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Research Papers

Calibration of Hyperelastic Material Properties of the Human Lumbar Intervertebral Disc under Fast Dynamic Compressive Loads

[+] Author and Article Information
Eric Wagnac1

 Laboratoire de Biomécanique Appliquée, UMRT 24 IFSTTAR-Université de la Méditerranée, Faculté de Médecine secteur Nord, Boulevard Pierre Dramard, F-13916, Marseille,Cedex 20, France;  Biomedical Engineering Institute, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville, Montreal, PQ, H3C 3A7, Canada e-mail address: eric.wagnac@polymtl.ca Laboratoire de Biomécanique Appliquée, UMRT 24 IFSTTAR-Université de la Méditerranée, Faculté de Médecine secteur Nord, Boulevard Pierre Dramard, F-13916, Marseille, Cedex 20, France Laboratoire de Biomécanique Appliquée, UMRT 24 IFSTTAR-Université de la Méditerranée, Faculté de Médecine secteur Nord, Boulevard Pierre Dramard, F-13916, Marseille, Cedex 20, France;  Biomedical Engineering Institute, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville, Montreal, PQ, H3C 3A7, Canada Laboratoire de Biomécanique Appliquée, UMRT 24 IFSTTAR-Université de la Méditerranée, Faculté de Médecine secteur Nord, Boulevard Pierre Dramard, F-13916, Marseille, Cedex 20, France;  Biomedical Engineering Institute, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville, Montreal, PQ, H3C 3A7, Canada Biomedical Engineering Institute, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville, Montreal (Quebec), H3C 3A7, Canada; Research Center, Sainte-Justine Hospital, 3175 Cote Sainte-Catherine Rd, Montreal, PQ, H3T 1C5, Canada

Pierre-Jean Arnoux, Anaïs Garo, Marwan El-Rich, Carl-Eric Aubin

 Laboratoire de Biomécanique Appliquée, UMRT 24 IFSTTAR-Université de la Méditerranée, Faculté de Médecine secteur Nord, Boulevard Pierre Dramard, F-13916, Marseille,Cedex 20, France;  Biomedical Engineering Institute, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville, Montreal, PQ, H3C 3A7, Canada e-mail address: eric.wagnac@polymtl.ca Laboratoire de Biomécanique Appliquée, UMRT 24 IFSTTAR-Université de la Méditerranée, Faculté de Médecine secteur Nord, Boulevard Pierre Dramard, F-13916, Marseille, Cedex 20, France Laboratoire de Biomécanique Appliquée, UMRT 24 IFSTTAR-Université de la Méditerranée, Faculté de Médecine secteur Nord, Boulevard Pierre Dramard, F-13916, Marseille, Cedex 20, France;  Biomedical Engineering Institute, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville, Montreal, PQ, H3C 3A7, Canada Laboratoire de Biomécanique Appliquée, UMRT 24 IFSTTAR-Université de la Méditerranée, Faculté de Médecine secteur Nord, Boulevard Pierre Dramard, F-13916, Marseille, Cedex 20, France;  Biomedical Engineering Institute, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville, Montreal, PQ, H3C 3A7, Canada Biomedical Engineering Institute, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville, Montreal (Quebec), H3C 3A7, Canada; Research Center, Sainte-Justine Hospital, 3175 Cote Sainte-Catherine Rd, Montreal, PQ, H3T 1C5, Canada

1

Corresponding author.

