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

TL;DR: A model for the annulus is presented 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 is presented.

About: This article is published in Journal of Biomechanics.The article was published on 1999-10-01 and is currently open access. It has received 95 citations till now. The article focuses on the topics: Annulus (firestop) & Constitutive equation.

Summary

Introduction

• Accurate tissue stress predictions for the annulus fibrosus are essential for understanding the factors that cause or contribute to disc degeneration and mechanical failure.
• To address this need the authors 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.
• Toward that goal, the foundation for improved understanding of disc degeneration requires the ability to precisely define the stress with the disc in vivo.
• Alternatively, the authors sought to simultaneously consider data from multiple experimental deformations to prescribe the requisite ma terial constants.

2.1. Fiber-reinforced continuum theory

• In the theory for a "ber-reinforced continuum, it is assumed that the anisotropic properties of the material are solely attributed to the presence of the families of "bers (Spencer, 1984).
• The constant � has dimensions of stress while the remaining constants are dimensionless.
• Hence, the authors chose another approximation to the strain energy func tion that places the dependence of the invariants in separ ate exponentials, instead of the same exponential as in (9).

2.3. Experimental data

• For (A) and (B), the authors set the principal stretches in the two directions that were normal to the loading direction equal to unity.
• For (C) and (D), the authors had to make assumptions concerning the principal stretches that were not directly measured.
• Hence, when invoking the traction-free boundary conditions for (C), the authors derived equations for the axial and circumferential stresses, ¹ and ¹ , in terms of the radial stretch, � , �� �� � and set these equations equal to zero and included them in the set of equations used in the simultaneous regres sion.
• The resulting stress vs. stretch equations pre dicted by the theory and the speci"c choices of the strain energy function are too long to present here.

2.4. Statistical analysis

• To prescribe the material coe$cients imbedded in their formulation, the authors conducted a simultaneous nonlinear regression to the mean response of the stress versus stretch data from the four experiments using a Leven berg}Marquardt nonlinear "tting algorithm (Mathe matica).
• This composite function was then "t to the four data sets to determine the material coe$cients.
• The authors repeated the simultaneous, nonlinear regression using the four equations mentioned above plus four additional equations corresponding to the trac tion-free boundary conditions for experiments (C) and (D) to determine a new set of the material constants.
• The authors rejected any solutions for the material constants for either strain energy function that predicted non-mono tonic behavior for any of the measured stress-stretch relationships in the range of experimentally reported stretches.

3. Results

• The strain energy given by (11) resulted in the lower �� value in the simultaneous regression to the four dir ectly measured stress}stretch relations.
• Solid curves"mean experimental response $1 standard deviation; open circles"theoretical prediction of measured stress}stretch response.
• The strain energy given by Eq. (11) resulted in the lower �� value in the simultaneous regression to the four directly measured stress}stretch relations plus the four additional equations corresponding to the traction-free boundary conditions for experiments (C) and (D).
• With these values, the predicted stress versus stretch composite function (13) was correlated to the experimental data with a �� value of 0.51 (Fig. 5).

4. Discussion

• The authors have developed a nonlinear, orthotropic constitut ive theory for the annulus that matches current experi mental data representing the broad range of stresses anticipated in vivo.
• Using a theory for an isotropic continuum reinforced with two families of "bers, the authors were able to specify a formulation whose predictions simultaneously fell within one stan dard deviation of the mean response of the annulus when measured in four di!erent experimental deformations.
• This "nding is especially noteworthy as the single exponential form of the strain energy function (11) used two less material constants than that incorporating multiple exponentials (9). a �� value of 0.08.
• The inclusion of the traction-free boundary conditions in the simultaneous regression resulted in a poorer over all "t to the experimental data.
• Since it has been shown that the "ber angle from the axial direc tion varies from $603 at the periphery to $453 at the innermost layers (Cassidy et al., 1989), the "ber-rein forced theory has the potential for separating the e!ects of heterogeneous collagen "ber orientation from other microstructural parameters (e.g., collagen "ber density and proteoglycan content).

Figures

Fig. 3. Simultaneous curve-"t results using the strain energy of Eq. (9) to match the four equations representing the measured stress}stretch data. The material constants (� , � , � , � , � , � , � , � , � , � , � ) equal to (0.204, 0.751, 0.075, !0.291, 0.647, 3.868, !0.399, 1.364, !0.084, 2.004, 1.045) � � � � � � � � � �� ��

Fig. 4. Simultaneous curve-"t results using the strain energy of Eq. (11) to match the four equations representing the measured stress}stretch data. The material constants (� , � , � , � , � , � , � , � , � ) equal to (0.001, !0.737, !3.289, !6.496, 0.409, 17.713, 0.683, !2.615, 0.262) resulted in � � � � � � � � ��

Fig. 1. Continuum idealization of the annulus "brosus in the reference con"guration. The basis vectors (E , E , E ) are aligned in the local radial, circumferential, and axial directions, respectively. The two familites of "bers are represented in this reference con"guration by the vectors a and b which lie in the E !E plane and are oriented at angles of plus and minus � degrees from the axial direction E .

Fig. 5. Simultaneous curve-"t results using the strain energy of Eq. (11) to match the four equations representing the measured stress}stretch data and the additional four equations representing the traction-free boundary conditions for experiments C and D. The material constants (� , � , � , � , � , � , � , � , � ) equal to (0.001, !1.716, !3.338, !4.574, 0.369, 12.955, 0.600, !2.911, 0.212) resulted in a �� value of 0.51. � � � � � � � � ��

Fig. 2. Schematic reqresentation of the applied stresses for the four experimental protocols: A * radial con"ned compression; B * axial con"ned compression; C * radial tension; D * circumferential ten sion. The basis vectors (E , E , E ) are aligned in the local radial, circumferential, and axial directions, respectively. The traction-free boundary conditions in experiment C are satis"ed by setting the cir cumferential stress ¹ and the axial stress ¹ to zero. The tractionfree boundary conditions in experiment D are satis"ed by setting the radial stress ¹ and the axial stress ¹ to zero.

Frequently asked questions

Q1. What contributions have the authors mentioned in the paper "Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus" ?

To address this need the authors 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. 

Q2. What have the authors stated for future works in "Application of a fiber-reinforced continuum theory to multiple deformations of the annulus fibrosus" ?

Also, in the future the in # uence of specimen speci '' c variability may be reduced by including addi tional parameters ( donor, disc level, and disc region ) into the statistical regression. For these reasons, the authors believe that the proposed `` ber-reinforced continuum the ory is a logical `` rst step in characterizing the strongly anisotropic behavior of the annulus, and future studies should include data from other deformations ( such as shear ), the precise measurement of all three stretches in tensile experiments, and specimen-speci '' c regressions. Since it has been shown that the `` ber angle from the axial direc tion varies from $ 603 at the periphery to $ 453 at the innermost layers ( Cassidy et al., 1989 ), the `` ber-rein forced theory has the potential for separating the e ! ects of heterogeneous collagen `` ber orientation from other microstructural parameters ( e. g., collagen `` ber density and proteoglycan content ).