finite element modelling of anular lesions in the lumbar ... · using these geometric data and...

Finite Element Modelling

of Anular Lesions

in the

Lumbar Intervertebral

Disc

J. Paige Little, B.E. (Mechanical)(Hons)

Submitted for the award of degree of Doctor of Philosophy in

The Centre for Built Environment and Engineering Research,

School of Mechanical, Manufacturing and Medical Engineering,

Queensland University of Technology.

iii

KKeeyywwoorrddss Spine, intervertebral disc, degeneration, anular lesions, finite element analysis,

hyperelastic material model, biomechanics

iv

AAbbssttrraacctt Low back pain is an ailment that affects a significant portion of the community.

However, due to the complexity of the spine, which is a series of interconnected

joints, and the loading conditions applied to these joints the causes for back pain are

not well understood. Investigations of damage or failure of the spinal structures from

a mechanical viewpoint may be viewed as a way of providing valuable information

for the causes of back pain. Low back pain is commonly associated with injury to, or

degeneration of, the intervertebral discs and involves the presence of tears or lesions

in the anular disc material. The aim of the study presented in this thesis was to

investigate the biomechanical effect of anular lesions on disc function using a finite

element model of the L4/5 lumbar intervertebral disc.

The intervertebral disc consists of three main components – the anulus fibrosus, the

nucleus pulposus and the cartilaginous endplates. The anulus fibrosus is comprised of

collagen fibres embedded in a ground substance while the nucleus is a gelatinous

material. The components of the intervertebral disc were represented in the model

together with the longitudinal ligaments that are attached to the anterior and posterior

surface of the disc. All other bony and ligamentous structures were simulated through

the loading and boundary conditions.

A high level of both geometric and material accuracy was required to produce a

physically realistic finite element model. The geometry of the model was derived

from images of cadaveric human discs and published data on the in vivo configuration

of the L4/5 disc. Material properties for the components were extracted from the

existing literature. The anulus ground substance was represented as a Mooney-Rivlin

hyperelastic material, the nucleus pulposus was modelled as a hydrostatic fluid in the

healthy disc models and the cartilaginous endplates, collagen fibres and longitudinal

ligaments were represented as linear elastic materials. A preliminary model was

developed to assess the accuracy of the geometry and material properties of the disc

components. It was found that the material parameters defined for the anulus ground

substance did not accurately describe the nonlinear shear behaviour of the tissue.

Accurate representation this nonlinear behaviour was thought to be important in

v

ensuring the deformations observed in the anulus fibrosus of the finite element model

were correct.

There was no information found in the literature on the mechanical properties of the

anulus ground substance. Experimentation was, therefore, carried out on specimens

of sheep anulus fibrosus in order to quantify the mechanical response of the ground

substance. Two testing protocols were employed. The first series of tests were

undertaken to provide information on the strain required to initiate permanent damage

in the ground substance. The second series of tests resulted in the acquisition of data

on the mechanical response of the tissue to repeated loading. The results of the

experimentation carried out to determine the strain necessary to initiate permanent

damage suggested that during daily loading some derangement might be caused in the

anulus ground substance. The results for the mechanical response of the tissue were

used to determine hyperelastic constants which were incorporated in the finite

element model. A second order Polynomial and a third order Ogden strain energy

equation were used to define the anulus ground substance. Both these strain energy

equations incorporated the nonlinear mechanical response of the tissue during shear

loading conditions.

Using these geometric data and material properties a finite element model of a

representative L4/5 intervertebral disc was developed.

When the measured material parameters for the anulus ground substance were

implemented in the finite element model, large deformations were observed in the

anulus fibrosus and excessive nucleus pressures were found. This suggested that the

material parameters defining the anulus ground substance were overly compliant and

in turn, implied the possibility that the stiffness of the sheep anulus ground substance

was lower than the stiffness of the human tissue. Even so, the mechanical properties

of the sheep joints had been shown to be similar to those of the human joint and it was

concluded that the results of analyses using these parameters would provide valuable

qualitative information on the disc mechanics.

To represent the degeneration of the anulus fibrosus, the models included simulations

of anular lesions – rim, radial and circumferential lesions. Degeneration of the

vi

nucleus may be characterised by a significant reduction in the hydrostatic nucleus

pressure and a loss of hydration. This was simulated by removal of the hydrostatic

nucleus pressure.

Analyses were carried out using rotational loading conditions that were comparable to

the ranges of motion observed physiologically. The results of these analyses showed

that the removal of the hydrostatic nucleus pressure from an otherwise healthy disc

resulted in a significant reduction in the stiffness of the disc. This indicated that when

the nucleus pulposus is extremely degenerate, it offers no resistance to the

deformation of the anulus and the mechanics of the disc are significantly changed.

Specifically, the resistance to rotation offered by the intervertebral disc is reduced,

which may affect the stability of the joint. When anular lesions were simulated in the

finite element model they caused minimal changes in the peak moments resisted by

the disc under rotational loading. This suggested that the removal of the nucleus

pressure had a greater effect on the mechanics of the disc than the simulation of

anular lesions.

The results of the finite element model reproduced trends observed in both the healthy

and degenerate intervertebral disc in terms of variations in nucleus pressure with

loading conditions, axial displacement of the superior surface and bulge of the

peripheral anulus. It was hypothesised that the reduced rotational stiffness of the

degenerate disc may result in overload of the surrounding innervated

osseoligamentous anatomy which may in turn cause back pain. Similarly back pain

may result from the abnormal deformation of the innervated peripheral anulus in the

vicinity of anular lesions. Furthermore, it was hypothesised that biochemical changes

may result in the degeneration of the nucleus, which in turn may cause excessive

strains in the anulus ground substance and lead to the initiation of permanent damage

in the form of anular lesions. With further refinement of the components of the model

and the methods used to define the anular lesions it was considered that this model

would provide a powerful analysis tool for the investigation of the mechanics of

intervertebral discs with and without significant degeneration.

vii

TTaabbllee ooff CCoonntteennttss Keywords .................................................................................................................... iii

Abstract .......................................................................................................................iv

Table of Contents ......................................................................................................vii

List of Tables .......................................................................................................... xvii

List of Figures ............................................................................................................xx

List of Symbols ......................................................................................................xxvii

List of Abbreviations .......................................................................................... xxviii

Statement of Originality ........................................................................................xxix

Acknowledgements .................................................................................................xxx

1 Introduction............................................................................................................1

1.1 Aims and Objectives of the Thesis ...................................................................4

1.2 Limitations of the Study ...................................................................................5

2 Literature Review ..................................................................................................6

2.1 Spinal Anatomy ................................................................................................6

2.1.1 The bony spinal column.........................................................................7

2.1.2 The intervertebral disc ...........................................................................8

2.1.2.1 Nucleus pulposus .....................................................................9

2.1.2.2 Anulus fibrosus ........................................................................9

2.1.2.3 Cartilaginous endplates ..........................................................11

2.1.3 Anatomy and attachment of the longitudinal ligaments ......................12

2.1.3.1 Cross-sectional area ...............................................................13

2.1.3.2 Lateral width ..........................................................................15

2.1.3.3 Pre-tension in the ligaments ...................................................16

2.2 Location of the Instantaneous Centres of Rotation during Physiological

Loading...........................................................................................................16

2.2.1 Flexion and extension ..........................................................................17

2.2.2 Axial Rotation......................................................................................17

2.2.3 Lateral bending ....................................................................................18

2.3 Degeneration and Anular Lesions ..................................................................19

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2.3.1 The mechanism of degeneration and the initiation of anular lesions...22

2.3.2 Relevance of studying anular lesions...................................................23

2.4 The Use of FEM to Study the Spine and in particular Anular Lesions ..........23

2.5 Shortcomings in Previous Models ..................................................................24

2.6 Mechanical Properties of Components in the Spine.......................................27

2.6.1 The intervertebral disc components .....................................................27

2.6.1.1 Nucleus pulposus ...................................................................27

2.6.1.2 Anulus fibrosus and the anulus fibrosus ground substance ...28

2.6.1.3 Cartilaginous endplate............................................................29

2.6.1.4 Collagen fibres .......................................................................30

2.6.2 Incompressibility of the intervertebral disc .........................................31

2.6.3 Functional behaviour of the anulus fibrosus and nucleus pulposus.....32

2.6.3.1 The inclination of collagen fibres ..........................................33

2.6.3.2 Uniaxial compression.............................................................33

2.6.3.3 Bending ..................................................................................34

2.6.3.4 Torsion ...................................................................................34

2.6.4 Mechanical properties of the longitudinal ligaments...........................35

2.6.4.1 Average elastic modulus and spring stiffness of the anterior

longitudinal ligament .............................................................38

2.6.4.2 Average elastic modulus and spring stiffness of the posterior

longitudinal ligament .............................................................39

2.7 Use of a Hyperelastic Model for the Anulus Ground Matrix .........................40

2.7.1 Rubber elasticity theories and continuum mechanics..........................40

2.7.1.1 Strain invariants (Reference: Williams, 1973, Chapter 1;

Ugural and Fenster, 1995)......................................................40

2.7.1.2 Stress components and the strain energy equation,

(Reference: Williams, 1973, Chapter 1; Ugural and Fenster,

1995) ......................................................................................46

2.7.2 Forms and applications of the strain energy equation .........................50

2.8 Experimental Testing of the Intervertebral Disc ............................................54

2.8.1 Types of testing carried out and material information available in

literature...............................................................................................54

2.8.2 Specimen handling...............................................................................55

2.9 Conclusions ....................................................................................................58

ix

3 Development of the Preliminary FEM..............................................................60

3.1 Basic description of the FE method................................................................61

3.2 Abaqus 6.3 Finite Element Modelling Software ............................................63

3.2.1 Specifics of finite element analysis carried out using Abaqus 6.3 ......64

3.3 Geometry of the Anulus Fibrosus and Nucleus Pulposus in the Transverse

Plane ...............................................................................................................66

3.3.1 Methods – anulus boundary .................................................................67

3.3.1.1 Measurements ........................................................................67

3.3.1.2 Development of equations .....................................................69

3.3.1.3 Disc area.................................................................................73

3.3.1.4 Validation of the anulus formulae..........................................73

3.3.2 Methods – nucleus ...............................................................................74

3.3.2.1 Measurement ..........................................................................75

3.3.2.2 Development of equations .....................................................75

3.3.2.3 Nucleus area ...........................................................................76

3.3.2.4 Validation of the nucleus equations .......................................76

3.3.3 Discussion concerning the anulus and nucleus boundaries .................80

3.4 Geometry of the Collagen Fibres....................................................................82

3.4.1 Cross-sectional area of the collagen fibres ..........................................83

3.4.2 Collagen fibre spacing .........................................................................85

3.4.3 Angle of inclination of the rebar elements within the layers of collagen

fibres....................................................................................................85

3.4.4 Embedding elements............................................................................86

3.5 Determination of Sagittal Geometry...............................................................87

3.6 Location of the Instantaneous Axes of Rotation During Rotation .................87

3.6.1 Flexion/Extension ................................................................................87

3.6.2 Axial rotation .......................................................................................88

3.6.3 Lateral bending ....................................................................................89

3.7 Fortran Programming .....................................................................................90

3.8 Description of the Finite Elements Used in the FEM.....................................90

3.8.1 Anulus fibrosus and cartilaginous endplate .........................................91

3.8.2 Collagen fibres .....................................................................................92

3.8.3 Nucleus pulposus .................................................................................93

3.9 Mesh Generation using Abaqus Input Files ...................................................94

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3.10 Material Properties.........................................................................................95

3.10.1 Collagen fibres .....................................................................................95

3.10.2 Cartilaginous endplate .........................................................................96

3.10.3 Nucleus pulposus .................................................................................97

3.10.4 Anulus fibrosus ground substance .......................................................97

3.11 Boundary Conditions and Loading...............................................................102

3.11.1 Professor Nachemson's research on spinal loading ...........................102

3.11.2 Nucleus pulposus pressurisation........................................................103

3.11.3 Modelling adjacent vertebrae.............................................................103

3.11.4 Musculature and posterior elements ..................................................105

3.11.5 Uniaxial compression loading for validating the preliminary model 106

3.11.6 Iteration to determine the initial sagittal geometry of the intervertebral

disc FEM ...........................................................................................107

3.12 Optimising the Mesh Density of the FEM...................................................108

3.13 Analysis of the FEM....................................................................................112

3.13.1 The effect of variation in the transverse profile of the anulus and

nucleus boundaries ............................................................................113

3.13.2 Response of the FEM (Specimen 50) to the 70kPa nucleus pulposus

pressure..............................................................................................120

3.13.3 Analysis of the FEM under compression...........................................121

3.13.4 Full forward flexion ...........................................................................124

3.13.4.1 Validation criterion for full flexion......................................125

3.13.4.2 Results of analysis of the FEM under full flexion ...............125

3.14 Assessment of the Accuracy of the FEM ....................................................133

4 Experimental Testing of the Anulus Fibrosus................................................137

4.1 Objectives for Testing the Anulus Fibrosus .................................................137

4.2 Mechanical Testing – Rationale and Description.........................................138

4.3 Specimen Harvesting....................................................................................141

4.4 Biaxial Compression Testing Methods and Equipment ...............................144

4.4.1 Principle of operation.........................................................................144

4.4.2 Design details and pressure vessel components.................................146

4.4.2.1 Maximum vessel pressure and design pressure ...................146

4.4.2.2 Vessel walls..........................................................................147

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4.4.2.3 Fasteners...............................................................................148

4.4.2.4 Viewing windows ................................................................149

4.4.2.5 Attachment of specimen to nylon cord ................................150

4.4.2.6 Leaking piston and bore insert .............................................150

4.4.2.7 Adjustment knob for accurate orientation of the specimens 154

4.4.3 Proof testing .......................................................................................155

4.4.4 Setup of equipment ............................................................................156

4.4.5 Measurement of biaxial compressive stress and strain ......................157

4.4.5.1 Choice of pressure regulator ................................................157

4.4.5.2 Profile projector ...................................................................158

4.4.5.3 Data acquisition - hydrostatic pressure and deformation.....159

4.4.6 Commissioning of pressure vessel.....................................................159

4.4.6.1 Force applied to the piston ...................................................160

4.4.6.2 Biaxial compression of EVA foam ......................................162

4.5 Uniaxial Compression and Simple Shear .....................................................164

4.5.1 Testing equipment..............................................................................164

4.5.1.1 Uniaxial compression...........................................................164

4.5.1.2 Simple shear .........................................................................165

4.5.2 Maximum strains applied during testing............................................167

4.6 Strain Rate during Uniaxial Compression and Simple Shear Loading.........167

4.6.1 Procedure for testing to determine the tissue response to varied strain

rates ...................................................................................................168

4.6.2 Results and discussion of strain rate experiments..............................169

4.6.2.1 Strain rate 0.001 sec-1...........................................................169

4.6.2.2 Strain rate 0.10 sec-1.............................................................170

4.6.2.3 Strain rate 0.01 sec-1.............................................................171

4.6.3 Discussion and justification for the choice of strain rate...................171

4.7 Results for Mechanical Testing of the Anulus Fibrosus Ground Substance 174

4.7.1 Results of initial and repeated loading – stress-strain tests................174

4.7.2 Statistical analysis..............................................................................179

4.7.2.1 Simple shear .........................................................................180

4.7.2.2 Uniaxial compression...........................................................181

4.7.2.3 Biaxial compression.............................................................182

4.7.3 Range of test data...............................................................................184

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4.7.4 Discussion..........................................................................................185

4.8 Pilot Study to Determine the Derangement Strain .......................................186

4.8.1 Rationale for carrying out additional experimentation ......................186

4.8.1.1 Fluid loss ..............................................................................186

4.8.1.2 Viscoelastic effects in the anulus fibrosus solid skeleton ....186

4.8.1.3 Derangement of the anulus fibrosus.....................................187

4.8.2 Testing to determine the derangement strain .....................................187

4.8.2.1 Procedure .............................................................................188

4.8.2.2 Results ..................................................................................188

4.8.2.3 Discussion of the range of derangement strain of the anulus

fibrosus ground substance....................................................192

4.8.2.4 An hypothesis for disc degeneration....................................193

4.9 Discussion of Regional Stiffness and Stiffening Mechanisms in the Anulus

Fibrosus Specimens ......................................................................................193

4.9.1 Uniaxial compression.........................................................................194

4.9.2 Simple shear.......................................................................................195

4.9.3 Biaxial compression...........................................................................196

4.9.3.1 Deformation mechanism in the radial and circumferential

regions..................................................................................196

4.9.3.2 Difference in regions of highest stiffness when measured

radially and circumferentially ..............................................197

4.9.3.3 Drop in stiffness between the initial and repeated loading and

derangement strains for biaxial compression.......................198

4.10 Discussion of Edge Effects..........................................................................199

4.11 Potential Sources of Error in the Results.....................................................201

4.12 Conclusion ...................................................................................................201

5 Determining Hyperelastic Parameters for the Anulus Fibrosus Ground

Substance ............................................................................................................203

5.1 Chapter Overview.........................................................................................203

5.2 Manipulation of Experimental Regression Lines to Obtain Input for the Strain

Energy Equations..........................................................................................205

5.2.1 Simple shear compared to pure shear (Treloar, 1975).......................205

5.2.2 Manipulating simple shear data to obtain pure shear data.................206

xiii

5.2.3 Principal extension ratios for Simple Shear deformation ..................209

5.2.4 Average biaxial compression data .....................................................212

5.3 Approach to Choosing Hyperelastic Models for the Anulus Fibrosus Ground

Substance ......................................................................................................212

5.3.1 Possible strain energy equations for the anulus fibrosus ground

substance ...........................................................................................213

5.3.1.1 Veronda and Westmann .......................................................214

5.3.1.2 Ogden ...................................................................................214

5.3.1.3 Extended Mooney equation .................................................215

5.3.1.4 Polynomial ...........................................................................215

5.3.2 Verification of the Abaqus algorithm used to determine hyperelastic

parameters .........................................................................................216

5.4 Strain Energy Equations Used for the Anulus Fibrosus Ground Substance.222

5.4.1 Inhomogeneous hyperelastic model for the ground substance ..........222

5.4.1.1 Explanation of the criterion used to select the hyperelastic

strain energy equation for the anterior, lateral and posterior

anulus during initial and repeated loading ...........................225

5.4.1.2 Inhomogeneous hyperelastic constants for initial and repeated

loading..................................................................................232

5.4.2 Homogeneous hyperelastic model for the ground substance.............232

5.5 Conclusion ....................................................................................................235

6 Implementation of the Improved Anulus Fibrosus Material Properties....236

6.1 Chapter Overview.........................................................................................236

6.2 Implementation of the Homogeneous Anulus Ground Substance into the FEM

..................................................................................................................237

6.3 Compatibility of the Material Stiffness of the Collagen Fibres and the Anulus

Fibrosus Ground Substance ..........................................................................240

6.4 Improved Element Configuration for the Hydrostatic Fluid Elements on the

Inner Anulus Fibrosus ..................................................................................241

6.4.1 Results of analysis of the Homogeneous FEM with improved

hydrostatic fluid element configuration ............................................245

6.4.2 Discussion..........................................................................................250

6.4.3 Summary ............................................................................................250

xiv

6.5 Improved Properties for the Collagen Fibres in the Anulus Fibrosus ..........251

6.5.1 Collagen fibre inclination ..................................................................252

6.5.2 Collagen fibre stiffness ......................................................................253

6.5.3 Results of the analysis of the Homogeneous FEM using improved

collagen fibre geometry and material properties...............................255

6.5.4 Discussion and conclusions ...............................................................257

6.6 Implementation of the Inhomogeneous Anulus Ground Substance into the

FEM ..............................................................................................................258

6.6.1 Results of the Inhomogeneous FEM..................................................259

6.6.2 Discussion and conclusions for the Inhomogeneous FEM................263

6.6.2.1 Posterior and posterolateral bulge of the anulus fibrosus ....263

6.6.2.2 Anterior translation and rotation of the superior surface of the

Inhomogeneous FEM...........................................................264

6.6.2.3 Compliance of the Inhomogeneous anulus fibrosus ground

substance ..............................................................................264

6.6.2.4 Method for applying compressive torso load.......................265

6.7 Discussion and Conclusions on Implementation of the Homogeneous and

Inhomogeneous Material Parameters for the Anulus Fibrosus Ground

Substance ......................................................................................................266

7 Modelling Anterior and Posterior Longitudinal Ligaments........................268

7.1 Method of Representing the Longitudinal Ligaments in the FEM...............269

7.1.1 Spring elements..................................................................................269

7.1.2 Anterior and posterior longitudinal ligament geometry.....................272

7.1.3 Crimp and pre-tension in the anterior and posterior longitudinal

ligaments ...........................................................................................272

7.1.4 Stiffness of the anterior and posterior longitudinal ligaments ...........274

7.2 Analysis of the Homogeneous FEM with Longitudinal Ligaments .............275

7.2.1 Results................................................................................................275

7.2.2 Discussion..........................................................................................279

7.3 Analysis of the Homogeneous FEM with Longitudinal Ligaments – Correct

Disc Heights .................................................................................................280

7.3.1 Results................................................................................................280

7.3.2 Discussion..........................................................................................284

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7.4 Analysis of the Inhomogeneous FEM with Longitudinal Ligaments...........285

7.4.1 Results and discussion of unsuccessful analyses of the Inhomogeneous

FEM...................................................................................................285

7.4.1.1 Effects of removing the 70kPa loading condition................289

7.4.2 Results of the successful analysis of the Inhomogeneous FEM using a

single loading condition of 500N compression.................................291

7.4.3 Discussion..........................................................................................295

7.5 Discussion of the Displacement Convergence Problems in the Unsuccessful

Analyses of the Inhomogeneous FEM..........................................................295

7.6 Analysis of the Inhomogeneous FEM with Longitudinal Ligaments – Correct

Disc Heights .................................................................................................296

7.6.1 Results for the Inhomogeneous FEM with longitudinal ligaments,

correct sagittal geometry and a single 500N compression loading

condition............................................................................................296

7.6.2 Conclusions with respect to the Inhomogeneous FEM......................300

7.7 Discussion of the Mechanical Properties of the Anulus Fibrosus Ground

Substance ......................................................................................................300

7.7.1 Strain rate ...........................................................................................301

7.7.2 Testing environment ..........................................................................301

7.7.3 Compatibility of sheep and human anulus fibrosus ground substance

...........................................................................................................302

7.7.4 Justification for continued use of the overly compliant anulus ground

substance ...........................................................................................303

7.8 Conclusions ..................................................................................................304

8 Simulation and Analysis of Anular Lesions in the FEM...............................306

8.1 Physiological Loading Simulated in the FEM..............................................306

8.2 Representing the degenerate disc..................................................................307

8.2.1 Use of initial loading parameters for the anulus ground substance ...307

8.2.2 Removal of the nucleus pulposus pressure ........................................308

8.3 Simulating Anular Lesions ...........................................................................309

8.3.1 Rim lesions.........................................................................................310

8.3.2 Radial lesion.......................................................................................310

8.3.3 Circumferential lesions ......................................................................311

xvi

8.3.4 Contact relationships..........................................................................312

8.4 Validation of the Degenerate Disc Model ....................................................316

8.4.1 Results................................................................................................317

8.4.1.1 Rim and radial lesion simultaneously represented in the

Degenerate FEM ..................................................................328

8.4.2 Simulation of a rim lesion in a disc FEM with a hydrostatic nucleus

pulposus.............................................................................................328

8.4.3 Discussion of validation analyses ......................................................334

8.4.3.1 Discussion of the Degenerate FEMs with circumferential

lesions...................................................................................334

8.4.3.2 Difficulties in obtaining a converged solution for the

validation analyses ...............................................................335

8.4.3.3 Discussion of the decrease in peak moment between the

Healthy FEM and the Healthy Anulus FEM........................337

8.4.3.4 Discussion of the validation results .....................................339

8.4.3.5 Discussion of the Rim Lesion FEM.....................................341

8.4.3.6 Discussion of the approach to simulating anular lesions .....343

8.5 Analysis of the Healthy and Degenerate FEM using Compressive and

Rotational Loading Conditions.....................................................................344

8.6 Conclusions ..................................................................................................346

9 Conclusions and Recommendations................................................................351

9.1 Recommendations for Further Work ............................................................356

9.1.1 Parameters for the disc components ..................................................356

9.1.2 Simulation of anular lesions...............................................................357

9.1.3 Compressive loading conditions ........................................................358

9.1.4 Simulation of a sheep intervertebral disc...........................................358

Appendices................................................................................................................360

Bibliography .............................................................................................................361

xvii

LLiisstt ooff TTaabblleess Table 2-1 Representative values of published data for the collagen fibre tilt angle in

the anulus fibrosus ...............................................................................................11

Table 2-2 Cross-sectional areas of ALL and PLL.......................................................14

Table 2-3 Linear elastic moduli used for collagen fibres in previous FEM studies....30

Table 2-4 Anterior longitudinal ligament – stiffness and limited geometric data .....36

Table 2-5 Posterior longitudinal ligaments – stiffness and limited geometric data ...37

Table 2-6 Details of specimen handling techniques employed by previous researchers

.............................................................................................................................56

Table 3-1 Abaqus output for convergence of analysis increments ............................65

Table 3-2 Comparison between the nucleus offset determined from the displacement

between the calculated centroids of the nucleus and the anulus and the nucleus

offset value stated in the experimental results .....................................................79

Table 3-3 Details of published material properties for the cartilaginous endplates...96

Table 3-4 A comparison of displacement and nucleus pulposus pressure with the

average experimental results for a 500N compressive load...............................101

Table 3-5 Summary of material properties used in the FEM...................................101

Table 3-6 Comparison of nucleus pressure and von Mises stress for a rigid superior

endplate and superior endplate modelled as cortical bone after the 500N

compression load ...............................................................................................105

Table 3-7 Comparison of FE and experimental results with the results from the FEM

for rotational stiffness under flexion..................................................................127

Table 4-1 R2 statistic for lines of best fit in simple shear ........................................180

Table 4-2 R2 statistic for lines of best fit in uniaxial compression ..........................181

Table 4-3 R2 statistic for lines of best fit in biaxial compression ............................182

Table 4-4 Comparison of stiffness between disc regions with experimental findings

for tensile loading ..............................................................................................194

Table 4-5 Potential sources of error in the experimental data..................................201

xviii

Table 5-1 Comparison of hyperelastic parameters determined by Abaqus and

determined using the Matlab algorithm for the polynomial, N=2 hyperelastic

equation..............................................................................................................221

Table 5-2 Summary of Inhomogeneous hyperelastic material parameters ..............230

Table 5-3 Specifications for the Ogden, N=3 hyperelastic parameters for the three

disc regions during initial and repeated loading ................................................232

Table 5-4 Polynomial, N=2 hyperelastic strain energy parameters for the

Homogeneous anulus under initial loading .......................................................234

Table 6-1 Radial variation of fibre stiffness (Shirazi-Adl et al., 1986) ...................254

Table 6-2 Radially varying elastic modulus of the rebar elements representing the

collagen fibres....................................................................................................255

Table 6-3 Inhomogeneous hyperelastic material parameters for the Ogden, N=3

strain energy equation ................................................................................................258

Table 7-1 Displacements and rotation of the superior surface of the FEM due to the

500N load...........................................................................................................276

Table 7-2 Comparison of von Mises stress in the anulus ground substance of the

FEMs with and without ALL and PLL..............................................................277

Table 7-3 Comparison of the results for the Homogeneous FEM loaded with both a

70kPa nucleus pressure and a 500N compression load and loaded with only a

500N compression load .....................................................................................290

Table 7-4 Comparison of the displacements observed in the Inhomogeneous FEM

with and without the ALL and PLL present. .....................................................291

Table 7-5 Comparison of the von Mises stress observed in the Inhomogeneous FEM

with and without the ALL and PLL present ......................................................292

Table 7-6 Stress in the FEM.....................................................................................298

Table 7-7 Water content (by total mass) in the anulus fibrosus of human and sheep

intervertebral discs. ............................................................................................302

Table 8-1 Angles of rotation for maximum physiological movements expressed in

degrees (SD – standard deviation) .....................................................................307

Table 8-2 Lesions present in the degenerate finite element models .........................309

Table 8-3 Percentage reduction in peak moment of the Healthy Anulus FEM

compared with the Healthy FEM.......................................................................323

xix

Table 8-4 Comparison of the change in peak moments in the Degenerate FEMs and

in the results of Thompson (2002) (The experimental values from Thompson,

2002 were average data) ....................................................................................325

Table 8-5 Percentage variation in the peak moment in the Degenerate FEM with a rim

lesion and in the Rim Lesion FEM. The values in brackets are the magnitude of

the increase or decrease in the peak moment.....................................................341

xx

LLiisstt ooff FFiigguurreess Figure 2-1 The lumbar spine (from Bogduk, 1997).....................................................7

Figure 2-2 Diagram of the saggital/frontal section of the intervertebral disc (Bogduk

1997) ......................................................................................................................8

Figure 2-3 Concentric layers of anulus fibrosus showing alternating angle θ (Bogduk

1997) ....................................................................................................................10

Figure 2-4 Schematic of the vertebra showing the location of the anterior and

posterior longitudinal ligaments (from Marieb,1998) .........................................12

Figure 2-5 Locations of ICRs during right and left lateral bending at various levels in

the lumbar spine viewed from the posterior (Rolander, 1966) ............................18

Figure 2-6 Transverse section of a healthy intervertebral disc showing a moist,

gelatinous nucleus pulposus and an anulus fibrosus with no apparent fissures...21

Figure 2-7 Transverse section of a degenerate intervertebral disc showing a fibrous,

granular and fissured nucleus pulposus and an anulus fibrosus with radial tears,

obvious circumferential separation of lamellae and vascular tissue growing into

the radial defect....................................................................................................21

Figure 2-8 General plane in a body showing the angles to a normal from the plane .41

Figure 2-9 General plane showing stress in that plane resolved in rectangular co-

ordinates...............................................................................................................41

Figure 2-10 Cube of unit length subjected to pure deformation to give side lengths of

λ1, λ2 and λ3 .................................................................................................................46

Figure 3-1 Picture of a sectioned cadaveric intervertebral disc. ................................67

Figure 3-2 Tangent lines creating the rectangular boundary in the transverse

sectioned view of a disc .......................................................................................68

Figure 3-3 Definition of anulus boundary points........................................................69

Figure 3-4 Cosine and sine curve showing angle over which the parametric equations

are chosen ............................................................................................................71

Figure 3-5 Comparison of total disc area with the results from Vernon-Roberts (1997)

.............................................................................................................................74

Figure 3-6 Percentage variation in disc area compared to the area values from

Vernon-Roberts et al. (1997) ...............................................................................74

Figure 3-7 Comparison of nucleus area ratio data ......................................................77

xxi

Figure 3-8 Percentage variation in nucleus area ratios ..............................................77

Figure 3-9 Definition of variables for centroid calculations.......................................78

Figure 3-10 Collagen fibre spacing in a lamellae .......................................................83

Figure 3-11 Schematic of lamellae in the intervertebral disc. ....................................84

Figure 3-12 Determining the average width of the circumferential element layers in

the FEM ...............................................................................................................84

Figure 3-13 Three dimensional continuum element with embedded rebar layer. ......85

Figure 3-14 The rectangular configuration for the rebar layer ...................................86

Figure 3-15 Approximate location of ICR for full flexion from upright standing.

Based on the calculations of Pearcy and Bogduk (1988) ....................................88

Figure 3-16 Location of the ICR for right and left axial rotation viewed from above

.............................................................................................................................89

Figure 3-17 Location of the ICR for right and left lateral rotation viewed from the

posterior disc........................................................................................................90

Figure 3-18 Three dimensional continuum elements in the model. A. Elements

representing the lamellae of the anulus fibrosus; B. Elements in the cartilaginous

endplates ..............................................................................................................91

Figure 3-19 Hydrostatic fluid elements modelling the nucleus pulposus..................93

Figure 3-20 Comparison of the nominal stress-strain response of a Mooney-Rivlin

hyperelastic material – analysed using a single element FEM ............................99

Figure 3-21 Iterative procedure to attain a final sagittal geometry comparable to in

vivo observations (NB. the deformations shown are exaggerated)....................108

Figure 3-22 Varied mesh density used to determine the optimum density for the

analysis of the FEM ...........................................................................................109

Figure 3-23 Comparison of analysis results from finite element models with differing

mesh densities ....................................................................................................111

Figure 3-24 Varied mesh density. A. Specimen 50; B. Symmetric mesh; C. Flattened

posterior curvature; D. Increased posterior curvature; E, F. Displaced nucleus

(endplates not shown) ........................................................................................115

Figure 3-25 Von Mises stress contours for varied mesh geometry (endplates not

shown) A. Specimen 50: B. Symmetric mesh: C. Flattened posterior curvature;

D. Increased posterior curvature; E, F. Displaced nucleus ................................118

Figure 3-26 Contour plot of anterior-posterior displacement ..................................121

xxii

Figure 3-27 Deformed shape of the FEM. Shaded grey: Deformed shape, Wireframe

outline: undeformed shape.................................................................................122

Figure 3-28 Contour plot of von Mises stress in the FEM loaded with 500N

compressive torso load.......................................................................................122

Figure 3-29 Comparison of FEA and experimental results for displacements, 500N

compression. Error bars are 1 standard deviation from the experimental mean.

(AB=anterior bulge, LB=lateral bulge, PB=posterior bulge, AD=axial

displacement) .....................................................................................................123

Figure 3-30 Comparison of the ratio of applied pressure to nucleus pressure for the

500N compression .............................................................................................124

Figure 3-31 Deformed shape of FEM with flexion applied......................................129

Figure 3-32 Contour plots of the fully flexed FEM showing A, B. Maximum

principal strain; C. Minimum principal strain; D, E. Von mises stress .............132

Figure 4-1 P-Q curve showing the potential stress states on a structure..................138

Figure 4-2 Compressive portion of the p-q curve ....................................................141

Figure 4-3 Sheep intervertebral disc set in a dental cement plug and mounted on an

aluminium bracket to allow for sectioning ........................................................142

Figure 4-4 Determining the specimen width required to ensure there were no

continuous fibres connecting the endplates in the specimen .............................142

Figure 4-5 A sectioned specimen..............................................................................143

Figure 4-6 The assembled biaxial testing rig A. With lid in place; B. With lid

removed. ............................................................................................................145

Figure 4-7 Dental cement plug for attaching specimen to nylon cord......................150

Figure 4-8 Schematic of piston attachment in pressure vessel (not to scale) ..........151

Figure 4-9 Ceramic piston with titanium cap glued to the end. ...............................153

Figure 4-10 Assembly of pressure vessel wall, bore insert and glass ceramic piston

...........................................................................................................................154

Figure 4-11 Adjustment knob assembly ..................................................................155

Figure 4-12 Assembled pressure vessel ....................................................................157

Figure 4-13 Measurement of specimen deformation during biaxial compression...159

Figure 4-14 Comparison of the improved measured force and the calculated force

which was manipulated to account for the calibration of the Hounsfield 500N

load cell..............................................................................................................161

Figure 4-15 Measuring the deformation during biaxial compression testing ..........162

xxiii

Figure 4-16 Pressure vs. minimum width for biaxial compression testing on EVA

foam ...................................................................................................................163

Figure 4-17 Hounsfield attachments to apply simple shear.....................................165

Figure 4-18 Anulus fibrosus showing potential directions of shear ........................166

Figure 4-19 Strain rate 0.001 sec-1 ...........................................................................169

Figure 4-20 Examples of stress-strain data for uniaxial compression ......................175

Figure 4-21 Examples of stress-strain data for simple shear. ..................................176

Figure 4-22 Examples of stress-strain data for biaxial compression – the stress is

measured in MPa. ..............................................................................................178

Figure 4-23 Simple Shear-Lines of best fit for response to initial and repeated

loading ...............................................................................................................180

Figure 4-24 Uniaxial Compression - Lines of best fit for response to initial and

repeated loading.................................................................................................181

Figure 4-25 Biaxial Compression - Lines of best fit for response to initial and

repeated loading. ................................................................................................183

Figure 4-26 Range of uniaxial compression test data for the anterior anulus under

initial loading .....................................................................................................185

Figure 4-27 Uniaxial compression loading. A. Derangement strain between 22 and

27%; B Derangement strain between 20 and 27% ............................................189

Figure 4-28 Simple shear loading. A. Derangement strain between 21 and 30%; B.

Derangement strain between 30 and 35%; C. Derangement strain between 24 and

27% ....................................................................................................................191

Figure 4-29 Anulus specimen viewed from the circumferential direction ..............196

Figure 4-30 Anulus specimen viewed from the radial direction..............................197

Figure 4-31 Deformation under biaxial compression loading. .................................199

Figure 4-32 Aspect ratio ...........................................................................................200

Figure 5-1 Shear deformation detailing the stretch ratios.........................................205

Figure 5-2 Simple shear loading on a cubic specimen..............................................207

Figure 5-3 Pure shear loading on a cubic specimen.................................................208

Figure 5-4 Unstrained circle and strain ellipse for pure shear loading ....................209

Figure 5-5 Simple shear deformation........................................................................210

Figure 5-6 Schematic of the deformation of the test specimen during simple shear

loading ...............................................................................................................211

xxiv

Figure 5-7 A comparison between the experimental data for uniaxial compression

and the theoretical stress calculated using hyperelastic constants obtained from

the least squared error algorithm. ......................................................................220

Figure 5-8 Comparison of the theoretical response calculated using Abaqus constants

and the theoretical response calculated using the Matlab algorithm with the

experimental data ...............................................................................................221

Figure 5-9 Comparison of the theoretical results from the Ogden, N=2, N=3, N=4 and

Polynomial, N=2 hyperelastic strain energy equations with the experimental

results .................................................................................................................224

Figure 5-10 Comparison of the experimental response and the theoretical

hyperelastic response for A. Biaxial compression loading – anterior anulus,

initial loading; B. Uniaxial compression loading – anterior anulus, repeated

loading; C. Planar shear loading – lateral anulus, repeated loading. ...............227

Figure 5-11 Uniaxial compression stress vs. strain for the anterior, lateral and

posterior anulus fibrosus ground substance .......................................................233

Figure 6-1 Comparison of uniaxial compression response for the Polynomial, N=2

and Mooney-Rivlin hyperelastic models ...........................................................238

Figure 6-2 Comparison of simple shear response for the Polynomial, N=2 and

Mooney-Rivlin hyperelastic models ..................................................................239

Figure 6-3 Attachment of 3 and 4 node fluid elements to the face of the continuum

elements on the inner anulus surface .................................................................241

Figure 6-4 The undeformed and deformed shape of one element on the inner anulus

surface, at the boundary of the anulus and nucleus ...........................................242

Figure 6-5 Improved hydrostatic fluid elements on the anulus wall.........................243

Figure 6-6 The undeformed and deformed shape of one element on the inner anulus

surface after a single 4 node hydrostatic element was attached to the continuum

element face .......................................................................................................244

Figure 6-7 Deformed shape of Homogeneous FEM – wireframe shows undeformed

shape and arrows define translation and rotation...............................................246

Figure 6-8 Posterior FEM demonstrating outward bulge of posterior anulus and

inward bulge of posterolateral anulus ................................................................248

Figure 6-9 The inferior surface of the intervertebral disc FEM viewed from an

anterior direction................................................................................................248

Figure 6-10 Von Mises stress contour in the Homogeneous FEM..........................249

xxv

Figure 6-11 Von Mises stress distribution for the Homogeneous FEM with improved

collagen fibre properties ....................................................................................256

Figure 6-12 Anulus regions in the Inhomogeneous FEM mesh ..............................259

Figure 6-13 Deformed shape of Inhomogeneous FEM (Wireframe shows

undeformed mesh) .............................................................................................259

Figure 6-14 Posterior anulus bulges outward, posterolateral anulus bulges inward......

