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.

References

References
1.
Cassidy
,
J. J.
,
Hiltner
,
A.
, and
Baer
,
E.
,
1989
, “
Hierarchical Structure of the Intervertebral Disc
,”
Connect. Tissue Res.
,
23
(
1
), pp.
75
88
.10.3109/03008208909103905
2.
Marchand
,
F.
, and
Ahmed
,
A. M.
,
1990
, “
Investigation of the Laminate Structure of Lumbar Disc Anulus Fibrosus
,”
Spine
,
15
(
5
), pp.
402
410
.10.1097/00007632-199005000-00011
3.
Nachemson
,
A.
,
1981
, “
Disc Pressure Measurements
,”
Spine
,
6
(
1
), pp.
93
97
.10.1097/00007632-198101000-00020
4.
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
5.
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
.10.1097/00002517-199203000-00001
6.
Canale
,
S. T.
, and
Beaty
,
J. H.
,
2008
,
Operative Orthopaedics
, Vol.
2
,
Mosby Elsevier
,
Philadelphia
.
7.
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
.10.1097/00007632-199605150-00009
8.
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
.10.1097/00007632-199406000-00002
9.
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
.10.1016/0021-9290(83)90128-8
10.
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
.10.1007/s007760200040
11.
Prud'Homme
,
D.
,
2008
, “
Mécanisme de la Hernie Discale: Modélisation Non-Linéaire
,” Master's thesis, École de Technologie Supérieure, Montréal.
12.
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
.10.1016/j.jbiomech.2007.11.019
13.
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
.10.1016/S0268-0033(97)00045-4
14.
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
.10.1002/jor.21232
15.
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
.10.1097/00007632-198003000-00004
16.
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
.10.1080/00140139308967940
17.
McNally
,
D. S.
, and
Arridge
,
R. G. C.
,
1995
, “
An Analytical Model of Intervertebral Disc Mechanics
,”
J. Biomech.
,
28
(
1
), pp.
53
68
.10.1016/0021-9290(95)80007-7
18.
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
.10.1016/j.jbiomech.2003.12.026
19.
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
.10.1111/j.1469-7580.2009.01076.x
20.
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
.10.1016/j.matbio.2011.03.009
21.
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
.10.1007/s00586-012-2308-x
22.
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
.10.3901/CJME.2010.05.582
23.
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
.10.1115/1.2172957
24.
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
.
25.
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
.
26.
Eijkelkamp
,
M. F.
,
2002
, “
On the Development of an Artificial Intervertebral Disc
,” Ph.D. thesis, University of Groningen, Netherlands.
27.
Adams
,
M. A.
,
McNally
,
D. S.
, and
Dolan
,
P.
,
1996
, “
‘Stress' Distributions Inside Intervertebral Discs
,”
J. Bone Jt. Surg.
,
78
(
6
), pp.
965
972
.10.1302/0301-620X78B6.1287
28.
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
.10.1007/s005860000140
29.
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
.10.1007/s10237-004-0053-8
30.
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
.10.1016/j.jbiomech.2007.01.007
31.
Matlab
,
2009
, version 7.8.0.347 (R2009a), MathWorks, Natick MA.
32.
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
.
33.
ANSYS
,
2007
, Release 11.0, ANSYS, Inc., Southpointe, PA.
34.
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
.10.1115/1.4002550
35.
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
.10.1016/j.clinbiomech.2007.10.006
You do not currently have access to this content.