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

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

Sylvie Nadeau

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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Cassidy, J. J., Hiltner, A., and Baer, E., 1989, “Hierarchical Structure of the Intervertebral Disc,” Connect. Tissue Res., 23(1), pp. 75–88. [CrossRef] [PubMed]
Marchand, F., and Ahmed, A. M., 1990, “Investigation of the Laminate Structure of Lumbar Disc Anulus Fibrosus,” Spine, 15(5), pp. 402–410. [CrossRef] [PubMed]
Nachemson, A., 1981, “Disc Pressure Measurements,” Spine, 6(1), pp. 93–97. [CrossRef] [PubMed]
Roaf, R., 1960, “A Study of the Mechanics of Spinal Injuries,” J. Bone Jt. Surg. Br., 42-B(4), pp. 810–823. Available at: http://www.bjj.boneandjoint.org.uk/content/42-B/4/810.abstract
Gunzburg, R., Hutton, W. C., Crane, G., and Fraser, R. D., 1992, “Role of the Capsulo-Ligamentous Structures in Rotation and Combined Exion-Rotation of the Lumbar Spine,” J. Spinal Disord., 5(1), pp. 1–7. [CrossRef] [PubMed]
Canale, S. T., and Beaty, J. H., 2008, Operative Orthopaedics, Vol. 2, Mosby Elsevier, Philadelphia.
Iatridis, J. C., Weidenbaum, M., Setton, L. A., and Mow, V. C., 1996, “Is the Nucleus Pulposus a Solid or a Fluid? Mechanical Behaviors of the Nucleus Pulposus of the Human Intervertebral Disc,” Spine, 21(10), pp. 1174–1184. [CrossRef] [PubMed]
Skaggs, D. L., Weidenbaum, M., Iatridis, J. C., Ratcliffe, A., and Mow, V. C., 1994, “Regional Variation in Tensile Properties and Biochemical Composition of the Human Lumbar Anulus Fibrosus,” Spine, 19(12), pp. 1310–1319. [CrossRef] [PubMed]
Klein, J. A., Hickey, D. S., and Hukins, D. W., 1983, “Radial Bulging of the Annulus Fibrosus During Compression of the Intervertebral Disc,” J. Biomech., 16(3), pp. 211–217. [CrossRef] [PubMed]
Goto, K., Tajima, N., Chosa, E., Totoribe, K., Kuroki, H., Arizumi, Y., and Arai, T., 2002, “Mechanical Analysis of the Lumbar Vertebrae in a Three-Dimensional Finite Element Method Model in Which Intradiscal Pressure in the Nucleus Pulposus was Used to Establish the Model,” J. Orthop. Sci., 7(2), pp. 243–246. [CrossRef] [PubMed]
Prud'Homme, D., 2008, “Mécanisme de la Hernie Discale: Modélisation Non-Linéaire,” Master's thesis, École de Technologie Supérieure, Montréal.
Heuer, F., Schmidt, H., and Wilke, H.-J., 2008, “The Relation Between Interverteral Disc Bulging and Annular Fiber Associated Strains for Simple and Complex Loading,” J. Biomech., 41(5), pp. 1086–1094. [CrossRef] [PubMed]
Wenger, K. H., and Schlegel, J. D., 1997, “Annular Bulge Contours From an Axial Photogrammetric Method,” Clin. Biomech. (Bristol, Avon), 12(7/8), pp. 438–444. [CrossRef] [PubMed]
OConnell, G. D., Vresilovic, E. J., and Elliott, D. M., 2010, “Human Intervertebral Disc Internal Strain in Compression: The Effect of Disc Region, Loading Position, and Degeneration,” J. Orthop. Res., 29(4), pp. 547–555. [CrossRef] [PubMed]
Hickey, D. S., and Hukins, D. W., 1980, “Relation Between the Structure of the Annulus Fibrosus and the Function and Failure of the Intervertebral Disc,” Spine, 5(2), pp. 106–116. [CrossRef] [PubMed]
Waters, T. R., Putz-Anderson, V., Garg, A., and Fine, L. J., 1993, “Revised NIOSH Equation for the Design and Evaluation of Manual Lifting Tasks,” Ergonomics, 36(7), pp. 749–776. [CrossRef] [PubMed]
McNally, D. S., and Arridge, R. G. C., 1995, “An Analytical Model of Intervertebral Disc Mechanics,” J. Biomech., 28(1), pp. 53–68. [CrossRef] [PubMed]