...........................................................................................................................260

Figure 6-15 Von Mises stress contours for the anulus fibrosus...............................261

Figure 6-16 Comparison of FEA and experimental results .....................................262

Figure 7-1 Spring elements connected to corner nodes. ..........................................271

Figure 7-2 Deformed shape of the Inhomogeneous FEM with the ALL and PLL

modelled.............................................................................................................276

Figure 7-3 Von Mises stress distribution in the anulus fibrosus ground substance of

the Homogeneous FEM with longitudinal ligaments modelled. .......................278

Figure 7-4 Deformed sagittal geometry of the Homogeneous FEM with the correct

disc heights. .......................................................................................................281

Figure 7-5 Shear stress in the anulus fibrosus due to the anterior translation of the

superior surface with respect to the inferior surface..........................................282

Figure 7-6 Sagittal view of the deformed nucleus pulposus. (Wireframe lines denote

the undeformed mesh) .......................................................................................283


the Homogeneous FEM with corrected sagittal dimensions .............................284

Figure 7-8 Nodes in the anulus fibrosus where difficulties were encountered in the

displacement algorithms ....................................................................................286

Figure 7-9 Deformed geometry of the circumferential element layer in the anulus

fibrosus where the nodes with the largest displacement correction were located.

...........................................................................................................................287

Figure 7-10 Orientation of rebar elements in outermost circumferential element layer

of anulus fibrosus...............................................................................................289

Figure 7-11 Deformed geometry of the Inhomogeneous FEM with the ALL and PLL

present. (Wireframe lines are the undeformed geometry) .................................292


the Inhomogeneous FEM with the ALL and PLL simulated.............................294

xxvi

Figure 7-13 Deformed sagittal geometry of the Inhomogeneous FEM with the ALL

and PLL present and a single 500N compression loading condition.................297

Figure 7-14 Von Mises stress distribution in anulus ground substance of the

Inhomogeneous FEM.........................................................................................299

Figure 8-1 Position of rim lesion in FEM viewed from right anterolateral direction

(Rim lesion surface in blue)...............................................................................310

Figure 8-2 Position of the radial lesion (Radial lesion surface shown in blue) .......311

Figure 8-3 Position of circumferential lesion in the FEM (Circumferential lesion

surface in blue)...................................................................................................312

Figure 8-4 Schematic of contact simulation for the radial lesion. ............................312

Figure 8-5 Two types of contact definitions offered by Abaqus. .............................314

Figure 8-6 Comparison of peak moments. A. Extension; B. Flexion; C. Left lateral

bending; D. Right lateral bending; E. Left axial rotation; F. Right axial rotation

...........................................................................................................................319

Figure 8-7 Comparison of peak moments in Degenerate FEMs with the peak moment

in the Healthy Anulus FEM ...............................................................................321

Figure 8-8 Right lateral bending moment for the Healthy FEM, the Healthy Anulus

FEM and the Degenerate FEM with a rim lesion present..................................322

Figure 8-9 Deformed geometry of the anulus fibrosus in the Degenerate FEM with a

rim lesion simulated and with a 200N compressive load applied – viewed from

the right lateral direction....................................................................................324

Figure 8-10 Deformed geometry of the Degenerate FEM with a radial lesion

simulated and with a 200N compressive load applied – viewed from the left

posterolateral direction (Wireframe shows undeformed shape) ........................324

Figure 8-11 Deformed geometry of the anulus ground substance. ..........................327

Figure 8-12 Comparison of peak moments...............................................................331

Figure 8-13 The peak moments in the Healthy Anulus FEM, the Degenerate FEM

with a rim lesion and the Healthy FEM with a rim lesion simulated are compared

with the peak moment in the Healthy FEM (Rim+Hydrostatic nucleus = rim

lesion simulated in the Healthy FEM) ...............................................................333

Figure 8-14 Nucleus pressure in the healthy disc FEM during rotational loading ...339

xxvii

LLiisstt ooff SSyymmbboollss l = direction cosine with respect to the x direction

m = direction cosine with respect to the y direction

n = direction cosine with respect to the z direction

S = total stress on general plane

Sx = x component of total stress on a general plane

Sy = y component of total stress on a general plane

Sz = z component of total stress on a general plane

Sn = Stress normal to general plane

TU = nominal axial stress

σ = normal stress

τ = shear stress

θ = angle of shear strain

γ = shear strain

λi = extension or stretch ratio; i=1, 2, 3 for principal directions

I = strain invariant

K = stress invariant

D = displacement

E = error using least-squared-error algorithm

K = curvature of a polynomial

W = work

U = strain energy density

F = force generating simple shear deformation

f = force generating pure shear deformation

Cij = material constants for the hyperelastic strain energy equations

αi = material constant for Ogden Nth order strain energy equation for

i = 1,…, N

µi = material constant for Ogden Nth order strain energy equation for

i = 1,…, N

δ = Denotes a virtual quantity

µ = co-efficient of friction

fh = design strength at test temperature

Ph = proof testing pressure

xxviii

LLiisstt ooff AAbbbbrreevviiaattiioonnss dof degrees of freedom

FEM Finite element model

ICR Instantaneous centre of rotation

kPa Kilopascal

MPa Megapascal

QUT Queensland University of Technology

xxix

SSttaatteemmeenntt ooff OOrriiggiinnaalliittyy “The work contained in this thesis has not been previously submitted for a degree or

diploma at any other higher educational institution. To the best of my knowledge and

belief, the thesis contains no material previously published or written by another

person except where due reference is made”

Signed: ………………………

Date: ………………………

xxx

AAcckknnoowwlleeddggeemmeennttss To Mark I say a huge thanks for being the sensible and good-natured overlord of my

PhD. You were supportive, optimistic and helped me to always try to see the big

picture. Thanks to John for the long and insightful discussions of the intricacies of

biomechanics. Your fountain of knowledge was much appreciated. To Graeme I say

thankyou for always being the one to ask the hard questions, but at the same time, for

always being the one to say “This is great, you’ve done a good job”. Clayton… you have been a constant inspiration to me and for that I will be eternally

grateful. Needless to say I would have given up long ago if not for your quick wit,

insightful comments and uncanny ability to make seemingly useless results

worthwhile – you are a Champion! In short, you are a great supervisor, a fantastic

researcher, an all round nice guy and when I grow up I want to be just like you (but

not a guy)! I must also say big big thanks to Mr Ocean. Your continual humour has made the

good days even better and the bad days more than bearable. I would also like to thank

the other postgraduate students in MMME for your willingness to help and good-

humoured nature. Lots of thanks to Greg T for his endless help with the design of the testing equipment

and advice on the testing protocols. Also, many thanks to Terry and Wayne for

manufacturing the biaxial compression rig – this was a fantastic effort on both your

parts and was much appreciated. Now to my family. My mum and dad, especially, have provided me with so much

support during my academic career. No matter what I’ve done they’ve believed in

both me and in my abilities. I owe a great deal of thanks to them both for their

constant support and love throughout my academic life. Thankyou. And last but certainly not least I say thankyou to my lovely husband. He has taken up

the slack that my thesis has made in our lives over the last few months and throughout

my candidature and done so willingly. He is a continual support for me and when

times have been tough he has provided me with endless love and encouragement and

helped me to believe in myself. Thankyou Bee!

Finite Element Modelling of Anular Lesions in the Lumbar Intervertebral Disc

Chapter 1: Introduction 1

CChhaapptteerr

11

IInnttrroodduuccttiioonn Low back pain, both chronic and acute, is a medical condition affecting a large

portion of the population. According to the Australian Bureau of Statistics, in 2001

back and intervertebral disc complaints were one of the most commonly reported long

term health issues with 21% of those interviewed complaining of pain (Australian

Bureau of Statistics, 2002). The National Health and Medical Research Council

reported that each year approximately 600,000 individuals present with lumbar back

pain (National Health and Medical Research Council, 2000). Lumbar back pain may

result from injury or degeneration of the spinal structures or from disorders of the

spinal nerves. The intervertebral discs are one possible source of back pain but the

relationship between disc degeneration and back pain requires clarifying.

While low back pain is a common ailment in both the young and elderly and the

expenses associated with its treatment are considerable, research to date is still

lacking in providing a causal relationship for this illness and the diagnosis of the

source of back pain is difficult. The aim of this study was to provide some insight

into the mechanisms through which low back pain originates, by using a finite

element model to study the effect of degeneration of the lumbar intervertebral disc on

the biomechanics of the spinal joint.

The degeneration of the intervertebral disc may be characterised by a loss of

hydration (Eyre, 1976), loss of disc height (Vernon-Roberts, 1988), a granular texture

in both the anulus fibrosus and the nucleus pulposus and the presence of anular

lesions. Anular lesions are defects in the anulus fibrosus and are frequently linked to



back pain. These lesions involve the failure of the bonds present within the anulus

and commonly manifest as radial lesions, circumferential lesions and rim lesions.

The precise aetiology of anular lesions has not been adequately described in previous

studies. It is as yet unclear whether anular lesions have a detrimental affect on the

biomechanics of the intervertebral disc. Hence the aim of this study was to determine

whether the presence of anular lesions results in a significant loss of the mechanical

ability of the intervertebral disc. It was postulated that the presence of abnormal disc

mechanics as a result of the presence of anular lesions may be related to the incidence

of back pain.

There has been much previous experimental research carried out with the intention of

gaining a better understanding of the initiation and progression of anular lesions

through the disc components. Additionally, several finite element studies have been

carried out to analyse the effects of degeneration on the mechanical capabilities of the

disc. Even so, a conclusive result on the precise causes and growth patterns of lesions

has not yet been provided. Also, there have been many previous studies carried out to

develop finite element models of the intervertebral disc alone or the disc and its bony

and muscular attachments. Several of these models incorporate novel approaches to

describe the viscoelastic nature of the disc components and complex finite element

codes to describe the material behaviour. Chapter 2 details a review of these studies

and provides details of the anatomy and function of the intervertebral disc and the

surrounding spinal structures. Evidence is provided for the suitability of the finite

element method to an investigation of the intervertebral disc mechanics.

In order to ensure the modelling techniques employed were capable of producing a

geometrically accurate representation of the intervertebral disc it was decided that a

Preliminary finite element model (FEM) would be developed. This model would

permit the assessment of the suitability of methods used to obtain the geometry for the

model, the accuracy of the material parameters employed to represent the disc

components and the suitability of the methods employed to simulate physiological

loading conditions. The development of this model is detailed in Chapter 3.

Further to the development of the Preliminary FEM it was apparent that the

mechanical properties of the anulus fibrosus ground substance, the methods used to



simulate the hydrostatic nucleus and collagen fibres and the level of anatomical

accuracy of the model required improvements. Experimental testing was carried out

on sheep anulus in order to obtain accurate data for the mechanical behaviour of the

anulus ground substance. Chapter 4 details this experimentation. It was apparent

from these data that the anulus fibrosus ground substance was circumferentially

inhomogeneous with different mechanical characteristics for the anterior, lateral and

posterior regions.

An improved mechanical description for the anulus fibrosus ground substance was

developed using these experimental data. These data were used to determine

parameters for a hyperelastic strain energy equation that better described the

behaviour of the material under simple shear loading. Chapter 5 is devoted to an

extensive description of the possible hyperelastic strain energy equations that could

have been applied to the anulus ground substance and the criteria used to select the

final parameters to describe the material. Both a homogeneous and an

inhomogeneous material model were defined.

Implementation of the improved homogeneous hyperelastic material description for

the anulus ground substance is detailed in Chapter 6. This chapter also provides an

explanation of the improvements to the methods for simulating the nucleus pulposus

and the material parameters describing the collagen fibres.

Subsequent to the implementation of the Homogeneous FEM and the improvements

to the nucleus pulposus and collagen fibres, the anterior and posterior longitudinal

ligaments were included in the mesh. This was considered to improve the anatomical

accuracy of the model due to the close relationship between longitudinal ligaments

and the intervertebral disc. The methods employed to simulate the ligaments are

detailed in Chapter 7. This chapter is divided into three main topics that detail the

simulation of the ligaments and the implementation of the homogeneous and

inhomogeneous material parameters.

The simulation of disc degeneration in the Homogeneous FEM is described in

Chapter 8. Subsequent to a description of the methods employed to simulate the

degeneration in the intervertebral disc, the results for analyses of the disc models with



varying degrees of degeneration are provided. An extensive discussion is given in

this chapter to provide possible causes for the difficulties encountered in obtaining a

converged solution for the analyses of the lesions and to provide useful deductions

with regard to the variation in disc mechanics observed subsequent to the simulation

of disc degeneration.

Finally, Chapter 9 is dedicated to emphasizing the objectives that were achieved in the

thesis and the most significant discoveries that were made in relation to the variation

in disc mechanics as a result of the presence of anular lesions. This chapter also

contains details of suggested future work that, if completed, would provide a powerful

analysis tool for the investigation of various loading conditions and further

exploration of the biomechanical effects of anular lesions.

1.1 Aims and Objectives of the Thesis

The aim of this thesis was to develop a finite element model of an L4/5 intervertebral

disc that accurately represented the geometry and material properties of the disc. This

model would be used to study the effects of degeneration on the mechanics of the

disc. Particular objectives in this project were the:

• Development of a preliminary model of the L4/5 intervertebral disc to ensure

the modelling methods are capable of generating a geometrically accurate

model. This model will also highlight areas for improvement in the geometry or

material descriptions for the disc components.

• Acquisition of accurate mechanical data for the anulus fibrosus ground

substance and the fitting of a hyperelastic strain energy equation to this data.

This equation must incorporate the nonlinear shear behaviour of the anulus

fibrosus.

• Implementation of the improved material parameters in the finite element model

of the intervertebral disc to obtain a model that is capable of simulating the

biomechanical effects of anular lesions

• Simulation of a healthy and a degenerate intervertebral disc to observe the

variation in mechanics as result of the presence of anular lesions



1.2 Limitations of the Study

For the objectives of the research to be achieved it was necessary to narrow the scope

of the project. In particular, the following features were not incorporated in the

model.

• Dynamic analyses: The analyses of the biomechanical effects of anular lesions

on the disc were static.

• Fracture mechanics: While investigations of anular lesions may have benefited

from an approach that incorporated the theories of fracture mechanics, this was

deemed to be outside the scope of the project. As such, analysis of the

biomechanical effects of lesions would not include an assessment of the stress

concentrations in the vicinity of the lesion. Rather an assessment of the overall

mechanics of the disc was carried out.

• Compressible structures: The structures within the intervertebral disc were

assumed to be single phase and incompressible. It was thought that this

assumption was reasonable on the basis of the simulation of physiological strain

rates.

• Bony anatomy: The adjacent vertebra and bony posterior elements were not

included in the model. Also the muscles and ligaments of the spine were not

modelled. These structures were simulated using specific loading and boundary

conditions.


Chapter 2: Literature Review 6

CChhaapptteerr

22

LLiitteerraattuurree RReevviieeww The literature review presented in the following sections provides information on the

physical and mechanical properties of the intervertebral disc and the surrounding

anatomy. Information on the nature of disc degeneration and the relevance of

conducting an investigation into the biomechanical effects of anular lesions on disc

mechanics is presented. Further to this, the applicability of using the finite element

method to investigate this topic was explored. Details of the continuum mechanics

relating to hyperelastic materials are presented.

2.1 Spinal Anatomy

The vertebral column is comprised of 24 separate vertebrae joined axially by the

intervertebral discs (Figure 2-1). The intervertebral discs and spinal muscles allow

for the functional capabilities of the spine such as bending or turning, while the

ligaments of the spine provide limitation on spinal movements, in order to prevent

damage to the soft or hard tissues. The lumbar spine comprises the five lowermost

vertebrae with their interconnecting intervertebral discs.



Figure 2-1 The lumbar spine (from Bogduk, 1997)

2.1.1 The bony spinal column

Each vertebra consists of two main regions – the vertebral body and the posterior

elements or arch (Bogduk, 1997; Marieb, 1998).

The vertebral body has a kidney shaped profile in the transverse plane and is curved

concavely on the axial faces when viewed in the saggital or frontal planes. The outer

surface of the vertebral body is comprised of cortical bone and the inner shell is

comprised of the comparatively less stiff cancellous bone.

The posterior elements consist of the following structures.

• the pedicles are rod-like structures projecting from the posterior surface of the

vertebral body;

• the laminae extend from the pedicles to fuse centrally on the posterior spine;

• the inferior articular processes extend inferiorly and posteriorly from the

laminae;

• the superior articular processes extend superiorly and posteriorly from the

junction of the laminae and pedicles;

halla

This figure is not available online. Please consult the hardcopy thesis available from the QUT Library



• the spinous process is a fin-like structure arising from the junction of the

laminae; and

• the transverse processes are bony masses that project laterally from the

junction of the laminae and pedicles (Bogduk, 1997; Marieb, 1998).

The superior articular processes of one vertebra articulate with the inferior articular

processes of the superiorly adjacent vertebra, creating a synovial joint called the

zygapophysial joint. The spinal cord runs along the axial cavity created by the neural

arch posteriorly and the posterior surface of the vertebral bodies and intervertebral

discs anteriorly. This canal is called the vertebral foramen. The neural arch is

comprised of the two pedicles and the two posteriorly fused laminae arising from

these pedicles. Spinal nerves issue from the openings between the pedicles of

adjacent vertebra called intervertebral foramen.

The bony protrusions and landmarks on the vertebra serve as attachment points for

various muscles and ligaments of the spine (Bogduk, 1997).

2.1.2 The intervertebral disc

The intervertebral disc consists of three structures, the anulus fibrosus, the nucleus

pulposus and the cartilaginous endplates (Figure 2-2).

Figure 2-2 Diagram of the saggital/frontal section of the intervertebral disc

(Bogduk 1997)

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2.1.2.1 Nucleus pulposus

In healthy, young discs, the nucleus pulposus is a gelatinous structure which exists

near the centre of the disc. A healthy nucleus pulposus contains 70-90% water

(Bogduk 1997). Proteoglycans constitute 65% of the dry weight of the nucleus and

irregularly dispersed collagen fibres comprise 15-25% of the dry weight of the

nucleus (Bogduk, 1997, Dickson et al., 1967, Pedrini et al. 1973). There are also

limited amounts of other selected proteins present.

Proteoglycans are long chain molecules consisting of subunits of glycosaminoglycans

and proteins linked to hyaluronic acid chains. These molecules are commonly found

in cartilage and serve to promote osmotic swelling in the cartilage and encourage

hydration of the tissue. This hydrating function is brought about due to the ionic

nature of the glycosaminoglycan chains, which electrically attract water molecules

(Bogduk, 1997). Of the 11 types of collagen observed in connective tissue, type I and

II are the main types found in the intervertebral disc. Type II collagen fibres

dominate the innermost nucleus (Eyre and Muir, 1977; Bogduk, 1997) whilst there is

some evidence of small amounts of type I collagen. Type II collagen is a

comparatively elastic material which is commonly observed in biological tissues that

are subjected to pressure (Bogduk, 1997).

2.1.2.2 Anulus fibrosus

The nucleus is enclosed peripherally by the anulus fibrosus which consists of a series

of concentric layers. These layers are comprised of collagen fibres embedded in a

ground matrix. The collagen fibres are regularly aligned at a specific angle to the

cranio-caudal axis through the disc (Horton, 1958). However, this angle varies

alternately for each successive lamella to create a criss-cross pattern between adjacent

lamellae (Figure 2-3).



Figure 2-3 Concentric layers of anulus fibrosus showing alternating angle θ

(Bogduk 1997)

Both type I and type II collagen are found in the anulus, but type I is the principal

form present (Bogduk, 1997, Eyre, 1976). Type I collagen is largely found in tissues

which experience tensile or compressive loading (Bogduk, 1997). Eyre and Muir

(1976) observed that the distribution of the type I and type II collagens was not

constant throughout the anulus fibrosus of pigs. It was found that the outermost

lamellae had very little type II collagen and were comprised of almost all type I

collagen. Testing of anulus regions successively closer to the nucleus pulposus

showed an increase in the amount of type II collagen. A ‘transition zone’ between the

anulus and nucleus was identified and in this zone Eyre and Muir (1976) observed

only type II collagen.

The collagen content of the young anulus fibrosus is quoted to be approximately 67%

by Pedrini et al. (1973) while other researchers give values between 50 and 60%

(Adams et al., 1977; Dickson et al., 1967).

It was determined by Adams et al. (1977) that the collagen content in the anterior

anulus fibrosus varied radially. The outer anulus exhibited the largest collagen

content and the inner anulus the lowest content. In the L4/5 disc, the outer anulus had

a collagen content of 58% of dry weight, the middle anulus collagen content was 48%

of dry weight and the inner anulus collagen content was 30% of dry weight. From the

results of Adams et al. (1977) it may also be observed that the collagen content in the

anulus fibrosus is higher in discs at lower spinal levels.

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Table 2-1 Representative values of published data for the collagen fibre tilt angle

in the anulus fibrosus

AUTHOR ANGLE Bogduk (1997) Average between 65-70º

Cassidy et al. (1989) 62º in outer anulus, 45º

in inner anulus Hukins et al. (1989) Average between 60-70º

Marchand & Ahmed (1990) Average 70º Natali & Meroi (1990) Average between 52-54º

Shirazi-Adl et al. (1986) Average 61º

It may be seen from Table 2-1 that there is a reasonable amount of variation in the tilt

angle of the collagen fibres. The majority of researchers use a single average value to

represent the tilt angle, θ, of the fibres within the entire anulus fibrosus, rather than

varying this angle radially through the anulus as Cassidy et al. (1989) observed.

Possibly, the broad range of values stated (52-70º) is a result of using experimental

techniques and modelling approaches which only permit the use of a single value for

the angle, rather than accounting for the possibility of variation in the collagen fibre

angle depending upon position in the anulus. On the basis of the angles listed in

Table 2-1 it may be assumed that the average angle of inclination of the collagen

fibres in the anulus fibrosus is between 60o and 70o.

2.1.2.3 Cartilaginous endplates

The cartilaginous endplates are thin layers of cartilage on the superior and inferior

surfaces of the disc. These structures are at the boundary between the intervertebral

disc and the adjacent vertebrae; however, there is no intimate connection between the

endplates and the vertebral bone (Inoue, 1981).

The collagen fibres of the inner 1/3 of the anulus fibrosus are inserted directly into the

cartilaginous endplates and Inoue (1981) stated that there was “a morphologic

interconnection of the lamellar fibres of the anulus with the fibres of the horizontally

aligned cartilage endplate”. Thus it was observed that the collagen fibres in the

cartilaginous endplate are aligned with the transverse plane through the disc.



However, it was not clear from the literature what orientation the collagen fibres

exhibit in the plane of the endplates. The fibres from the remaining outer 2/3 of the

anulus insert directly into the subchondral layer of the vertebra and are referred to as

Sharpeys Fibres (Inoue, 1981; White and Panjabi, 1978).

The cartilaginous endplates are permeable structures which permit the diffusion of

nutrients from the bone marrow of the adjacent vertebral bone. This facilitates the

nutrition of the intervertebral disc (Bogduk, 1987).

2.1.3 Anatomy and attachment of the longitudinal ligaments

The ligaments attached to the spine provide limits on the physiological motion of the

spine, protect the spinal cord by preventing motion of the spine outside these limits

and aid the spinal muscles in providing stability during physiological motions (White

and Panjabi, 1978). The mechanical behaviour of the ligaments is most pronounced

when they are loaded in the direction of the collagen fibres. The spinal ligaments

which are most intimately related to the function of the disc are the anterior

longitudinal ligament, ALL, and the posterior longitudinal ligament, PLL. The ALL

and PLL extend the length of the spine and pass over the anterior and posterior

surfaces of the vertebral column, respectively (Figure 2-4).

Figure 2-4 Schematic of the vertebra showing the location of the anterior and

posterior longitudinal ligaments (from Marieb,1998)

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Collagen fibres in both these structures attach to the bone of the vertebral bodies and

extend over as few as one or as many as five intervertebral joints (Bogduk, 1997).

The PLL also exhibits an intimate connection with the anulus fibres in the posterior

surface of the intervertebral discs (Haughton et al. 1980; White and Panjabi 1978;

Bogduk 1997), however, the ALL does not display a similar connection and is not

firmly attached to the anterior anulus fibrosus (Neumann et al., 1992; White and

Panjabi, 1978).

Bogduk (1983) reported that the ligaments of the lumbar spine and in particular the

ALL and PLL were innervated. He suggested the innervation of these and other

ligaments of the lumbar spine as well as the peripheral regions of the intervertebral

disc highlighted “potential sources of primary low-back pain”.

2.1.3.1 Cross-sectional area

White and Panjabi (1978) mention the difficulty involved in determining quantitative

information on the dimensions and properties of the ligaments. This was due to

variability in techniques used for testing and also difficulties in delineating the

boundaries of the ligaments from surrounding soft tissue in the spine. They consider

these factors explain the variability observed in the morphology and properties of the

spinal ligaments.

The significant variation in cross-sectional area of the ligaments is likely to be a result

of the difficulty encountered in defining the boundaries of the ligaments, either when

viewed in vitro or when viewed using medical imaging technology. Additionally, due

to the viscoelastic nature of the tissue there are difficulties encountered when the

dimensions of the ligaments are measured (Neumann et al., 1993, 1994).



Table 2-2 Cross-sectional areas of ALL and PLL

Author ALL - Area (mm2) PLL - Area (mm2)Tkaczuk (1968) Lateral width=20mm Lateral width = 14.5mm

Thickness in anter-posterior Thickness in anter-posteriordirection = 0.8 mm direction = 1.35 mmArea of ellipse = 13 Area of ellipse = 14.4

Ohshima et al. Lateral width = 17mm(1993) Using thickness from Tkaczuk (1968)

Area of ellipse = 18.0Pintar et al. 32.4 ± 10.9 5.2 ± 2.4

(1992) range = 10.6 - 52.5 range = 1.6 - 8.0Neumann et al. 38.2 ± 3.5

(1992)Shirazi-Adl et al. 24 14.4

(1986)Chazal et al. 65.6 30.8

(1985)Panjabi et al. 75.9 ± 20.9 51.8 ± 6.6

(1991)White & Panjabi 53 16

(1990)McGill & 30

Norman (1986)

The average cross-sectional area of the ALL calculated using the results outlined in

Table 2-2 was 43.2mm2. The average cross-sectional area of the PLL was 25.2mm2.

Pintar et al. (1992) obtained data on the geometry of the spinal ligaments of 8

cadaveric lumbar spines with ages between 31 and 80 years. The ‘normal’ spinal

curvature was maintained post-mortem and a cryosectioning technique used to freeze

the spine immediately post-mortem. Once the frozen spine was mounted, 1mm tissue

slices were removed and the exposed surface extensively photographed. Pintar et al.

(1992) considered that the use of a cryomicrotome table was a preferred method for

determining ligament cross-sectional and sagittal geometry because the tissues were

frozen in their natural state, ensuring the anatomy and position of the ligament was

maintained. Also, the ligaments were more easily distinguished due to the lack of

fluid loss into tissues surrounding the ligaments. However, the cross-sectional area

which was found for the PLL was significantly lower than the results from other

studies. Upon calculation of the linear elastic modulus using this cross-sectional area,

the result was an order of magnitude larger than the values from other studies (Section

2.1.3.2). On this basis, the calculation of the average cross-sectional area of the PLL

did not include the value determined by Pintar et al. (1992).



2.1.3.2 Lateral width

In the lateral directions, the PLL is wider across the intervertebral disc and thinner

across the vertebral bodies (Bogduk, 1997; Tkaczuk, 1968; White and Panjabi, 1978)

while the opposite is true for the ALL (White and Panjabi, 1978).

Tkaczuk (1968) reported that the width (lateral dimension) of the ALL over the L5

vertebral body was 20mm and over the L3 vertebral body was 25mm and the

thickness of the ligament at this location was 1.9mm. It was reported that the

ligament thickness reduced over the anulus fibrosus and was measured to be 0.8mm

over the L4/5 intervertebral disc.

Tkaczuk (1968) stated that the width of the PLL was 14mm at the L5/S1 disc level

and 15mm at the L3/4 disc level. Therefore, the approximate width over the L4/5 disc

would be 14.5mm. The thickness over the L4 and L5 vertebral bodies were 1.4mm

and 1.3mm, respectively.

Few researchers since Tkaczuk (1968) have investigated the anatomy of the ALL for

the purpose of specifically quantifying the width and thickness over the intervertebral

disc. In a study investigating posterior herniation of the intervertebral disc, Ohshima

et al. (1993) stated that the average width of the PLL was 17mm.

On the basis of the above discussion:

• the width of the ALL over the L4/5 intervertebral disc is assumed to be 20mm;

• the thickness of the ALL is assumed to be 0.8mm;

• the lateral width of the PLL is assumed to be 15.75mm;

• the thickness of the PLL is assumed to be 1.35mm;

• the cross-sectional profile of the PLL is represented as an ellipse (Tkaczuk,

1968) and the ALL cross-section represented as an ellipse.



2.1.3.3 Pre-tension in the ligaments

It has been reported that the ligaments of the lumbar spine exist in a state of prestress

when in vivo (Tkaczuk, 1968). Prestress in the ligament is the force per unit cross-

sectional area present when the spine is in the neutral position. This prestress is

evident since, depending on ligament age and type, they retract by between 7.1% and

13.4% when cut. The amount of prestress is directly dependant on the magnitude of

intradiscal pressure (Tkaczuk, 1968).

Nachemson and Evans (1968) stated that the pre-tension present in the ALL and PLL

was one-tenth that of the ligamentum flavum. They stated that the pre-tension in the

ligamentum flavum of young discs (< 20 years) was 18N and in old discs (> 70 years)

was 5N and there was a near linear variation in this pre-tension force with age. The

tensile nature of these forces indicated that the spine was in compression when in the

neutral position. Based on these pre-tension forces for the ligamentum flavum, the

pre-tension in the ALL and PLL would be 1.8N in young discs and 0.5N in old discs.

In order for a computational model of the intervertebral disc to represent the

physiological condition, it would be necessary for the elements modelling the ALL

and PLL to reach a state of tension after loading conditions simulating relaxed

standing were applied. Also, the age of the disc modelled would be taken into

account to determine the tensile force present in the ligaments.

2.2 Location of the Instantaneous Centres of Rotation during Physiological

Loading

An instantaneous centre of rotation, ICR, is a point about which pure rotation occurs

when the system is loaded. When discussing three dimensional structures this is

sometimes referred to as an instantaneous axis of rotation. Due to the functional

differentiation of different regions within the anulus there is not a unique location for

the ICR in the intervertebral disc during bending (Klein and Hukins, 1983). That is,

due to the varying contributions from the regions of the intervertebral disc, from the

bony anatomy and from the musculature of the spine, the location of the ICR varies

with the motion carried out.



2.2.1 Flexion and extension

Pearcy and Bogduk (1988) used lateral radiographs of 10 subjects in order to

determine the ICR during flexion and extension. This axis was determined for all

lumbar joints and was based on movement from a relaxed upright position to a fully

flexed or extended position. The locations of the ICRs from each subject were

normalized for the size of the subject’s vertebra. The results were given as x and y

co-ordinates measured from an origin at the superior, posterior corner of the lower

vertebra in the joint. These co-ordinates were quoted as a proportion of the depth and

height of this vertebra. As a result of the limited range of motion of the L4/5 joint in

extension, it was not possible to calculate a location for the ICR for full extension

from upright.

2.2.2 Axial Rotation

On the basis of experimentation on cadaveric joints, Cossete et al. (1971) found the

location of the ICR during axial rotation was:

• in the posterior intervertebral disc;

• was located near the median line; and

• tended to move toward the side corresponding to the direction of rotation.

That is, during right axial rotation (clockwise when viewed from the cranial

direction), the ICR was located in the right side of the disc and moved further into the

right side of the disc as the rotation increased.

Adams and Hutton (1981) loaded cadaveric lumbar intervertebral joints in combined

torsion and compression. They found the centre of rotation during torsion was in the

neural arch or posterior anulus similar to the findings of Cossete et al. (1971).

The axial location of the ICR was not clear from the literature.



Thompson (2002) carried out rotational tests on sheep lumbar intervertebral joints

using a robotic testing facility. The location of the ICR for axial rotation was based

on the findings of Cossete et al. (1971). The ICR was initially located at the median

line of the disc and moved laterally to a final location of ¼ of the lateral width of the

disc when the full axial rotation angle was applied. In the axial direction the ICR was

located at mid disc height.

2.2.3 Lateral bending

In testing on human cadaveric intervertebral joints Rolander (1966) made

observations of the ICR during lateral bending. There were no specific details

provided for this location, rather a schematic of the positions observed for various

joints tested under right and left lateral bending was provided (Figure 2-5). This

schematic showed the locations viewed from the frontal plane. Tests were carried out

on both healthy and degenerate discs.

Figure 2-5 Locations of ICRs during right and left lateral bending at various

levels in the lumbar spine viewed from the posterior (Rolander, 1966)

The scatter of ICR locations for the non-degenerate discs indicated that during right

lateral bending, the ICR was located in the left disc and during left lateral bending it

was located in the right disc. The scatter of the ICR locations appears to be less for

the non-degenerate discs.

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In the rotational testing carried out by Thompson (2002) the location of the ICR

during lateral bending was defined on the basis of the work of Rolander (1966). The

ICR location used in this study moved from the centre of the disc, when viewed in

both the frontal and transverse planes, to a final location halfway between the centre

of the disc and the peripheral disc when full lateral bending was applied.

Details of the location of the lateral bending ICR in the axial direction through the

disc are not readily available. Rolander (1966) gave only pictorial information on the

axial location of the ICR but from the locations in Figure 2-5 Thompson (2002)

placed the ICR at the mid disc height level.

2.3 Degeneration and anular Lesions

As the disc ages it undergoes various changes including a reduction in the fluid

content of the anulus and nucleus (Eyre, 1976), a decrease in disc height and

development of osteophytes at the vertebra-disc junction (Benzel 1995; Vernon-

Roberts 1988). The boundary between the anulus and nucleus becomes even less

distinguishable and the nucleus tends to become more fibrous with age (Eyre, 1976).

These changes in the disc structure are attributed to disc degeneration.

The degeneration process is related to abnormal disc behaviour under physiological

loading (Benzel 1995). Natarajan et al. (1994) noted that the mechanism for initiation

of degeneration in the disc is not as yet understood.

Anular lesions involve failure of the bonds present within the anulus fibrosus and are

found in degenerate discs or in discs that have experienced trauma. There are three

types of lesions, which may develop and these include:

• Radial lesions – these lesions develop in a radial direction through the anulus

(Figure 2-6 and Figure 2-7);



• Rim lesions – these lesions are a failure of the anulus material parallel to the

cartilaginous endplate, often near either the superior or inferior surface of the disc;

and

• Circumferential lesions – resulting in a separation of the lamellae in a

circumferential path around the anulus fibrosus (Figure 2-6 and Figure 2-7).

Depending on the type of lesion and how large the tear has grown, nuclear material

may protrude through the fissure to the outer anulus. This process results in disc

herniation and the protruded material may impinge on structures outside the disc, such

as the nerve roots or spinal cord. In the case of the lumbar spine, back pain and/or

irritation of the nerves feeding the lower torso and lower limbs may be experienced.

The discs, which most commonly develop lesions are in the lower lumbar and lower

cervical spine and disc herniations are predominantly seen on the posterior aspect of

the disc (Armstrong 1958). Armstrong (1958) noted that these disc locations correlate

with regions of a higher degree of movement of the spine.

Vernon-Roberts (1988) found that there is frequently a growth of granular and

vascular material into the lesions. Additionally, if the defect is a rim lesion, the

granulation material may develop between the bone of the vertebral body and the

remaining anulus material, to restore some stability to the joint.

A comparison of Figure 2-6 and Figure 2-7 shows a significant difference between the

healthy and degenerate disc. Figure 2-7 shows obvious signs of degeneration in the

form of radial and circumferential lesions and the dry, granular texture of both the

anulus and nucleus.

For the purposes of this thesis degeneration of the intervertebral disc will include:

• The presence of anular lesions;

• The reduction in hydration of the anulus and nucleus; and

• The loss of a hydrostatic fluid pressure in the nucleus.



Figure 2-6 Transverse section of a healthy intervertebral disc showing a moist, gelatinous nucleus pulposus and an anulus fibrosus with no apparent fissures

Figure 2-7 Transverse section of a degenerate intervertebral disc showing a fibrous, granular and fissured nucleus pulposus and an anulus fibrosus with

radial tears, obvious circumferential separation of lamellae and vascular tissue growing into the radial defect

Circumferential

lesion

Radial

lesion



2.3.1 The mechanism of degeneration and the initiation of anular lesions

The chronology of the development of the lesions in relation to the degenerative

process is not clear from the literature. Do these lesions develop as a result of the

degenerative changes in the disc or do the lesions develop within the disc as a result

of excessive stresses and in turn, initiate the process of disc degeneration?

Also, it is not clear precisely why discs degenerate. Is degeneration a result of

mechanical and biochemical processes, which manifest as degeneration of the disc

and result in abnormal discal loading? Or is degeneration secondary to existing

abnormal spinal movements and excessive stresses that result in degenerative

biochemical changes in the disc and more pronounced aberrant movements?

It is possible that the mechanism by which anular lesions develop in the disc involves

an overlap of the mechanical and biochemical processes.

The intervertebral disc is a dynamic structure. Goel and Weinstein (1990) note that

when the disc is subjected to excessive or abnormal loading patterns, it is likely to

demonstrate structural change in an effort to reduce the internal stresses. Vernon-

Roberts (1988) states that the degenerative changes in the disc may be secondary to

other changes in the spine. It may be that due to variations in other structures within

the spine, such as a reduction in the integrity of the zygapophysial joints, the response

of the disc is altered and leads to degenerative changes in the disc components.

However, it has also been found in previous studies that as age increases, the water

content of the disc decreases (Pearce et al., 1987). This in turn lends strength to the

argument that the degeneration of the disc is related to aging and anular lesions

develop as result of the degeneration of this structure.

Yet another factor to be considered in order to understand the mechanism of anular

lesion development is the clinical history of the patient and any previous spinal

trauma.



It is proposed that the development of an accurate computational model of the

intervertebral disc would allow meaningful deductions to be made in relation to the

mechanism by which anular lesions and degeneration occur.

2.3.2 Relevance of Studying Anular Lesions

Development of an accurate finite element model of the disc will enable the

relationship between degenerative changes in the disc and altered mechanics to be

characterised. The altered mechanics of the disc could then be used to predict

consequent overload of the intervertebral joint components that might cause damage

leading to back pain. Hence this will assist in the understanding of the mechanisms

leading to back pain.