J Biomech Eng 133(10), 101007 (Oct 31, 2011) (10 pages) doi:10.1115/1.4005224 History: Received April 07, 2011; Accepted September 24, 2011; Published October 31, 2011; Online October 31, 2011

Under fast dynamic loading conditions (e.g. high-energy impact), the load rate dependency of the intervertebral disc (IVD) material properties may play a crucial role in the biomechanics of spinal trauma. However, most finite element models (FEM) of dynamic spinal trauma uses material properties derived from quasi-static experiments, thus neglecting this load rate dependency. The aim of this study was to identify hyperelastic material properties that ensure a more biofidelic simulation of the IVD under a fast dynamic compressive load. A hyperelastic material law based on a first-order Mooney-Rivlin formulation was implemented in a detailed FEM of a L2-L3 functional spinal unit (FSU) to represent the mechanical behavior of the IVD. Bony structures were modeled using an elasto-plastic Johnson-Cook material law that simulates bone fracture while ligaments were governed by a viscoelastic material law. To mimic experimental studies performed in fast dynamic compression, a compressive loading velocity of 1 m/s was applied to the superior half of L2, while the inferior half of L3 was fixed. An exploratory technique was used to simulate dynamic compression of the FSU using 34 sets of hyperelastic material constants randomly selected using an optimal Latin hypercube algorithm and a set of material constants derived from quasi-static experiments. Selection or rejection of the sets of material constants was based on compressive stiffness and failure parameters criteria measured experimentally. The two simulations performed with calibrated hyperelastic constants resulted in nonlinear load-displacement curves with compressive stiffness (7335 and 7079 N/mm), load (12,488 and 12,473 N), displacement (1.95 and 2.09 mm) and energy at failure (13.5 and 14.7 J) in agreement with experimental results (6551 ± 2017 N/mm, 12,411 ± 829 N, 2.1 ± 0.2 mm and 13.0 ± 1.5 J respectively). The fracture pattern and location also agreed with experimental results. The simulation performed with constants derived from quasi-static experiments showed a failure energy (13.2 J) and a fracture pattern and location in agreement with experimental results, but a compressive stiffness (1580 N/mm), a failure load (5976 N) and a displacement to failure (4.8 mm) outside the experimental corridors. The proposed method offers an innovative way to calibrate the hyperelastic material properties of the IVD and to offer a more realistic simulation of the FSU in fast dynamic compression.

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Copyright © 2011 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Functional spinal unit L2-L3 of the Spine Model for Safety and Surgery (SM2S)

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Figure 2

Segmentation of the vertebra in 9 components of different cortical thicknesses

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Figure 3

Nonlinear load-displacement curve used to model the mechanical behavior of a single collagenous fiber element

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Figure 4

Pareto charts of the main standardized effects (t-values). (a) KCOMP (b) FFAIL (c) DFAIL . Any effect that extends past the dashed lines represents a statistically significant effect (p ≤ 0.05).

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Figure 5

(a) Surface response for the compressive stiffness KCOMP , (b) contour plot of KCOMP obtained by the projection of its surface response on the XY plane, (c) contour plot of the failure force FFAIL, (d) contour plot of the displacement to failure DFAIL . Dot numbers represent simulation IDs. The dotted lines represent sets of material constants for which simulations would result in the average KCOMP , DFAIL and FFAIL measured experimentally.

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Figure 6

Final domain of solution obtained by superimposing the KCOMP , FFAIL and DFAIL contour plots and by intersecting their individual domain of solution. The black dot (simulation #35) represents C10-Annulus and C10-Nucleus values derived from quasi-static experiments [31].

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Figure 7

Types of fracture observed on the 34 simulations: (a) vertebral body collapse of L2 (type 1) (b) superior endplate fracture of L3 (type 2) (c) inferior endplate fracture of L2 (type 3). Fractures are represented by a black zone.

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Figure 8

Fracture pattern per simulation. Simulations that lied within the final domain of solution (gray area) resulted in superior endplate fractures of L3 (•), as observed experimentally. Other simulations either resulted in superior endplate fractures of L3 (•), vertebral body collapse of L2 (▴) or inferior endplate fracture of L2 (▪).

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Figure 9

Simulated load-displacement curves versus experimental corridor. Simulations #10 and #21 used calibrated material constants (C10-Annulus and C10-Nucleus within the final domain of solution) while simulation #35 used material constants derived from quasi-static experiments [31].

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