of an artery segment, computed the residual stress distribution, and compared it to that from a similar, but circular segment. Our findings suggest that segment length must be ten times the wall thickness for it to be “long” enough. We also found that the circularity assumption may be a reasonable approximation for typical arteries.

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.

Nonlinear anisotropic stress analysis of anatomically realistic cerebral aneurysms.

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 100 mm Hg. 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 0.30 MPa in a small aneurysm to as high as 1.06 MPa 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.