Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus

Abstract

Accurate tissue stress predictions for the annulus fibrosus are essential for understanding the factors that cause or contribute to disc degeneration and mechanical failure. Current computational models used to predict in vivo disc stresses utilize material laws for annular tissue that are not rigorously validated against experimental data. Consequently, predictions of disc stress resulting from physical activities may be inaccurate and therefore unreliable as a basis for defining mechanical-biologic injury criteria. To address this need we present a model for the annulus as an isotropic ground substance reinforced with two families of collagen fibers, and an approach for determining the material constants by simultaneous consideration of multiple experimental data sets. Two strain energy functions for the annulus are proposed and used in the theory to derive the constitutive equations relating the stress to pure stretch deformations. These equations are applied to four distinct experimental protocols and the material constants are determined from a simultaneous, nonlinear regression analysis. Good agreement between theory and experiment is achieved when the invariants are included within multiple, separate exponentials in the strain energy function.

A Bimodular Theory for Finite Deformations: Comparison of Orthotropic Second-order and Exponential Stress Constitutive Equations for Articular Cartilage

“…For example, in Huang et al (1999) the Poisson's ratios measured at 16% strain for UT in the 1 and 2 directions were 1.31 and 1.33, respectively, whereas the 6-parameter second-order model here resulted in constant Poisson's ratios of 0.499. In contrast to earlier studies with exponential strain energy functions (Klisch and Lotz 1999;Wagner and Lotz 2004), in this study an exact solution to UT can be obtained for the second-order models. The stress-strain curves for the exact solutions were nearly identical to those obtained in the nonlinear regression; consequently, the uncertainties in the Poisson's ratios do not seem to affect the models' ability to describe the UT stresses.…”

“…Lotz and colleagues proposed several exponential strain energy functions for the annulus fibrosus with the aim of obtaining a fit of the experimental data to within one standard deviation of the mean response. In particular, a 9-parameter model was used to fit two UT and two CC experiments (Klisch and Lotz 1999) and a 6-parameter model was used to fit three UT, two CC, one unconfined compression, and one biaxial tension experiment (Wagner and Lotz 2004). Other models for the Poisson's ratios were constant for the 6-PAR and 8-PAR models but not for the other models.…”

“…The unit vec tor E 1 is parallel to the local split-line direction, the unit vector E 3 is perpendicular to the articular surface, and the unit vector E 2 is per pendicular to the split-line direction and parallel to the surface. The cylinders labeled P 11 , P 22 , and P 33 represent specimens loaded in ten sion or compression along the E 1 , E 2 , and E 3 directions, respectively oped by Klisch and Lotz (1999), in which the LevenbergMarquardt method is used to minimize the sum of squared differences using the theoretical and experimental stress val ues. Four second-order models were studied, with 6, 7, 8, and 9 parameters (Table 2), and one exponential model was studied, with 7 parameters.…”

“…Lotz and colleagues developed an orthotropic finite deformation model for the annulus fibrosus using an exponential strain energy function; however, maximum errors between the the oretical and experimental stresses in uniaxial tension (UT) were approximately 50% (Klisch and Lotz 1999;Wagner and Lotz 2004). For articular cartilage, there has been no finite deformation model presented that accurately describes its orthotropic response for multiple experimental protocols including tension and compression.…”

“…Recently, several exponential models for finite deformations that allow for different mechanical properties in tension and compression have been used for the arterial wall (Holzapfel et al 2004) and the annulus fibro sus (Baer et al 2004). One reason that an exponential strain energy function is often used may be due to its ability to model the highly asymmetric tension-compression response without invoking the bimodular feature (Almeida and Spilker 1997;Klisch and Lotz 1999;Wagner and Lotz 2004).…”