Iatridis, J. C., and ap Gwynn, I., 2004, “Mechanisms for Mechanical Damage in the Intervertebral Disc Annulus Fibrosus,” J. Biomech., 37(8), pp. 1165–1175. [CrossRef] [PubMed]
Schollum, M. L., Robertson, P. A., and Broom, N. D., 2009, “A Microstructural Investigation of Intervertebral Disc Lamellar Connectivity: Detailed Analysis of the Translamellar Bridges,” J. Anat., 241(6), pp. 805–816. [CrossRef]
Smith, L. J., and Elliott, D. M., 2011, “Formation of Lamellar Cross Bridges in the Annulus Fibrosus of the Intervertebral Disc is a Consequence of Vascular Regression,” Matrix Biol., 30(4), pp. 267–274. [CrossRef] [PubMed]
Gregory, D. E., Bae, W. C., Sah, R. L., and Masuda, K., 2012, “Anular Delamination Strength of Human Lumbar Intervertebral Disc,” Eur. Spine J., 21(9), pp. 1716–1723. [CrossRef] [PubMed]
Yuan, G., Liu, H., and Wang, Z., 2010, “Optimum Design for Shrink-Fit Multi-Layer Vessels Under Ultrahigh Pressure Using Different Materials,” Chin. J. Mech. Eng., 23(5), pp. 582–589. [CrossRef]
Jahed, H., Farshi, B., and Karimi, M., 2006, “Optimum Autofrettage and Shrink-Fit Combination in Multi-Layer Cylinders,” ASME J. Pressure Vessel Technol., 128(2), pp. 196–200. [CrossRef]
Naga, S. A., and Mokhtar, M., 2005, “An Analytical and Finite Element Analysis Study of Multilayered Pressure Vessels Under Thermal Conditions,” Pressure Vessels and Piping Division Conference, 3, pp. 173–179.
Adali, S., Verijenko, V. E., Tabakov, P. Y., and Walker, M., 1995, “Optimization of Multilayered Composite Pressure Vessels Using Exact Elasticity Solution,” Pressure Vessels and Piping Division Conference, 302, pp. 203–212.
Eijkelkamp, M. F., 2002, “On the Development of an Artificial Intervertebral Disc,” Ph.D. thesis, University of Groningen, Netherlands.
Adams, M. A., McNally, D. S., and Dolan, P., 1996, “‘Stress' Distributions Inside Intervertebral Discs,” J. Bone Jt. Surg., 78(6), pp. 965–972. [CrossRef]
Zhou, S. H., McCarthy, I. D., McGregor, A. H., Coombs, R. R. H., and Hughes, S. P. F., 2000, “Geometrical Dimensions of the Lower Lumbar Vertebrae - Analysis of Data From Digitised CT Images,” Eur. Spine J., 9(3), pp. 242–248. [CrossRef] [PubMed]
Holzapfel, G. A., Schulze-Bauer, C. A. J., Feigl, G., and Regitnig, P., 2005, “Single Lamellar Mechanics of the Human Lumbar Anulus Fibrosus,” Biomech. Model. Mechanobiol., 3(3), pp. 125–140. [CrossRef] [PubMed]
Little, J. P., Adam, C. J., Evans, J. H., Pettet, G. J., and Pearcy, M. J., 2007, “Nonlinear Finite Element Analysis of Anular Lesions in the l4/5 Intervertebral Disc,” J. Biomech., 40(12), pp. 2744–2751. [CrossRef] [PubMed]
Matlab, 2009, version (R2009a), MathWorks, Natick MA.
Demers, S., Bouzid, A.-H., and Nadeau, S., 2012, “Stress Analysis of the Intervertebral Disc Using a Developed Multi-Shell Model,” Twelfth Pan American Congress of Applied Mechanics (Proceedings of PACAM XII), pp. 1–7.
ANSYS, 2007, Release 11.0, ANSYS, Inc., Southpointe, PA.
Chagnon, A., Aubin, C.-E., and Villemure, I., 2010, “Biomechanical Inuence of Disk Properties on the Load Transfer of Healthy and Degenerated Disks Using a Poroelastic Finite Element Model,” ASME J. Biomech. Eng., 132(11), p. 111006. [CrossRef]
Heuer, F., Schmidt, H., Claes, L., and Wilke, H.-J., 2008, “A New Laser Scanning Technique for Imaging Intervertebral Disc Displacement and Its Application to Modeling Nucleotomy,” Clin. Biomech. (Bristol, Avon), 23(3), pp. 260–269. [CrossRef] [PubMed]


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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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