2.4 The Use of FEM to Study the Spine and in particular Anular Lesions

Previous studies of the intervertebral disc and associated structures have used three

main approaches, namely:

1. Experimentation on individual discs, spinal motion segments or lumbar spine

segments (Thompson et al., 2000; Osti et al., 1990; Markolf and Morris, 1974; Adams

and Hutton, 1981; Nachemson, 1960; Hirsch, 1955);

2. Development of analytical/mathematical models (Hickey and Hukins, 1980;

McNally and Arridge, 1995); or

3. Development of finite element models (Kumaresan et al., 1999; Natarajan et al.,

1994; Shirazi-Adl et al., 1984, 1996; Goel et al., 1995).

Experimental studies carried out on disc units provide worthwhile data on the overall

behaviour of the disc in terms of forces, displacements and pressures present.

However, it is not easy to define the internal stress state of the disc using experimental

techniques. Whilst analytical models provide greater accuracy in representing the

material properties of the disc materials (McNally and Arridge, 1995) they do not

permit as much accuracy in representing the disc geometry.



The approach adopted in the current study is the use of the finite element (FE) method

to model the intervertebral disc. The finite element method is outlined in Chapter 3.

The advantages of the finite element method are:

• It permits a high degree of control over the loading applied – the load

magnitude and method of application can be defined to accurately represent

physiological loading and coupled motions;

• It allows various defects to be introduced and monitored in the disc by

performing several solution runs with the same model; and

• It permits the internal stress state of the disc to be determined.

However, it must be noted that a difficulty inherent in the use of the finite element

method is the ability to correctly define the true material nature of the disc. The

components of the disc are comprised of very complex materials. Additionally, the

mechanical properties of the disc components have not yet been rigorously defined.

As such, all finite element models are limited to some degree by the material

representation employed.

2.5 Shortcomings in Previous Models

Two important aspects of a finite element model are the geometric description and the

material properties used to describe the components. Many previous finite element

models of the intervertebral disc incorporate simplified descriptions for both the

geometry and material properties. The geometry of the disc has been simplified as an

axisymmetric structure (Shirazi-Adl et al., 1986; Natali and Meroi, 1990) and the

transverse geometry has been simplified as an idealised “kidney shape” without

accurate definitions for the axial dimensions (Shirazi-Adl et al., 1986; Natali and

Meroi, 1990). A linear elastic material formulation has been used to describe both the

bulk response of the anulus fibrosus and the response of the anulus ground substance

(Kurowski and Kubo, 1986; Shirazi-Adl et al., 1986; Goel et al., 1995; Ueno and Liu,

1987; Kumaresan et al., 1999); however, this material behaves nonlinearly under

loading (Acaroglu et al., 1995; Best et al., 1994; Fujita et al., 1997).



Natarajan et al. (1994) carried out a finite element analysis of a spinal motion

segment in order to determine the most likely loading mode to initiate endplate

fractures or anular lesions in the disc. This study provided data on the compressive

and flexion/extension loading necessary to initiate fracture in the endplates and anulus

of a healthy disc. However, an extensive analysis of the stress state within the disc as

a result of the presence of anular lesions was not provided. Also, the material

properties of the model in which fracture was initiated were those of a healthy disc.

The possibility of degenerate disc material properties prior to lesion growth was not

considered.

Natarajan et al. (1994) found that the presence of radial or circumferential lesions in

the anulus had no effect on the flexion/extension moments necessary to initiate

failure. This result is debatable and no explanation or validation from previous

studies was provided. This questionable result may be related to the method of

application of the flexion/extension loading. This loading was applied as a linearly

varying load over the superior surface with the maximum loads applied on the

extreme anterior and posterior edges. Such a loading method does not appear to take

into account the instantaneous axis of rotation of the spinal segment (Pearcy and

Bogduk, 1988) or the physiological limits of rotation exhibited by the vertebra-disc-

vertebra unit (Pearcy, 1985).

A common approach to the modelling of degeneration of the disc materials is to vary

the material properties of the anulus and/or nucleus. Natali and Meroi (1990) and

represented degeneration of the disc as an increase in the compressibility of the

nucleus alone. However, there was no variation in the properties of the anulus

material which is known to decrease in water content, become more granular

(Bogduk, 1997) and vary in stiffness (Acaroglu et al., 1995; Iatridis et al., 1999).

Shirazi-Adl et al. (1986) represented the degenerate disc by reducing the pressure in

the nucleus.

Simon et al. (1985) developed a poroelastic finite element model of the disc and

simulated degeneration using a decrease in the permeability of the matrix. However,

such a decrease in the permeability of the disc materials may not be sufficient to



represent the complex material modifications that have resulted in the varied material

stiffness.

Belytschko et al. (1974) developed a finite element model of the intervertebral disc in

order to investigate the behaviour of the disc under axial loading. They simulated

degeneration and development of anular lesions in the disc by reducing the elastic

modulus of the anular material. However, the anular lamellae were simplified as

anisotropic, homogenous materials without considering the presence or path of

collagen fibres. Also, the method of representing the anular lesions could be

improved to provide a more sophisticated analysis of their effects on the disc.

In research carried out by Natali (1991) the nucleus pulposus was represented as a

hyperelastic material. It was believed that the material formulation in this study was

well suited to the incompressible nature of the material. Natali and Meroi (1990)

stated that a more accurate FEM was achieved by specifically modelling the material

of the nucleus pulposus rather than using a hydrostatic pressure boundary condition.

Natali and Meroi (1990) represented all the disc material as a hyperelastic material

citing a similar advantage of this representation to simulate incompressibility.

A limitation in this approach to modelling the nucleus pulposus is that in order to

obtain parameters for the material it would be necessary to carry out experimentation

on the nucleus to determine its mechanical behaviour. Such experimentation is

difficult and no evidence was found in the literature for this research. Without this

work, it would be difficult to accurately define the parameters of the nucleus. Also,

these studies represented the anulus fibrosus as a series of fibre layers, rather than

defining the material response of the ground substance separate to the response of the

collagen fibres. Modelling the anulus as fibre layers may limit the accuracy of the

model because the relationship between the fibres and the ground matrix in which

they are embedded plays an important role in the mechanical function of the disc.

This was shown by the observations of Klein and Hukins (1983) on the functional

differentiation in the spine. Also, if the bulk behaviour of the anulus fibrosus is

modelled, it is not possible to obtain data for the stress/strain state of the collagen

fibres and anulus ground substance individually.



2.6 Mechanical properties of components in the spine

Several material descriptions have been used by other authors to define the behaviour

of the intervertebral disc components. These descriptions include:

• Linear elastic materials which exhibit linear, elastically recoverable

mechanical behaviour.

• Hyperelastic materials that exhibit nonlinear, elastically recoverable behaviour

under the application of large strains. Classic linear elastic material theories

apply to small strains of approximately 5 – 10%. Hyperelastic theory deals

with material strains greater than these and is a material description commonly

applied to large strain materials such as rubbers. Hyperelastic materials are

also incompressible or near incompressible.

• Viscoelastic materials which show elastically recoverable mechanical

behaviour, but the stress-strain relationship is dependent upon a third variable

of time. Varied loading rates result in varied stiffness for viscoelastic

materials.

• Poroelastic materials combine elastic or plastic behaviour for a solid matrix

with porous fluid flow through this matrix. The load bearing ability of these

materials is a result of the increase in pore fluid pressure and the mechanical

stiffness of the solid matrix.

2.6.1 The intervertebral disc components

The following sections provide data for the mechanical properties of the nucleus

pulposus, the anulus fibrosus ground substance, the cartilaginous endplates and the

collagen fibres.

2.6.1.1 Nucleus pulposus

Nachemson (1960) reported that the nucleus pulposus behaved as a hydrostatic

material. When loaded, it will behave as an incompressible material (Bodgduk,

1997). Nachemson (1963) reported that there was a constant relationship between the

pressure which was applied to the superior surface of the disc and the pressure present



within the nucleus pulposus. The pressure within the nucleus was 1.5 times the

pressure applied to the superior surface.

Researchers have represented the nucleus pulposus as both a linear elastic material

(Goel et al., 1995) and as a hyperelastic material (Natali and Meroi, 1990; Natali,

1991). However, it is considered that this material would be better represented as an

incompressible fluid in keeping with the observations of Nachemson (1960). The use

of an elastic material description (Belytschko et al., 1974; Kurowski and Kubo, 1986;

Shirazi-Adl et al., 1986) would require the approximation of material parameters that

may not accurately represent the incompressible, hydrostatic nature of the tissue.

2.6.1.2 Anulus fibrosus and the anulus fibrosus ground substance

The overall response of the anulus fibrosus has been represented in models using a

linear elastic material description (Kurowski and Kubo, 1986). Also, the anulus

fibrosus ground substance has been represented separately as a linear elastic material

(Shirazi-Adl et al., 1984, 1986, 1987; Kumaresan et al., 1999; Goel et al., 1995; Rao

and Dumas, 1991; Ueno and Liu, 1987).

Natarajan et al. (1984) and Belytschko et al. (1974) modelled the bulk response of the

anulus fibrosus as an orthotropic linear elastic material. The orthotropy accounted for

the action of the collagen fibres and the linear elastic modulus of the anulus was

varied radially through the disc to simulate the inhomogeneity of the collagen fibres

in the anulus.

It is considered that the use of a linear elastic material description for the anulus

fibrosus is a significant simplification of the tissue behaviour since it does not

accurately represent its nonlinear nature.

Several researchers have described the anulus using a viscoelastic material description

(Kelley et al. 1983; Burns et al., 1984; Kaleps et al., 1984). A limitation of these

studies is their use of lumped parameter models or Kelvin models to describe the disc

response, rather than defining the individual contribution of the disc materials. Also,



the creep behaviour of the anulus fibrosus which is observed during testing to

quantify the viscoelastic nature of the materials is partly a result of the fluid

movement within the anulus. This is contrary to the classic mechanism of

viscoelasticity which is related to the sliding of micro-chains in the microstructure of

the viscoelastic material.

Previous researchers have argued that a more accurate representation of the anulus

fibrosus requires consideration of its poroelastic nature (Pangiotacopulos et al., 1987;

Laible et al., 1993, 1994; Klisch and Lotz, 1999). The poroelastic nature of the

material was referred to by Laible et al. (1994) where they state that for a biological

tissue both the solid structure and the fluid are incompressible but when combined

behaved as a compressible material. However, it is considered that testing methods to

define the poroelastic parameters may not have been consistent between studies and a

definitive value for the permeability of the solid matrix is difficult to obtain.

It is considered that the discussion of Natali (1991) with respect to the suitability of

the hyperelastic material description for modelling incompressible materials may in

fact be more relevant to the anulus fibrosus. The anulus displayed nonlinear elastic

behaviour, is strained to large strains and may be assumed to behave as an

incompressible material when analysed at physiological loading rates because there is

no time for fluid expression. The hyperelastic material description fulfils these

criteria and has been shown to perform well for biological tissues (Bischoff et al.,

2002; Jemiolo and Telega, 2001; Natali and Meroi, 1990; Natali, 1991).

2.6.1.3 Cartilaginous Endplate

The cartilaginous endplates may be represented as separate entities in the FEM (Natali

and Meroi, 1990; Kumaresan et al., 1999; Ueno and Liu, 1987; Rao and Dumas,

1991) or their mechanical contribution may be combined with the cortical bone of the

vertebra (Shirazi-Adl et al., 1984, 1986, 1987; Kurowski and Kubo, 1986; Goel et al.,

1995; Belytschko et al., 1974; Simon et al., 1985). In both cases these structures are

modelled as isotropic linear elastic (Shirazi-Adl et al., 1984, 1986, 1987; Kumaresan

et al., 1999; Kurowski and Kubo, 1986; Goel et al., 1995; Belytschko et al., 1974;



Ueno and Liu, 1987; Simon et al., 1985) or orthotropic linear elastic materials (Natali

and Meroi, 1990; Natali, 1991).

2.6.1.4 Collagen fibres

Collagen fibres have been represented as both linear elastic materials (Kumaresan et

al., 1999; Goel et al., 1995; Ueno and Liu, 1987) and nonlinear elastic materials

(Shirazi-Adl et al., 1984, 1986, 1986b, 1987). Some values for the linear elastic

modulus are listed in Table 2-3. The results of these FEM studies which use the

linear elastic material description showed good agreement with the known disc

response.

Table 2-3 Linear elastic moduli used for collagen fibres in previous FEM studies

Elastic Modulus Author

450MPa Goel et al., 1995

500MPa Kumaresan et al., 1999

500MPa Ueno and Liu, 1987

Shirazi-Adl et al. (1984, 1986, 1987) based the nonlinear material law used for the

collagen fibres on the findings of several previous studies and fit an exponential

equation to these data. Also, Shirazi-Adl et al. (1984, 1986, 1987) varied the elastic

modulus of the fibres with radial location in the nucleus such that the stiffness at the

innermost lamellae was 65% of the stiffness in the outermost lamellae. This was

based on the findings of Eyre (1976) and was an effective way to represent the

variation in the distribution and mechanical characteristics of the different collagen

types present within the anulus fibrosus. A similar approach was adopted by

Natarajan et al. (1994) who varied the orthotropic elastic modulus of the collagen

fibres radially.

Collagen content in the anulus fibrosus has been represented in previous FE models as

between 16 and 20% (Shirazi-Adl et al., 1984, 1986, 1987; Kumaresan et al., 1999;

Goel et al., 1995) which were similar to the findings of Marchand and Ahmed (1990).



Betsch and Baer (1986) report results for the stress-strain response of rat tendons.

Their discussion suggests that the failure strain of the tissue is strain rate independent

but the stiffness is strain rate dependent. Viidik (1973) states that the maximum

tensile deformation of collagenous tissues is 10-15% but notes that values as high as

40% have been reported. This is evidenced by the findings of Morgan (1960) who

determined a stress-strain response of collagen fibres which showed the fibres still

bearing a load at a strain of 25%. The collated results from Shirazi-Adl et al. (1984,

1986, 1987) also show the tissue bearing a load at a strain of 25%. The failure strain

of collagen fibres may reasonably be expected to fall in the range of 10-25%.

2.6.2 Incompressibility of the intervertebral disc

When the intervertebral disc is loaded it may lose fluid from either the nucleus

pulposus or the anulus fibrosus. This is a result of the poroelastic nature of the

materials. Testing has been conducted on specimens of anulus fibrosus to quantify

the mechanical response of the tissue without any frictional effects of fluid flow. The

strain rates for these tests range from 0.00009 sec-1 to 0.0001 sec-1 (Skaggs et al.,

1994; Acaroglu et al., 1995), suggesting that in order for fluid to flow unhindered

from the tissue, slow strain rates are necessary. At strain rates greater than these,

some fluid may be trapped within the tissue.

Whilst no definite statement has been found in the literature for the range of the

physiological strain rate, it could reasonably be expected to be higher than the range

quoted above.

Higginson et al. (1976) developed an analytical model of cartilage and carried out

experiments on bovine knee cartilage to validate this model. The analytical model

accounted for the solid matrix stress and fluid flow through the pores of the matrix in

the cartilage. They demonstrated that when cartilage was loaded at a similar

frequency to that occurring during walking (1Hz), fluid movement through the matrix

of the cartilage had a negligible effect on the strain. They stated that fluid flow in the

cartilage was only relevant to the long term strain in the material when subjected to

creep loading.



It has been demonstrated that when the intervertebral disc is loaded at increasing

strain rates, it shows an increasingly stiffer mechanical response (Duncan et al.,

1996). The mechanical behaviour of the intervertebral disc components are a result of

the interaction between the ground matrix and the water trapped within the pores of

this matrix and as such show similarities to classic consolidation theory. When the

tissue is loaded, the total stress is resisted by both solid stress in the matrix and fluid

pressure in the pore fluids. The ability of the tissue to resist stress will be directly

related to how much fluid is trapped within the pores of the elastic matrix and how

easily this fluid can escape from these pores. This load bearing mechanism is

characteristic of all cartilaginous structures and therefore, the findings of Higginson et

al. (1976) may be applied to the intervertebral disc behaviour under physiological

loading.

On the basis of this discussion, it is thought that at strain rates similar to physiological

loading, the fluid flow within the intervertebral disc tissues is negligible. Therefore,

an assumption of incompressibility for the anulus fibrosus would be acceptable for

loading over short time periods. Previous FE studies have successfully modelled the

intervertebral disc components as incompressible materials (Natali and Meroi, 1990;

Natali, 1991; Belytschko et al., 1974) although the motivation for this assumption was

not as clearly defined in these papers.

2.6.3 Functional behaviour of the anulus fibrosus and nucleus pulposus

Bogduk (1997) outlined the ability of the nucleus to generate a horizontal force on the

inner anulus and an axial force on the inner surface of the endplates when an axial

force is applied to the superior surface of the disc.

Yu et al. (2002) found evidence of elastin fibres in an organised configuration within

the nucleus pulposus and anulus fibrosus. In the transverse plane the elastin fibres in

the anulus fibrosus lamellae were observed in an orientation parallel to the lamellae

and also in a radial direction through the lamellae to form “cross-bridges”. In the

sagittal plane the dense populations of elastin fibres were present between the



lamellae. Yu et al. (2002) suggest that the presence of elastin fibres between the

lamellae may contribute to the sliding of adjacent lamellae during mechanical loading

of the disc, by allowing this deformation to be reversed and the undeformed

configuration of the anulus to be restored.

The anatomy and microstructure of the spine and in particular the intervertebral disc

are intimately linked to how they operate in vivo and how they distribute loads to the

adjacent musculature and bony anatomy. As such, some of these components are

considered below.

2.6.3.1 The inclination of collagen fibres

Since collagen fibres are primarily active in carrying tensile loads, the orientation of

the collagen fibres is a reflection of the directions in which the anulus fibrosus

experiences tensile strains (Hukins et al. in Hukins and Nelson, 1989). On the basis

of calculations using a disc model which incorporated disc bulge and volume changes

Hickey and Hukins (1980) determined that the inclination of the collagen fibres in the

anulus fibrosus ensured that the fibres would resist the hoop stresses introduced into

the anulus during compressive loading.

2.6.3.2 Uniaxial Compression

When the disc is subjected to uniaxial compression this causes an increase in the

pressure in the nucleus pulposus (Nachemson, 1960). This pressure is resisted by

hoop stress in the anulus fibrosus which is similar to the behaviour of a pressure

vessel (Naylor et al., 1954). Klein and Hukins (1983) state that compressive loading

creates minimal strain in the outer lamellae therefore the inner lamellae must have a

higher strength during compression.

This may be related to the observed higher concentration of type II collagen in the

inner anulus regions. This collagen type is considered to be most active in regions of

the body which experience pressure loading.



2.6.3.3 Bending

Bending of the spine will create compression and tension in different regions of the

anulus fibrosus (Klein and Hukins, 1983). Because the axis of rotation of the disc is

generally located within the disc in the transverse plane (refer Section 2.2), this will

create a larger bending moment at the outer anulus. Therefore, these will be the

regions which must be best equipped to resist the compressive or tensile loading

generated during bending. The higher collagen content in the outer surface of the

intervertebral disc suggests it is well suited to resisting this loading since collagen

fibres carry only tensile loading (Klein and Hukins, 1983).

During bending, the outer anulus will compress and bulge considerably. The collagen

fibres in the lamellae exist at an angle of between 25o and 35o to the transverse plane

through the disc. Nachemson (1981) observed that the in vivo nuclear pressure during

bending was higher than that observed during relaxed standing. When the disc

compresses and bulges during bending in the sagittal or frontal planes, the collagen

angle will reduce and better orientate the fibres to resist the increased circumferential

hoop stresses created by the higher nuclear pressure present during bending.

2.6.3.4 Torsion

Klein and Hukins (1983) made a comparison between torsional loading applied to a

cylindrical rod and torsion loading in the spine. This loading applied to a cylindrical

rod causes the largest torque on the outer surface. Similarly, in the spine, Klein and

Hukins (1983) state that the largest torque is generated in the cortical bone of the

vertebra which is in turn attached only to the outer lamellae of the anulus fibrosus.

Conversely, the inner lamellae will experience less torque loading due to their

location in relation to the axis of rotation and also because these lamellae are

connected to the cartilaginous endplates which do not have as rigid a connection to

the vertebra (Inoue, 1981). Klein and Hukins (1983) considered that the higher torque

in the outer lamellae may be related to the higher collagen concentration observed in

these layers (Adams et al., 1977).



The angle of inclination of the collagen fibres in each successive lamellae alternates

to create a criss-cross pattern as described previously (Section 2.1.2.2). This variation

in inclination of the fibres between adjacent lamellae means that the anulus is equally

equipped to resist both right and left axial torsion of the spine. If the fibres were not

arranged in this pattern and were all inclined at a common angle, the torsional

resistance of the disc would favour one direction for rotation.

2.6.4 Mechanical properties of the longitudinal ligaments

White and Panjabi (1978) collated the results of several researchers (Chazal et al.

1985; Dvorak et al., 1988; Goel et al., 1986; Nachemson and Evans, 1968; Tkaczuk,

1968) to give an average ultimate tensile strength for the ALL of 11.6MPa (range =

2.4-21MPa) and for the PLL of 11.5MPa (range = 2.9-20MPa). The average ultimate

tensile strain for the ALL was 36.5% (range = 16-57%) and for the PLL was 26.0%

(range = 8-44%). These data give useful information on the failure and damage of the

ligaments in the FEM.

Pintar et al. (1992) reported that the stiffness of the spinal ligaments showed limited

variation between levels of the spine. Therefore, it was reasonable to use the values

for ligament stiffness calculated for other lumbar levels in determining the properties

at the L4/5 level.

Table 2-4 and Table 2-5 outline the stiffness and associated dimensional data for the

ALL and PLL.



Table 2-4 Anterior longitudinal ligament – stiffness and limited geometric data

Author Spring Elastic Area Length Tested Segment orStiffness Modulus Intact; Level; Details

N/mm MPa mm2 mmPintar et al. 40.5 ± 14.3 46.375 32.4 ± 10.9 37.1 ± 5.0 L4/5, disc level;

(1992) *** intact ligamentNeumann - 759 38.2 ± 3.5 30 all lumbar levels;

et al .(1992) intact ligamentNeumann 87 - - - lumbar levels;

et al . (1994b) intact ligamentNeumann 78 ± 32 Couldn't - - lumbar levels;

et al . (1993) calculate intact ligamentSchendel 14.3 29 - - L1/2 disc level;

et al . (1993) intact ligamentShirazi-Adl - 1.12 * 24** - FE study; data fromet al . (1986) review of other

researchers**Roberts et 33.9 37.17 - calculate as L1 vertebra level;al. (1998) *** ≈ 39.8 intact ligament

Hukins et al. - 1.785 * - 10 Lumbar levels;(1990) excised specimens

with undefineddimensions

Chazal et 21.34 65.6 12.3 Avge for lumbaral. (1985) levels; intact ligaments

* Calculated as the slope of the linear (elastic) portion of the stress-strain curve

** Based on findings of among others Nachemson and Evans (1968), Farfan (1973),

Rissanen (1960), Tkaczuk (1968)

*** Calculated using either the area and length dimensions from the study or the

average cross-sectional area of 43.2mm2 and the average anterior height of the

vertebra (Panjabi et al., 1992)



Table 2-5 Posterior longitudinal ligaments – stiffness and limited geometric data

Author Spring Elastic Area Length Tested segment orStiffness Modulus intact; level

N/mm MPa mm2 mmPintar et al. 25.8 ± 15.8 165 ** 5.2 ± 2.4 33.3 ± 2.3 L4/5 disc level;

(1992) intactRoberts 15.6 20.9 ** - calculate as L1 vertebra level;

et al. (1998) ≈ 31.8mm intactChazal et 70.9 25.7 13.9 Average for lumbar;al. (1985) intact ligaments

* Calculated as the slope of the linear (elastic) portion of the stress-strain curve

** Calculated using either the area and length dimensions from the study or the

average cross-sectional area of 25.2mm2 and the average posterior height of the

vertebra (Panjabi et al., 1992)

The experimental work carried out by Tkaczuk (1968) was a key study into the

morphology and functionality of the longitudinal ligaments of the spine. Even so, it

was not possible to obtain useful information on the mechanical stress-strain response

of the longitudinal ligaments based on the data provided by Tkaczuk (1968). He did

not find an average stiffness for the ALL and PLL, rather quoted values for the

deformation of the ligament when loaded, over 3 successive tests, to 500gm force.

These deformations were normalized against the maximum deformation observed

during the third test. Because no values were stated for the average maximum

deformation observed during the third test, these normalized values were of no use in

calculating stiffness, and were only relevant for comparative purposes.

Roberts et al. (1998) conducted tensile testing on the ALL once the L1 vertebra,

T12/L1 and L1/2 intervertebral discs had been removed from a cadaveric lumbar

spine (Table 2-4). The length of these ligaments was not measured. However, it was

estimated using the average in vivo height of the L1 vertebra and the anterior disc

height of the L1/2 intervertebral disc (Panjabi et al., 1992; Tibrewal and Pearcy,

1985). The approximate length of the ALL tested by Roberts et al. (1998) was

39.8mm and the length of the PLL was estimated to be 31.8mm (Table 2-4 and



Table 2-5).

2.6.4.1 Average Elastic Modulus and Spring Stiffness of the Anterior

Longitudinal Ligament

The elastic modulus used to determine the spring stiffness of the ALL in the model

was an average of published values. However, the results from several studies were

not included.

The stiffness results found by Neumann et al. (1992) were high in comparison to the

results of other studies and therefore were not included. They considered this was due

to the age of the specimens tested – four of the six specimens were below 30 years of

age. Tkaczuk (1968) observed an inverse relationship between age and ultimate

loads, deformations and strength of ALL.

Hukins et al. (1990) provided a stress-strain curve using specimens of ALL with a

thickness which was obtained by reducing the specimen size until the microscopy

imaging techniques could be used accurately. Additionally, the specimen length was

unclear. The cross-sectional areas were then calculated using density calculations. It

was not clear whether these methods used to determine the specimen dimensions were

suitable or that the extremely small thickness of the specimens tested would not have

resulted in variability and inaccuracy in the results. The ligaments have a complex

fibre composite nature and as a result may be better tested intact rather than as

segmented samples. Therefore, the elastic modulus obtained from this study was not

included in the calculation of an average elastic modulus for the ALL.

The elastic modulus estimated from the average stress-strain curve used by Shirazi-

Adl et al. (1986) was not used to calculate the spring stiffness for the FEM since some

of the studies used to determine this average mechanical response were carried out on

sectioned specimens of ligament rather than intact ligaments. This may have resulted

in less accuracy in results due to the complexity of the ligamentous tissue and could

explain why the average elastic modulus obtained from the results of Hukins et al.

(1990) was similar to that of Shirazi-Adl et al. (1986).



The average elastic modulus for the ALL was 32.7MPa.

A comparable spring stiffness, k, may be calculated using the elastic modulus, E, the

average area of the ALL, A, and the anterior height of the L4/5 disc, l, which is the

length over which the ALL would act (Eqn 2-1).

lAEk .

=

Eqn 2-1 Spring Stiffness

This average spring stiffness is 103.4 N/mm.

2.6.4.2 Average Elastic Modulus and Spring Stiffness of the Posterior

Longitudinal Ligament

It is important to note that a limitation of the study carried out by Pintar et al. (1992)

was the method used to determine values for the engineering stress and strain of the

ligament specimens. Pintar et al. (1992) used force-displacement data for 132

samples of spinal ligaments tested in a study carried out previously by Myklebust et

al. (1988). When Pintar et al. (1992) carried out their study, no data on the

dimensions of the ligaments was recorded. To determine the ligament dimensions

used in calculating stress and strain Pintar et al. (1992) obtained the average ligament

dimensions from eight recently acquired cadaveric lumbar spines. Whilst the

dimensions of ligaments in both studies would be similar, a more precise method

would have involved the use of measurements from the actual ligaments tested.

The elastic modulus for the PLL as determined by Pintar et al. (1992) was

considerably higher than that of other studies. However, the cross-sectional area of

the PLL determined in this study was notably smaller than the average cross-sectional

area which was calculated from previous studies – 25.2mm2 (Table 2-2). Since the

specimen dimensions used by Pintar et al. (1992) were not those of the specimens

tested then calculation of an elastic modulus using the average cross-sectional area of

the PLL that was determined in Section 2.1.3 would be justified. This results in an

elastic modulus of 34.09MPa which is a similar order of magnitude to the results of

Roberts et al. (1998) and Chazal et al. (1985).



An average elastic modulus for the PLL of 42MPa was determined using the results

from Roberts et al. (1998), Chazal et al. (1985) and the manipulated results of Pintar

et al. (1992). Using the elastic modulus, the average cross-sectional area of the PLL

and the posterior height of the L4/5 intervertebral disc, the spring stiffness of the PLL

may be calculated (Eqn 2-1) to be 192.44 N/mm.

2.7 Use of a Hyperelastic Model for the Anulus Ground Matrix

The usefulness of the hyperelastic material description for defining the mechanical

behaviour of the anulus ground substance was established in Section 2.6.1.2. There

are various forms for the equations governing the behaviour of these materials and

these are described in the following sections. In order to fully understand these

equations it is necessary to understand the laws for the state of stress in a structure.

2.7.1 Rubber Elasticity Theories and Continuum Mechanics

A description of the laws of continuum mechanics relating to the hyperelastic strain

energy equation is provided in the following sections.

2.7.1.1 Strain Invariants (Reference: Williams, 1973, Chapter 1; Ugural and

Fenster, 1995)

In order to understand these variables it is necessary to understand the nature of the

state of stress and strain on a general plane within a material (Figure 2-8).



Figure 2-8 General plane in a body showing the angles to a normal from the

plane

The orientation of this general plane may be expressed in terms of direction cosines, l,

m, and n (Eqn 2.6-1). In this equation xOP, yOP and zOP are the vector co-ordinates of

a vector between the origin of the co-ordinate system, O, and a point on the general

plane, P.

rxl OP== )cos(α ,

rym OP== )cos(β ,

rzn OP== )cos(γ

Eqn 2-2 Direction cosines for a general plane in space

Because the direction cosines aren’t mutually exclusive a relationship is defined for

them (Eqn 2-3).

l 2 + m 2 + n 2 = 1

Eqn 2-3 Relationship between the direction cosines

Figure 2-9 General plane showing stress in that plane resolved in rectangular co-

ordinates

α βγ

Normal to the plane

Normal to the plane

Sz

Sy

Sx

Z

Y

X

Z

Y

X

O

P



Let S be the total stress on a general plane, ABC. If this stress is resolved in the

rectangular co-ordinate system (Figure 2-9), it may be expressed as shown in Eqn 2-4.

SSSS zyx

2222 ++=

Eqn 2-4 Equation for the total stress on a general plane

If the equilibrium of forces in each of the 3 directions is considered, expressions for

the individual stress components in three orthogonal directions, Sx, Sy, and Sz, are

determined (Eqn 2-5).

mml

nml

nml

zzzyzxz

yzyyyxy

xzxyxxx

SSS

...

...

...

στττστττσ

++=

++=

++=

Eqn 2-5 Stress components

Where, σ = a stress normal to a plane

τ = a shear stress on a plane

The subscripts on the normal and shear stresses in Eqn 2-5 are interpreted as follows:

• The first subscript is the direction of the normal to the plane in which the stress

acts; and

• The second subscript is the direction in which the stress acts.

The x, y and z planes are normal to the x, y and z axes, respectively.

The stress in a direction normal to the general plane may be determined by resolving

the stress components in Eqn 2-5 in the normal direction to give Eqn 2-6. In this

equation Sn is the stress normal to the general plane.

lnnmmlnml zxyzxyzzyyxxnS ...2...2...2... 222 τττσσσ +++++=

Eqn 2-6 Expression for stress normal to a general plane in the structure

This is essentially the equation for transformation of stresses between co-ordinate

systems of varied orientation.



There are 3 planes within the stressed system which are mutually perpendicular and

on which there is a zero shear stress acting. The normal stresses acting on these 3

planes are called principal stresses. The first stress is a maximum, σ1, the second is

an intermediate value, σ2, and the third is a minimum value, σ3.

The principal stresses may be determined using the knowledge that in order for a

normal stress to be a maximum on a plane, then the derivative with respect to the

direction cosines of the expression for the normal stress (Eqn 2-6) must be zero.

Finding this derivative gives the expression stated in Eqn 2-7.

σ pzyx

nmlSSS ===

and so, lS px .σ= ; mS py .σ= ; nS pz .σ=

where, σ p = principal stress

Eqn 2-7 Expression for the principal stress in terms of the x, y and z stress

components and the direction cosines

This expression is then substituted into Eqn 2-5 to obtain a system of equations which

may be solved to determine the direction cosines for the plane in which the principal

stresses act (Eqn 2-8).

nmlnmlnml

pzzzyzx

yzpyyyx

xzxypxx

)...0

.).(.0

..).0

(

(

σστττσστττσσ

−++=

+−+=

++−=

Eqn 2-8 System of equations which may be solved to determine the direction

cosines to the planes of prinicipal stresses

In order to find a nontrivial solution for the system of equations outlined in Eqn 2-8

the determinant of the stress matrix must be zero (Eqn 2-9).



0=−

−−

σστττσστττσσ

pzzzyzx

yzpyyyx

xzxypxx

Eqn 2-9 Determinant of the stress matrix

If this determinant is expanded, the result is a cubic equation which may be solved to

determine the principal stresses in terms of the stress invariants, Ki=1,2,3 (Eqn 2-10,

Eqn 2-11).

0.. 322

13 =−+− KKK ppp σσσ

Eqn 2-10 Cubic equation for principal stress

where

τστστστττσσσστττστττσ

τττσσσσσσσσσ

222

3

2222

1

......2.

...

. xyzzzxyyyzxxzxyzxyzzyyxx

zzzyzx

yzyyyx

xzxyxx

xzyzxyxxzzzzyyyyxx

zzyyxx

K

KK

−−−+=

=

−−−++=

++=

Eqn 2-11 Expressions for the three stress invariants, K, for a general state of

stress

The importance of the stress invariants is that they are independent of the direction

cosines and are therefore, independent of the orientation of the co-ordinate system.

This is advantageous when the principal stresses are being calculated as these values

do not have any dependence on the orientation of the general plane in question.

These derivations which have been used to describe the state of stress on a general

plane may also be used to describe the state of strain and in particular, to express the

straining of an arbitrary line on the general plane. As an example, the expression for

the extension ratio of an arbitrary line on the general plane is defined in Eqn 2-12.



2222222222 ....2....2....2... lnnmmlnml xxzzzzyyyyxxzzyyxxr λλλλλλλλλλ +++++=

where, λ = the extension/stretch ratio

= o

f

dd

and d f = final dimension

d o = initial dimension

Eqn 2-12 Extension ratio of an arbitrary line (NB. The first subscript on the λ

term is the plane in which the extension occurs and the second subscript is the

direction of the extension)

A comparison of Eqn 2-12 and Eqn 2-6 shows that these expressions are comparable

if the statements in Eqn 2-13 are correct.

myyxxxy ..2 λλλ = ; nzzyyyz ..2 λλλ = ; lxxzzzx ..2 λλλ =

Eqn 2-13

Using similar derivations to those employed for the stress invariants, it may be seen

that the three strain invariants, Ii=1,2,3, are expressions similar to Eqn 2-11 with the

stress variables replaced by the squared extension ratios in a similar direction (Eqn

2-14).

λλλλλλλλλλλλλλλλλλλλλ

λλλ

4242422222223

4442222222

2221

2 xyzzzxyyyzxxzxyzxyzzyyxx

zxyzxyxxzzzzyyyyxx

zzyyxx

III

−−−+=

−−−++=

++=

Eqn 2-14 Strain invariants for a general state of strain

The strain invariants give a direction independent measure of the strain/stretch within

the body. The first strain invariant, I1, gives a measure of how the dimensions of the



body change. The second strain invariant, I2, is a measure of how the overall area of

the element changes and the third strain invariant, I3, indicates how the volume of the

body has changed.

2.7.1.2 Stress components and the strain energy equation, W (Reference:

Williams, 1973, Chapter 1; Ugural and Fenster, 1995)

Strain energy in a body will be independent of the orientation of the structure.

Therefore in finding an expression for strain energy, it is desirable to express this

quantity in terms of strain parameters which have no dependence on the body’s

orientation. Accordingly, the strain energy may be expressed as a function of the

strain invariants (Eqn 2-15).

( )IIIfW 321 ,,=

where, W = strain energy equation

Eqn 2-15 The general form of the strain energy equation

1

1

σ2

σ3

λ3

1 λ2

λ1

1

2

3

Undeformed

Deformed

σ1

Figure 2-10 Cube of unit length subjected to pure deformation to give side lengths of λ1, λ2 and λ3.



Consider the application of a stress to a unit cube such that the final structure

experiences pure deformation and no shear. This deformation results in final edge

lengths of λ1, λ2 and λ3. Because there is no shear, these edge lengths are the

principal extensions. For this deformation, the strain invariants will not include any

terms for shear (Eqn 2-16).

λλλλλλλλλ

λλλ

2223

2222222

2221

..

...

zzyyxx

xxzzzzyyyyxx

zzyyxx

III

=

++=

++=

Eqn 2-16 Strain invariants for a body subjected to pure deformation

The force acting in direction 1 in Figure 2-10, F1, is defined in Eqn 2-17 in terms of

the stress acting in this direction, σ 1 and the area of the face on which this stress acts.

λλσ 3211 ..=F

Eqn 2-17

The displacement caused by this force, F1 is d λ1. Work is performed when a force

acts over some displacement, therefore, it may be seen that the above force, F1, does

work (Eqn 2-18).

λλλσ 13211 ... dW =

Eqn 2-18 Work performed by force, F1

An expression for a small change in the energy of the complete structure will include

the effects of the strain energies, Wi, due to the stresses applied in the 3 principal

directions (Eqn 2-19).



λλλσλλλσλλλσ 321323121321 ......... ddddW ++=

Eqn 2-19 Expression for a small change in the stored energy in the structure

In order to determine a stress component such as σ1, equation Eqn 2-18 may be

rearranged to give Eqn 2-20.

λλλσ132

1 ..1

∂∂

=W

Eqn 2-20

Given that the strain energy equation for a material is a function of the three strain

invariants, Ii=1,2,3, (Eqn 2-15) the expression in Eqn 2-20 may be determined by

finding an expression for the partial derivative (Eqn 2-21).

λλλλ 3

3

32

2

21

1

11

...∂

∂

∂∂

+∂

∂

∂∂

+∂

∂

∂∂

=∂∂ I

II

II

IWWWW

Eqn 2-21

Substituting Eqn 2-21 and Eqn 2-16 into Eqn 2-20 results in an expression for the

stress in a compressible, isotropic structure that is subjected to a pure deformation

(Eqn 2-22).

∂∂

+∂∂

+∂∂

−∂∂

=IIIII

III

WWWW

33

22

22

1

3

1

2

15.03

1 .....2

λλσ

Eqn 2-22 Expression for a component of stress in a compressible, isotropic

material

The partial derivatives in Eqn 2-22 are the elastic property functions for the material.

Given the above expression for the state of stress in a compressible material, it is now

relevant to find a similar expression for the state of stress in an incompressible



material. In order to do this, it is necessary to consider the effect a hydrostatic

pressure will have on the material. Since incompressible materials do not generate a

change in volume under load, the 3rd strain invariant, I3, will be equivalent to 1 (Eqn

2-16). Expressions for the strain invariants for incompressible materials may be

generated using the unity of I3 (Eqn 2-23).

1

111

3

2

3

2

2

2

12

2

3

2

2

2

11

=

++=

++=

I

II

λλλ

λλλ

Eqn 2-23 Strain invariants for an incompressible material

where the variables λ1, λ2 and λ3 are the principal extension ratios.

