Toward a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: identification of a finite strain constitutive model and evaluation of its applicability
Introduction
Abdominal aortic aneurysm (AAA) is a prevalent disease commonly localized to the abdominal aorta. If left untreated, the aneurysm diameter will enlarge, typically at the rate of 0.4 cm/yr (Cronenwett et al., 1990), until rupture. Since surgical resection of AAA involves major abdominal surgery with a mortality rate of 4–5% (Katz et al., 1992), it is prudent to repair the aneurysm only when it is felt that rupture is imminent. Hence, predicting the susceptibility of a particular AAA to rupture can significantly aid in the clinical management of these patients. A reliable predictor for AAA rupture does not presently exist. In the current clinical management of AAA patients, the maximum transverse dimension of the aneurysm is often used as the primary indicator of potential for rupture. Usually, when this dimension reaches 5 cm, the risk of rupture is assumed to justify elective surgical resection. However, this is not a fail-safe criterion since some AAA rupture at a size smaller than 5 cm and other have grown to as large as 8 cm without rupturing (Darling et al., 1977). It has been suggested that increasing wall stresses (Dobrin, 1989; Inzoli et al., 1993; Mower et al., 1993; Stringfellow et al., 1987; Elger et al., 1996; Vorp et al., 1998) and/or decreasing wall strength (Raghavan et al., 1996; Vorp et al., 1996) could be the ultimate cause of AAA rupture. Therefore, a method for the reliable, noninvasive estimation of AAA wall stresses may be a useful clinical tool for the assessment of rupture potential.
To perform accurate stress analysis of AAA, an appropriate constitutive theory for the AAA wall material is necessary. Most of the previously reported stress analyses of AAA have used either the law of Laplace (Dobrin, 1989) or the theory of linearized elasticity (Inzoli et al., 1993; Mower et al., 1993; Stringfellow et al., 1987; Elger et al., 1996; Vorp et al., 1998). However, these models are not appropriate for the AAA wall since ex vivo experiments show that the aneurysmal tissue is materially nonlinear and undergoes large strains of the order of 20–40% prior to failure (He and Roach, 1994; Raghavan et al., 1996). Therefore, a finite strain constitutive theory is necessary to reliably model AAA. Yamada et al. (1994) utilized a hyperelastic material model (and parameters) previously developed for normal aortic wall for AAA stress analysis. However, recent studies (He and Roach, 1994; Raghavan et al., 1996) have shown that the onset of aneurysm significantly affects the constitutive behavior and mechanical properties of the abdominal aortic wall. Therefore, it is more appropriate to develop a constitutive model particularly suited to AAA rather than using material models of normal abdominal aorta.
In this paper we present the development of a finite strain constitutive theory for AAA material based on known experimental data. We then attempt to answer two important questions related to the clinical utility of this model. First, how do the model parameters for AAA vary within the patient population? Secondly, does this variation lead to significant differences in computed stress distribution? We address the former question to help understand inherent differences in mechanical properties of AAA from patient to patient within a sample population. Experimental data from mechanical testing of 69 human AAA specimens were used to estimate the material parameters in each case by nonlinear regression. We address the latter question to assess the practicality of our constitutive model for patient-specific, noninvasive AAA stress analysis. In a clinical setting, the material model parameters specific to particular patient will not be known. Ex vivo (destructive) mechanical testing for the direct determination of these parameters is not possible for an intact, unruptured AAA, and alternative means of estimation are necessary. For example, one approach would be to utilize mean values from a large sample population. Another approach could be to use the known in vivo deformation of the AAA wall under physiological conditions to inverse-estimate the material parameters — a method that could become a reality with improved imaging capabilities. Irrespective of the alternate methodology used, there will be some degree of error in the predicted material parameters. It is therefore important to answer this second question by studying the sensitivity of computed AAA wall stresses to physiologic uncertainty in the constitutive model parameters. Towards this end, we perform stress analyses of hypothetical, asymmetric, three-dimensional (3D) models of AAA to determine the variation in the computed wall stress distribution due to variation in material parameters within their 95% confidence interval for the patient population.
Section snippets
Methods
We have previously suggested a uni-dimensional mathematical model for AAA wall material which conforms to the known microstructure of the AAA tissue while also accounting for its load-dependent response (Raghavan et al., 1996). Although the model provided structure-based interpretations of the mechanical characteristics of aneurysm tissue, it is not possible to expand it to the three-dimensional case and hence cannot be used for stress analyses of intact AAA. Our present approach, therefore, is
Results
The nonlinear regression for the parameter estimation converged for all specimens studied, with a minimum R2 value of 0.9 (see Fig. 2 for a representative fit of the model (7) to the experimental data). The estimated parameters had no dependence on the initial values, demonstrating that the acquired values are the true best-fits for the data based on global minimizations of the sum of square errors. We did not find a significant difference in or β(p>0.1) between the longitudinal and
Discussion
The knowledge of in vivo wall stress distribution in AAA may lead to more accurate clinical predictions of their risk of rupture, thereby serving to aid in the decision for elective repair. One necessary component of a method that would allow reliable, noninvasive estimates of wall stress in intact AAA is a suitable constitutive model that describes the mechanical behavior of the aneurysm wall. To the best of our knowledge, this is the first report to develop and evaluate a finite strain
Acknowledgements
This work was supported in part by a Biomedical Engineering Research Grant from the Whitaker Foundation. The authors would like to thank Drs. Marshall Webster, David Steed, Michel Makaroun, Satish Muluk, Robert Rhee and Jeffery Trachtenberg from the Division of Vascular Surgery, Department of Surgery, University of Pittsburgh, for supplying AAA specimens and providing useful insight to our work. We would also like to acknowledge Prof. K.R. Rajagopal for his input with constitutive model
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