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Research Papers

Anisotropic Multishell Analytical Modeling of an Intervertebral Disk Subjected to Axial Compression

[+] Author and Article Information
Sébastien Demers

Department of Mechanical Engineering,
École de Technologie Supérieure,
1100 Notre-Dame West,
Montréal, QC H3C 1K3, Canada
e-mail: sebastien.demers.3@ens.etsmtl.ca

Sylvie Nadeau

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

Abdel-Hakim Bouzid

Fellow ASME
Department of Mechanical Engineering,
École de Technologie Supérieure,
1100 Notre-Dame West,
Montréal, QC H3C 1K3, Canada
e-mail: hakim.bouzid@etsmtl.ca

Manuscript received May 15, 2015; final manuscript received January 21, 2016; published online February 25, 2016. Assoc. Editor: James C. Iatridis.

J Biomech Eng 138(4), 041004 (Feb 25, 2016) (10 pages) Paper No: BIO-15-1241; doi: 10.1115/1.4032628 History: Received May 15, 2015; Revised January 21, 2016

Studies on intervertebral disk (IVD) response to various loads and postures are essential to understand disk's mechanical functions and to suggest preventive and corrective actions in the workplace. The experimental and finite-element (FE) approaches are well-suited for these studies, but validating their findings is difficult, partly due to the lack of alternative methods. Analytical modeling could allow methodological triangulation and help validation of FE models. This paper presents an analytical method based on thin-shell, beam-on-elastic-foundation and composite materials theories to evaluate the stresses in the anulus fibrosus (AF) of an axisymmetric disk composed of multiple thin lamellae. Large deformations of the soft tissues are accounted for using an iterative method and the anisotropic material properties are derived from a published biaxial experiment. The results are compared to those obtained by FE modeling. The results demonstrate the capability of the analytical model to evaluate the stresses at any location of the simplified AF. It also demonstrates that anisotropy reduces stresses in the lamellae. This novel model is a preliminary step in developing valuable analytical models of IVDs, and represents a distinctive groundwork that is able to sustain future refinements. This paper suggests important features that may be included to improve model realism.

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Figures

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Fig. 2

Free body diagrams: (a) force equilibrium on a membrane element isolated from lamella j, at equilibrium plane i, shown with geometrical parameters and (b) force equilibrium along vertical axis z, using the compressive force Fcomp, the force induced by intradiscal pressure pNP, and the vertical component of the longitudinal membrane force Nφ,i,j in each lamella

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Fig. 3

Boundary conditions for the application of the beam-on-elastic-foundation theory. At the endplates, the displacements δ[m] induced by pressure are counteracted by δ[d] induced by edge forces F to cancel the displacements. At the midtransverse plane, the displacement δ[m] is counteracted by δ[d] to yield the final the displacement δ[f].

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Fig. 5

Contact pressure between adjacent lamellae at the midtransverse plane

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Fig. 6

(a) Final circumferential stress obtained analytically in the innermost lamella, and divided into membrane and discontinuity stresses and (b) comparison of the final analytical circumferential stress to that of FE model

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Fig. 8

Initial and deformed shape of the lamellae obtained bythe analytical and FE modeling. The initial shape and the analytical deformed shape are only shown for the innermost and outermost lamellae.

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Fig. 7

Stresses through the AF thickness at the midtransverse plane

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