Hydrostatic stresses will result in no change in the strain energy. This may be seen by

manipulation of Eqn 2-19 when the stress components are set to σH and the

relationship λ1 . λ2 . λ3 = 1 is included. Therefore, the expression for the stress

components in an incompressible material will be similar to Eqn 2-20 with an

additional term for hydrostatic pressure (Eqn 2-24).

pW+

∂∂

=λλλσ

1321 .

.1 , where =p hydrostatic pressure

Eqn 2-24 Expression for stress in direction 1 in an incompressible material

If the hydrostatic pressure term was not included in Eqn 2-24, then it would not

accurately predict the stresses present when only a hydrostatic pressure was applied.

In this instance, the W∂ term would not predict any stress and the presence of the

hydrostatic pressure would not be evident.

On the basis of Eqn 2-24 expressions for the 3 stress components may be derived for

an incompressible material.



pWW

pWW

pWW

II

II

II

+∂∂

−∂∂

=

+∂∂

−∂∂

=

+∂∂

−∂∂

=

22

31

2

33

22

21

2

22

22

11

2

11

.2..2

.2..2

.2..2

λλσλλσλλσ

Eqn 2-25 Stress components for an incompressible material

The strain energy equation, W , is a function of the strain invariants and satisfies the

condition that 0=W when, λ1 = λ2 = λ3 = 1.

2.7.2 Forms and Applications of the Strain Energy Equation

Mooney (1940) stated that the primary problem in elastic theory was to find a strain

energy equation which accurately described the material in question. He notes that if

the material is subjected to small strains and is isotropic and homogeneous, then an

expression for the strain energy of the material may be derived on the basis of the

elastic modulus and rigidity modulus. However in the case of rubber, the strains

observed are too large for the materials mechanical behaviour to be accurately

modelled using classic small strain theory. Mooney (1940) noted the necessity for the

development of a relationship which could suitably describe the nonlinear, elastic,

large strain behaviour of rubbers.

Mooney (1940) observed that under uniaxial loading, the mechanical response of

rubber is nonlinear, while under shear loading, the mechanical response follows

Hooke’s law. Also, rubbers behave as near incompressible materials. On the basis of

these criteria, Mooney (1940) developed two strain energy equations to describe

rubber mechanics. The first equation assumed that the material was linear in shear

and incompressible (Eqn 2-26) and the second equation assumed the material was

nonlinear in shear and incompressible. However, the second equation was later

discounted as incorrectly representing the behaviour of rubbers (Rivlin, 1984). An



additional assumption associated with the Mooney equation that assumed linear shear

behaviour was that the material was isotropic.

)3111()3(),,( 2

3

2

2

2

12

2

3

2

2

2

11321 −+++−++=λλλλλλλλλ CCW

where, C1 and C2 are material constants

Eqn 2-26 Mooney Strain Energy Equation

The expression in Eqn 2-26 was later altered by Rivlin (1984) to incorporate the strain

invariants (Eqn 2-27).

λλλ 2

3

2

2

2

11 ++=I ; λλλ 2

3

2

2

2

12

111 ++=I ; λλλ 2

3

2

2

2

13 ..=I

Eqn 2-27 Strain Invariants

This equation was then referred to as the Mooney-Rivlin equation (Eqn 2-28) where

the expressions for I1 and I2 were substituted into Eqn 2-26. For incompressible

materials, I3 is equivalent to 1.

)()( 33 2211 −+−= ICICW

Eqn 2-28 Mooney-Rivlin strain energy equation

Several researchers have used hyperelastic material formulations to represent the

material behaviour of biological tissues.

The Mooney-Rivlin strain energy equation has been applied to nonlinear biological

tissues in previous studies (Crisp, J. D. C in Fung et al., 1972; Weiss et al., 2001;

Bilston et al., 2001; Vossoughi, 1995; George et al., 1988) as both a complete model

for the tissue and as a base model which was further developed to include

viscoelasticity or poroelasticity of the tissue. Natali and Meroi (1990) and Natali



(1991) represented the behaviour of the nucleus pulposus and of the disc material in

general using a Mooney-Rivlin material description. However, it is considered that

the hyperelastic material law would be better applied to the behaviour of the anulus

fibrosus ground substance rather than the nucleus pulposus given the low shear

stiffness and semi-fluid nature of the nucleus pulposus.

Crisp (in Fung et al., 1972) states that the regular implementation of the Mooney

strain energy equation was due to its simplicity – most other strain energy equations

hitherto developed were quite complex and not readily comprehended by researchers

unfamiliar with the mathematics that forms their basis. An additional attraction

offered by the Mooney-Rivlin equation was the ease with which the constants could

be determined from experimental data – these constants were the gradient and

intercept of the best fit curve. The Mooney-Rivlin equation may have provided an

adequate fit for experimental data at very low strains, but at high strains, the

inaccuracy in the model became apparent (Crisp in Fung et al., 1972).

One of the integral assumptions made in the derivation of the Mooney-Rivlin equation

was the linear relationship between the shear stress and the shear strain. The results

from experimental testing on the anulus fibrosus ground substance (Chapter 4)

suggested that this material was in fact nonlinear under shear loading. Similar

observations may be made for other biological tissues (Yamada, 1970).

There are a considerable number of hyperelastic strain energy equations which have

been developed to incorporate nonlinear shear behaviour as well as more complex

behaviour such as viscoelasticity, poroelasticity or anisotropy.

Tschoegl (1971) extended the Mooney-Rivlin strain energy equation to include higher

orders of the strain invariants and to represent nonlinear shear behaviour. The general

form of these polynomial models is expressed in Eqn 2-29. It was observed that these

higher order combinations provided a more accurate representation of the

experimental results, in particular at high strains.



jiN

jiij IICW )3()3( 21

1

−−= ∑=+

where, Cij are material constants

Eqn 2-29 General form of the polynomial strain energy equations

Ogden (1972) developed a strain energy equation which involved the assumption of

nonlinearity of the material response under shear loading (Eqn 2-30).

)3.(.2321

12 −++= −−−

=∑ iii

N

i i

iW ααα λλλαµ

where, µi and αi are material constants

Eqn 2-30 The Ogden strain energy equation

The Ogden strain energy equation (Eqn 2-30) has been applied to both mechanical

engineering situations (Andra et al. 2000; Jemiolo and Turteltaub 2000; Salomon et

al. 1999) and to simulate biological tissues (Miller and Chenzei, 2002; Jemiolo and

Telega, 2001; Zobitz et al., 2001; Tang et al. 1999).

Other researchers have proposed strain energy equations to describe biological

tissues. Bischoff et al. (2002) developed a strain energy equation which incorporated

orthotropy to represent the mechanical contribution of the fibres in soft tissues. Weiss

et al. (2001) modelled the mechanical behaviour of ligaments and due to the

difficulties encountered in fitting the nonlinear shear response of the ligaments with

the Mooney-Rivlin equation they implemented a strain energy equation that modelled

the material as a fibre reinforced composite (Veronda and Westmann, 1970). A more

complex model was proposed by Rubin and Bodner (2002) which incorporated both

an elastic component and a dissipative component. The elastic component accounted

for the anisotropy of the fibres and for dilation and distortion in the material. The

model also accounted for the recovery of the deformed shape with time. Criscione et

al. (2000) developed a strain energy equation to model isotropy in a finitely

deforming material.



2.8 Experimental Testing of the Intervertebral Disc

Details of relevant previous studies that have involved experimental testing of the

intervertebral disc are outlined in the following section. The literature was reviewed

to obtain information on the mechanical behaviour of the anulus fibrosus in terms of

its constituents (i.e. the collagen fibres and the ground substance).

2.8.1 Types of Testing Carried out and Material Information Available in

Literature

Experimental testing of the intervertebral disc has been carried out to determine the

behaviour of either the disc as a complete entity or to quantify the behaviour of the

individual components, specifically the anulus fibrosus. Testing on the anulus

fibrosus was for the purpose of determining the overall response of the tissue rather

than for quantification of the response of the individual materials from which it is

comprised.

Researchers have carried out static loading, impact loading, relaxation and vibrational

testing on isolated cadaveric intervertebral discs in order to quantify the mechanical

properties of the structure (Brown et al., 1957; Hirsch, 1955; Virgin, 1951).

Testing has been carried out on specimens of anulus fibrosus under various loading

conditions. Acaroglu et al. (1995), Skaggs et al. (1994), Fujita et al. (1997) and Wu

and Yao (1976) carried out tensile testing on dumb-bell or rectangular shaped

specimens of anulus fibrosus. The specimens used by Acaroglu et al. (1995), Skaggs

et al. (1994) and Fujita et al. (1997) were region specific. This allowed for details of

the inhomogeneous tensile response of the anulus fibrosus to be quantified. Evidence

was provided for this inhomogeneity of the anulus in both a radial and circumferential

direction. Acaroglu et al. (1995) carried out this testing on both healthy and

degenerate specimens in order to evaluate the effects of aging on the tensile properties

of the tissue.



Best et al. (1994) carried out experiments on plugs of anulus fibrosus from the

anterior and posterolateral anulus fibrosus to determine the swelling response and the

compressive creep response. The orientation of these specimens was such that the

compressive load was applied in a radial direction to cylinders with a 5mm diameter.

This research provided data on the hydraulic permeability of the anulus and the elastic

modulus of the anulus fibrosus solid matrix (ie. all structures of the anulus fibrosus

except the water and electrolytes). Best et al. (1994) found evidence for the

inhomogeneity of the anulus fibrosus both radially and circumferentially.

Iatridis et al. (1999) demonstrated the shear response of the anulus fibrosus by testing

cylindrical specimens under torsional loading at various amplitudes and frequencies.

Fujita et al. (2000) also carried out shear testing on specimens of anulus fibrosus with

various orientations. The stress-strain results from these tests were not published and

only values of the shear modulus were stated. They found the shear modulus in the

outer anulus was 3-5 times larger than the inner anulus.

There was no evidence found in the literature for experimentation that has been

carried out to determine the mechanical behaviour of the anulus fibrosus ground

substance. All previous researchers aimed to quantify the response of the composite

tissue. Additionally, there was no evidence for the mechanical response of the tissue

under biaxial compression.

2.8.2 Specimen Handling

The details of techniques used to maintain fluid content in disc material and the

freezing temperatures employed are listed in Table 2-6.



Table 2-6 Details of specimen handling techniques employed by previous

researchers

Adams et al.

(1994)

- Spinal specimens were placed in a sealed plastic bag and frozen

at -17oC.

- On the day of dissection the spines were thawed at 3oCfor 12

hours

- The discs were generally tested on the same day and if not were

stored overnight for testing the next day. The specimens were

stored in a vacuum sealed bag

- While the specimens were tested they were protected from fluid

loss by a polythene film

Pearcy and

Hindle (1991)

- Spinal specimens were stored at -20oC until they were to be

dissected

- For dissection the specimens were thawed and the fat and

muscles were removed

- The soft tissue was kept moist during the dissection and testing

using Ringers solution

Ebara et al.

(1996)

- Lumbar spines were stored in a sealed plastic bag and frozen at

-20oC

- For dissection the spines were partially thawed, the disc removed

and cut in half and then these specimens refrozen in double-

sealed plastic bags at -20oC until the day of testing

- Ebara et al. (1996) carried out a pilot study to determine the

most effective environmental condition for the specimens and

determined that immersion in 0.15M saline solution for 15

minutes brought the hydration of the tissue to 96% of the final

equilibrium value

- On the day of testing the frozen disc portion was sectioned into

blocks of anulus. These blocks of anulus were still frozen and

were sectioned into smaller portions, soaked in saline solution

for 15 minutes to equilibrate and then sectioned into test

specimens



Iatridis et al.

(1999)

- The spines were sealed in plastic and frozen at -20oC

- The intervertebral discs were removed from the thawed spines

and the discs refrozen at -80oC

- On the day of testing blocks were removed from the anulus,

mounted on a freezing stage and test specimens cut from these

blocks

- Specimens were frozen or wrapped in plastic during all

preparation steps to maintain the hydration level

- The specimens were tested in an environmental chamber filled

with 0.15M saline solution

Best et al.

(1994)

- These researchers attempted to employ a technique which

limited dehydration and proteoglycan leaching

- Intact motion segments were frozen at -20oC

- The intervertebral discs were dissected from the spine in a

humidity chamber which was maintained at room temperature

and > 95% R.H.. The discs were then refrozen at -80oC

- Blocks were cut from the frozen discs

- The blocks were frozen onto a stage at -20oC and remained

frozen while the test pieces were removed

- The test specimens were stored frozen at -80oC until the day of

testing

Skaggs et al.

(1994)

- On the day of dissection the lumbar spines were cut mid-

sagittally, sealed in plastic and frozen at -20oC until the day of

testing

- On the day of testing the hemidiscs were thawed and the disc cut

from the bone. During this procedure the tissue was kept moist

with saline soaked gauze

- The specimens were tested immersed in 0.15 M saline solution

Markolf and

Morris (1974)

- The intervertebral discs were removed at autopsy and tested soon

afterward

- The discs were kept moist during testing using saline solution



From Table 2-6 it appears that common practice for specimen handling when testing

intervertebral discs and in particular the anulus fibrosus involves freezing of the entire

spine at -20oC until the dissection of the disc or specimens is to take place. The

spines are commonly wrapped in plastic to avoid fluid loss and when thawing is

desirable, this is done gradually. However, the test specimens are generally sectioned

from the discs or spine while they are frozen in order to avoid unnecessary fluid loss.

During the sectioning procedure, the tissue fluid level is maintained using Ringers

solution or saline solution, either by wrapping the specimen in fluid soaked gauze or

by applying the solution directly.

2.9 Conclusions

From the review of literature it was concluded that an investigation into the

biomechanical effects of anular lesions on the disc mechanics would provide valuable

information. This information would serve to improve the current state of

understanding of the mechanisms for degeneration of the intervertebral disc and the

necessity and suitability of current techniques for treatment of back pain.

In order to carry out this investigation a finite element model of the intervertebral disc

was proposed. This technique for developing a computational model of a mechanical

structure provides a unique method for determination of both the external and internal

stress state present within the materials that comprise the structure. Determination of

the internal stress state of the intervertebral disc components in a model of both a non-

degenerate and a degenerate disc would provide useful information on the change in

stiffness and deformation in the disc as a result of anular lesions. Representation of

the individual components of the intervertebral disc, in particular the anulus fibrosus

ground substance and the collagen fibres would provide a broader description of the

loaded stress state in the intervertebral disc.

Representation of the anulus ground substance using a hyperelastic material

formulation would provide a close representation for the highly nonlinear behaviour

of the material. Additionally, it was desirable to simulate the condition of the

intervertebral disc at strain rates simulating common physiological loading conditions,



therefore, the assumption of incompressibility inherent in the hyperelastic material

laws was ideal. Initially, a Mooney-Rivlin strain energy equation was used to

represent the ground substance. Even though it was stated that this material

formulation involved an assumption of linearity during shear loading, several

previous researchers had used this equation to represent biological tissues and the

determination of material parameters was relatively straight-forward.

A preliminary finite element model of the intervertebral disc was developed. This

model was used to determine the suitability of the material formulations and

parameters employed to describe the disc components and to refine the loading

conditions applied to the model to simulate the physiological condition. Details of

development and analysis of this model are the subject of Chapter 3.

Finite Element Analysis of Anular Lesions in the Lumbar Intervertebral Disc

Chapter 3: Development of the Preliminary FEM 60

CChhaapptteerr

33

DDeevveellooppmmeenntt ooff tthhee PPrreelliimmiinnaarryy

FFEEMM

The review of previous studies that have analysed the biomechanics of the

intervertebral disc suggested that the use of the finite element method was the most

appropriate means to analyse the effects of anulus lesions on the disc mechanics.

Several of the finite element models developed by previous researchers to investigate

loading on the intervertebral disc included the adjacent vertebrae, the posterior

elements and/or some spinal muscles. Since the primary structure of interest in the

current study was the intervertebral disc, it was decided that simulation of the

structures external to the disc using specific loading and boundary conditions would

permit a more computationally efficient analysis to be performed. It was desirable to

obtain an extremely detailed and accurate finite element model of the intervertebral

disc in order to better understand the effects of lesions on this structure.

Chapter 3 details the development and analysis of a preliminary finite element model.

The finite element model consisted of the anulus fibrosus, the cartilaginous endplates

and the nucleus pulposus. There was no bony anatomy, musculature or ligaments

included in the model. The majority of the details of this model were maintained in

the FEM which was developed for the final analyses (chapter 8 and 9). Development

of the preliminary model was carried out to determine the acceptability of the

modelling methodology employed for the components of the disc.



3.1 Basic description of the FE method

The finite element method was described by Zienkiewicz (1980). This method

involves the subdivision of a continuous structure into finite regions. These regions

are in 1, 2 or 3 dimensional space and are referred to as elements. The finite elements

are connected by common points called nodes. When all these regions are connected

the resulting arrangement is referred to as a mesh.

Degrees of freedom exist at the nodes and are mutually independent variables of

displacement and/or rotation which define the node’s position and orientation in

space. For a mechanical system a relationship exists between the displacement and

the force at the nodes (Eqn 3-1).

uKf .=

where, f = externally applied force and moments

u = displacements and rotations

K = the stiffness of the system

Eqn 3-1 Force-displacement equation

In a linear system the stiffness is calculated directly from the geometry of the

structure and the elastic stiffness characteristics of the material. For nonlinear

systems, constants describing the mechanical behaviour at each node are collated into

a system of equations for the entire structure and this is expressed in matrix form

as K , the stiffness matrix. This matrix is dependent on the material properties and on

the geometry of the structures being modelled.

Basic features of any structure analysed using the finite element method are the

boundary and loading conditions. These conditions are in the form of forces/moments

and displacements/rotations. Forces and moments applied to the nodes of the model

are defined in the matrix f and any known displacements and rotations are defined in

the matrix, u .



The combined finite element mesh, loading and boundary conditions and material

relationships for the structure are collectively called the finite element model, FEM.

Once the finite elements have been generated, common nodes between them

prescribed and the boundary and loading conditions entered, it is possible to solve the

system of equations governing the mechanical response of the structure to find the

unknown values of displacements.

fKu .1−=

Eqn 3-2 The displacements are determined using the inverse of the stiffness

matrix.

For a nonlinear FEM, this is achieved by matrix manipulation to determine the inverse

of the stiffness matrix. It is an iterative procedure requiring complex matrix algebra.

Analysis of linear systems requires the direct solution of equation Eqn 3-2 since the

stiffness of the system can generally be calculated directly.

An advantage of the finite element method is that it can be enlisted to solve problems

of fluid dynamics or mechanics that involve complicated or geometrically nonlinear

structures. Irrespective of geometry, these structures may be subdivided into elements

to create a mesh and a solution obtained. The only limit on the structures and systems

which can be solved using the finite element method is the processing power available

to solve the complex iterative algorithms involved in this method.

Whilst all aspects of the finite element model control the final accuracy of the

solution, an integral component of the FEM is the relationship that is prescribed

between the displacement and the force at the nodes in the mesh. Inaccuracy in the

material properties for the model components can have drastic effects on the results of

the model and it is essential that the material properties prescribed for the FEM

components mimic the realistic response of the structures.



Commercial software packages are available which facilitate the development and

solution of finite element problems. Examples of these packages are HKS Abaqus

(Worley Advanced Analysis) and ANSYS (Leap Australia Pty Ltd).

3.2 Abaqus 6.3 Finite Element Modelling Software

Commercially available finite element software, Abaqus/Standard 6.3 was used for

the generation of the finite element model. The Abaqus finite element products were

originally developed in the USA by David Hibbitt, Bengt Karlsson and Paul Sorenson

as a tool for structural analysis in engineering applications. This package was

employed as it was robust modelling software, which had experienced widespread use

in both mechanical and biomechanical applications. Additionally, Abaqus was well

suited to the analyses of the intervertebral disc which were static problems involving

nonlinear material properties and nonlinear geometry.

Abaqus 6.3 provided an extensive suite of material descriptions including the

hyperelastic material model employed for the development of the intervertebral disc.

The software was capable of modelling this tissue using a variety of classic models

such as the Mooney equation (Mooney, 1940) and the Ogden model (Ogden, 1972) as

well as providing for a user defined material description. If the material constants for

the model were known, they could be input directly. However, if the material being

modelled did not have documented material constants the user could input raw test

data from specific experimental testing on material samples and Abaqus 6.3

calculated the necessary constants using a least squared error algorithm (Abaqus

Theory Manual, § 4.6.2).

The Abaqus 6.3 software incorporated user-friendly preprocessor input commands

and postprocessing facilities. There was an extensive element library and methods

available for nodal and element constraints were well suited to the current application.



3.2.1 Specifics of finite element analysis carried out using Abaqus 6.3

The FEM could be generated using the graphic user interface, Abaqus/CAE or using a

data file containing the necessary information on the model geometry, mesh, material

properties and the boundary and loading conditions. The latter option was employed

for the development of the FEM as it allowed for greater control over the form of the

mesh which was generated and the model details prescribed. With the use of the data

file, or input file, it was possible to generate finite element models of varied mesh

density and mechanical properties.

Abaqus organised the input of the boundary and loading conditions into specific

analysis phases called steps. The loading that was being simulated was organised into

specific events which were then analysed successively by the software as individual

steps. Each step was associated with a time frame. By default the time for any given

step was 1.

For nonlinear analyses such as the intervertebral disc FEM, the steps were subdivided

into increments. An increment corresponded to the application of a portion of the

total load and boundary condition for a particular step. The length of the increment

was dependent on the time frame over which the step occurred and on convergence

difficulties encountered by the software. If the time frame for the increment was too

long, then the change in the displacement and force condition over this time would be

too large and Abaqus may not have been able to converge on a solution for that

period. In this instance, the software automatically reduced the time period for that

increment – perform a time-cutback - and attempted to resolve. Abaqus would only

carry out time-cutbacks 5 times. If any more attempts were required the software

considered that the solution was diverging and a valid solution could not be obtained.

If the time period for the increment was too small, Abaqus would carry out

unnecessary calculations and thus inefficiently use processing time. In order for a

timely solution to be obtained, the user could designate a minimum and maximum

time period for the increments and Abaqus attempted to find a solution within this

range.



In order to determine whether a FEM analysis was converging on a solution in any

given increment, Abaqus analysed the residuals of the fluxes and the corrections to

the displacements. Both the residuals and the corrections were errors in the

force/displacement or moment/rotation equations. The residuals related to errors in

the fluxes – force and moments – and the corrections related to errors in the nodal

variables – displacement and rotation.

If the largest residual force or moment was more than 0.5% of the time average force

or moment, then Abaqus did not consider that equilibrium had been achieved for that

variable and would initiate further iteration. If the largest correction to any nodal

variable was greater than 1% of the largest change in the nodal variable then the

equilibrium of that variable was not achieved. Abaqus would provide information on

the nodes at which the residuals or corrections were too large and the associated

degree of freedom (Table 3-1).

Table 3-1 Abaqus output for convergence of analysis increments

Abaqus required that the residuals or corrections in each iteration of a step, converge

quadratically in accordance with the Newton-Raphson criteria. If they did not

decrease in accordance with this criterion Abaqus warned that the solution was either

converging too slowly or diverging.

EQUILIBRIUM ITERATION 1 AVERAGE FORCE 2.32 LARGEST RESIDUAL FORCE -3.207E-02 AT NODE 165510 DOF 2 LARGEST INCREMENT OF DISP. -3.825E-03 AT NODE 119570 DOF 2 LARGEST CORRECTION TO DISP. -1.179E-02 AT NODE 119570 DOF 2 FORCE EQUILIBRIUM NOT ACHIEVED WITHIN TOLERANCE AVERAGE MOMENT 0.00 LARGEST RESIDUAL MOMENT -8.656E-04 AT NODE 9999999 DOF 6 LARGEST INCREMENT OF ROT’N -2.201E-07 AT NODE 10 DOF 4 LARGEST CORRECTION TO ROT’N -3.246E-08 AT NODE 10 DOF 6 THE MOMENT IS ZERO EVERYWHERE BUT THE MOMENT RESIDUAL OR THE ROTATION CORRECTION IS NON-ZERO AVERAGE CAV. VOL. 1.97 LARGEST RESIDUAL CAV. VOL. 1.473E-05 AT NODE 2 DOF 8 LARGEST INCREMENT OF H. PRESS. 1.513E-05 AT NODE 2 DOF 8 LARGEST CORRECTION TO H. PRESS. -1.984E-05 AT NODE 2 DOF 8 H PRESS. CORRECTION TOO LARGE COMPARED TO H. PRESS. INCREMENT EQUILIBRIUM ITERATION 2 …



3.3 Geometry of the Anulus Fibrosus and Nucleus Pulposus in the Transverse

Plane

The spinal level modelled was the L4/5 lumbar level. This decision was based on

reports from clinicians of the prevalence of anulus lesions in this level of the spine.

The finite element model developed consisted of only the L4/5 intervertebral disc.

There was no musculature, ligaments or bony attachments. The actions of these

structures were simulated through the boundary and loading conditions applied to the

model.

In order to develop a dimensionally accurate FE model of the disc, it was necessary to

accurately represent the outer disc profile. Existing literature in the area provided

only gross dimensions for the full anterior-posterior depth and for the lateral width.

There was very little information available on the precise curvatures and form of the

outer anulus and nucleus transverse boundaries. This was largely due to the

significant variation in these dimensions between different specimens.

One approach to obtaining the necessary dimensions would have been to measure the

dimensions from a series of specimens and attempt to develop a standard dimensional

data series for the different sections of the disc. However, this would have proven to

be time consuming and more importantly, due to the high degree of variation in the

shape and area of different discs from the same level of the spine, it may have been

prone to a high level of error.

Alternatively, the approach adopted was to develop a series of formulae to

appropriately represent the anulus and nucleus boundaries. These formulae would be

applied on the basis of a sequence of twelve significant points on the outer boundaries

and could be manipulated to represent any disc shape.

The co-ordinates of the boundary nodal points were defined using the formulae. To

make the identification and input of the nodal points more straight-forward, it was

desirable for them to be input directly into the FE input file using Matlab execution

codes.



The formulae were developed using data from the images obtained by Vernon-

Roberts et al. (1997) for the L4/5 lumbar intervertebral discs (Figure 3-1).

Additionally, other morphological data from this study was used for validation of the

final formulae.

Figure 3-1 Picture of a sectioned cadaveric intervertebral disc.

3.3.1 Methods – anulus boundary

The following section details the development of formulae to map the outer boundary

of the anulus fibrosus and the criteria used to determine the accuracy of these

formulae.

3.3.1.1 Measurements

Photographs of sectioned intervertebral discs were used to obtain tracings of the outer

anulus boundaries in the L4/5 disc of 18 specimens from various age groups.

In order to define the regions, over which each formula was to apply, the disc was

divided into 6 separate sectors. These sectors were defined using 4 tangent lines and

6 intersection points on the anulus boundary.

The lines and points were defined as shown in (Figure 3-2 and Figure 3-3).



• Points 5 and 1 – the 2 intersections of a line (line A) drawn tangent to the most

posterior points on the anulus boundary

• Points 2 and 4 – defined by the intersection of 2 lines drawn perpendicular to line

A (called lines B and D, respectively) and tangent to the lateral-most points

• Point 3 – defined by a line (line C) parallel to A and intersecting the most anterior

point

• Point 0 – intersection of a line (line E) parallel to line A and intersecting the most

anterior point in the posterior concavity of the disc

Figure 3-2 Tangent lines creating the rectangular boundary in the transverse

sectioned view of a disc

Radial lines from the geometric centre to these points were denoted r0, r1, …, r6. The

lengths of these radii and the angle (θ) from the x axis to the radii were measured

(Figure 3-3).

The disc centre was the geometric centre of a rectangle, defined by lines A through D.

The rectangular co-ordinate axes were defined in relation to lines A and B with the x

axis passing through the geometric centre, parallel to line B and the y axis through the



centre, parallel to line A – so orientating the disc with its left lateral side down (Figure

3-3).

Figure 3-3 Definition of anulus boundary points

3.3.1.2 Development of equations

Each point on the anulus boundary was defined using parametric equations and the

basic form of the equations for sectors 1 to 4 was an ellipse (Eqn 3-3).

θθ

coscos

y

x

ryrx

==

Eqn 3-3 Parametric equations for an ellipse

The typical form for the equations developed is shown in Eqn 3-4.



).sinsin().()(

).coscos().()(

PcurveofperiodnecessaryangleinitialradiisuccessiveofvaluesyindifferenceyofvalueinitialY

PcurveofperiodnecessaryangleinitialradiisuccessiveofvaluesxindifferencexofvalueinitialX

++=

++=

radiiboundingthebetweenrotationangulartotalradiifirstfromrotationangularP =

Eqn 3-4 Typical form for the equations to plot sectors 1 to 4

The value of θ was increased through each sector to generate the series of bounding

points.

A cosine or a sine term could be applied because for a specific angular period, the y

variation of these terms was similar to the variation of the difference between the radii

in any given segment. Since the points 0 to 5 were located at the tangent points to

perpendicular lines, the gradients at each point were readily defined as zero or

infinity. If Eqn 3-4 was applied piecewise to each sector, then the gradients at the

ends of each of these sectors could be manipulated to ensure continuity. This was

achieved by selecting the ‘necessary period of the sine and cosine curves’ to be

continuous between each sector. This was shown in the choice of the period for the

curves in Eqn 3-5.

The parametric equations used to represent each sector were:

Sector 1

( )

( )

−−

−+=

−−

−+=

12

1112211

12

1221122

.2

sin.sin.sin.sin.

.2

cos.cos.cos.cos.

θθθθπθθθ

θθθθπθθθ

rrry

rrrx for 21 θθθ <<

Sector 2

( )

( )

−−

+−+=

−−

+−+=

23

2332233

23

2332222

.22

sin.sin.sin.sin.

.22

cos.cos.cos.cos.

θθθθππθθθ

θθθθππθθθ

rrry




Sector 3

( )

( )

−−

+−+=

−−

+−+=

34

3443333

34

3334444

.2

sin.sin.sin.sin.

.2

cos.cos.cos.cos.

θθθθππθθθ

θθθθππθθθ

rrry


Sector 4

( )

( )

−−

+−−=

−−

+−+=

45

4554455

45

4445544

.22

3sin.sin.sin.sin.

.22

3cos.cos.cos.cos.

θθθθππθθθ

θθθθππθθθ

rrry


Eqn 3-5 Parametric equations to plot sectors 1 to 4

The choice of the ‘initial angle’ and the ‘necessary period of the curve’ for each sector

were dependent upon whether the x and y co-ordinates were increasing or decreasing

over the sector. For example, in sector 4 it may be seen from Figure 3-3 that the x co-

ordinate will increase over the trajectory from r3 to r4 and the y co-ordinate will

decrease over the trajectory from r3 to r4. Figure 3-4 shows that the region on both the

cosine and sine curves where this occurred was from π to 23π . Hence the initial

angle was π and the period of the curve was 2π . In this way the x co-ordinate

varied from 33 cos. θr to 44 cos. θr and the y co-ordinate will vary from 33 sin. θr to

44 sin. θr .

-1.5

-1

-0.5

0

0.5

1

1.5

0 3.14159 6.28318Angley

cos(θ) sin(θ)

Figure 3-4 Cosine and sine curve showing angle over which the parametric equations are chosen



The form of the equations for sectors 0 and 5 varied from the other sectors, due to the

inflection point present on their boundaries. An attempt was made to fit a cosine or

an arcsine equation for these sectors, but the inflection points produced with these

were too severe resulting in an inaccurate and exaggerated representation of the

posterior concavity. Instead, a cubic equation was used for these segments.

The general forms of the 4 constants for the cubic equation (Eqn 3-6) were

determined using the commercial mathematical program, Maple V. These are

detailed in Eqn 3-7.

dcxbxaxy +++= 23

Eqn 3-6 Cubic equation

3223

2323

..3..3

...3....3.

.).(.6

)(.2

)).((.3

llnlnnKK

lnmnmonlold

Knlomc

Komb

Klnoma

−+−=

−++−=

−=

−=

+−−=

where, l = x co-ordinate of first point in sector path

m = y co-ordinate of first point in sector path

n = x co-ordinate of second point in sector path

o = y co-ordinate of second point in sector path

Eqn 3-7 Defining constants for the cubic equations modelling sectors 0 and 5

All the sector equations were implemented using Matlab code and a separate file

developed containing the data values for the radii and associated angles for each

specimen (Appendix A).



3.3.1.3 Disc area

For the purposes of validation of the formulae, it was necessary to develop a method

for determining the area of all the sectors of the entire disc surface.

This was achieved by using numerical integration. Traditional methods of numerical

integration such as Simpson's rule or the Trapezium rule could not be applied as this

was a finite area within a distinct outer boundary rather than a curve on the two

dimensional plane. Therefore, a method involving the cumulative sum of the areas of

a series of triangles was used.

The two long sides of the triangle intersected at the geometric centre and the smaller

side was the distance between any two adjacent points on the outer boundary. A

Matlab code was developed to calculate the final sum (Appendix A).

3.3.1.4 Validation of the anulus formulae

Two criteria were used to validate the final formulae:

• Firstly, a visual validation was used. If the general shape of the disc matched

that of the original specimen, in terms of curvature, gradients and turning

points, then the formulae were considered to be suitable.

• Secondly, the area of the overall disc shape was determined. This area was

compared with the discal area data provided by the study carried out by

Vernon-Roberts et al. (1997).

In terms of the visual validation, the formulae developed produced plots with outer

boundaries very similar to the specimens from which they were taken. However, each

sector was slightly more curved outward than the original tracing of the disc. This

was due to the initial assumption that the sector formulae were of an elliptical form.

Even so, the discrepancy between the areas calculated during this study and those

provided by Vernon-Roberts et al. (1997) were relatively low (Figure 3-5 and Figure

3-6). Also, this error was generally negative, indicating that the areas calculated from



the formulae were conservative and that the over-representation of the curvature of

the boundary did not greatly impact on the final disc shape.

0

500

1000

1500

2000

2500

3000

8 12 16 29 33 37 38 39 40 41 43 44 45 48 50 56 57 58Specimen number

Are

a (m

m2)

Current study Vernon-Roberts et al. (1997) Figure 3-5 Comparison of total disc area with the results from Vernon-Roberts et

al. (1997)

-15

-10

-5

0

5

10

15

8 12 16 29 33 37 38 39 40 41 43 44 45 48 50 56 57 58

Specimen number

Perc

enta

ge

Percentage Variation

Figure 3-6 Percentage variation in disc area compared to the area values from Vernon-Roberts et al. (1997)

3.3.2 Methods – nucleus

This section details the development of formulae to map the outer profile of the

nucleus pulposus and the criteria used to validate this technique.



3.3.2.1 Measurement

Measurements taken for the nucleus boundary were similar to those obtained for the

anulus boundary. A local x-y co-ordinate system was constructed using the same

approach as was outlined for the anulus.

However, there were 3 additional dimensions obtained, which allowed the nucleus

location and orientation relative to the anulus to be defined. These additional

dimensions included:

• the radial distance from the geometric centre of the anulus to the geometric

centre of the nucleus;

• the angle from the x axis of the anulus to a line between the anulus geometric

centre and the nucleus geometric centre; and

• the angle between the x-axis of the anulus and the x-axis of the nucleus.

Fewer specimens were used to determine the formulae for the nucleus profile because

only specimens with a distinct boundary between the anulus and nucleus were chosen.

3.3.2.2 Development of equations

The equations used to define the nucleus boundary were similar to those for the

anulus boundary.

Additional lines of code were implemented to apply a rotation matrix to the complete

x and y matrices of the nucleus in order to translate it and rotate it relative to the

anulus. If the local x-y co-ordinate system of the nucleus was similar to that of the

anulus then this was incorporated into the code.

Based on a visual comparison of the original specimens, it was apparent that the

nucleus was subject to a much greater degree of variation in its form than the outer

anulus boundary. It was important to ensure that the formulae developed were robust

when applied to these varying shapes.



A special case for the nucleus was encountered for one specimen (specimen 12) in

which the nucleus was essentially circular, creating a nucleus with only four sectors.

Because the equations for both the nucleus and the anulus generated points for 6

sectors, it was necessary to define two arbitrary points on the nucleus boundary which

were very close to the posterior-most point ‘0’.

The final calculated area of the nucleus for this specimen was very similar to that

determined by Vernon-Roberts et al. (1997) (Figure 3-7 and Figure 3-8). This

process provided evidence for the flexibility of the formulae.

3.3.2.3 Nucleus area

The nucleus area was determined using the same approach as that adopted for the

anulus.

3.3.2.4 Validation of the nucleus equations

Three criteria were used for the validation of the nucleus. Both the visual and the area

validations outlined for the anulus were applied to the nucleus and additionally, a

measure of nucleus displacement was employed.

Visual validation

Visually, the shape, location and orientation of the nuclei were very similar to that of

the original specimens. Again, the only discrepancy was the increased curvature on

the computed nucleus, which was attributed to the elliptical formulation applied.

However, this increased curvature was not as pronounced over the small sector

lengths of the nucleus.

Area validation

The nucleus areas determined by Vernon-Roberts et al. (1997) were expressed in

terms of the ratio of the nucleus area to the total discal area. In presenting this ratio

for the computed areas, the computed nucleus areas were compared to the total disc



areas from the results of Vernon-Roberts et al. (1997). This avoided inclusion of the

error present in the computed total disc area (Figure 3-7 and Figure 3-8).

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

8 12 37 38 39 50Specimen number

Are

a ra

tio

Computed ratioExperimental ratio (Vernon-Roberts et al., 1997)

Figure 3-7 Comparison of nucleus area ratio data

-10

-5

0

5

10

15

20

8 12 37 38 39 50

Specimen number

Perc

enta

ge

Percentage variation

Figure 3-8 Percentage variation in nucleus area ratios

Given how sensitive the ratios were to variation, there was reasonable correlation

between the experimental results from Vernon-Roberts et al. (1997) and the computed

ratios. It was considered that the variation between the experimental and computed

ratios was due to the difficulty in defining the precise location of the nucleus

boundary. The nucleus boundaries that were traced in this study may have been

considerably different to those depicted in the experimental study.



Offset between the centroid of the nucleus and the anulus profiles

The nucleus offsets provided by Vernon-Roberts et al. (1997) were most likely

measured as the distance from the centroid of the disc area to the centroid of the

nucleus area. However, the centre locations that had been referenced in the current

study were the geometric centres, based on the specific boundary points.

When comparing the nucleus offsets from each study, it was believed that these two

centre locations were not in the same position.

It was considered that in order to validate the nucleus in terms of the nucleus offset, a

different method of determining the centre of the regions needed to be developed.

This method involved determining the centroid of the anulus and nucleus using the

standard sum of area method.

For ease of calculation the anulus and nucleus profiles were rotated by 90°, so that the

extreme posterior surface and the line A overlapped with the x-axis. The anulus was

divided into a series of rectangular areas of width ∆x and varying height according to

the x location through the disc (Figure 3-9).

Figure 3-9 Definition of variables for centroid calculations



The formulae used to determine the location of the centroid of the anulus and nucleus

are stated in Eqn 3-8.

