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Research Papers

On the Modeling of an Intervertebral Disc Using a Novel Large Deformation Multi-Shell Approach

[+] Author and Article Information
Sébastien Demers

e-mail: sebastien.demers.3@ens.etsmtl.ca

Abdel-Hakim Bouzid

Professor
Fellow ASME
e-mail: hakim.bouzid@etsmtl.ca

Sylvie Nadeau

Professor
e-mail: sylvie.nadeau@etsmtl.ca
Department of Mechanical Engineering,
École de Technologie Supérieure,
1100 Notre-Dame West,
Montreal, QC, H3C 1K3, Canada

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received September 21, 2012; final manuscript received March 26, 2013; accepted manuscript posted April 4, 2013; published online April 24, 2013. Assoc. Editor: James C. Iatridis.

J Biomech Eng 135(5), 051003 (Apr 24, 2013) (8 pages) Paper No: BIO-12-1426; doi: 10.1115/1.4024133 History: Received September 21, 2012; Revised March 26, 2013; Accepted April 04, 2013

The objective of this study is to develop an analytical model to predict the stresses and displacements in the lamellae of the intervertebral disc subjected to a compressive force. This is achieved by developing a model based on membrane theory combined to large deformation multishell structural behavior. Equations for longitudinal and circumferential stresses are formulated for each lamella of the anulus fibrosus. Multilamellae interaction is a statically indeterminate problem, which requires equations of compatibility of the displacements of adjacent lamellae to be resolved. The large deformation inherent to soft tissue is considered and the solution is obtained using an iterative process. Elastic interactions with a large deformation is a novelty in analytical modeling of soft tissues. This provides model realism and offers the possibility for new and in-depth investigations. Results are given for longitudinal and circumferential stresses and displacements as well as contact pressures for every lamella of the anulus fibrosus. The analytical results are compared to those of two finite element models. The results suggest that the most highly stressed zone is located on the innermost lamella. Stresses decrease through disc thickness and are at a maximum at the innermost lamella. Circumferential stress is predominant and the difference is less than 5% at any point of the anulus fibrosus when the analytical model is compared to the finite element model using coupled degrees of freedom at the lamellae interface. When compared to the finite element model using contact elements, the difference is below 11%. Contact pressures from the inside to the outside of the anulus fibrosus are shown to decrease nonlinearly. The model presented in this study has demonstrated that it is possible to analytically simulate the complex mechanical behavior of a multishell intervertebral disc subjected to compression, provided some simplifications. Further improvements are suggested to increase model realism and recommendations are given for future experimentation necessary to support both the analytical and numerical models.

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References

Figures

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Fig. 1

Representation of the intradiscal pressure pNP generated by a uni-axial load acting on the lumbo-sacral junction

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Fig. 2

Geometry of a single lamella and illustration of an infinitesimal membrane element with its corresponding r-θ-φ coordinate system

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Fig. 3

Free body diagram of the axisymmetric intervertebral disc, shown in the saggital section view (adapted from Ref. [32])

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Fig. 4

Compatibility of displacements

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Fig. 5

Convergence of the analytical model with respect to the FEM

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Fig. 6

Membrane stresses: (a) longitudinal stresses along the height of the disc, (b) circumferential stresses along the height of the disc, (c) longitudinal stresses across the lamellae at the transverse plane, and (d) circumferential stresses across the lamellae at the transverse plane

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Fig. 7

Contact pressures between adjacent lamellae

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Fig. 8

Radial displacements along the height of the disc

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