( )∑ ∆+∆=i

iixxxLAX 2..1

∑ ∆=i

ii

LxLAY2

...1

Eqn 3-8 Formulae to determine the co-ordinates of the centroid

Where, A = total area of anulus or nucleus calculated previously

Li = length of rectangle = y value at xi

X = x value of centroid

Y = y value of centroid

Using the values for the centroids of each specimen, improved results for the nucleus

offsets were obtained (Table 3-2).

Table 3-2 Comparison between the nucleus offset determined from the

displacement between the calculated centroids of the nucleus and the anulus and

the nucleus offset value stated in the experimental results

Specimen Computed

Offset

(mm)

Experimental

Offset

Vernon-

Roberts et al.

(1997)

(mm)

Absolute

Error in the

Offset

Compared to

Experimental

(mm)

8 5.45 4.02 1.43

12 1.25 3.19 1.94

37 0. 844 2.65 1.81

38 1.94 3.48 1.54

39 1.80 1.16 0. 638

50 0. 392 1.02 0. 628

Mean 1.33

Standard Deviation 0. 521



The error in the offset between the centroids of the nucleus and anulus ranged from

0.628mm to 1.94mm with a mean of 1.33mm. The low magnitude of the standard

deviation in the error was attributed to consistency in the determination of the nucleus

offset.

It was considered that the use of a centroid for the determination of the nucleus

displacement in the current study was a more accurate approach than the use of

geometric centres.

3.3.3 Discussion concerning the anulus and nucleus boundaries

Modelling the anulus boundary using the six formulae developed yielded acceptable

levels of variation for the anulus area. Also resultant plots of the anulus were visually

similar to the specimens on which they were based.

A major cause for the discrepancies between the nucleus area ratios was the location

of the nucleus boundary in both the experimental study and in the current computer

study.

There were several causes for this obscuring of the boundary:

• the precise boundary between the nucleus and the anulus was generally

difficult to discern in healthy discs, as the constituents of each tended to

‘blend’ into one another;

• many of the discs sampled in this study showed various degrees of

degeneration, which in some discs caused the nucleus and anulus to no longer

exhibit distinct or definable boundaries; and

• the tracings from which the dimensions were attained in this study were taken

from photocopies of the original images, which would have been a much

higher resolution.

It was considered that the reason for the variation between the nucleus offset in the

experimental study and the current study may have been related to the difficulty



encountered in determining the precise location of the outer boundary of the nucleus.

Since the nucleus area ratios compared well with the results of Vernon-Roberts et al.

(1997) this indicated that the general area of the nucleus was defined accurately but

the precise location of the nucleus was not. The location of the nucleus centroid was

directly related to the nucleus boundary, which was depicted during the initial tracing

of the disc components. If at this stage the general form of the nucleus was defined

with reasonable accuracy, but the precise shape and curvature of the nucleus boundary

was slightly inaccurate then centroid calculations would yield a centroid location that

was imprecise. It must also be noted that the variation in the nucleus displacements in

absolute measurements ranged from 0.6 to 1.9mm. These values were comparatively

small when considering that on some disc tracings, the region over which the nucleus

boundary could have reasonably been interpreted was approximately 2.5 to 3.5mm.

Given that the potential error in the location of the nucleus was higher than the

maximum observed error of 1.9mm, this difference between the experimental results

and the results of the current study was not of concern.

The six formulae developed were used to determine the profile of the nucleus

pulposus and anulus fibrosus. The rationale for this decision was based on a

comparison of this approach to that which had been traditionally adopted. In previous

studies the nucleus had been placed centrally in the frontal plane of the disc and

slightly posteriorly in the transverse plane. This transverse placement was based on

reports that the anulus was thicker anteriorly and the central placement was possibly

based on an assumption of lateral symmetry. However, perusal of only a few of the

disc specimens obtained in the study carried out by Vernon-Roberts et al. (1997)

showed that very few discs exhibited a lateral symmetry of the nucleus. In fact, the

nucleus was commonly both displaced and rotated from a central location.

If the nucleus were assumed to be placed in a position similar to previous studies, then

the error in the values of nucleus offset would be much higher than their current

values. Therefore, even though the nucleus offset values attained in the current study

did exhibit error when compared to the results of Vernon-Roberts et al. (1997), this

error was considered to be low in comparison to that introduced if the nucleus

location mimicked traditional approaches.



Therefore, based on the comparatively low percentage variation between the

experimental and the current computer studies of the anulus and nucleus data, it was

considered that the formulae developed modelled the true form of the anulus and

nucleus to a reasonable level of error. In total 27 parameters were measured from the

transverse pictures of the intervertebral discs. The formulae were used to generate a

series of data points on the anulus and nucleus boundary which became the nodal co-

ordinates in the FEM.

A limitation for the use of these formulae in defining the profile of the nucleus

pulposus was the necessity to trace photographs of human intervertebral discs with a

well defined outer boundary for the nucleus pulposus. This would likely only occur

in relatively healthy intervertebral discs or discs with minimal degeneration of the

nucleus.

The specimen geometry that was used for the development of the final FEM was

specimen 50. This choice was due to the regularity of the anulus and nucleus

boundaries in this specimen. It was considered that incorporation of skewed nuclei

and extremely non-symmetric geometries would be of more benefit once the analysis

of the symmetrical, non-skewed geometry was carried out.

3.4 Geometry of the Collagen Fibres

In order to define the geometry of the elements representing the collagen fibres in the

FEM, it was necessary to prescribe:

• the cross-sectional area of the fibres;

• the fibre spacing in the fibre layer and in a radial direction within individual

lamellae;

• the angle of inclination of the fibres within the elements; and

• the elements in which the fibres were located.

The elements used to define the collagen fibres were rebar elements. These will be

further defined in Section 3.8.2.



3.4.1 Cross-sectional area of the collagen fibres

Marchand and Ahmed (1990) provided detailed information on the morphology of the

collagen fibres in the anulus fibrosus (Figure 3-10). The average dimensions for the

geometric parameters were:

SB = Fibre spacing

= 0.23mm

tB = Thickness of one bundle of fibres

= 0.14mm

WB = Width of one bundle of fibres

= 59% of tL

tL = Thickness of an individual lamellae

= 0.17mm

The cross-sectional shape of the collagen bundles in the FEM was assumed to be an

ellipse. The cross-sectional area of the fibres was calculated using Eqn 3-9.

2.

2. BB tWArea π=

where, Bt = Thickness of one bundle of fibres, 0.14mm

BW = Width of one bundle of fibres, 59% of tL

Lt = Thickness of individual lamellae

= average thickness of a circumferential

element layer in the FEM

Eqn 3-9 Equation to determine cross-sectional area of the collagen fibres

On the basis of Eqn 3-9 the cross-sectional area of the fibre bundles was 0.01268mm2.

However, this value was for the fibre bundles in an anulus fibrosus in vivo, which

contains an average of 20 lamellae (Marchand and Ahmed, 1990). The FEM of the

intervertebral disc did not necessarily contain the same number of circumferential

element layers as there were lamellae in the in vivo disc. The number of element

layers in the FEM was reduced if it was considered that a high mesh density had little

effect on the results of the model. Therefore, the cross-sectional area of the rebar

Figure 3-10 Collagen fibre spacing in a

lamellae

tL

SB

tB

Fibre bundle

WB



elements in the model was varied according to the number of element layers present

and the comparative number of lamellae represented (Figure 3-11). This avoided the

use of an inaccurate collagen fibre content in the FEM.

Figure 3-11 Schematic of lamellae in the intervertebral disc demonstrating increased collagen fibre cross-sectional area when the number of circumferential element layers was less than the number of lamellae in the intervertebral disc in

vivo.

The radial width of the collagen fibre bundles within the lamellae is 59% of the radial

dimension of the lamellae (Marchand and Ahmed, 1990). Therefore, the sum of the

radial widths of the collagen fibre bundles would be 59% of the total radial width of

the anulus fibrosus. In the FEM, this width varied with circumferential position so an

average radial dimension, Ranulus was obtained from the lateral anulus (Figure 3-12).

Using the average width of the anulus, an average radial dimension for the lamellae

was determined as NRR anuluslamellae = where N was the number of lamellae

modelled (Figure 3-12).

Figure 3-12 Determining the average width of the circumferential element layers in the FEM

Ranulus

Rlamellae

Area of the rebar elements representing the fibre bundles in FEM

Fibre bundle in intervertebral disc in vivo

20 lamellae in the intervertebral disc in vivo

Circumferential element layer representing lamellae in disc



The value of Rlamellae was used as the lamellae thickness, tL, in the calculation of the

cross-sectional area of the collagen fibres using Eqn 3-9. The preliminary FEM

contained 8 element layers and the calculated cross-sectional area of the rebar

elements was 0.1199mm2. The fibre density in this FEM was approximately 17% by

volume which was comparable to the value determined from the results of Marchand

and Ahmed (1990).

3.4.2 Collagen fibre spacing

The fibre bundles were assumed to be positioned halfway through the thickness of the

lamellae in the radial direction. A value of 0.23mm was used to define the fibre

bundle spacing within the lamellae (Marchand and Ahmed, 1990) (Figure 3-10).

3.4.3 Angle of inclination of the rebar elements within the layers of collagen

fibres

In order to define the angle of inclination of the rebars, Abaqus required the use of

isoparametric directions. The isoparametric directions were in relation to the local co-

ordinates of the element and were different to the dimensions of the elements in

physical space. The use of isoparametric co-ordinates and directions were especially

important when meshes were distorted in relation to a set of orthogonal axes. The

element configuration of the continuum elements in the anulus fibrosus (Figure 3-13

A) needed to be converted into an isoparametric configuration (Figure 3-13 B).

Figure 3-13 Three dimensional continuum element with embedded rebar layer. A. Configuration in the FEM. B. Configuration for an isoparametric cube

L

W

BA

z



It was necessary to define the orientation of the rebar layer in relation to the

isoparametric directions. For example, in Figure 3-14 the orientation of the rebars in

the rectangle, α, needed to be converted to an angle in relation to the isoparametric

directions, β.

Figure 3-14 The rectangular configuration for the rebar layer, A, must be converted to a cubic configuration, B, to obtain the isoparametric collagen fibre

inclination angle

This conversion was achieved using the relationship in Eqn 3-10.

)tan(.)tan( αβWL

=

Eqn 3-10 Converts angle in physical space to angle in isoparametric space

In the case of the anulus lamellae, the angle α, was an average of 30o.

The radial and circumferential measurements of the elements in the anulus were

similar. However the height of the elements varied in an antero-posterior direction

due to the wedge shape of the disc. Rather than determining the isoparametric

orientation of the rebar elements in each individual element, 8 separate elements were

selected as having element dimensions in the z direction that were representative of

the entire mesh. The L and W dimensions of these elements were used to determine

the orientation of the rebar elements in the anulus fibrosus lamellae.

3.4.4 Embedding elements

The anulus fibrosus continuum elements were divided into circumferential element

sets such that each lamella could be defined. A rebar element command was

BA L

W

βα

z



prescribed for each individual lamella in order to reproduce the alternating collagen

inclination between each lamella in the anulus.

3.5 Determination of Sagittal Geometry

The axial dimensions for the wedge shaped disc were obtained from the study carried

out by Tibrewal and Pearcy (1985). They stated values of 14mm for the anterior disc

height and 5.5mm for the posterior disc height. The axial disc height was varied

linearly in an antero-posterior direction.

3.6 Location of the Instantaneous Axes of Rotation During Rotation

Details of the instantaneous centre of rotation (ICR) are found in Chapter 2. This

point was used to define an axis about which rotation in the three orthogonal

directions occurred. These locations in the FEM were based on the findings of

previous researchers.

3.6.1 Flexion/Extension

Using the average dimensions of the L5 vertebra as determined by Panjabi et al.

(1992) the depth of the upper surface of the L5 vertebra was 34.7 ± 1.17 mm and the

height of L5 was 22.9 ± 0.95 mm. Using the results from Pearcy and Bogduk (1988),

the location for the ICR in full flexion and extension was (0, -2.918mm, -9.9729 mm)

(Figure 3-15), measured from the origin of the intervertebral disc FEM. This was

located on the superior surface of the disc and was at the centroid of the disc surface

in the transverse plane. The flexion and extension loading on the intervertebral disc

were defined as a rotation about a medio-lateral axis passing through the ICR.



Figure 3-15 Approximate location of ICR for full flexion from upright standing. Based on the calculations of Pearcy and Bogduk (1988)

3.6.2 Axial rotation

Determination of a location for the ICR under axial rotation was based on the findings

of Cossette et al. (1971), Adams and Hutton (1981) and Thompson (2002). The ICRs

were located in the posterior anulus, ½ way through the posterior disc and a distance

of ¼ of the total lateral disc width from the extreme lateral edges (Figure 3-16).

Axially, the ICRs were level with the superior surface of the disc. Under right axial

rotation, the ICR was located in the right disc and under left axial rotation the ICR

was located in the left disc. In specimen 50, the location of the ICR for left axial

rotation was (-11.45mm, -12.671mm, 0) and the location of the right axial rotation

ICR was (11.45mm, -12.671mm, 0) in relation to the origin of the intervertebral disc

FEM.

Axial rotations were defined as a rotation about an axis that passed through the ICR in

a caudo-cephalic direction.

Origin according to Pearcy and Bogduk (1988)

X

Y

Lower vertebra in joint, (L5)

Upper vertebra in joint, (L4)

Intervertebral disc

Depth

Height ICR

Origin of the intervertebral disc FEM



Posterior Disc

Figure 3-16 Location of the ICR for right and left axial rotation viewed from above

3.6.3 Lateral bending

On the basis of the work of Rolander (1966) and Thompson (2002) the ICRs during

lateral bending were located mid-way between the lateral edge of the disc and the

centre of the disc. Thompson (2002) varied the lateral location of the ICR during

lateral bending. Initially the ICR was located in the centre of the disc when viewed in

the frontal plane and at full lateral rotation it was located mid-way between the lateral

edge of the anulus and the disc centre. It was not possible to incorporate a

rotationally varying location for the ICR during lateral bending in the FE – rotational

degrees of freedom of the model components were specified from fixed nodal

locations. Therefore, the final location of this axis as defined by Thompson (2002)

was employed.

In an antero-posterior direction the ICR were level with the disc centre and axially,

they were located at the mid-disc height. Because the sagittal geometry of the disc

was a wedge shape and there was an assumed linear variation between the disc

heights posteriorly and anteriorly, the axial location of the ICRs was defined as the

average of the anterior and posterior disc heights. Under right lateral bending, the

ICR was located in the left disc and under left lateral bending the ICR was located in

the right disc (Figure 3-17).

1/41/4 1/4 1/4

ICR for right rotation

Origin1/4

1/4

1/4

1/4

Left Right



Figure 3-17 Location of the ICR for right and left lateral rotation viewed from the posterior disc

In relation to the centroid of specimen 50, the final location of the left lateral bending

ICR was (12.5mm, 0, -4.875mm). The location of the ICR for right lateral bending

was (-12.5mm, 0, -4.875mm). Lateral bending rotations were applied to the FEM

about an axis passing through the ICR in the antero-posterior direction.

3.7 Fortran Programming

In order to improve the efficiency of generation of input files for the FEM a Fortran

executable file was developed. This file enabled FE meshes displaying differing

mesh sizes to be generated with minimum effort on the part of the operator.

The Fortran file required user input for the number of lamellae and the number of

elements in the anulus in a radial direction. On the basis of these parameters, an input

file was generated. The data input for the nodal co-ordinates of the anulus and

nucleus profiles were obtained from data files created by the Matlab executable files

outlined in Section 3.3. In this way, with the combination of the Matlab and Fortran

executables, it was possible to readily create a FE mesh on the basis of 27 parameters

measured from a transverse image of an intervertebral disc.

3.8 Description of the Finite Elements Used in the FEM

The following sections detail the elements used to represent the components of the

intervertebral disc FEM.

1/41/4 1/4 1/4

ICR for left lateral rotation

ICR for right lateral rotation

Left Right 1/2

1/2



3.8.1 Anulus fibrosus and cartilaginous endplate

Nonlinear, 20 node continuum elements – C3D20RH – were used to model the

cartilaginous endplates and the anulus ground substance. These elements were

“hybrid” elements which were intended for use with nonlinear materials. They also

employed reduced integration techniques for the stiffness matrix to limit the size of

the analysis.

The continuum elements were arranged in the anulus fibrosus such that one concentric

layer of elements around the FEM anulus represented one concentric “layer” of

lamellae in the intervertebral disc. However, it must be noted that in the physical disc

the lamellae were not circumferentially continuous structures. Marchand and Ahmed

(1990) reported that the number of incomplete layers was region dependent and was a

maximum posterolaterally, with 53% of the lamellae being discontinuous in this

region. The anterior anulus had the minimum number of discontinuous layers with

43% incomplete layers. Figure 3-18 shows the FEM representation of the anulus

lamellae.

Figure 3-18 Three dimensional continuum elements in the model. A. Elements representing the lamellae of the anulus fibrosus; B. Elements in the cartilaginous

endplates

The nodes at the interface between each circumferential layer of elements in the

anulus fibrosus were duplicated. This duplication allowed for the introduction of

contact definitions in future analyses for the purpose of investigating circumferential

lesions and the interlaminar stress/strain state of the anulus fibrous during various

physiological loading conditions. Until analyses were carried out which required

A. B.



information on the properties of these interfaces, these nodes were “tied” together

using specific constraint definitions in Abaqus. These constraints caused all degrees

of freedom in a pair of duplicate nodes to be equal, thereby creating a continuum of

elements and negating the effects of the additional nodes.

3.8.2 Collagen fibres

Tension only elements were incorporated to model the collagen fibres in the anulus

fibrosus. These elements, called rebar elements were continuous fibre reinforcements

embedded within continuum elements. Abaqus required the user to identify:

• the continuum elements in which the rebar elements existed;

• the three dimensional orientation of the rebar element within the continuum

element;

• the material characteristics for the rebar elements; and

• the rebar cross-sectional area.

The use of the rebar elements was a convenient means for modelling the collagen

fibres embedded at alternating angles in successive lamellae of the anulus fibrosus.

The fibres could be precisely orientated within the layers of continuum elements

modelling the lamellae and the prescription of a cross-sectional area for the rebars

ensured the fibre density in the modelled anulus was similar to experimental reports.

While single fibres were not simulated in the model the rebar reinforcements in the

model represented continuous reinforcement in the ground substance.

No localised interaction existed between the rebar elements and the ground substance

elements in which they were embedded. The rebar elements provided a summative

stiffness to the underlying ground substance.

It was desirable to include the discontinuity of the anulus lamellae using the

designation of the rebar elements. A possible mechanism for including this

discontinuity was to designate different element groups or element sets for different

circumferential regions of each lamella and then define specific rebar groups with

similar inclinations for each of these regions. This was intended to introduce a



discontinuity in the collagen fibres of the lamellae. However, Abaqus viewed these

separate rebar groups as the same group if their inclination was similar. Therefore, it

was not possible to model the discontinuity of the anulus lamellae through the

designation of the rebar elements.

The only other method to introduce the discontinuity of the lamellae was to apply a

contact definition in a circumferential direction between elements of a single lamellae.

This method was not employed at this stage of the modelling as it would have

introduced an increased level of complexity into the model which was intended to be

a preliminary analysis.

3.8.3 Nucleus pulposus

The nucleus pulposus was modelled using hydrostatic fluid elements – F3D3 and

F3D4. These elements were three dimensional, linear elements. The hydrostatic fluid

element geometry was defined using the anulus fibrosus ground substance elements

which formed the walls that enclosed the hydrostatic fluid. In this way, the

hydrostatic fluid elements of the nucleus were defined using the elements on the inner

anulus wall and the inner cartilaginous endplate surfaces (Figure 3-19). It was

necessary to use both the 3 and the 4 node elements because the continuum elements

on the inner walls of the anulus were 20 node elements, thereby having 8 nodes

requiring constraint on each face.

Figure 3-19 Hydrostatic fluid elements modelling the nucleus pulposus



In vivo, the superior and inferior margins of the nucleus pulposus are rounded which

is in contrast to the distinct corners on the hydrostatic fluid elements used to model

the nucleus (Figure 3-19). In fact, the boundary between the nucleus and the anulus

fibrosus in vivo is not clearly delineated and is marked by a transition zone. However,

the representation of this physical structure using straight-sided elements necessitated

the assumption of a distinct boundary between the anulus and the nucleus.

User input for the hydrostatic elements required only a material density for the fluid,

however, this material constant would only be used for analyses of fluid flow. In the

FEM of the intervertebral disc, while Abaqus required the input of fluid density, the

fluid pressure was determined using the initial and deformed nucleus volume. This

material property was superfluous to these calculations.

3.9 Mesh Generation using Abaqus Input Files

Finite element modelling packages such as Abaqus 6.3 provided for automatic mesh

generation which was the automatic generation of nodes and elements on the basis of

prescribed geometry. This automatic mesh generation used both tetrahedral and

hexahedral elements and gave a mesh density which was reasonably well controlled

by the user. However, the automatically generated meshes, especially for circular or

elliptical-type structures, often incorporate both types of elements and the mesh

generated was lacking in order. Also, for highly irregular structures the software had

difficulty in obtaining a mesh with elements of acceptable shape, size and aspect ratio.

It was decided that the mesh generated for the intervertebral disc should possess a

high level of order. Specifically it was desirable to organise the elements in the

anulus fibrosus in a similar manner to the lamellae in the physical disc. That is, the

FEM would be configured into a series of concentric layers of continuum elements

with tension only rebar elements embedded within these layers.

Such a high level of order could not be achieved using automatic meshing techniques.

Therefore, the FEM mesh and in fact all preprocessing was achieved using an Abaqus



data input file that incorporated the data for the disc sagittal and transverse geometry

detailed in Sections 3.3, 3.4 and 3.5

Nodal co-ordinates for all nodes in the anulus fibrosus and cartilaginous endplate

were determined and using these, elements defined in an ordered configuration

resembling the physical disc. For the technique used to determine these nodal co-

ordinates refer section to Section 3.3.

The input file also contained material descriptions for the disc components and

loading and boundary constraints applied to the model. The methods used to generate

the input file provided a versatile and efficient means to generate a suite of analyses

of the intervertebral disc with an ordered mesh configuration.

3.10 Material Properties

The material properties for the collagen fibres, cartilaginous endplates, nucleus

pulposus and the anulus ground substance were initially determined from the

literature. Preliminary validation analyses were carried out using these material

properties.

3.10.1 Collagen fibres

The collagen fibres in the anulus were modelled as linear elastic isotropic materials.

Material properties for the collagen fibres were based on published values from

previous studies. These studies showed reasonably varied values for the fibre

material properties. Natarajan et al. (1994) used orthotropic values with a modulus in

the strongest direction of only 66MPa while Morgan (1960) found an elastic modulus

of 600MPa. Ueno and Liu (1987) and Kumaresan et al. (1999) used an elastic

modulus of 500MPa and Kumaresan et al. (1999) applied a Poisson’s ratio of 0.3.

The final value assigned for the elastic modulus in the model was 500MPa as both

studies using this value provided good correlation with experimentation in their

results.



3.10.2 Cartilaginous endplate

The endplate was a cartilaginous material consisting of collagen fibre bundles

(Bogduk, 1997). The specific arrangement of collagen fibres within the endplate

could cause the structure to exhibit a degree of orthotropy/anisotropy under load.

However, the exact path of fibres within the individual endplates was not known. If

an orthotropic material description were defined for the endplates, this could create an

imprecise description of the material behaviour and introduce inaccurate bias in the

strength of the endplates under certain loading conditions. Given this limitation in

current knowledge of the microstructure of the endplates, an isotropic linear elastic

material formulation was utilised to describe the cartilaginous endplates.

Table 3-3 Details of published material properties for the cartilaginous

endplates

Author Elastic Modulus (MPa)

Poisson’s Ratio

Kumaresan et al. (1999) 600 0.3 Ueno and Liu (1987) 23.8 0.4 Natarajan et al. (1994) 24 Belytschko et al. (1974) 24.3 0.4 Yamada (1970) 24 Wu and Chen (1996) 330 0.25

The majority of the studies found provided an elastic modulus for the cartilaginous

endplates of 23.8 to 24MPa (Table 3-3). However, two studies provided an elastic

modulus an order of magnitude greater. It was considered that the latter studies may

have overestimated the stiffness of the endplates, perhaps due to the close relationship

between these structures and the comparatively stiffer cortical bone of the vertebra.

Therefore, the elastic modulus for the cartilaginous endplates was defined as 24MPa.

The value of the Poisson’s ratio showed considerable variation (Table 3-3). Variation

for this parameter from 0.25 to 0.4 indicated a considerable variation in the properties

of the material and it was postulated that these values may have been obtained without

consideration of the strain rates applied to the material. As such, in keeping with the

assumption of incompressibility of the intervertebral disc and therefore the



assumption of higher strain rates, a Poisson’s ratio near incompressibility was defined

for the cartilaginous endplates – ν = 0.46.

3.10.3 Nucleus pulposus

The input for the hydrostatic nucleus material definition was a density. Given the

comparatively high fluid content of the material this density was assumed to be

slightly higher than the density of water – 1125 kg/m3 – to account for the

proteoglycan chains in the nucleus. While this parameter was required by Abaqus, it

was not used in the analysis since there was no fluid flow and the nucleus pulposus

was considered to be incompressible.

3.10.4 Anulus fibrosus ground substance

An isotropic hyperelastic material description was used to define the ground matrix in

the anulus fibrosus (Fung et al., 1972). The advantage in using this type of material

definition was that hyperelastic materials could readily accommodate

incompressibility while it could be difficult to achieve a converged solution when

using elastic materials with a Poisson’s ratio of 0.5. Also, hyperelastic materials were

well suited to the description of materials that demonstrated nonlinear behaviour and

that exhibited the high levels of strain displayed by biological tissues (Fung in Fung et

al., 1972). On the basis of deductions made by Fung in Fung et al. (1972) a Mooney-

Rivlin strain energy function using two hyperelastic constants was utilised. While it

had been shown that this material description may be less accurate in terms of shear

behaviour, given the prevalence of use of the Mooney-Rivlin equation for modelling

biological materials and the ease of determination of the constants, it was considered a

good preliminary attempt.

Natali and Meroi (1990) used a Mooney-Rivlin hyperelastic equation to represent the

disc material. The constants employed in this representation were 0.7 and 0.2 for C10

and C01, respectively. These researchers did not state the source from which these

parameters were determined. While these parameters provided acceptable results for



the finite element model analysed, it was considered prudent to ascertain the accuracy

of these constants for the preliminary FEM.

A sensitivity analysis was carried out to determine acceptable hyperelastic

parameters. This analysis initially utilised a single 3D-continuum element model

under uniaxial compression. This model was used to ascertain the effect of variations

in the Mooney-Rivlin constants on the nominal stress-strain response of the material.

The hyperelastic constants, C10 and C01 that were used in these analyses were:

C10 = 0.07, C01 = 0.002

C10 = 0.50, C01 = 0.50

C10 = 0.50, C01 = 1.00

C10 = 0.70, C01 = 0.20 (Constants used by Natali and Meroi, 1990)

C10 = 1.00, C01 = 0.50

C10 = 10, C01 = 5

C10 = 80, C01 = 20

The choice of constants was based on the parameters provided by Natali and Meroi

(1990). Sets of parameters were selected to be orders of magnitude higher or lower

than the values of C10 = 0.70 and C01 = 0.20. Also parameter sets were selected such

that that C10 and C01 were equivalent or so their magnitude was reversed (i.e. C10 <

C01). The results for these analyses of the unit element FEM are shown in Figure

3-20. In general, an increase in the magnitude of one or both of the hyperelastic

parameters resulted in a stiffer material.

The results presented in Figure 3-20 gave useful information on the relationship

between the Mooney-Rivlin constants and the stiffness of the material. However,

they did not provide sufficient information to determine the correct constants to

represent the anulus fibrosus ground substance in the FEM. Since there was no

experimental data for the mechanical response of this material to compressive

loading, no comparison could be made with the results in Figure 3-20.



05

1015202530354045

0 0.2 0.4 0.6 0.8 1Compressive Strain

Com

pres

sive

Str

ess

(MPa

)

0.07, 0.002 0.5, 0.5 0.5, 1.0 0.7, 0.21.0, 0.5 10, 5 80, 20

Figure 3-20 Comparison of the nominal stress-strain response of a Mooney-Rivlin hyperelastic material – analysed using a single element FEM. Each curve corresponds to a particular set of hyperelastic constants – the first number in the

set is C10 and the second is C01.

Analyses of the entire disc were carried out using various hyperelastic constants.

These analyses incorporated a 500N compressive torso load and the details of this

loading condition and the boundary conditions in the model are detailed in Section

3.11. The results of axial deformation and disc bulge were compared with those of

existing experimental and finite element studies (Markolf and Morris, 1974; Natali

and Meroi, 1990; Shirazi-Adl et al., 1984). The ratio between the pressure in the

nucleus pulposus in the FEM and the applied pressure was compared with the

experimentally determined value of 1.5 (Nachemson, 1960). The hyperelastic

constants used in these analyses were:

C10 = 0.70, C01 = 0.20 (Constants used by Natali and Meroi, 19901)

C10 = 1.00, C01 = 0.50

C10 = 0.50, C01 = 1.00

C10 = 0.07, C01 = 0.02

The results for anterior, lateral and posterior bulge, axial displacement and nucleus

pulposus pressure were compared with the average experimental results for these

parameters (Table 3-4). A completed solution was not obtained when C10 = 0.07 and

C01 = 0.02. The deformation in the anulus fibrosus was too high and the maximum

load which was applied to the FEM was 343N. The results presented in Table 3-4 for

this combination of constants are for a reduced compressive load of 343N.



The constants obtained from the study by Natali and Meroi (1990) provided the best

correlation with the experimental results. The nucleus pulposus pressure, axial

displacement and posterior bulge demonstrated excellent agreement. Preliminary

analyses incorporated the material constants used by Natali and Meroi (1990) into an

initial finite element model. These constants were C10=0.7, C01=0.2.

While the above sensitivity analysis was an accepted approach for determination of

material constants in computational models, there were disadvantages inherent in this

method. One disadvantage in determining material constants from the FEM was that

the constants obtained would only be entirely accurate for that particular FEM. The

constants which were determined would have incorporated any error in the geometry

of the model or the material descriptions for the other components and as such, could

only be applied in an FEM which displayed similar geometric or material

inconsistencies. Another disadvantage in this method was that the analyses were

carried out using only compressive loading, which was one loading mode of many to

which the disc was subjected in vivo. However, for the purposes of preliminary

analysis this method for determination of the ground matrix material constants was

acceptable in order to obtain qualitative results.



Table 3-4 A comparison of displacement and nucleus pulposus pressure with the average experimental results for a 500N compressive load. Displacement results were compared with the findings of Markolf and Morris (1974), Natali and Meroi (1990) and Shirazi-Adl et al. (1984). The nucleus pulposus pressure was compared with the results of Nachemson (1960).

Anterior Bulge (mm)

Lateral Bulge (mm)

Posterior Bulge (mm)

Axial Displace-

ment (mm)

Ratio of Nucleus Pulposus

pressure to Applied Pressure

Average Experimental

0.81 0.37 0.40 0.48 1.5

C10 = 0.70, C01 = 0.20

0.46 0.22 0.42 0.50 1.56

C10 = 1.00, C01 = 0.50

0.33 0.15 0.24 0.34 1.47

C10 = 0.50, C01 = 1.00

0.12 0.05 0.05 0.10 1.18

C10 = 0.07, C01 = 0.02 *

1.05 0.488 1.02 1.47 1.92

* Compressive load of 343N

A summary of the material parameters used in the FEM is given in Table 3-5.

Table 3-5 Summary of material properties used in the FEM

Intervertebral Disc Structure Material Property

Cartilaginous endplate Elastic modulus = 24.3MPa, υ=0.46

Nucleus Pulposus Density=1125 kg/m3 Anulus Fibrosus Ground Substance C10=1.0, C01=0.5 Collagen Fibres Elastic modulus=500MPa

υ=0.30



3.11 Boundary Conditions and Loading

The boundary and loading conditions defined in the FEM served to simulate the

actions of the muscles, ligaments and bony anatomy surrounding the intervertebral

disc during physiological loading of the joint.

3.11.1 Professor Nachemson's research on spinal loading

The compressive loads applied to the disc FEM were derived from the studies of

Nachemson (1992). He carried out extensive research into the in vivo loading in the

intervertebral disc. He used a pressure transducer mounted in a needle and inserted

the needles directly into the intervertebral disc. When the needle was rotated in the

three directions of principal stress, it was shown that the nucleus of the healthy disc

behaved hydrostatically (Nachemson, 1960).

In order to quantify the disc loading, the needle was orientated within the disc so the

pressure measured was in a direction perpendicular to the superior endplate. A

variety of common activities were performed by the subject, such as standing, sitting,

flexing with and without weights and even sneezing and coughing. Nachemson

(1960) found that the pressure in the nucleus was 50% higher than the pressure

applied to the superior surface of the disc.

Wilke et al. (1999) attempted to repeat the tests carried out by Nachemson (1960,

1963, 1964, 1981) and found similar results for most activities. However, Wilke et al.

(1999) reported contradictory results for the loading on the disc during sitting. There

was no apparent explanation for the disagreement between the results from these

studies.

Studies carried out by Nachemson to determine the in vivo disc loads were published

in 1960, 1963, 1964, 1966, 1981 and 1992. The results from the most recent of these

studies were stated to have been obtained using an improved testing method. The

results from the studies carried out by Nachemson were still the definitive work in the

area of in vivo loading on the intervertebral discs. These studies were used to define

the compressive loads on the disc during relaxed standing and to provide details of the



pressure in the nucleus pulposus during flexion/extension, lateral bending and axial

rotation.

3.11.2 Nucleus pulposus pressurisation

Nachemson (1963) reported that the removal of the posterior elements and ligaments

in cadaveric human spines resulted in a nucleus pulposus pressure of zero in the

unloaded segment. However, when the posterior elements and ligaments remained

intact the unloaded disc demonstrated a nucleus pressure of 70kPa. This pressure was

simulated in the FEM.

The initial loading step on the FEM involved the introduction of a 70kPa pressure into

the nucleus. This was achieved by loading the 'cavity node'. This was the node which

Abaqus used to measure and control the pressure within the hydrostatic fluid. It was a

node used to directly apply pressure boundary conditions and to measure pressure

variations during analyses.

In order to obtain a disc stress state and geometry which was similar to the in vivo

conditions, the nucleus pressure was held at 70kPa for only the first step. This fluid

essentially established a volume in the nucleus. In subsequent loading steps there was

no specific nucleus pressure boundary condition applied. The pressure was permitted

to increase and decrease in accordance with the stress state within the anulus and

endplates.

3.11.3 Modelling adjacent vertebrae

The adjacent vertebrae were represented as rigid bodies.

All nodes on the inferior surface of the disc were constrained in all 3 degrees of

freedom (dof). All the nodes on the superior surface of the disc were constrained to a

reference node using a rigid beam constraint. This effectively created a rigid surface

on the superior disc. The rigid beam constraint was achieved using the Abaqus

modelling function, ‘multi-point constraints’ or MPCs. With the use of the rigid



beam MPC to link all the nodes on the superior surface to one reference node, this

allowed for the efficient application of loading to the disc. Both rotational and

translational dof for the superior disc surface could be constrained at the reference

node.

Modelling the adjacent vertebrae as rigid bodies was a simplification of the

mechanical properties of these materials. The cortical and cancellous bone are

deformable materials, with elastic moduli of approximately 12,000MPa and 100MPa,

respectively (Shirazi-Adl et al., 1984). This elastic stiffness was high in comparison

to the equivalent elastic modulus of 4.7MPa (Kumaresan et al., 1999) for the anulus

fibrosus ground substance. Therefore, it was assumed that the vertebral bone could be

modelled as a rigid structure given the significantly compliant behaviour of the

anulus.

In order to support this assumption a pilot analysis was carried out on the FEM in

which a thin layer of cortical bone was modelled adjacent to the superior disc surface.

The boundary conditions used in the model are outlined in Section 3.11.2 and Section

3.11.3. These included a 70kPa nucleus pressure and a 500N compressive torso load

to the superior surface of the cortical bone. The inferior surface of the model was

held in all degrees of freedom. Results for the nucleus pulposus pressure and the

maximum von Mises stress in this FEM were compared to the results of the FEM with

the rigid superior surface.

A comparison of the nucleus pressure and von Mises stresses observed in the two

models (Table 3-6) showed that simulation of a layer of vertebral bone on the superior

surface of the FEM resulted in an increase in the magnitudes of these variables. The

maximum variation between the results of the FEM with the cortical bone present and

the FEM with the rigid vertebral bone was 4%. This percentage was considered to be

an acceptable level of error and justified the representation of the adjacent vertebrae

as rigid bodies. It was believed that this assumption would not compromise the

accuracy of the results.



Table 3-6 Comparison of nucleus pressure and von Mises stress for a rigid

superior endplate and superior endplate modelled as cortical bone after the 500N

compression load

Analysis Variable

Measured

Measured Value Percentage

variation from the

rigid endplate

FEM

Rigid superior

vertebra

Nucleus pressure 0.656MPa

Von Mises Stress

in the anulus

fibrosus

Maximum in inner

posterior anulus

1.859-2.390MPa

Linear elastic

superior vertebra

Nucleus pressure 0.668MPa 2% higher

Von Mises Stress

in the anulus

fibrosus

Maximum in inner

posterior anulus

2.283-2.490MPa

4% higher

3.11.4 Musculature and posterior elements

The muscles attached to the posterior elements of the spine are the primary

mechanism by which the in vivo spine is actively loaded in flexion, extension and

lateral bending. The muscles and ligaments also provide a stabilising role for the

spine. Because the FEM did not include either the posterior elements or the

musculature and ligaments, it was important that the loading and motion constraint

offered by these structures be simulated by alternative means.

Simulation of the musculature and posterior elements was achieved by applying the

flexion/extension, lateral bending and axial rotation loading through the ICRs for

these respective motions.

The ICR for a bending motion was a constructed point for the entire spinal level and

was therefore, intimately related to the spinal structures external to the intervertebral



disc. All bending loads that were applied to the FEM, other than torso compressive

loads, were applied as boundary conditions (i.e. rotation angles) instead of pressures

or concentrated loads. Therefore, the loading on the FEM was such that the final

configuration of the model was comparable to in vivo observations for

flexion/extension, lateral bending and axial rotation angles. Because the FEM loads

were applied as rotations that produced a desired deformed shape and since these

rotations were applied about an ICR which was comparable to the axis of rotation for

the joint, the loading on the FEM was capable of simulating the loading observed on

the intervertebral disc in vivo.

3.11.5 Uniaxial compression loading for validating the preliminary model

In order to represent the loading on the disc due to relaxed standing, a 500N

compressive load was applied perpendicular to the superior surface of the disc. This

value estimated the torso load above the L3/4 intervertebral joint of a 70kg individual

and was based on measurements obtained by Nachemson (1992). In the absence of

data on the load above the L4/5 intervertebral disc, a 500N compression load was

assumed in the FEM. It should be noted that the results obtained by Nachemson and

colleagues during their early studies of the nucleus pressures, overestimated the

values. An improved method was employed in his later studies and in the study

published in 1981 it was stated that the results were more accurate. As such, all

values obtained from the work of Nachemson were based on the results published in

the 1992 study.

The 500N compressive load was applied to a reference node, which was located

0.8mm above the geometric centre of the superior endplate and to which all nodes on

the superior surface of the disc were rigidly constrained.

The choice of compressive loading was based on the ease with which values for axial

displacement, radial bulge and nucleus pulposus pressure could be obtained for this

loading type. Data on displacements were not as readily available for disc loading

which involved bending. The choice of this load was based on the previous in vitro

experimental studies carried out on the intervertebral disc and the availability of



displacement data for validation purposes (Markolf and Morris, 1974, Brown et al.,

1957, Shirazi-Adl et al., 1984, Virgin, 1951).

In summary, the loading steps applied to the model for the purpose of validation were:

Step 1 Introduce a 70kPa pressure into the nucleus pulposus to

represent the intrinsic pressure in the unloaded disc as a result

of the presence of the spinal ligaments

Step 2 Apply a 500N compressive load to simulate torso loading

3.11.6 Iteration to determine the initial sagittal geometry of the intervertebral

disc FEM

Data to define the sagittal geometry of the FEM was obtained from the study by

Tibrewal and Pearcy (1985). The anterior and posterior heights of the L4/5

intervertebral disc during relaxed standing were stated. These dimensions were

utilized to model the sagittal dimensions of the FEM. However, these were the

dimensions for a loaded disc, while the FEM of the intervertebral disc was initially an

unloaded structure. Therefore, it was necessary for an initial sagittal configuration for

the FEM to be determined such that the deformed shape under the 500N compressive

torso load was comparable to the dimensions stated by Tibrewal and Pearcy (1985).

In order to determine the correct sagittal dimensions of the disc under torso loading,

an analysis was carried out using the sagittal geometry for the torso loaded disc. The

translation of the superior surface of the FEM was determined. This translation was

used as a correction to the initial geometry of the FEM (Figure 3-21). The FEM was

then reloaded under torso compression and the anterior and posterior height

determined. If these heights were not comparable to the correct in vivo dimensions

for the disc, the initial geometry of the disc was again corrected and the FEM

reanalysed under torso loading. This iterative process was carried out until the

anterior and posterior heights of the FEM under torso loading were 14mm and

5.5mm, respectively.



Figure 3-21 Iterative procedure to attain a final sagittal geometry comparable to in vivo observations (NB. the deformations shown are exaggerated)

3.12 Optimising the Mesh Density of the FEM

The mesh density for a finite element model must be selected to ensure a

computationally efficient analysis is performed with analysis results of a suitable

accuracy. A coarser finite element mesh results in less precision of the results due to

the reduced number of nodes at which a solution may be found and the necessity for

interpolation of results across large distances between nodes. However, extremely

fine finite element meshes significantly increase the degrees of freedom of the

solution and therefore result in extremely long solution times. In certain models, this

high mesh density may be necessary to achieve a suitable level of accuracy in regions

of a structure which experience high stresses/strains or demonstrate notable

discontinuities. However, the use of an unnecessarily fine mesh density will result in

very high solution times with little benefit in terms of improved accuracy of results.

To determine a suitable mesh density for the intervertebral disc FEM, various meshes

were analysed using the loading conditions outlined in Section 3.11.5 – a 70kPa

pressure was introduced in the nucleus during the first loading step and the second

loading step applied a 500N compressive load to the superior surface of the FEM.

The number of circumferential element layers and the number of axial element layers

were varied in these models and the results for nucleus pressure and axial

displacement of the anterior edge of the FEM were compared. Six different mesh

densities were analysed.

Corrected initial geometry of FEM to be reloaded with 500N

Original sagittal geometry of the FEM Deformed geometry of FEM under 500N 14m

5.5m

500 N

The original sagittal geometry is the same as the desired final sagittal geometry under the 500N torso load



• 4 circumferential element layers, 2 axial element layers (4 x 2)






The 4 x 2 mesh and the 10 x 7 mesh are shown in Figure 3-22 A and B.

A

B

Figure 3-22 Varied mesh density used to determine the optimum density for the analysis of the FEM A. 4 circumferential element layers, 2 axial layers; B. 10

circumferential element layers, 7 axial layers

Generating the various mesh densities was automated using the Fortran code detailed

in Section 3.7.

The results of these analyses are shown in Figure 3-23.



A

0

0.1

0.2

0.3

0.4

0.5

0.6

4 x 2 3 x 3 6 x 4 7 x 5 8 x 6 10 x 7Mesh Density

Dis

plac

emen

t (m

m)

Displacement After 70kPa (cephalic)Displacement After 500N (caudal)

B

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


Nuc

leus

Pre

ssur

e (M

Pa)

500N Nucleus pressure

C

0

0.5

1

1.5

2

2.5

3

3.5


Von

Mis

es S

tres

s (M

Pa)

Von Mises Stress



D

0

50

100

150

200

250


Tim

e (m

inut

es)

Solution Time

Figure 3-23 Comparison of analysis results from finite element models with differing mesh densities (mesh density expressed as circumferential element

layers x axial element layers). A. Axial displacement of superior surface of FEM; B. Nucleus pulposus pressure; C. Anulus ground substance maximum von Mises

Stress; D. Analysis solution time

These results showed very low variation between the nucleus pressure and

displacement; however, there was a general trend for the variables to decrease with

increasing mesh density. The maximum variation in the axial displacement resulting

from the 500N compressive load (Figure 3-23 A) was 0.0209mm between the 4 x 2

and the 8 x 6 mesh density. The maximum variation in the nucleus pressure (Figure

3-23 B) was 0.0198MPa between the 4 x 2 and the 6 x 4 mesh density. Increases in

the mesh density resulted in considerable increases in the von Mises stress in the

anulus ground substance and the solution time for the analysis (Figure 3-23 C). While

the von Mises stress in the anulus ground substance increased by 15% from the 7 x 5

mesh to the 8 x 6 mesh, there was only a 2.5% variation between the maximum stress

in the 8 x 6 mesh and the 10 x 7 mesh. Furthermore, the location of the peak stress in

all the meshes was located on the inner surface of the posterior anulus. Considering

the limited variation in the von Mises stress, axial displacement and nucleus pressure

when the mesh density was increased from an 8 x 6 mesh to a 10 x 7 mesh, the

increase in solution time from 54 minutes to 210 minutes was excessive. The 8 x 6

mesh was considered to be an optimal model for the intervertebral disc.

Generally, a finer mesh density will provide results that more closely simulate the

physical condition modelled. Ideally, the mesh density used in the model would have



included 20 circumferential element layers to represent the average number of

lamellae in the anulus (Marchand and Ahmed, 1990) but such a fine mesh density

would have resulted in excessive analysis times. So, the choice of final mesh density

was a compromise between the simulation of the physical structure and computation

time. The results for the 8 x 6 finite element mesh were considered to provide an

accurate solution with an efficient solution time. This mesh was used for subsequent

analyses of the intervertebral disc FEM.

3.13 Analysis of the FEM

The FEM was analysed under compressive loading to ascertain the effect of variations

in the transverse profile of the anulus and nucleus boundaries. In separate analyses

the results of compressive loading and forward flexion were examined to validate the

FEM with in vitro and in vivo experimental studies. A 70kPa pressure was introduced

into the nucleus pulposus of the FEM in the first step of each of these analyses.

Knowledge of the mechanism of failure of a material should be used to determine

whether an assessment of maximum principal stress or the maximum von Mises stress

is most appropriate. If the failure of the material is a ductile failure then von Mises

stresses should be used as the material failure criteria. If the material failure is brittle

in nature then the maximum principal stresses should be used to assess the material

stress state. In the case of the anulus fibrosus, it was not clear which type of failure

would occur. There was no evidence of prior experimental testing of this material to

determine the failure mechanisms. Von Mises stresses were used to assess the stress

state in the intervertebral disc components. The location of the maximum von Mises

stress is similar to the location of the maximum principal stresses, therefore, this was

not considered to compromise the results of the analyses.



3.13.1 The effect of variation in the transverse profile of the anulus and nucleus

boundaries

In order to determine the sensitivity of the FEM results to the transverse geometry of

the disc, several FEM meshes were developed with differing curvatures on the

posterior anulus and different locations for the nucleus (Figure 3-24 A-F).

A

B



C

D

E



F

Figure 3-24 Varied mesh density. A. Specimen 50; B. Symmetric mesh; C. Flattened posterior curvature; D. Increased posterior curvature; E, F. Displaced

nucleus (endplates not shown)

These analyses were carried out to provide information on the inaccuracy introduced

into the results of FEM of the intervertebral discs that use simplified transverse

geometry for this structure. Analysis of the symmetric model demonstrated the

variation in results due to the assumption of symmetry of the intervertebral disc.

Analysis of finite element meshes with an increased or decreased posterior curvature

provided information on the potential inaccuracy in the results of analysis which do

not accurately represent the curvature of the posterior anulus. The analysis of the

finite element meshes in Figure 3-24 D and E was carried out to observe the potential

inaccuracy in the FEM results due to imprecision in locating the nucleus pulposus

boundary from photographs of human discs.

All 6 mesh geometries used were based on the geometry of Specimen 50. These

models were analysed using a 70kPa nucleus pressure and a 500N compressive

loading condition.

• The unaltered finite element mesh for Specimen 50 was analysed and all

additional geometries were compared to these results (Figure 3-24 A).

• A symmetric transverse geometry was represented (Figure 3-24 B). This mesh

was based on the right lateral geometry of Specimen 50.

• The posterior concavity on the anulus fibrosus outer profile was decreased by

80%. This was achieved by decreasing the distance between the posterior-



most node on the anulus profile and the anterior-most node in the posterior

concavity (Figure 3-24 C).

• The posterior concavity on the anulus fibrosus outer profile was increased by

80% (Figure 3-24 D). This increased concavity was achieved in a similar

manner to the decreased concavity in Figure 3-24 C.

• The location of the nucleus with respect to the geometric centre of the anulus

in the transverse plane was altered. In Section 3.3 the maximum variation

between the experimentally measured nucleus displacements by Vernon-

Roberts et al. (1997) and the nucleus displacements determined from the

formulae used to map the nucleus profile was 1.9mm. This variation was used

as the displacement of the nucleus from the anulus geometric centre. The

nucleus was displaced posteriorly (Figure 3-24 E).

• The nucleus pulposus was displaced by 1.9mm in the anterolateral direction

(Figure 3-24 F).

The peak stresses and stress contours for the varied mesh geometries were compared

to the stress contours for Specimen 50 (Figure 3-25 A). The maximum stress in this

FEM was 3.18MPa. The results from these analyses showed that the peak stresses

observed in the posterior anulus were increased by up to 42% to 4.52MPa, when the

posterior curvature was increased (Figure 3-25 D). Approximately 1% of this

increase was a result of the decreased cross-sectional area of the disc. Flattening the

posterior anulus curvature resulted in a 43% decrease in the peak stress in the

posterior anulus (1.81MPa) of which 1% was related to the increased cross-sectional

disc area (Figure 3-25 C). The maximum stress in the meshes with a reduced and an

increased posterior curvature was in the inner posterior anulus. This location was the

same as the unchanged mesh for Specimen 50 (Figure 3-25 B). The symmetrical

mesh showed very similar results to the results of Specimen 50 (Figure 3-25 A).



A

B

C



D

E

F

Figure 3-25 Von Mises stress contours for varied mesh geometry (endplates not shown) A. Specimen 50: B. Symmetric mesh: C. Flattened posterior curvature;

D. Increased posterior curvature; E, F. Displaced nucleus



When the error in the location of the nucleus resulted in a reduction in the posterior

radial width of the anulus the maximum stress was increased by 50% to 4.75MPa

(Figure 3-25 E). Displacement of the nucleus in an anterolateral direction resulted in

a decrease in the maximum stress of 21% to 2.52MPa and the location of the

maximum stress was in the inner anterolateral anulus (Figure 3-25 F).

These results showed that variation in the outer profile of the anulus fibrosus or in the

location of the nucleus pulposus caused a variation in the peak von Mises stresses

observed in the FEM. The maximum variation in the peak stress as a result of

variation in the posterior concavity of the anulus was 43%. This error was determined

from analyses of meshes with an inaccurately defined profile of only one region of the

outer anulus and with an associated error in the cross-sectional area of 1%. It was

reasonable to expect that inaccuracy in defining the outer profile of the anulus around

the entire perimeter could cause up to 5% variation in the cross-sectional area of the

disc FEM and therefore, significantly higher errors would be observed.

Variations in the maximum von Mises stresses as a result of the varied location of the

nucleus were dependant on the direction in which the nucleus was displaced.

Comparison of the stress contours for the different locations of the nucleus indicated

that the location of the nucleus did affect the magnitude and location of the peak

stress in the FEM. The nucleus in the varied meshes in Figure 3-24 E and F was

displaced by 1.9mm which was the error between the results of Vernon-Roberts et al.

(1997) and the location determined using the mathematical algorithm presented in

Section 3.3.2.4. It was not clear from the results of Vernon-Roberts et al. (1997)

which direction the displacement of the nucleus pulposus centre with respect to the

overall disc centre was measured. Therefore, the direction of the 1.9mm inaccuracy

in the nucleus location could not be determined. Displacement of the nucleus

posteriorly resulting in a reduction in the radial width of the posterior anulus was

likely a worst case scenario since it would increase the existing stress concentration in

the posterior anulus. This resulted in a 50% increase in the maximum von Mises

stress (Figure 3-25 E).

These results indicated that it was necessary to incorporate an accurate transverse

profile of the anulus fibrosus in order to obtain accurate results. Simplifications of



this profile using overall anterior-posterior and lateral dimensions of the intervertebral

disc could cause significant inaccuracy in the cross-sectional area of the FEM mesh

and result in overstated or understated results. The assumption of a centrally located

nucleus could also result in inaccurate peak stresses since the nucleus in the

intervertebral disc was commonly skewed from the disc centre by between 1 and

4mm.

The inaccuracy in the location of the nucleus in Specimen 50 compared to the

experimental results of Vernon-Roberts et al. (1997) resulted in an error in the peak

von Mises stress between 21 and 50%. These errors were comparable to or lower

than the potential errors introduced due to inaccurate or simplified geometric

definitions for the anulus boundary. It was thought that while the representation of

the transverse disc geometry in the current study may have produced some error due

to the differing nucleus location between the results of Vernon-Roberts et al. (1997)

and the results of the mathematical algorithm, this error was similar to or less than the

error that would be introduced with a simplified anulus profile. Since this method

allowed for the geometric representation of actual discs it was preferable to other

techniques that used idealised anulus profiles or averaged dimensions.

3.13.2 Response of the FEM (Specimen 50) to the 70kPa nucleus pulposus

pressure

The model of Specimen 50 was used as an example of a real L4/5 intervertebral disc.

In response to the 70kPa nucleus pressure, the superior surface of the FEM displaced

axially in a cephalic direction and the peripheral surface of the anulus fibrosus bulged

both outward and inward. The axial displacement of the superior surface ranged

between 1.80 x 10-2 and 2.30 x 10-2mm. The outward radial bulge of the anterior

anulus was very low and ranged between 9.54 x 10-4 and 1.01 x 10-2mm.



Figure 3-26 Contour plot of anterior-posterior displacement. Posterolateral

anulus bulged radially inward and mid-posterior anulus bulged radially outward.

Figure 3-26 shows the contour pattern of the radial bulge on the posterior surface of

the FEM. This pattern showed an interesting trend – the mid-posterior anulus bulged

outward while the posterolateral anulus bulge radially inward. This inward

deformation of the posterolateral anulus was thought to be a result of the cephalic

displacement of the superior surface of the FEM and the resulting decrease in the

radial width of the incompressible anulus fibrosus ground substance in the

posterolateral anulus. An outward bulge of the mid-posterior anulus resulted from the

inflation of the nucleus pulposus and the bulge of the comparatively thinner posterior

anulus.

3.13.3 Analysis of the FEM under compression

The FEM was analysed under a 500N compressive load and the deformed shape

(Figure 3-27) and peak stresses observed.

Inward bulge of posterolateral anulus – 1.36 x 10-2 – 2.08 x 10-2mm

Outward bulge of mid-posterior anulus – 5.90 x 10-2 – 6.63 x 10-2mm



Figure 3-27 Deformed shape of the FEM. Shaded grey: Deformed shape, Wireframe outline: undeformed shape

The peak stress range observed under 500N compression was 2.91-3.18MPa on the

inner posterior anulus surface (Figure 3-28). Overall the FEM was not highly

stressed. This was reasonable given the loading was torso compression.

Figure 3-28 Contour plot of von Mises stress in the FEM loaded with 500N

compressive torso load

The analysis output for the rebar elements which modelled the collagen fibres was for

the force in the rebar. Abaqus did not provide data for either the true stress or the

instantaneous cross-sectional area of the rebars. Therefore, it was not possible to

determine the true stress in the collagen fibres. Under the 500N compressive load the

Peak stress range 2.91-3.18 MPa

Posterior anulus



maximum nominal principal stress in the rebar elements was 0.77MPa and the

maximum strain was 0.15%. The failure strain of collagen fibres ranged from 10-

15% (Viidik, 1973). Therefore, as expected the strained state of the rebar elements

under torso compressive load would not have initiated failure in the collagen fibres.

The FEM results for axial displacement, disc bulge and nucleus pulposus pressure

were compared with the average of the results from previous studies (Markolf and

Morris, 1974, Brown et al., 1957, Shirazi-Adl et al., 1984, Virgin, 1951, Nachemson,

1992) (Figure 3-29 and Figure 3-30). An average value for each of the parameters

was determined across the three studies and the standard deviation of the data

determined. These results were compared for the 500N compressive load.

00.2

0.40.6

0.81

1.21.4

1.6

AB LB PB AD

Dis

plac

emen

t (m

m)

Experimental FEA

Figure 3-29 Comparison of FEA and experimental results for displacements, 500N compression. Error bars are 1 standard deviation from the experimental

mean. (AB=anterior bulge, LB=lateral bulge, PB=posterior bulge, AD=axial displacement)



11.11.21.31.41.51.61.71.81.9

2

500N Load

Rat

io o

f App

lied

Pres

sure

to N

ucle

ar P

ress

ure

Nachemson (1960) FEA

Figure 3-30 Comparison of the ratio of applied pressure to nucleus pressure for the 500N compression

The anterior and posterior bulge and the axial displacement in the finite element

model showed good agreement with the experimental average and were within the

range of the first standard deviation from the experimental results (Figure 3-29). The

lateral bulge was outside the first standard deviation range. The nucleus pressure

reached a peak value of 0.66MPa (Figure 3-30). Nachemson (1960) stated that the

nucleus pressure was 1.5 times the applied pressure on the superior disc surface. The

disc analysed had a surface area of 1200mm2. Therefore, the 500N load equated to a

0.417MPa pressure. Using these pressures, the ratio between the applied pressure and

the nucleus pressure in the model was 1.57.

The results of loading with the torso load showed excellent correlation for the ratio of

the nucleus pressure to the applied pressure. The nucleus pressure was considered to

be an important validation parameter because the pressurisation of the nucleus

provided the majority of loading on the anulus and in particular, the collagen fibres.

To further test the hyperelastic material formulation employed to represent the anulus

ground substance, further analyses were carried out on the FEM under flexion loads.

3.13.4 Full forward flexion

The FEM was analysed under forward flexion of 13o simulating the full range of

motion of the L4/5 intervertebral disc (Pearcy, 1985).



3.13.4.1 Validation criterion for full flexion

Nachemson (1992) provided data for the in vivo nucleus pulposus pressure during a

forward flexion of 40o. It was stated that this rotation was associated with a

compressive load perpendicular to the superior surface of the L3/4 intervertebral disc

of 1000N. This was a compressive stress of 0.833MPa and would have resulted in a

nucleus pulposus pressure of 1.25MPa (Nachemson, 1960). However, it was not clear

how this angle of rotation was measured – were the subjects prevented from hip

rotation therefore, measuring the pure spinal rotation or were the subjects free to

rotate their hip joints and as such, did this angle represent the trunk rotation? These

details were not clear therefore, it was assumed the 40o angle measured a trunk

rotation, including motion of the hips. It was considered that this assumption would

provide representative values for comparison of the results of the FEM analysis.

A forward rotation of 40o is approximately half the full forward flexion of the trunk.

Full forward flexion of the L4/5 intervertebral disc is 13o. It was assumed that the

trunk rotation of 40o was associated with a rotation of the L4/5 intervertebral disc of

6.5o and a nucleus pulposus pressure of 1.25MPa.

The second validation criteria used for the flexion loading was the rotational stiffness

of the FEM. Schultz et al. (1979) and Schmidt et al. (1998) determined the flexion

moment and flexional stiffness of the lumbar spine joints by experimenting on

cadaveric joints. Shirazi-Adl (1986) carried out a finite element analysis of a spinal

joint and stated the flexion moment observed in the FEM for rotations from zero

degrees up to full flexion. The results from these studies were compared to the

rotational stiffness from the FEM.

3.13.4.2 Results of analysis of the FEM under full flexion

When a 13 degree rotation was applied to the FEM the nucleus pulposus pressure at

the end of the loading was 2.14MPa. When the rotation was 6.5o, the pressure in the

nucleus was 1.04MPa.



The pressure in the nucleus pulposus reached 1.25MPa at a rotation angle of 10.4o

which was high in comparison to the expected value of a 6.5o rotation. A possible

explanation for this disagreement was that the 40o rotation angle measured by

Nachemson (1992) did not correspond to rotation of the trunk, rather it corresponded

to rotation of the spine with no corresponding hip motion. A flexion rotation of 10.4o

in the L4/5 intervertebral disc would be compatible to approximately 75% of full

forward motion in this joint (Pearcy, 1985). Pure flexion of the lumbar spine with no

contribution from the hip joints, results in a forward rotation of 51o (Pearcy, 1985).

Therefore, the 40o rotation was approximately 80% of full forward motion in the

lumbar spine. This suggested that the results for the nucleus pulposus pressure in the

flexed FEM were acceptable.

Data for the torsional stiffness in the FEM were compared with the results of both

experimental (Schultz et al., 1979; Miller et al., 1986; Schmidt et al., 1998) and

mathematical studies (Shirazi-Adl et al., 1986; McGill, 1988). These data are listed

in Table 3-7.



Table 3-7 Comparison of FE and experimental results with the results from the

FEM for rotational stiffness under flexion

FE and Experimental result

Variable FEM

Results

Shirazi-Adl et

al. (1986)

Schultz et al.

(1979)

McGill (1988)

Miller et al.

(1988)

Schmidt et al.

(1998)

5.89o rotation

8.6Nm moment

generated

15.5Nm moment

4.7Nm moment

5.93o rotation

8.7Nm moment

generated

15.5Nm moment

10.6Nm moment

6.5o rotation

9.7Nm moment

20Nm moment

12o rotation

22.14Nm 60Nm

moment

98Nm moment

11.7 – 13.8o,

average of 12.75o

23.96Nm

70Nm

moment

Torsional stiffness

for healthy disc *

2.1Nm/ degree rotation

measured up to 6.6o

flexion

1.8Nm/ degree rotation averaged

up to 6.6o

flexion *this stiffness was calculated as the variation in the rotation moment divided by the

corresponding increase in the rotation angle

Comparison of the flexion moments in the FEM for rotations between 5.89 and 12o

demonstrated significantly lower rotational moments to those observed by Shirazi-Adl

et al. (1986) (Table 3-7). The results of Miller et al. (1988) and McGill (1988) also

demonstrated significantly higher moments than the preliminary FEM for flexion

rotations in the order of 12o. Moments generated in the FEM were between 35 and

60% of those found in the FEM developed by these researchers. This suggested that



the response of the FEM was too compliant in comparison to the in vivo response of

the intervertebral disc.

The results of Schultz et al. (1979) were unusual because a doubling of the flexion

moment applied to the cadaveric joints resulted in no real increase in the sagittal

rotation (Table 3-7). This result was not repeated in the FEM, nor was it observed in

the study carried out by Shirazi-Adl et al. (1986) (Table 3-7) and was considered to be

an artefact of the loading methods used by Schultz et al. (1979) to apply rotation.

These methods did not consider the instantaneous axis of rotation during flexion and

thereby may have produced anomalous results.

A similar rotational stiffness per degree of flexion was observed in the FEM and the

experimental results of Schmidt et al. (1998). The rotational stiffness stated by these

researchers was 1.8Nm/degree of flexion. Therefore, a flexion rotation of 12o would

result in moment of 21.6Nm. This value was significantly lower than the results of

Shirazi-Adl et al. (1986), McGill (1988) and Miller et al. (1988). The experimental

technique employed by Schmidt et al. (1998) did not take into account the

physiological loading condition of the intervertebral disc and the location of the ICR

during flexion. It was postulated that this loading method resulted in inaccuracy in

the results and was the cause for the reduced stiffness.

Schmidt et al. (1998) noted that the flexional stiffness of the intervertebral disc

specimens increased with increasing flexion rotation up to 6.6o flexion. At rotations

in the FEM up to 5.85o, the flexional moment per increment of rotation ranged from

1.7-1.9Nm/degree. With increasing angles of rotation up to 12.72o this stiffness

increased to 2.6Nm/degree. However, for rotation angles between 12.72 and 13o, the

rotational stiffness reduced to values as low as 1.1Nm/degree. A similar trend was

not observed in the experimental study carried out by Schmidt et al. (1998). It was

postulated that this significant reduction in flexional stiffness may have been related

to the extreme deformation of the inferior region of the anterior anulus fibrosus. At

the completion of the 13o rotation, this region of the anulus had deformed to such an

extent that it extended over and below the inferior surface of the disc (Figure 3-31 A).

However, when the sagittal deformation of the anterior anulus was observed at a

rotation angle of 12.72o (Figure 3-31 C), this inferior, anterior deformation of the



A

B

C

Figure 3-31 Deformed shape of FEM with flexion applied. A. Sagittal view during full flexion of 13o; B. Posterior anulus – showing outward bulge at mid-posterior during full flexion; C. Sagittal view during forward flexion of 12.72o –

reduced deformation of inferior, anterior anulus (undeformed mesh in wireframe)

Inward posterolateral bulge

Outward mid posterior bulge

Extreme deformation of inferior, anterior anulus

Reduced deformation of inferior, anterior anulus at forward rotation 12.72o



anulus was not as extreme. It was considered that forward rotations of the FEM

exceeding approximately 12.7o, caused the resistance to rotation offered by the

inferior FEM to be reduced due to the protrusion of the anulus material anterior to this

surface. This indicated that the model was not a true representation above 12.7o

flexion.

Logarithmic strains were used as measurements of strain in the FEM. Logarithmic

strain is the same as true strain and is calculated using Eqn 3-11. εNominal is the

standard engineering strain calculated as the change in dimension divided by the

original dimension.

εTrue = ln(1 + εNominal)

Eqn 3-11 Equation for true strain in terms of nominal strain

The highest maximum principal logarithmic or true strain in the flexed FEM was

between 1.32-1.58 and was on the inferior margin of the outer posterolateral anulus

(Figure 3-32 A, B). This was comparable to an engineering strain of 2.74-3.85. This

location for the highest principal strain was reasonable given the highly deformed

posterolateral anulus observed in Figure 3-31. A region of high strain was observed at

the inferior margin of the anterior anulus and ranged from 0.99-1.21. This region of

high strain was a result of the extremely large deformations observed in this region of

the anulus (Figure 3-31 A). The maximum principal logarithmic strain in the mid-

posterior anulus fibrosus ranged from 0.44-0.55 and in general the anterior and lateral

anulus was not highly strained (Figure 3-32 A, B).

The highest minimum principal logarithmic strain in the disc range from -1.50 to -

2.00 on the anterior margin of the inferior surface of the anulus (Figure 3-32 C) which

was comparable to an engineering strain of -3.38 to -6.38. At this location, the

anterior disc was very highly compressed under flexion and it was reasonable to

observe high strains.

The maximum von Mises stress was observed in similar locations to the highest

maximum and minimum principal strains. On the inferior margin of the posterolateral



anulus the stress ranged from 10.87 to 13.04MPa and on the inferior margin of the

anterior anulus the stress ranged from 13.04 to 15.65MPa (Figure 3-32 D, E). A

diamond pattern was observed in the von Mises stress contour on the inner posterior

wall of the anulus fibrosus. The stress in this region ranged from 3.30-5.46MPa,

however, these stress magnitudes were likely an artefact of the inaccurate deformation

of the inner anulus due to the method of defining the hydrostatic elements and were

later improved (see discussion of Chapter 6).

A

B

Region of high strain in anterior anulus – 0.99-1.21

Maximum log strain – 1.32-1.58



C

D

E

Figure 3-32 Contour plots of the fully flexed FEM showing A, B. Maximum principal strain; C. Minimum principal strain; D, E. Von mises stress



The maximum nominal stress in the rebar elements was 34.74MPa and the maximum

nominal strain was 6.95%. These occurred in the left lateral anulus, near the inferior

surface. Specifically, this maximum strain was in the circumferential element layer

second in from the peripheral surface. This location of maximum rebar stress was due

to the orientation of the rebar elements in this layer – forward flexion introduced a

state of tension into the anteriorly inclined rebar elements. Given that the failure

strain of collagen fibres ranged from 10-15% (Viidik, 1973) this peak strain was not

sufficient to initiate damage in the rebar elements which represented the collagen

fibres.

The deformed sagittal geometry of the FEM showed an inward bulge of the

posterolateral regions of the anulus fibrosus while the anterior, lateral and posterior

anulus bulged outward (Figure 3-31). This was an interesting finding as it had not

previously been reported.

The inward bulge of the posterolateral anulus whilst unexpected was not

unreasonable, considering the highly strained state of the anulus under full forward

flexion. Given the increase in the posterior height of the disc under flexion and the

incompressibility of the anulus ground substance, a corresponding decrease in the

radial dimension of the anulus could be expected. It was postulated that an MRI scan

of a human spine in full flexion may have revealed a similar posterior curvature of the

L4/5 intervertebral disc to that which was observed in the FEM. However, available

MRI facilities did not permit for subjects to be in a fully flexed position while being

scanned. Additionally, the resolution of the scans was not sufficient to accurately

define the posterior margin of the intervertebral disc in vivo. Therefore, this unusual

posterior deformation of the FEM could not be validated. It was also questioned

whether this anomalous posterior deformation was an indication of the inaccuracy of

the material parameters employed to represent the anulus fibrosus in the FEM.

3.14 Assessment of the Accuracy of the FEM

Results for the rotational stiffness of the FEM during flexion indicated that the overall

disc behaviour was significantly more compliant than had been demonstrated by



previous researchers. While the experimental results of Nachemson (1992) could be

interpreted to provide a similar nucleus pulposus pressure to that observed in the

FEM, the precise method employed by Nachemson to define the flexion angle during

experimentation was not clear. Consequently, there was a lack of confidence in the

validation of the FEM under flexion loads using the results of Nachemson (1992). In

light of the increased compliance of the FEM compared to in vitro observations, the

correlation established between the nucleus pressure in the FEM and the in vivo

condition may not have been correct.

Since, the method for defining the loading and boundary conditions on the FEM was

based on in vivo observations of physiological loading on the disc, these conditions

were not considered to be causes for the high compliance of the FEM in flexion.

Therefore, the increased compliance of the FEM was a result of either inaccuracy in

the geometry employed to represent the intervertebral disc or the use of incorrect

parameters to represent the components of the disc.

The geometry of the FEM was of an acceptable level of accuracy. The transverse

geometry was obtained from images of cadaveric discs and the algorithms employed

to map the profiles of the anulus and nucleus showed good correlation to experimental

data. The sagittal geometry of the FEM was obtained from in vivo measurements of

the L4/5 intervertebral disc during relaxed standing and the sagittal dimensions of the

model were iterated until the deformed shape of the model under torso loading was

comparable to the dimensions observed in vivo.

Material properties for the disc components were obtained from experimental data in

the literature. The material parameters selected and the assumption of linear elasticity

for the cartilaginous endplates and the collagen fibres were considered to be

reasonable representations for these structures. Several previous FEM studies had

used similar approaches. The stresses/strains observed in the rebar elements

representing the collagen fibres were reasonable given the loading applied, therefore,

this material was not considered to have resulted in incorrect results in the FEM.



The hydrostatic nature of the nucleus pulposus had been extensively studied and this

material representation applied by numerous previous researchers. This was not

considered to be a source of inaccuracy in the FEM.

All stress contours of the FEM exhibited a diamond stress pattern on the inner surface

of the anulus. This pattern was an artefact of the method of defining the hydrostatic

fluid elements. As outlined in section 3.8.3, hydrostatic fluid elements were defined

on the surface of the continuum elements at the boundary of the nucleus pulposus

fluid to model the fluid. These continuum elements were at the inner surface of the

anulus and were 20 node 3D elements. The hydrostatic fluid elements were either 3

or 4 node elements, therefore, it was not possible to attach one element to each

continuum element of the anulus without midside nodes on these elements remaining

unattached to the fluid. Therefore, 5 hydrostatic fluid elements were connected to

each continuum element – one 4 node fluid element and four 3 node fluid elements –

whereby, the nodes defining the 4 node element were all midside nodes and this

element was positioned diagonally across the face of the continuum element. As there

was no mention of inconsistencies in analyses as a result of attachment of the

hydrostatic fluid elements to midside nodes, this method of defining the fluid

elements was considered to be acceptable.

Consequent to the uniaxial compression and flexion analyses carried out on this

preliminary model, it was evident that this method of defining the hydrostatic fluid

elements was not acceptable and may have been resulting in erroneous stress contours

and peak stresses. An improved method of defining the hydrostatic fluid elements is

outlined in Chapter 6. The results for the nucleus pulposus were analysed in light of

the potential overestimation of the peak stress.

While the hyperelastic material formulation was considered to be well suited to the

behaviour of the anulus fibrosus ground substance, the method for determination of

the hyperelastic material parameters was considered to be lacking. These parameters

were obtained from the finite element study of Natali and Meroi (1990). Their

suitability for the FEM was established using analysis of a single element model and

by comparing the response of the preliminary FEM when various constants were used

to represent the anulus ground substance. This technique was an accepted modelling



technique; however, the results showed that the parameters used were limited in their

ability to model the tissue. Also, the Mooney-Rivlin hyperelastic equation that was

employed to represent the anulus ground substance was developed using the

assumption that the behaviour of the material was linear under simple shear loading.

Previous experimental studies had demonstrated that this was not the case for the

anulus fibrosus.

As such, the Mooney-Rivlin equation was not considered to be ideal as a

representation of the mechanical response of the anulus ground substance in an FEM

that included only the intervertebral disc structures. Such a model required an

extremely high level of accuracy to simulate the behaviour of the disc components

because the primary output of this model was the stress/strain state of these materials.

Therefore, it was decided to conduct experiments to obtain the values required to

determine improved parameters for the anulus ground substance. The details of this

experimentation are the subject of Chapter 4 and the determination of improved

hyperelastic parameters to represent the anulus ground substance is detailed in

Chapter 5.


Chapter 4: Experimental Testing of the Anulus Fibrosus 137

CChhaapptteerr

44

EExxppeerriimmeennttaall TTeessttiinngg ooff tthhee

AAnnuulluuss FFiibbrroossuuss

In the previous chapter the preliminary FEM results suggested that an improved

hyperelastic equation was required for representation of the anulus fibrosus ground

substance. In order to determine improved hyperelastic parameters accurate data on

the mechanical behaviour of the anulus fibrosus ground substance was required.

Specifically, the response of the material to uniaxial loading, biaxial loading and pure

shear loading were required in order to describe its comprehensive mechanical

behaviour. There was no information in the literature on the mechanical response of

the anulus fibrosus ground substance. Previous experimental studies carried out

testing on specimens of anulus fibrosus or on entire disc specimens. These studies did

not provide the data required for the FEM. An experimental procedure was developed

to determine the required values. (All stress and strain measurements presented in

this chapter are nominal values.)

4.1 Objectives for Testing the Anulus Fibrosus

In the current research, anulus fibrosus specimens were loaded under uniaxial

compression, biaxial compression and simple shear loading and a typical stress-strain

response for the anulus fibrosus ground substance to repeated loading was

determined. Responses to both the initial load application and repeated loading were

determined. Subsequent to these experiments, uniaxial compression and simple shear

tests were carried out to ascertain a range of strains at which damage was initiated in



the anulus fibrosus ground substance and to determine the response of the tissue to

varied strain rates. A set of representative stress-strain curves were obtained for the

anterior, lateral and posterior anulus ground substance. A significant difference was

observed between the response of the tissue in the different regions and the response

to the initial and repeated load applications. These mechanical data were essential for

development of the hyperelastic parameters to be implemented in the FEM for the

anulus ground substance.

4.2 Mechanical Testing – Rationale and Description

The p-q curve provides a representation of the full range of stresses which a structure

could experience as a result of the applied loading (Figure 4-1). Any loading applied

to a structure results in either hydrostatic stress or pure shear/deviatoric stress within

the structure. Commonly a combination of both these stresses would be experienced.

The p axis on the p-q curve represents the hydrostatic stress component of stress and

the q axis represents the pure shear component of the stress. The application of an

unconfined uniaxial load to a structure results in a line with a gradient of three and a

length which is dependent upon the magnitude of the loading applied. Equibiaxial

loading on a structure would result in a line with a gradient of 3/2 and a length

dependent upon the magnitude of the loads.

Figure 4-1 P-Q curve showing the potential stress states on a structure

When a material is subjected to three dimensional loading the state of stress on the

material may be expressed in terms of principal stresses in three orthogonal directions

Q

P

Pure shear

Triaxial compression Triaxial tension

Uniaxial tension Uniaxial compression

Biaxial compression Biaxial tension

Lines depicting specific loading conditions



(Section 2.6). When the solid material that is subjected to a general state of stress is

rotated into the orientation of the principal stress state, the shear stresses become zero

and the normal stresses become maximum and minimum values and are referred to as

principal stresses. The deviatoric and hydrostatic stresses on the material are

calculated from the principal and shear stresses. Eqn 4-1 shows the equation for the

general state of stress in a three dimensional material in terms of the hydrostatic and

deviatoric stresses. The deviatoric stress is the difference between the actual stress in

the system and the hydrostatic stress. It is the deviatoric stress that is responsible for

distortion in a loaded structure. The hydrostatic stress is defined in Eqn 4-2.

−−

−+

=

=

σσσσσσσσσσσσ

σσ

σ

σσσσσσσσσ

σHydrzzzyzx

yzHydryyyx

xzxyHydrxx

Hydr

Hydr

Hydr

zzzyzx

yzyyyx

xzxyxx

ij

000000

General Stress Hydrostatic Stress Deviatoric Stress

Eqn 4-1 Equation for general state of stress in a material

3σσσσ zzyyxx

Hydr

++=

Eqn 4-2 Equation for the hydrostatic stress

Using the p-q curve it is apparent that a comprehensive image of the mechanical

behaviour of a material can be obtained by applying various loading conditions. The

Abaqus software will determine hyperelastic parameters for a material based on user

input of test data from biaxial tension/compression, uniaxial tension/compression and

shear loading. It may be seen from Figure 4-1 that these three loading conditions

provide information on a considerable portion of the p-q curve. Since the Abaqus

input requires only these three loading types in order to define a complete hyperelastic

material model, biaxial compression, unconfined uniaxial compression and shear

loading were applied to the anulus fibrosus ground substance in order to quantify its

mechanical response.



Ideally the loading modes used to determine the hyperelastic parameters would be

based on the stress states within the anulus ground substance during physiological

loading. However, extensive data on these stresses was not found in the literature and

since the accuracy of the Mooney-Rivlin hyperelastic material parameters used in the

preliminary FEM was unclear, data for the state of stress in the anulus ground

substance could not be obtained from the FEM. Even so, the biaxial, uniaxial and

shear loading conditions would provide sufficient information to quantify the

hyperelastic parameters.

Testing was carried out on specimens of anulus fibrosus such that the mechanical

behaviour of the anulus ground substance was obtained. In order to achieve this, the

loading applied to the specimens did not apply tension to the collagen fibres in the

anulus. These fibres were tension-only components, so it was acceptable for them to

be loaded in compression. Unconfined uniaxial compression loading was the most

appropriate method to obtain information on the uniaxial loading behaviour of the

anulus fibrosus ground substance.

Pure shear loading was a difficult state of loading to apply to a biological material.

The shear loading which was applied to the anulus ground substance was simple

shear. This was a loading state which involved the translation of two parallel surfaces

of a material in opposite directions. The extension ratio in the maximum principal

direction increased, in the minimum principal direction decreased and in the third

principal direction the extension ratio remained at one. The Abaqus software

specified that the shear loading data was to be pure shear and therefore, the simple

shear stress data was converted to pure shear stress data (Section 5.3).

In the past, biaxial loading on biological tissues has generally been carried out under

tension. In the case of the anulus ground substance the application of tensile loading

would result in load bearing in the collagen fibres. Therefore, biaxial compression

was carried out to quantify the biaxial behaviour of the material.

The anulus fibrosus in the in vivo intervertebral disc experiences both compressive

and tensile stresses during physiological loading. However, it was thought that the

use of compressive rather than tensile loading modes in the experimental testing



would better reflect the stress state within the anulus ground substance. This type of

loading is in the right half of the p-q curve (Figure 4-2).

Figure 4-2 Compressive portion of the p-q curve

4.3 Specimen Harvesting

Sheep discs have been shown to exhibit similar kinematic and biochemical properties

to human discs (Wilke et al., 1997, Reid and Meakin, 2002). Seven intervertebral

discs were sectioned from the frozen lumbar spines of five sheep - two L3/4, one L4/5

and four L6/7 discs were obtained. The posterior elements, spinal cord and

surrounding musculature were removed. A 1-3mm layer of cartilaginous endplate and

vertebral bone on the superior and inferior surfaces of the disc were preserved. While

sectioning the discs from the spines they were kept moist with Ringers solution and

the room temperature was maintained at 20oC. Generally the disc had not thawed by

the time it was isolated from the spine. The discs were then surrounded with Ringers

soaked muslin, sealed in air-tight bags and refrozen to -20oC. Once frozen the

individual discs were set in dental cement in preparation for sectioning into test

specimens (Figure 4-3). The mold for the dental cement was formed from plasticine.

The mold had been frozen at -20oC before placing the disc and uncured dental cement

inside and it was placed in cold water while the uncured dental cement was poured

over the disc. This procedure was carried out to create a heat sink for the exothermic

reaction involved in the curing of the dental cement. The cement was cured for one

hour at room temperature, and then the hardened dental cement with embedded disc

was frozen at -20oC for a further 20 hours.

Compressive region of the p-q curve Q

P



Figure 4-3 Sheep intervertebral disc set in a dental cement plug and mounted on an aluminium bracket to allow for sectioning. The bracket was attached to a

rotating arm to permit cuts of necessary depth and width to be made.

In order to obtain mechanical data for the anulus ground substance in isolation, it was

necessary to ensure that no continuous collagen fibres coupled the two endplates of

the specimens which were tested. In this way, the mechanical response of the

specimen would give information on the response of the ground substance with the

collagen fibres embedded but not actively bearing a load. To ensure there were no

continuous fibres in the specimens, a maximum specimen width was determined using

the average height of the sheep discs and the average angle of inclination of the

collagen fibres in the anulus (Figure 4-4). The required cubic cross-sectional edge

length of the specimens was determined to be 3mm.

Figure 4-4 Determining the specimen width required to ensure there were no continuous fibres connecting the endplates in the specimen

Even though the collagen fibres in the specimens would not actively bear a load they

would still provide some resistance to the applied strain through the frictional

relationship between the fibres and the surrounding soft tissue. This was a desirable

artefact in the stress-strain response of the specimens. If data on the mechanical

α

No continuous fibres

Average Disc

height

α = 30o

Specimen width = 3mm

Endplate and vertebral bone

Anular Saw Blade

The dental cement plug was

attached to a bracket to hold

the disc for sectioning

Sheep intervertebral disc set in dental cement plug



response of the ground substance without any collagen fibres present was obtained,

then it would be necessary to designate a frictional relationship between the ground

substance and the collagen fibres in the FEM. The mechanical data obtained from the

testing of the specimens incorporated this relationship. As highlighted in Section

3.8.2 the reinforcing rebar elements representing the collagen fibres did not allow for

the incorporation of a relationship between these elements and the underlying

continuum elements representing the ground substance. However, since the results of

the mechanical testing included the effects of the interaction between the fibres and

the ground substance, the input of information to define this relationship was not

necessary.

Figure 4-5 A sectioned specimen.

A precision anular microsaw with diamond tipped blade (Figure 4-3) was employed to

section test pieces from the intact disc embedded in dental cement. Cuts were made

parallel to the sagittal and frontal planes through the disc such that the full disc height

and superior and inferior bone layers were preserved in the test specimens (Figure

4-5). The saw mechanism permitted the blade to be advanced after each cut and the

distance advanced could be controlled with an accuracy of 1mm. The blade was

advanced by 3mm after each successive cut in a plane. This produced test specimens

with a cubic cross-sectional width of 3 ± 0.2mm. The disc tissue was kept moist with

Ringer solution during the cutting process.

Once sectioned from the disc the test specimens that contained nucleus material were

discarded. Curing of polymethylmethacrylate was an exothermic reaction which had

been shown to cause tissue necrosis (Lieu, Nguyen and Payant, 2001) and could

Superior cartilaginous endplate

Inferior cartilaginous endplate

Inferior vertebral bone

Radial Direction

Anulus Fibrosus

Superior vertebral bone



involve temperature increases as high as 35oC. To ensure damaged specimens weren't

used for testing, anulus specimens from the peripheral disc were discarded as these

were in direct contact with the dental cement during curing. Anulus specimens were

labelled according to disc region (anterior, lateral or posterior), wrapped in Ringers

soaked muslin, sealed in clip seal bags and frozen to -20oC. Specimens were frozen

for a maximum of five days before being tested.

The number of specimens obtained from the disc regions varied:

• four - six specimens obtained from the anterior disc;

• four - six specimens obtained from the lateral disc; and

• three - five specimens obtained from the posterior disc.

Uniaxial compression, biaxial compression and simple shear tests were carried out on

specimens from each region of the disc. All specimen dimensions were measured

before testing. Stress and strain was calculated based on the unstressed dimensions.

4.4 Biaxial Compression Testing Methods and Equipment

The following sections provide details of the equipment and procedures employed for

the biaxial compression experimental testing. A testing rig was designed, built and

commissioned to carry out biaxial compression loading on the sheep anulus

specimens. This section provides a description of the design rationale and details, on

the proof testing of the biaxial compression equipment and the methods employed to

obtain biaxial compression data for the anulus fibrosus specimens.

4.4.1 Principle of operation

A novel testing rig was designed and built to carry out biaxial compression. The

design objective for the rig was to apply a hydrostatic pressure to the specimen and

then unload it along one axis to obtain a state of biaxial compression.



A B

Figure 4-6 The assembled biaxial testing rig A. With lid in place; B. With lid removed.

The developed rig was a stainless steel rectangular vessel, which could be filled with

Ringers' solution and pressurised (Figure 4-6). Two viewing windows were inserted

in two opposite walls of the vessel. The remaining walls provided attachment sites

for durable nylon thread, the ends of which were glued to the bone surfaces on the

specimen. Thus the specimen was suspended in the centre of the vessel and could be

viewed through the windows. It was necessary for the vessel sides to be planar so the

viewing windows could be inserted and to ensure the specimen was aligned parallel to

the direction of viewing.

One of the nylon threads connected the specimen directly to a wall. The other piece

of nylon connected the specimen to the end of a glass ceramic piston running in a well

polished bore in the opposite wall of the box. The cross-sectional area of the piston

was equal to the bone surface area of the specimen. The clearance between the bore

and piston was sufficient to allow the fluid in the box to leak when the fluid pressure

increased above gauge pressure. A low piston weight allowed it to be readily

suspended on a layer of fluid when the pressure in the box was increased. The polish

on the bore and piston surfaces and the use of Ringers' solution as lubricant meant

there was limited frictional resistance between bore and piston. A pressure inlet in the

lid of the box was connected to an air compressor through a high precision pressure

regulator which ensured accurate control of the pressure in the box.

When the pressure to the box was increased, this pressurised a 10mm air gap in the

Viewing windowsAttachment sites for nylon thread



top of the sealed box and in turn pressurised the solution. The pressure on the six

faces of the specimen and on the end of the piston was equivalent to the water

pressure. Because the area of the specimen bone face and the piston end were equal

and they were connected by an inextensible nylon thread, the forces on each face

would be equal in magnitude and opposite in direction. Thus, there would be no

compressive force acting on the specimen in the axis of the piston. However, the

compressive force in the other two axes would not be affected.

4.4.2 Design details and pressure vessel components

The walls of the pressure vessel were manufactured from 10mm stainless steel and the

viewing windows were 19mm thick standard glass. PVC brackets were used to hold

the glass viewing windows in position.

Refer to Appendix B for detailed engineering drawings and three dimensional solid

models of the pressure vessel components.

4.4.2.1 Maximum vessel pressure and design pressure

The relevant Australian Standard for non-serially produced pressure vessels was

AS1210-1997. This standard required that a maximum vessel pressure and design

pressure be determined. The maximum vessel pressure was the pressure which could

reasonably be expected to be reached during operation of the vessel and the design

pressure was to be greater than the maximum pressure and smaller than the pressure

setting on any pressure relief valves in the vessel.

Because biaxial compression loading had not previously been carried out on

specimens of anulus fibrosus ground substance it was difficult to determine a

maximum vessel pressure for the testing rig. Therefore, three pilot experiments were

carried out under uniaxial compression loading in order to determine the order of

magnitude of the pressures which would be applied to the material. From these tests

it was found that the maximum uniaxial compressive stress applied to the tissue was

approximately 0.28MPa. It was considered that the tissue would be stiffer under



biaxial compression and therefore, a scaling factor of 2.5 was applied to this

maximum pressure in order to determine an approximate maximum vessel pressure.

The maximum pressure was 0.7MPa.

A design pressure of 120psi or 0.827MPa was employed for all design calculations on

the box. This pressure was the supply pressure from the laboratory compressor.

4.4.2.2 Vessel walls

Originally the vessel walls were to be manufactured from perspex to allow the

deformation of the specimen to be viewed. However, this material selection was later

altered to 316 stainless steel. It was considered that the stainless steel demonstrated

more favourable mechanical properties than the perspex and additionally it could not

be assured that the specimen deformation viewed through the Perspex would not be

distorted. The steel wall thickness was 10mm.

The dimensions of the pressure vessel ensured the fluid pressure above the specimen

did not cause a high prestress and that there was sufficient fluid above the specimen

for the duration of the loading. The head of fluid above the specimen was

0.444x10-3MPa. This prestress on the specimen was considered to be negligible.

To seal the lid of the pressure vessel, a rubber gasket was cut to size.

The pressure vessel was designed in accordance with AS1210-1997. The relevant

requirements of this standard were the design material strength, the minimum wall

thicknesses and that the pressure vessel contained a pressure relief device.

• Design tensile strengths for materials were to be ¼ the specific tensile strength

of the material. The design tensile strength of the steel was 104MPa.

• The minimum wall thickness was calculated for a pressure vessel with non-

circular ends. This was 6.66mm. The 10mm thick stainless was considered to

be acceptable.



• A minimum wall thickness was calculated for the walls with viewing windows

inserted. This was 9.07mm. The 10mm thick stainless steel walls were

acceptable.

• A pressure relief valve was incorporated into the pressure line connecting the

vessel to the compressor.

It was not possible to carry out a stress analysis of the pressure vessel using standard

theories of thin-walled pressure vessels. The vessel was not thin-walled; it was not a

continuous vessel, rather was an assembly of parts; and it was neither circular nor

elliptical. Stress analysis of the pressure vessel was carried out using Roark’s (1989)

formulae for stresses in plates with varied geometries and boundary conditions.

Equations for the stress in a flat rectangular plate, with fixed edges and a uniform

surface pressure were employed to determine the maximum bending stress in the

vessel walls. The bending stress in the walls with the viewing windows was increased

by a stress concentration factor to account for the holes in the faces.

The maximum stress in the long walls, the lid and the base was 45MPa and in the

viewing window walls, the maximum stress was 96MPa. These values were

compared with the design tensile strength, 104MPa, and the yield strength, 170MPa,

of the steel. It was apparent that the bending stress in the walls was acceptable.

4.4.2.3 Fasteners

Fasteners for the pressure vessel were M6 x 1.0 steel socket cap screws. High tensile

strength socket cap screws were used to fasten the lid of the vessel to the walls but

stainless steel screws were used in all other locations on the vessel.

The design calculations for the fasteners in the pressure vessel included calculation of

the minimum engagement depth of the screws in the vessel walls to avoid thread

stripping, calculation of shear stresses across the shaft of the screw due to the wall

junctions and determination of the necessary number of fasteners on each wall. The

threaded length of the cap screws was 25mm and the engagement length in the vessel



walls was approximately 15mm which was significantly higher than the calculated

minimum length.

The number of fasteners on the edges of each side of the vessel was determined on the

basis of calculations of the stress states in the bolts with a safety factor of 1.5. A

comparison of these stresses with the yield strength and shear strength of stainless

steel and high tensile strength steel ensured that the working stress in the cap screws

was within the strength range of the materials.

4.4.2.4 Viewing windows

The viewing windows were standard glass plugs with a thickness of 19mm. They

were fastened into two opposite walls of the vessel and sealed with an O-ring.

To ensure there were no discontinuities or disruptions in the glass plugs that could

result in inaccuracies in the image viewed through the windows, the plugs were

rotated while viewing a straight line drawn on the wall. If there were discontinuities

in the glass, a distorted view of the straight line would have been found. There was

no distortion of the line in any orientation of either of the glass plugs. Calculations

were carried out to determine whether there would be any distortion in the image

viewed through the viewing windows once the specimen was in the assembled vessel.

These calculations were based on Snell’s Law for reflected/refracted light. It was

found that no distortion of the image would occur if the surfaces of the glass viewing

windows were parallel in the assembled vessel and if the windows were also parallel

to the projection screen for the image. A precision steel metrology rod with a

diameter of 10.00mm was measured whilst positioned in the assembled vessel. The

diameter of this rod was measured on a Sigmoscope (Section 4.4.5.2) with three sets

of ten measurements. The average error in the measured diameter was 0.005mm.

This was a good agreement between the measured diameter and the correct diameter.

The shear and bending stresses on the glass in the pressurised vessel were calculated

and compared with the tensile and shear strength of glass ceramics. There was no risk

of failure of the glass windows.



4.4.2.5 Attachment of specimen to nylon cord

The specimens were attached to the nylon cord using cylindrical dental cement plugs

(Figure 4-7) with an outer diameter of 3mm. One end of a piece of nylon cord was set

in the cement and the other end was fastened either in the titanium cap of the ceramic

piston or on the locator at the end of the adjustment knob (Figure 4-11).

Figure 4-7 Dental cement plug for attaching specimen to nylon cord

4.4.2.6 Leaking piston and bore insert

The piston diameter was selected such that the cross-sectional area of the piston was

equivalent to the cross-sectional area of the bone faces on the specimens. The

equivalence of these surface areas ensured that the force acting on each face was

equal in magnitude but opposite in direction and thereby unloaded the specimen along

the longitudinal axis acting through the two bone faces of the specimen. This

unloaded state of the specimen in the axial direction is demonstrated using a free-

body-diagram of the specimen in the assembled pressure vessel (Figure 4-8). The

tension in the right nylon cord at the fixed end was the same as the tension in the

opposite end of the cord at the face of the specimen. This tension was created by the

pressure acting on the face of the specimen. Also, the tension in the left nylon cord

which acted on the face of the piston was of the same magnitude (but opposite

direction) as the tension acting at the specimen face to which it was attached. Thus,

the specimen experienced no tensile or compressive stress in the axial direction. The

bone faces of the specimens were cubes with edge lengths of 3mm. Therefore, in

order for the cross-sectional area of these faces to be equal the piston diameter needed

to be 3.385mm. This component was manufactured with a diameter of 3.40mm.



Figure 4-8 Schematic of piston attachment in pressure vessel (not to scale). ‘P’ represents the pressure on the face of the specimen and piston due to the

pressurised fluid. ‘T’ represents tension in the nylon cord and is an equal but opposite force at the ends of the specimen, at the wall of the vessel and at the face

of the piston.

The piston design and the piston-bore clearance were selected to ensure that there was

sufficient pressurised fluid to surround the piston but that the flow rate did not deplete

the fluid volume in the vessel too quickly. Potential piston designs had incorporated

circumferential and longitudinal grooves that were intended to encourage fluid to

surround the piston and separate it from the walls of the bore. However, it was

suspected that the longitudinal grooves may create asymmetrical loading of the piston

once it was under fluid pressure. The pistons with circumferential grooves were not a

practical design. The groove corners created significant stress concentrations which

made manufacture on the lathe difficult and increased the potential for piston

breakage during setup of the testing rig. Therefore, the final piston design was a

straight sided shaft (Figure 4-9).

As the biaxial pressure on the specimen increased, the radial and circumferential

dimensions of the specimen decreased and the axial dimension increased. Therefore,

it was necessary for the piston to move in the bore so that the nylon cord connecting

the piston and specimen remained tensioned. It was imperative that the movement of

the piston through the bore be as near to frictionless as possible. This was achieved

by including a small clearance in the bore such that fluid leaked from the pressure

vessel during loading. The piston was manufactured from Macor Machinable® Glass

Ceramic. This was a material which possessed good machinability for high precision

components, had zero porosity, was non-corrosive and was capable of being

Leaking Piston

Vessel Walls

T

T T

T

P P Leaking Fluid through bore clearance

Axial direction of specimen

P

P

P



machined to high tolerances. Therefore, it was possible to obtain a low roughness

surface finish – Ra = 0.7µm – to encourage a low friction relationship between the

piston and the leaking fluid.

Using the theory of laminar fluid flow between two parallel plates (Eqn 4-3), the

bore-piston clearance was determined.

Lp

lQ a

..12.3

µ∆

=

Where, a = distance between the plates

p∆ = pressure variation along the length of the plate

L = depth of the plate

l = width over which the plates are facing

µ = viscosity

Q = volume flow rate

Eqn 4-3 Laminar fluid flow between parallel plates

In the case of the piston and bore,

a = clearance between the piston and bore

p∆ = pressure variation along between the average pressure in the

box and atmospheric pressure. This average pressure was half

the design pressure.

= average pressure, 0.4135 – atmospheric, 0.101MPa

= 0.3125x106 N/m2

l = the circumference of the piston, πD

= 10.68x10-3 m

L = length of the bore – even though the piston was capable of

moving through the bore, these calculations were based on the

assumption that the piston was completely inside the bore

= 0.025 m

µ = viscosity of Ringers solution which was approximated as the

viscosity of water

= 8x10-4 kg/ms



Q = the volume flow rate was selected to ensure that there would

be sufficient fluid in the vessel for a biaxial test to be carried

out for 30 minutes. The height of saline solution above the

specimen initially was 35mm.

= (0.035x0.110x0.150) m3 / 45 minutes

= 2.139x10-7 m3/s

Using these variables, the calculated clearance, a , was 0.022mm. Therefore, the

necessary bore diameter was 3.44mm. Such a small diameter reamed over a length of

25mm required notable manufacturing expertise to ensure concentricity of the bore

over its length. This was achieved by the QUT mechanical workshop. The bore was

reamed to ensure a high quality surface finish.

The assumption of laminar flow required that the fluid flow possessed a Reynolds

number < 1400 and that it was fully developed. It was reasonable to expect that with

a sufficiently small clearance, the fluid velocity would be low and the Reynolds

number below 1400. To make certain the flow was fully developed, it was ensured

that the clearance was very much smaller than the length of the cylinder.

In order to attach the nylon cord, a titanium cap was manufactured and fastened to the

end of the ceramic piston (Figure 4-9). The material choice for the cap was based on

the low corrosive properties of the titanium and its high strength-to-weight ratio. The

surface finish obtained on the cap was 0.2µm and it was attached to the ceramic piston

using LOCTITE® 324 acrylic adhesive and 7075 activator.

Figure 4-9 Ceramic piston with titanium cap glued to the end.

The bore insert was manufactured from stainless steel and inserted into one wall of

the pressure vessel (Figure 4-10).

Titanium Cap

Ceramic piston

The specimen was glued to the flat face of this dental cement plug



A B

C

Figure 4-10 Assembly of pressure vessel wall, bore insert and glass ceramic piston A. Solid model assembly; B. Sectioned view of assembly, sectioned along a vertical axis through the centre of the piston; C. Ceramic piston and bore insert

in the assembled pressure vessel – viewing inside vessel

4.4.2.7 Adjustment knob for accurate orientation of the specimens

In order to control the orientation of the specimen while it was suspended in the

Ringers solution, an adjustable fixture was placed on the inside of the wall to which

the nylon cord was fixed (Figure 4-11). This fixture was rotated using an adjustment

knob on the outer wall of the vessel. This permitted deformation measurements to be

taken in both the radial and circumferential directions of the anulus specimens and

ensured the specimen could be orientated such that either the radial or circumferential

direction was perpendicular to the viewing windows.

Pressure vessel wall

Bore insert

Piston



A B

C

Figure 4-11 Adjustment knob assembly A. Adjustment knob viewed from outside the vessel; B, C. Adjustable fixture viewed from inside the vessel

4.4.3 Proof testing

The Australian standard AS1210-1997 stated that the suitability and safety of pressure

vessels which were not adequately dealt with in the standard could be established by

either:

• Demonstrating successful performance of a prototype pressure vessel

subjected to similar conditions;

• Carrying out rigorous mathematical stress analysis, including FE analysis;

and/or

• Carrying out a proof test of the vessel. (Pressurise the vessel to a test pressure

of twice the design pressure for 15 seconds and examine the vessel for leakage

or signs of deformation.)

An extensive stress analysis was carried out during the design of the vessel

components. The proof test was carried out once the pressure vessel components

Attachment site for nylon cord

Teflon washer for low friction during rotation

Adjustment knob



were manufactured and assembled. AS1210 stated that single walled vessels should

be loaded to a pressure Ph during the test.

ffP h

hP ×= 5.1

where, P = design pressure = 150psi = 1.03MPa

hf = design strength at test temperature

f = design strength at design temperature

Therefore, Ph = 225psi = 1.55MPa

Eqn 4-4

The pressure vessel was subjected to a maximum pressure of 300psi (2.07MPa) and

held constant at this pressure for 30 seconds. At 150 psi and 225 psi, the condition of

the vessel was assessed – there was no visible leakage from the vessel and all

components of the vessel were undeformed and intact. At 300 psi there was a small

leakage from the screws on the lid, but this was eliminated by further tightening. All

components of the vessel were undeformed and intact.

It was concluded that the proof testing of the pressure vessel was successful and it

was acceptable for use in the biaxial compression testing of the anulus fibrosus

specimens. The hazard rating of the pressure vessel was obtained from AS4343-1999

and was a hazard level E which was classified as ‘negligible’.

4.4.4 Setup of equipment

Until the time of testing, the specimens were frozen in Ringers soaked muslin and

sealed in plastic bags. Assembly of the pressure vessel required 30 minutes during

which time the anulus specimen was wrapped in Ringers soaked muslin or immersed

in Ringers solution in the vessel. By the time the biaxial compression tests were

carried out the specimen was completely thawed. The assembled pressure vessel was

located on the table of the Sigmoscope profile projector (Herbert Controls and

Instruments Ltd, Letchworth, England) (Figure 4-12) ensuring the viewing windows

were parallel to the projector screen and the specimen was orientated in the vessel so



that the deformation would be measured in the desired axis of the anulus – that is,

radial or circumferential deformation.

Figure 4-12 Assembled pressure vessel

4.4.5 Measurement of biaxial compressive stress and strain

Details of the equipment used to measure the biaxial compressive stress and the

resulting deformation of the anulus fibrosus specimens are provided in the following

sections.

4.4.5.1 Choice of pressure regulator

The compressive stress applied to the specimen was equivalent to the pressure of the

Ringers solution surrounding it. This pressure was determined using a digital Druck

pressure calibrator (Model: DPI 705, GE Druck Ltd, Leicester, UK) and the vessel

was pressurised using a precision pressure regulator. It was important to use a

pressure regulator with a maximum pressure similar to the maximum operating

pressure during the testing. If the maximum pressure on the regulator was too high

then its accuracy was reduced and low pressures of 0 to 10psi would not be

adequately controlled. Additionally, because the pressure vessel was designed to leak

during operation, the regulator needed to maintain a constant pressure in the box

despite the loss of fluid.



A Norgren zinc alloy precision regulator (Model:11-818, IMI Norgren Ltd,

Staffordshire, UK) with a maximum pressure of 60 psi and an accuracy of 0.435 psi

was employed. This regulator provided excellent control over the pressure in the

vessel for the full range of pressures applied during testing. (The pressure regulator

was manufactured using imperial measuring units, therefore, pressure units of psi are

used in Sections 4.4.6.1 and 4.4.6.2 to convey details of pressures measured using this

device. 1psi = 0.0689MPa)

4.4.5.2 Profile projector

To measure the deformation of the specimen under load, a Sigmoscope profile

projector was used. This equipment was capable of providing high precision

measurements of specimen dimensions by shining a light source across the item to be

measured and projecting the shadow of the item onto a viewing screen. By moving

the stage on which the item was positioned it was possible to obtain a digital readout

for the necessary linear dimension.

In the case of the biaxial compression testing, the light source was projected through

the viewing windows of the vessel (Figure 4-13 A) and a projected image of the

deformed specimen was obtained (Figure 4-13 B). The specimen was orientated

using the positioning knob, such that either the radial or circumferential direction in

the anulus was perpendicular to the viewing windows, enabling these dimensions to

be measured (Figure 4-13 B). There was no distortion of the image as a result of the

light beam passing through the Ringers solution so long as the solution was

homogeneous.



A B

Figure 4-13 Measurement of specimen deformation during biaxial compression; A. Specimen through viewing window; B. Image projected onto the Sigmoscope

screen. Moving the machine table allowed the linear dimensions to be determined

4.4.5.3 Data acquisition - hydrostatic pressure and deformation

Data on the anulus fibrosus ground substance response to both initial loading and

repeated loading on a single specimen was obtained. A single specimen was tested

seven times. The specimen was permitted to recover for 15 minutes between each

test. Data for the initial specimen width in either the radial or circumferential

direction was recorded. It was expected that the deformation in these directions

would differ but it was not practical to measure the deformation in both directions on

a single specimen. Orientating the specimen in both directions at a single pressure

resulted in the specimen remaining at the pressure for too long. Creep in the tissue

caused its deformation to continue to vary at this constant pressure and it was not

possible to obtain reliable and repeatable results. Of the 24 specimens tested with

biaxial compression, 14 were orientated so the deformation was measured

circumferentially.

4.4.6 Commissioning of pressure vessel

Commissioning of the pressure vessel involved ensuring the pressure applied to the

inner face of the piston was accurate and ensuring the testing technique was

repeatable.

Deformed width measured

Loose tissue from specimen – not measured



4.4.6.1 Force applied to the piston

In order to ensure that the force acting on the inner face of the piston was accurate, the

vessel was assembled with the outer face of the piston in contact with a 500N

Hounsfield load cell (Hounsfield Test Equipment, Red Hill, England). Water in the

vessel was incrementally pressurised and the force output from the load cell recorded.

Five sets of pressure measurements were obtained and the pressure was increased by

10 psi between 0 and 96.85 psi in each set. The pressure was measured with the

digital Druck pressure calibrator.

The results were graphed as force vs. applied pressure. A line of best fit was

determined. An average error of 2.10% existed between the measured force and the

force which was calculated on the basis of the pressure applied and the cross-sectional

area of the piston. The inaccuracy between these results was firstly a result of the

shear stress on the walls of the piston and bore and secondly, due to the calibration of

the Hounsfield load cell.

The wall shear stresses were determined over the range of pressures using Eqn 4-5.

yu∂∂

= µτ and

( )cylp

yu

−

∆=

∂∂ 2..

21µ

where, u = velocity of the fluid

p∆ = pressure variation along length of piston

l = length of piston

µ = viscosity of the fluid

c = clearance between the piston and bore

y = distance measured across the clearance c

Eqn 4-5 Shear stress and the velocity profile for flow between infinite parallel plates

The shear force at each pressure was subtracted from the measured force. This

improved the average error between the measured force and the calculated force to a

value of 0.93%. The error between the calculated force and the improved measured



force increased with increasing pressure and was a maximum of 0.081N at 96.85 psi.

This average error between the forces was improved further by determining the

calibrated force from the Hounsfield load cell.

A series of weights between 1 and 500g were placed on the Hounsfield 500N load cell

and the force readout recorded. The variation between the calculated forces

(calculated using gravity) and the measured forces were determined. Over this load

range the relationship in Eqn 4-6 was found.

HounsfieldCorrect ForceForce ×= 988.0

Eqn 4-6 Relationship between the force output from the 500N Hounsfield load cell and the correct force

This result indicated that the Hounsfield load cell slightly overestimated the force

applied to the piston by the pressurised fluid (Figure 4-14). Using this relationship

the error between the measured forces and the calculated forces was improved to an

average of 0.37%.

01234567

0 20 40 60 80 100 120Pressure (psi)

Forc

e (N

)

Measured Force - Shear Force Hounsfield corrected calculated force

Figure 4-14 Comparison of the improved measured force and the calculated force which was manipulated to account for the calibration of the Hounsfield

500N load cell.

It was concluded from these measurements that the piston would apply an acceptably

accurate load to the specimen during the biaxial compression loading.



4.4.6.2 Biaxial compression of EVA foam

A cubic piece of closed cell EVA foam was tested to determine the repeatability of

the testing technique. During nine separate biaxial experiments the test piece was

loaded to a maximum pressure of 30 psi in approximate pressure increments of 5 psi.

A recovery time for EVA foam was not known therefore, the specimen was permitted

to relax for 1.75-2.0 hours between tests.

The pressure was increased incrementally because the measurement of the test piece

deformation was not automated. The testing procedure for biaxial compression

involved obtaining a pressure with the regulator, the specimen deformation being

measured on the Sigmoscope, the deformation recorded and the pressure then

manually increased further.

The deformation recorded for the biaxial compression testing was the minimum width

of the test piece at each pressure. (Figure 4-15)

Figure 4-15 Measuring the deformation during biaxial compression testing

This deformation was normalised with the width measured at the gauge pressure, do to

obtain the extension ratio. This value was referred to as the extension ratio even

though the deformation of the specimen involved compression. The use of the term

extension ratio was in keeping with the terminology used in Chapter 5 to define the

constitutive equations for the hyperelastic strain energy equation.

Increasing pressure

Length increase

do

dP1

dP2

P1 applied

P2 applied

Gauge

Dental cement plug with nylon cord attached

EVA foam



The change in specimen width was used to determine the biaxial strain in the

specimen. The deformed edges of the test piece were not regular and the minimum

width measured was generally not at the midpoint along the specimen. However, the

observed minimum points on each edge tended to remain in the same location

between tests, demonstrating some repeatability of the testing technique. Extension

of the test piece in the direction of the nylon cord was observed as a result of the

compression in the other two axes; however, this deformation was not measured.

2.52.7

2.93.13.3

3.53.7

0 10 20 30Pressure (psi)

Min

imum

Wid

th (m

m)

1st Test 2nd Test 3rd Test4th Test 5th Test 6th Test7th Test 8th Test 9th Test

Figure 4-16 Pressure vs. minimum width for biaxial compression testing on EVA foam

Nine sets of measurements were made for the EVA foam. These tests showed a

reduction in stiffness for the first and second load cycles. It was considered that the

variation in the response of the foam in the first and second tests was a result of the

consolidation of the material. When subjected to repeated loading, foams are known

to precondition which involves a reduction in the stiffness of the material until a

repeatable response is obtained (Nusholtz et al., 1996). The data for the EVA foam

showed a repeatable trend for the final seven tests (Figure 4-16). This implied that the

response of the EVA foam specimens was similar to the reported response for foams

and indicated that the biaxial compression testing method and measurement

techniques could provide repeatable results. On the basis of the tests on EVA foam, it

was apparent that the biaxial compression measurement techniques could produce a

repeatable response and therefore, they were employed to test specimens of anulus

fibrosus.



4.5 Uniaxial Compression and Simple Shear

Fixtures were designed to attach to existing materials testing equipment in order to

carry out uniaxial compression and simple shear tests. Details of the testing methods

and equipment used for the uniaxial compression and simple shear tests are provided

in the following sections.

4.5.1 Testing equipment

The following sections detail the test equipment used to carry out the uniaxial

compression and simple shear experiments.

4.5.1.1 Uniaxial compression

Uniaxial compression of the anulus fibrosus specimens was carried out on a

Hounsfield testing machine using a 500N load cell. The full scale deflection of the

load cell was set to 5%, therefore, the maximum load applied was 25N. Accuracy of

this load cell for small load values was outlined in section 4.4.6 and the relationship

between the force output from the load cell and the correct force was defined in Eqn

4-6. This relationship was taken into account in the analysis of the uniaxial

compression results.

The bone faces on the superior and inferior surfaces of the specimen were glued to the

fixtures on the load cell and on the crosshead of the machine with Loctite® 401. This

specimen orientation created compressive loading in the axial direction. To determine

the mechanical response of the tissue to both initial and repeated loading, five tests

were carried out on each specimen and the specimen was permitted to relax for 5

minutes between each test. The specimen was kept hydrated between tests and during

testing using a squirt bottle of Ringers solution and by wrapping it in Ringers soaked

muslin and a plastic sheet.



4.5.1.2 Simple shear

Simple shear loading was carried out using the Hounsfield testing machine equipped

with a 5N load cell. Accuracy of this load cell was tested using a spring of known

spring constant. The spring constant measured using the 5N load cell was 0.623%

higher than the correct value. This was considered to be an acceptable level of

accuracy.

Figure 4-17 Hounsfield attachments to apply simple shear

The load cell could measure only tensile loads, therefore, fixtures were designed and

manufactured to apply simple shear (Figure 4-17).

The fixtures were manufactured from aluminium alloy to minimise weight. The

upper fixture was suspended from the load cell therefore, it was necessary that its

weight be sufficiently low to be tared by the load cell. Anulus specimens were

attached to the opposing faces of the fixtures using Loctite® 401. When the

crosshead was driven downward a state of simple shear was created in the specimen.

Attachment to 5N load cell

Upper shear fixture

Lower shear fixture

Specimen



The shear force could have been applied to the specimens in either a circumferential

direction or a radial direction (Figure 4-18). Fujita et al. (2000) carried out shear

testing in all three anulus directions and found that the shear modulus in the radial and

circumferential directions were similar and in the axial direction it was twice the

modulus in the other two orientations. It was postulated in the current study, that this

significant variation in shear modulus with specimen orientation was a result of the

loading of continuous collagen fibres. Such significant variations in stiffness would

not be expected if the ground substance was tested.

Figure 4-18 Anulus fibrosus showing potential directions of shear

If the simple shear force on the specimens was aligned with the radial direction

(Figure 4-18), this may have resulted in separation of the lamellae. Thus the simple

shear response obtained would be partially dependent on the bonding strength of the

adjacent lamellae. Simple shear loading in either the axial direction or the

circumferential direction would not have caused separation of the lamellae; however,

it was not possible to load the specimens in the axial direction because of the layer of

vertebral bone that was preserved on the superior-most and inferior-most faces of the

specimens. It was considered that application of the simple shear load to the

specimens in a circumferential direction would be most appropriate.

In order to determine the response of the tissue to initial and repeated loading, the

specimens were loaded five times, with a five minute recovery period between each

test. The specimens were kept hydrated during the tests and the recovery time using

Ringers solution, Ringers soaked muslin and plastic wrap.

Axial directionCircumferential direction

Radial direction



4.5.2 Maximum strains applied during testing

In order to obtain a close fit between the hyperelastic model representing the anulus

ground substance and the experimental response of the material under load, it was

necessary to perform experiments over the same range of strains as would be expected

to occur physiologically. Pilot FEM analyses showed that hyperelastic parameters fit

to stress-strain data below the maximum strain observed in the disc, exhibited a good

fit for the experimental response only to strains for which the hyperelastic parameters

had been fit. At higher strains, the hyperelastic model did not display good agreement

with the experimental data.

The maximum strains observed in the preliminary FEM were for the full flexion

loading. Engineering strains as high as 4.2 were found. The maximum strain applied

during the compressive experimental testing could not reasonably exceed 60-70%

without introducing some tension in the collagen fibres of the anulus. Simple shear

loading was carried out to maximum strains of 50-80% and the maximum biaxial

compressive strain was 30%.

4.6 Strain Rate during Uniaxial Compression and Simple Shear Loading

The material testing carried out on anulus specimens aimed to simulate physiological

strain rates. These rates would result in no fluid loss or volume change in the

material. This was in keeping with the assumption of incompressibility employed for

the hyperelastic material description in the anulus fibrosus ground substance. No

evidence was found for the value for physiological loading rates. Several

experimental studies had been carried out on specimens of anulus fibrosus which

employed strain rates ranging from 0.00009 sec-1 and 0.005 sec-1 (Fujita et al., 1997;

Ebara et al., 1996; Acaroglu et al., 1995; Best et al., 1994; Skaggs et al., 1994; Wu

and Yao, 1976). The lower strain rates were intended to ensure the fluid drag through

the matrix of the anulus fibrosus was limited. Therefore, to ensure no fluid movement

within the specimens tested in the current study, a strain rate well above the range

0.00009 sec-1 to 0.00012 sec-1 was employed. Studies which employed strain rates

between 0.00012 sec-1 and 0.005 sec-1 were not intending to simulate the



physiological condition and made no mention of quantifying the mechanical response

of the material for the in vivo condition.

In order to determine a suitable strain rate to employ in the mechanical testing,

experiments were carried out under uniaxial compression and simple shear at 0.10

sec-1, 0.01 sec-1 and 0.001 sec-1.

4.6.1 Procedure for testing to determine the tissue response to varied strain

rates

Three specimens of anulus fibrosus were specifically tested to investigate the effects

of various strain rates. Individual specimens were tested under uniaxial compression

and simple shear. It was not possible to test the specimens using biaxial compression

as they could not be loaded at a higher strain rate than 0.01 sec-1. A similar procedure

to that used for the stress-strain experiments was employed (section 4.5.1). The

maximum strain in the strain rate tests was approximately 20% as this was the lower

threshold of the derangement strain. For the purpose of this study, the derangement

strain has been defined as any damage to the anulus fibrosus ground substance that

causes a reduction in the observed stiffness of the material, but which does not

prevent the tissue from bearing a load during subsequent loading. Further results for

these strains are provided in Section 4.8.

The specimens were tested four times at each strain rate. The first and second tests –

test a and b – were carried out 5 minutes apart in keeping with the procedure

employed for the stress-strain experiments. The specimen was then permitted to

recover for one hour in order to remove any viscoelastic effects in the tissue and to

permit absorption of any pore fluid lost during the first tests. After recovery, the

specimen was retested two times, 5 minutes apart – test c and d. Results for three

strain rates were compared – 0.001 sec-1, 0.01 sec-1 and 0.10 sec-1. The specimens

tested at 0.001 sec-1 were retested an additional two times, 5 minutes apart, after a one

hour recovery – test e and f.



4.6.2 Results and discussion of strain rate experiments

The following sections detail the results for the experimentation at varied strain rates.

A discussion of these results is given.

4.6.2.1 Strain rate 0.001 sec-1

At low strain rates the sensitivity of the 500N load cell was a limiting factor. The

results for this loading rate showed considerable noise, especially at strains below 5%,

but were repeatable between tests (Figure 4-19). The specimen exhibited the same

response during the first two tests carried out 5 minutes apart (Figure 4-19 test a and

test b). This suggested that there was no derangement of the anulus tissue when a

maximum strain of 20% was applied; that there were no remaining viscoelastic effects

in the tissue after 5 minutes of recovery time; and that there was no reduction in

stiffness due to loss of pore fluid from the matrix. The latter of these conclusions was

extremely important. Slower strain rates permitted the fluid to flow from the matrix.

If the fluid lost was not replaced after 5 minutes, the stiffness of the specimen would

have been reduced due to the lack of potential pore pressure. The repeatable

mechanical response after 5 minutes of specimen recovery suggested that this time

period was sufficient for any lost pore fluid to be imbibed.

-0.02

0

0.02

0.04

0.06

0.08

0.1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Strain

Stre

ss (M

Pa)

Test a Test b Test cTest d Test e Test f

Figure 4-19 Strain rate 0.001 sec-1



When the specimen was permitted to recover for 1 hour and re-tested to a maximum

strain of 30% a response similar to the previous tests was obtained. However, the

experiments indicated that allowing a recovery time of 5 minutes after loading to 30%

strain was not sufficient to restore the specimen stiffness (Figure 4-19). This

suggested that applying a higher maximum strain resulted in inability of the anulus

ground substance to imbibe sufficient fluid to restore fluid pore pressure. A recovery

time of 1 hour restored the stiffness of the specimen to the previous response which

suggested that 30% strain did not cause derangement of the anulus at this strain rate

(Figure 4-19 test e).

The negative stress values at low strains could have resulted from the lack of

sensitivity of the load cell or may have been due to residual tensile stress in the tissue

(Figure 4-19). It was likely that both of these were the cause. The residual stress

would be generated as a result of the viscoelastic nature of the ground substance or

may be due to some fluid motion in the tissue.

The observation of variation in the mechanical properties of the anulus specimens due

to pore fluid variations indicated that 0.001 sec-1 was not an appropriate loading rate

to ensure incompressibility of the anulus ground substance.


The specimens exhibited the stiffest response when tested at 0.10 sec-1; however, this

strain rate also showed the most significant drop in stiffness when the loading was 5

minutes apart. Using a 0.10 sec-1 strain rate, after one hour of recovery the specimen

did not regain its original stiffness if the maximum strain reached in the previous

loading was 23%. However, when the maximum strain in the test prior to the one

hour of recovery was 13%, the specimen stiffness was regained. This suggested that

some derangement was present in the anulus ground substance at strains between 13

and 23% when the strain rate was 0.10 sec-1. This was significantly lower than the

range determined for a rate of 0.01 sec-1 and was in keeping with the observations of

Morgan (1960) that varying strain rates caused a variation in the rupture stress of

collagenous tissue.



Physiological strains when standing would be in the range 5-15% for an average

individual. This suggested that if physiological strain rates were in the order of 0.10

sec-1, some derangement would be present in the anulus during low loading daily

activities such as standing. This was unlikely.


Tests at 0.01 sec-1 showed a drop in stiffness when they were carried out 5 minutes

apart, but the stiffness was recovered when the specimen was tested an hour later. In

light of the results of testing at 0.001 sec-1, it was concluded that this reduction in

stiffness was not due to pore fluid flow from the material. Fluid lost from the anulus

ground substance could potentially be recovered during a 5 minute period. Also, the

recovery of the tissue stiffness after one hour demonstrated that the loss of stiffness

was not a result of derangement in the material. Therefore, the stiffness decrease was

likely due to viscoelasticity of the anulus materials.

4.6.3 Discussion and justification for the choice of strain rate

It is important to make a distinction between the behaviour of the pore fluid and the

elastic skeleton in the anulus ground substance. The elastic skeleton forms the

boundaries of the pores and allows for the entrapment of the pore fluid and the build

up of pore fluid pressure under loading. The combined mechanical behaviour of these

materials creates the bulk response of the ground substance. Applied loads will be

resisted by the increasing pore pressures resulting from the entrapment of fluid in the

pores of the elastic skeleton and by the mechanical strength of the elastic skeleton

itself. The relative contribution to load bearing that is provided by the pore pressure

and elastic skeleton stress is dependent on the strain rate applied to the tissue. If the

strain rate is sufficiently slow, the fluid drag provided by the elastic skeleton is

diminished and fluid is released from the tissue. Consequently, the majority of the

load applied to the tissue is resisted by the elastic skeleton. Conversely, higher strain

rates cause entrapment of the pore fluid due to the increased fluid drag forces and the

loads on the tissue are resisted largely by the increased pressure of the pore fluid.



The initiation of damage in the ground substance at strains lower than those observed

in vivo when the tissue was tested at 0.1 sec-1 was thought to be a result of the

increased pore fluid pressure in the tissue under the faster loading rate. This lower

derangement strain was not an indication that the specific derangement strain of the

ground matrix skeleton had been altered. Rather the matrix skeleton was exposed to

higher stresses/strains as a result of the increased pore pressure. This caused the

strain at which derangement was initiated in the skeleton to be exceeded. This resulted

in the overall response of the ground matrix demonstrating a reduced derangement

strain which was low in comparison to the strains observed in the anulus fibrosus in

vivo.

Given the relationship between the elastic skeleton and the pore fluid in the anulus

ground substance, the application of a uniaxial compressive load to the tissue would

not result in pure uniaxial compression of either of these components. It is the overall

ground substance specimen which is subjected to this type of loading. Instead, the

stress state in the elastic skeleton would be multi-axial due to the pressure from the

pore fluid. However, it was necessary to introduce this stress state into the skeleton in

order to ensure there was no fluid loss from the tissue during loading. In this way,

uniaxial compression was a difficult loading condition to achieve on a material such

as the ground substance. In order to obtain a purely uniaxial compressive state in the

elastic skeleton, the loading rate applied to the tissue would need to be sufficiently

slow to permit all pore fluid to be released with a minimum of fluid drag forces

applied. This loading condition; however, would result in experimental data that was

not comparable with the assumption of incompressibility. The 0.001 sec-1 loading

rate demonstrated loss of fluid and as such would likely provide results that were

closer to a uniaxial stress state in the elastic skeleton. Conversely, the 0.1 sec-1 strain

rate resulted in excessive pore fluid pressures which would have created considerable

tri-axial stresses in the elastic skeleton.

Thus it was not possible to obtain a purely uniaxial compressive state on the elastic

skeleton of the ground substance without negating the assumption of

incompressibility in the results. On the basis of this discussion it was apparent that



the 0.01 sec-1 loading rate would be a reasonable compromise in order to obtain

adequate mechanical data for uniaxial compression of the tissue.

While a discussion of the exact stress/strain state in the components of the ground

substance was relevant, it was still noted that the overall loading condition applied to

the anulus ground substance was uniaxial compression. A distinction was made

between the mechanical response of the individual components of the ground

substance and the ground substance itself. The varying stress/strain states of the

components of the ground substance combined to provide the bulk response of the

material to uniaxial compression. The material parameters which were sought were

for the mechanical response of the ground substance rather than the results for

uniaxial compression of its components.

In summary, experiments carried out at 0.10 sec-1 demonstrated a derangement strain

for the anulus material which was too low in comparison to the physiological strains

evident in the intervertebral disc in vivo. Experimentation at 0.001 sec-1 indicated that

there was a loss of pore fluid from the anulus ground substance. Therefore, the

incompressibility of the material was not maintained at this loading rate. The results

of the tests carried out at 0.01 sec-1 demonstrated a recoverable loss of stiffness in the

anulus. Because this stiffness was recovered only after a one hour period, it suggested

that the stiffness loss was not a result of pore fluid effects but was due to the

viscoelastic nature of the anulus ground substance.

From the experimentation to determine the effects of varied strain on the mechanical

response of the anulus fibrosus ground substance it was concluded that a strain rate of

0.01 sec-1 would be applied to the specimens. It was desirable to maintain consistency

in the strain rate for the uniaxial compression, biaxial compression and simple shear.

However, it was not possible to obtain a strain rate above 0.01 sec-1 during the biaxial

compression tests due to the incremental method of measuring the deformation.

Prior to both the uniaxial compression and simple shear tests the specimen was

preconditioned for five cycles at 0.4Hz. Five preconditioning cycles at 1.5Hz were

carried out on the specimens tested in biaxial compression. It was not possible to



achieve a faster preconditioning rate during biaxial compression due to the

incremental method of applying the pressure.

4.7 Results for Mechanical Testing of the Anulus Fibrosus Ground Substance

The results for the response of the anulus fibrosus ground substance to repeated

loading are presented in Section 4.7.1 and details of the statistical analysis of these

results are presented in Section 4.7.2.

4.7.1 Results of initial and repeated loading – stress-strain tests

The uniaxial compression and simple shear tests were carried out using the

Hounsfield testing machine and the loading was controlled by the deformation

applied. Therefore, strain was plotted on the x axis for the uniaxial compression and

simple shear data (Figure 4-20 and Figure 4-21). Simple shear strain was calculated

as the ratio of the shear displacement of the Hounsfield crosshead to the axial height

of the anulus fibrosus in the specimens. The biaxial compression loading was

controlled by the pressure that was applied. The biaxial compression data was

plotted with the controlled variable of pressure on the x axis (Figure 4-22) and the

results for biaxial compression were expressed diagrammatically as extension ratio vs.

pressure. The biaxial compression extension ratio is defined in Section 4.4.6.2.



A

00.5

11.5

2

0 0.2 0.4Strain

Stre

ss (M

Pa)

Test a Test b Test cTest d Test e

B

0

0.5

1

1.5

2

0 0.2 0.4 0.6Strain

Stre

ss (M

Pa)


C

0

0.51

1.52

2.5

0 0.2 0.4 0.6Strain

Stre

ss (M

Pa)


Figure 4-20 Examples of stress-strain data for uniaxial compression A. Characteristic response; B. Very close agreement for repeated loading, tests b to

e; C. Short term drop in stiffness in test e

Short term drop in stiffness in test e may be due to lamellae separation



A

00.050.1

0.150.2

0.250.3

0 0.2 0.4 0.6Strain

Stre

ss (M

Pa)


B

0

0.05

0.1

0.15

0.2

0 0.2 0.4 0.6Strain

Stre

ss (M

Pa)


C

00.020.040.060.080.1

0 0.5Strain

Stre

ss (M

Pa)


Figure 4-21 Examples of stress-strain data for simple shear. A, B and C demonstrate the 3 characteristic responses observed from the specimens.

Results of the initial loading are similar to the results of the repeated loading for tests a to d



A

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2Stress

Exte

nsio

n ra

tio

Test a Test b Test c Test dTest e Test f Test g

B

0.60.7

0.80.9

11.1

0 0.1 0.2 0.3Stress

Exte

nsio

n ra

tio


C

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2 0.3Stress

Exte

nsio

n ra

tio




D

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2Stress

Exte

nsio

n ra

tio


E

0.6

0.7

0.8

0.9

1

1.1

0 0.1 0.2Stress

Exte

nsio

n ra

tio


Figure 4-22 Examples of stress-strain data for biaxial compression – the stress is measured in MPa. A, B, C Circumferential measurement; D, E Radial

measurement

A maximum compressive load of 25N was applied to the uniaxial compression

specimens. Characteristic specimen responses to the uniaxial compression (Figure

4-20 A) demonstrated a notably stiffer response to the initial test a compared to the

subsequent tests b to e. Repeated loading on the specimen resulted in a compliant

response up to a strain of 20-40% followed by a significant increase in stiffness. The

response to the repeated loading was reproducible and some specimens showed

exceptional agreement between tests (Figure 4-20 B). At higher uniaxial compressive

strains, separation of several lamellae was observed in some specimens. It was

postulated that the observed stepped shape in the stress strain response of some



specimens (Figure 4-20 C) was a short term drop in stiffness due to an increased

compliance in the anulus at the point of separation of the lamellae.

The results of the simple shear loading demonstrated three characteristic responses

(Figure 4-21 A, B, C). The first response (Figure 4-21 A) was similar to the uniaxial

compression data – the specimen was stiffer during the initial loading and

demonstrated a more compliant behaviour upon repeated loading. This compliant

behaviour was repeatable. The second response involved a reduction in stiffness of

the specimen compared to the stiffness during the initial loading, but this reduced

stiffness was not reproducible (Figure 4-21 B). The repeated loading stress-strain

data for these specimens showed a sustained decrease in stiffness. The third response

demonstrated no reduction in stiffness between the results of the initial loading and

the results of several of the subsequent tests (Figure 4-21 C).

Maximum biaxial compressive stresses of approximately 0.24 MPa were applied to

the specimens (Figure 4-22). This pressure resulted in a biaxial strain of between 10

and 35%. The response to biaxial compression was dependent on the orientation of

the specimen. Of the 14 specimens measured in the circumferential direction, ten

specimens showed a stiffer response initially and a drop in stiffness for the repeated

loading. A large drop in stiffness was observed in eight specimens (Figure 4-22 A)

but two specimens showed only a slight drop in stiffness (Figure 4-22 B). There was

no appreciable stiffness variation observed between the initial and repeated loading on

four specimens measured circumferentially (Figure 4-22 C) and on six of the nine

specimens measured radially (Figure 4-22 D). A slight drop in stiffness was observed

in three of the specimens measured radially (Figure 4-22 E).

4.7.2 Statistical analysis

The statistical analysis was carried out using the statistical software SPSS 11.0 (SPSS

Australasia Pty Ltd). Lines of best fit were calculated for the initial and repeated

loading in each region. If there was no difference between these responses for a

specimen, then the experimental curves were used to determine the regression line for

repeated loading in that region. It was not possible to fit a regression line for several



regions. In this case, a line of best fit was estimated and the R2 value calculated.

While the R2 values for several of these curves were low, the line was considered to

be an acceptable representation of the experimental data. Table 4-1, Table 4-2 and

Table 4-3 detail the R2 values for the lines of best corresponding to the initial and

repeated loading on the three regions of anulus ground substance.

4.7.2.1 Simple shear

For both the initial and repeated loading the anterior was the stiffest region and the

posterior was the most compliant region under simple shear loading (Figure 4-23).

Similar trends were found for the curves fit to each region – the initial loading was

reasonably linear and the repeated loading curve was nonlinear with an initial region

of increased compliance.

Table 4-1 R2 statistic for lines of best fit in simple shear

Region Anterior,

Initial

Anterior,

Repeated

Lateral,

Initial

Lateral,

Repeated

Posterior,

Initial

Posterior,

Repeated

R2 .590 .743 .590 .663 .993 .840

00.020.040.060.080.1

0.120.140.160.18

0 0.2 0.4 0.6Strain

Stre

ss (M

Pa)

LoBF - Anterior init LoBF - Lateral Init LoBF - Posterior InitLoBF - Anterior Rep LoBF - Lateral Rep LoBF - Posterior Rep

Figure 4-23 Simple Shear-Lines of best fit for response to initial and repeated loading



The response of the anulus ground substance to the repeated shear loading was more

compliant at low strains than the initial shear loading behaviour. At higher strains,

above approximately 50%, the repeated loading response demonstrated a higher

stiffness than the initial loading response. In general, the repeated loading response of

the tissue was more nonlinear than the initial loading response.

An analysis of variance (ANOVA) was carried out on the results for simple shear

loading. On the basis of both the F and P values, a significant difference existed

between the initial and repeated loading in each anulus regions and there was a

significant difference between the results for each anulus region.

4.7.2.2 Uniaxial compression

The posterior anulus was the stiffest and the lateral was the most compliant for both

initial and repeated uniaxial compression loading (Figure 4-24).

Table 4-2 R2 statistic for lines of best fit in uniaxial compression

Region Anterior,

Initial

Anterior,

Repeated

Lateral,

Initial

Lateral,

Repeated

Posterior,

Initial

Posterior,

Repeated

R2 .887 .838 .901 .800 .866 .802

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Strain

Stre

ss (M

Pa)

LoBF - Anterior Contd LoBF - Lateral Contd LoBF - Posterior ContdLoBF - Anterior Init LoBF - Lateral Init LoBF - Posterior Init

Figure 4-24 Uniaxial Compression - Lines of best fit for response to initial and repeated loading

Anterior Rep Lateral Rep Posterior Rep



Under uniaxial compression loading the repeated loading response was very

compliant in comparison to the initial loading response at strains below 30-40%.

Overall, the response of the anulus ground substance to both initial and repeated

loading was nonlinear but the repeated loading response demonstrated the most

significant variation in stiffness with increasing strain. At strains above 30-40% the

repeated loading response was notably stiffer than the initial loading response.

ANOVA was carried out on regression lines for the uniaxial compression results. A

comparison of the calculated F and P values with the critical values indicated that

there was a significant difference between the initial and repeated loading for all

anulus regions and the results for each anulus region.

4.7.2.3 Biaxial compression

When the deformation was measured radially the anterior anulus was the stiffest and

the lateral anulus was the least stiff during the initial and repeated loading (Figure

4-25 A). The posterior anulus was the stiffest and the anterior anulus the least stiff

when measurements were taken in the circumferential direction during the initial

loading (Figure 4-25 B). Repeated loading demonstrated a similar stiffness for the

anterior and lateral anulus when circumferential deformation was recorded.

Table 4-3 R2 statistic for lines of best fit in biaxial compression

Region Anterior,

Initial

Anterior,

Repeated

Lateral,

Initial

Lateral,

Repeated

Posterior,

Initial

Posterior,

Repeated

R2 –

Circumferential

measurement

.850 .800 .636 0.311 0.737 .700

R2 – Radial

measurement

N/A .790 .879 .923 .834 .965



A

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.05 0.1 0.15 0.2Strain

Str

ess

(MP

a)

Lat. Init. Rad. Post. Init. Rad. Ant. Cont. Rad.Lat. Cont. Rad. Post. Cont. Rad.

B

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.05 0.1 0.15 0.2 0.25 0.3Strain

Stre

ss (M

Pa)

Ant. Init. Circ. Lat. Init. Circ. Post. Init. Circ.Ant. Cont. Circ. Lat. Cont. Circ. Post. Cont. Circ.

Figure 4-25 Biaxial Compression - Lines of best fit for response to initial and repeated loading. A. Radial measurements; B. Circumferential measurements

Figure 4-25 A shows the mechanical response for the anterior anulus only during

repeated loading. ANOVA analysis indicated that there was no significant difference

between the response of the initial and repeated loading under biaxial compression

when the strain was measured radially.

The anterior and lateral anulus ground substance was stiffer when measured in the

radial direction. Radial and circumferential measurements of the response of the

posterior anulus showed a similar stiffness. Both the radial and circumferential

measurements showed a nonlinear response; however, in comparison to the uniaxial

compression and simple shear data there was less variation in the stiffness of the

Lat. Rep. Rad Post. Rep. Rad

Lat. Rep. CircAnt. Rep. Circ Post. Rep. Circ



material with increasing strain. The significant compliance observed at low strains

during repeated simple shear and uniaxial compression loading was not observed

under biaxial compression.

ANOVA were carried out on the biaxial compression results. With the exception of

the regression line for the anterior, repeated, radial response, there was a significant

difference between the initial and repeated loading in all regions for both radial and

circumferential measurements. Also, there was a significant difference between the

radial and circumferential measurements when compared across regions and loading

states.

Finally, the regression lines for each region were compared within the 3 loading

conditions of simple shear, uniaxial compression and biaxial compression. It was

determined that there was a significant difference between the regression lines for

each region.

4.7.3 Range of test data

Envelopes of behaviour were delineated by considering the response of the stiffest

and most compliant specimens under a specific loading type. For example, it may be

seen from Figure 4-26 that under uniaxial compression, the confidence thresholds for

the anterior anulus during the initial loading application were obtained from the

stress-strain data for specimens 0190-67-A-UC-02 and 0188-67-A-UC-01.

For details of the confidence boundaries on the regression lines fit to each region refer

to Appendix C.



00.5

11.5

22.5

3

0 0.2 0.4 0.6 0.8Strain

Stre

ss

LoBF - Anterior Init Confidence Limits

Figure 4-26 Range of uniaxial compression test data for the anterior anulus under initial loading

Figure 4-26 shows considerable variation between the upper and lower limits of the

maximum and minimum stress for a constant strain. With increasing strain, this

variation in stress increased.

4.7.4 Discussion

The mechanical response of the ground substance demonstrated a repeatable response

when tested after the initial load application. Possibly this repeatability was

attributable to the elastically recoverable nature of the elastin present in the anulus

fibrosus (Yu et al., 2002). While the elastin fibres present in the anulus fibrosus and

nucleus pulposus of the disc were not specifically represented in the FEM, the effects

of these fibres were incorporated in the experimental data for the ground substance.

It was initially thought that the reduction in stiffness between the initial and repeated

loading was a result of the viscoelasticity of the collagenous tissue. It was suggested

that the initial loading cycle was comparable to preconditioning the tissue and the five

minute period between the tests was not sufficient time for the tissue to recover to the

initial stiffness. However, it was not clear why some shear specimens did not show a

reduction in stiffness from the initial load to the repeated loading and maximum

strains of 40% were reached before the stiffness reduced (Figure 4-21 C). Further

experimentation was carried out to determine the cause for the reduced stiffness.

Upper limit: Specimen 0190-67-A-UC-02

Lower limit: Specimen 0188-67-A-UC-01

Stre

ss (M

Pa)



4.8 Pilot Study to Determine the Derangement Strain

These experiments aimed to acquire information on the derangement strain of the

anulus fibrosus ground substance. The derangement strain was defined in Section

4.6.1.

4.8.1 Rationale for carrying out additional experimentation

Further to the results of the stress-strain experiments on the anulus specimens, it was

postulated that the observed reduction in stiffness between the initial and repeated

loading on the majority of the specimens may have been the result of:

• Fluid loss from the matrix of the anulus fibrosus;

• The lack of sufficient recovery time for the viscoelastic solid skeleton in the

anulus fibrosus ground substance; or

• The initiation of damage in the anulus fibrosus which was sufficient to cause a

reduction in the stiffness of the material but not sufficient to prevent the

material from bearing a load upon subsequent loading.

4.8.1.1 Fluid loss

The choice of strain rate for the experimentation was intended to simulate average

physiological loading rates and to ensure that there was no fluid movement out of the

specimen. If the choice of strain rate was accurate for this purpose then the reduction

in stiffness observed in the stress-strain experiments would not have been due to fluid

loss. The accuracy of the strain rate used during the experimentation was determined

by carrying out uniaxial compression and simple shear testing at varied strain rates.

Details of these experiments are in Section 4.6.

4.8.1.2 Viscoelastic effects in the anulus fibrosus solid skeleton

It was reasonable to expect that some loss of tissue stiffness could be attributed to the

viscoelasticity of the anulus. The experimental procedure was intended to simulate

repeated loading on the anulus fibrosus and a reduction in stiffness of collagenous



tissues had been reported during repeated loading (White and Panjabi, 1978).

However, it was not clear whether this was the only cause for the reduced stiffness.

4.8.1.3 Derangement of the anulus fibrosus

It was feasible that the reduced stiffness of the anulus specimens was due to the

initiation of damage in the tissue. This damage was referred to as ‘derangement’.

Derangement of the tissue would compromise the stiffness of the material but would

not prevent it from bearing a load upon subsequent load application.

The maximum strain applied during the uniaxial compression and simple shear

loading was 50-80%. This choice of strain was based on the observed strains in the

preliminary FEM and on the necessity to obtain extensive mechanical data in order to

achieve the most accurate fit for the hyperelastic tissue model (Section 4.5.2). This

strain range was significantly higher than those observed in previous experimental

studies of the anulus fibrosus. Previous studies that used tensile and shear loads

applied strains in the range 10-30% and these maximum strains were selected based

on the potential physiological strains. On the basis of simple mathematical

calculations of strain in the anulus during flexion, it was considered that the maximum

strains experienced by the intervertebral disc in vivo could be at least 50%.

The maximum strains applied in the stress-strain experiments were higher than the

potential physiological strains in the intervertebral disc. It was hypothesised that a

likely cause for the reduction in stiffness between the initial and repeated loading was

the derangement of the anulus material. This hypothesis was tested by further

experimentation on anulus specimens under uniaxial compression and simple shear.

4.8.2 Testing to determine the derangement strain

Specimens of anulus fibrosus were obtained from an L6/7 sheep disc. Five specimens

were tested to specifically determine the strain required to initiate damage. These five

tests were intended to be a supplement to the main experimental investigation in order



to further explain the strain required to initiate damage in the anulus ground

substance.

4.8.2.1 Procedure

Individual specimens were tested under either uniaxial compression or simple shear

using the procedure outlined in section 4.5.1. It was not possible to carry out the

derangement strain tests using biaxial compression loading. A strain rate of 0.01 sec-1

was employed. The specimens were subjected to repeated loading with load cycles

one hour apart. The tissue was kept hydrated between tests with Ringers solution,

Ringers soaked muslin and plastic wrapping. Tests were carried out one hour apart to

permit the specimens to recover, ensuring that any observed reductions in stiffness

were a result of derangement in the tissue and not viscoelastic effects.

The maximum strain applied during each test was increased by approximately 5% in

the subsequent test. A maximum strain of 20% was reached in the first test as this

strain had been employed in previous experimental studies on the anulus with no

confounding effects due to tissue damage. When a non-recoverable reduction in the

stiffness of the specimen was observed it was considered that the derangement strain

had been exceeded.

4.8.2.2 Results

Plots of the stress and strain in the individual specimens were compared (Figure 4-27

and Figure 4-28). A range of strains was determined for the derangement strain of the

anulus fibrosus. This range was in the same order of magnitude for uniaxial

compression and simple shear.



A

0

0.5

1

1.5

2

0 0.05 0.1 0.15 0.2 0.25 0.3Strain

Str

ess

(MP

a)

Test a Test b Test c Test d

B

0

0.050.1

0.15

0.2

0.250.3

0.35

0.4

0 0.1 0.2 0.3Strain

Str

ess

(MP

a)

Test a Test b Test c Test d

Figure 4-27 Uniaxial compression loading. A. Derangement strain between 22 and 27%; B Derangement strain between 20 and 27%

Derangement strains for the specimens were determined by noting the maximum

strain reached before a reduction in stiffness was observed. For example, consider the

specimen in Figure 4-27 A. The results for test a and b were similar. Test c showed a

notable reduction in stiffness. Given that the specimen was permitted to recover for

one hour between testing, this reduction in stiffness was not a result of viscoelastic

effects in the tissue. Therefore, loading the specimen to a strain between 20 and 27%

resulted in derangement of the tissue and caused a reduction in stiffness upon repeated

loading. This reduced stiffness response was reproducible, which indicated that

although the matrix was deranged it had not failed.



The derangement strain under uniaxial compression was between 20 and 27% (Figure

4-27).

Simple shear loading demonstrated a derangement strain between 21 and 35% (Figure

4-28).

0.0027

0.0032

0.0037

0.0042

0.0047

A

0

0.005

0.01

0.015

0.02

0 0.1 0.2 0.3 0.4 0.5Strain

Str

ess

(MP

a)


Tests a and b showed a

similar response

Drop in stiffness in test c



0.009

0.014

0.019

0.024

0.029

B

00.010.020.030.040.050.060.070.080.09

0 0.1 0.2 0.3 0.4 0.5 0.6Strain

Str

ess

(MP

a)

Test a Test b Test c Test d Test eTest f Test g Test h Test i

0.002

0.022

0.042

0.062

0.082

0.102

C

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 0.1 0.2 0.3 0.4 0.5Strain

Stre

ss (M

Pa)


Figure 4-28 Simple shear loading. A. Derangement strain between 21 and 30%; B. Derangement strain between 30 and 35%; C. Derangement strain between 24

and 27%

Drop in stiffness in test f Test a to e

showed a similar

response

Test a and b

showed a similar

response

Drop in stiffness in test c



4.8.2.3 Discussion of the range of derangement strain of the anulus fibrosus

ground substance

The results for uniaxial compression showed a repeatable response both before and

after the derangement strain had been exceeded. However, the simple shear response

was only repeatable before the derangement strain was exceeded. This may have

been a result of the repeated derangement of the tissue as it was loaded to higher shear

strains and therefore, be a characteristic of the shear response of the anulus fibrosus

ground substance. Perhaps at strains above the derangement strain, the discontinuous

collagen fibres “pulled-through” the ground substance. Because these fibres were not

continuous between the endplates of the test specimens, there was no mechanism for

them to return to their original location in the un-deranged anulus. As higher shear

strains were reached, the fibres were displaced further from their original location and

therefore, a reproducible mechanical response could not be achieved.

In the stress-strain experiments the specimens were strained to values much higher

than the observed range of the derangement strains but showed a similar response to

those from the derangement strain tests. A reduction in stiffness occurred after the

initial test but then the attained characteristics were reproducible upon repeated

loading (Figure 4-20, Figure 4-21 and Figure 4-22). This suggested that the matrix

had been deranged during the first loading cycle but not failed. The biaxial

compression tests showed similar changes but did not result in such pronounced

losses in stiffness (Figure 4-22). Therefore, the mechanical data obtained from the

series of stress-strain experiments would provide information on the response of the

anulus fibrosus in an undamaged condition and also its response when some

derangement was present.

Knowledge of the material characteristics of the anulus fibrosus ground substance up

to 20% strain and following exposure to higher strains provided information on the

potential for anulus derangement in the FEM. Material parameters for the response of

the anulus fibrosus ground substance to the initial loading would be used in FEMs

simulating physiological loading. If the strains observed in these FEM exceeded the

range of derangement strains then material parameters for the repeated loading

behaviour of the anulus ground substance would be utilised in the FEM and the



resultant analyses would simulate physiological loading on an intervertebral disc with

some derangement present.

4.8.2.4 An hypothesis for disc degeneration

On the basis of simple strain calculations during full flexion, physiological strains in

the L4/5 intervertebral disc could be in the order of 50%. This value was based on

calculations of the maximum deformation observed in vivo. The current results for

the range of derangement strains, 20-35%, demonstrated that the expected

physiological strains would cause some permanent damage to the anulus ground

matrix. However, the matrix would still be capable of carrying stress upon repeated

loading.

Thompson et al. (2000) found that people over the age of 35 all exhibited signs of

disc degeneration. It was hypothesised that the regenerative ability of the anulus

ceased to function effectively with age and the continual damage caused to the anulus

tissue by daily activities could lead to the degenerative changes seen in the anulus.

4.9 Discussion of Regional Stiffness and Stiffening Mechanisms in the Anulus

Fibrosus Specimens

The varied stiffness in the regions of the anulus ground substance under the three

loading conditions is discussed in the following sections. This variation is shown in

Table 4-4. Possible causes for the regions of the anulus displaying stiffer or more

compliant behaviour are suggested.



Table 4-4 Comparison of stiffness between disc regions with experimental

findings for tensile loading

Loading Type Highest Stiffness Mid Stiffness Lowest Stiffness Uniaxial

Compression Posterior Anterior Lateral

Biaxial Compression:

Measured radially Anterior Posterior Lateral

Biaxial Compression:

Measured Circumferentially

Posterior Lateral Anterior

Simple Shear Anterior Lateral Posterior Tension (Acaroglu et al., 1995, Skaggs

et al., 1994, Galante, 1967)

Anterior Posterolateral

4.9.1 Uniaxial compression

The posterior anulus was the stiffest region, followed by the anterior and lateral

regions (Table 4-4).

Skaggs et al. (1994) determined there was no significant variation in the collagen

content and hydration of the anulus with circumferential region. Therefore, all disc

specimens tested should have possessed comparable collagen contents and any

variations in stiffness were not attributed to a higher density of collagen in certain

regions. Also, uniform hydration of the anulus implied that all regions demonstrated

a similar propensity to imbibe fluid. Therefore, the increased stiffness posteriorly

could not be attributed to an increased density of collagen fibres or a higher pore fluid

pressure in the posterior ground substance.

Marchand and Ahmed (1990) found the largest number of incomplete lamellae

occurred in the posterolateral anulus, and the least occurred in the anterior anulus.

They also found the lamellae were thickest in the lateral anulus and thinnest in the



posterior anulus. Yu (2001) found elastin fibres ‘densely distributed between the

lamellae’ of the anulus. The specimens in the current study were divided into three

regions, and as such specimens from the posterolateral disc according to the regional

divisions used by Marchand and Ahmed (1990) would have been classified as

posterior specimens. Of the specimens tested, the posterior specimens would contain

the most incomplete layers and the thinnest lamellae.

The increased number of incomplete lamellae posteriorly would result in a higher

number of lamellae interfaces and a higher concentration of elastin fibres in this

region. Since the lamellae were thinner posteriorly these specimens would contain

more lamellae than the anterior and lateral specimens which would also result in a

greater number of interlamellar interfaces in the posterior specimens. This indicated

that there was a greater amount of elastin fibres in the posterior specimens compared

to the anterior and lateral specimens. The observed maximum stiffness in the

posterior anulus may have been related to the higher concentration of elastin fibres.

During the uniaxial compression tests it was observed that at higher strains the

lamellae began to separate, resulting in a cleft in the anulus specimen similar to a

circumferential lesion. It was possible that the elastin fibres contributed some

resistance to this separation of lamellae under loading and therefore, contributed to

the stiffness of the anulus under uniaxial compression.

4.9.2 Simple shear

An approximate shear modulus was calculated for the disc regions during initial and

repeated loading up to 45% strain which was a near linear region on the curves. This

modulus ranged between 17 and 93kPa. This range was of a similar magnitude to the

findings of Fujita et al. (2000) who determined the shear modulus of the anulus

matrix in the axial direction ranged from 25 to 56kPa.

Under simple shear loading the anterior anulus demonstrated the highest stiffness and

the posterior anulus the lowest stiffness (Table 4-4). This was similar to the results of



tensile testing carried out by Acaroglu et al. (1995), Skaggs et al. (1994) and Galante

(1967).

The correlation between the results of tensile loading and shear loading may have

been a result of the load bearing function of the collagen fibres under these loading

modes. In tension, the collagen fibres would ultimately provide the anulus with the

majority of tensile stiffness. During shear loading, the collagen fibres performed a

similar role, however, the stiffness provided was a result of the pull-out strength of the

fibres from the anulus ground substance rather than strength due to a continuous

connection between the cartilaginous endplates. At very low strains the shear

stiffness was provided by the ground substance, but as the strain increased beyond the

laxity range of the collagen fibres in the anulus, these fibres began to “slide” through

the ground substance and provide some shear stress resistance.

4.9.3 Biaxial compression

The following sections provide a discussion of the deformation observed in the anulus

specimens under biaxial compression loading.

4.9.3.1 Deformation mechanism in the radial and circumferential regions

A B

Figure 4-29 Anulus specimen viewed from the circumferential direction. A. Undeformed specimen – axially aligned collagen fibres; B. Deformed specimen,

discontinuous fibres resist compression of lamellae

Strain in the radial direction was a measurement of how much the lamellae

compressed. When the specimen was viewed circumferentially and therefore,

measured radially the collagen fibres were orientated axially through the specimen

(Figure 4-29). These fibres were not continuous but the compression of the anulus in

Collagen

Radial

Collagen fibres increased in length due to curved deformed shape



a radial direction would have resulted in some stretching of the fibres due to their

axial orientation. This stretching would have increased the biaxial compressive

stiffness. This increase was due to the pull-out strength of the collagen fibres from

the anulus ground substance rather than the fibre tensile strength. The pull-out

strength of the fibres was due to the friction created between the fibres and the ground

substance as they realigned in this material.

A B

Figure 4-30 Anulus specimen viewed from the radial direction. A. Undeformed specimen – collagen fibres orientated at 30o to endplates; B. Deformed specimen

– collagen fibres become more axially aligned

Conversely, when the specimen deformation was measured in the circumferential

direction, the fibres were orientated diagonally through the lamellae of the specimen

(Figure 4-30). Compression of the tissue in circumferential direction would result in

the collagen fibres becoming more axially aligned due to the compression of the

ground substance. In this orientation, the pull-out strength of the collagen fibres

would not have provided a significant contribution to the biaxial stiffness of the

specimen as it was the compression of the ground substance rather than the stretching

of the fibres which resulted in their increasingly axial orientation.

4.9.3.2 Difference in regions of highest stiffness when measured radially and

circumferentially

When strain was measured in the radial direction, the anterior anulus was the stiffest

and the lateral anulus was the least stiff. This was in contrast to the results from the

circumferential measurements where the posterior anulus was the stiffest and the

anterior anulus was the least stiff. The prevalence of complete lamellae in the anterior

anulus resulted in less disruptions to the fibres in this region. In keeping with the

previous explanation of the radial anulus deformation, it was postulated that the lower

concentration of disrupted fibres permitted a greater resistance to compression of the

Collagen fibres

Circumferential



lamellae in the anterior anulus compared to the remaining disc regions. This resulted

in a higher biaxial compressive stiffness observed anteriorly when the biaxial

compression strain was measured in the radial orientation.

Marchand and Ahmed (1990) observed a fibre angle as high as 70o in the posterior

anulus. A steeper fibre inclination in the undeformed specimen in Figure 4-30 A

would result in a deformed specimen with collagen fibres inclined more axially. The

observed stiffness of this deformed specimen would be a result of both the

compressive strength of the anulus ground substance and a contribution due to the

pull-out of the more axially aligned collagen fibres. The more axially aligned

collagen fibres provided opposition to compression by creating resistance as they

were drawn through the ground substance.

4.9.3.3 Drop in stiffness between the initial and repeated loading and

derangement strains for biaxial compression

Half the specimens measured radially did not demonstrate a drop in stiffness between

the initial and repeated loading. If the specimen did show a decrease in stiffness it

was not of a high magnitude and did not extend for the full range of strains applied.

However, the results of the circumferentially measured specimens showed that 70%

of the specimens demonstrated a drop in stiffness between the initial and repeated

loading and the remaining specimens showed no difference.

Due to the higher stiffness of the anulus when measured radially, the maximum

strains reached in the initial testing on these specimens were 10-20%. However,

strains of 10-33% were reached during the initial loading on specimens measured

circumferentially. It was postulated that the reduction in stiffness observed in the

circumferentially measured specimens may have been because the derangement strain

of the material was exceeded. This theory was supported by the observation of a

maximum initial loading strain of 18-33% in the circumferentially measured

specimens that had shown a decrease in stiffness and a maximum initial strain of 7-

12% in the specimens which did not show a drop in stiffness between the initial and

repeated loading. This suggested that the derangement strain when the biaxial



compression was measured circumferentially was between 12 and 18% which created

an associated axial strain of 30-50%.

The radially measured specimens that demonstrated a drop in stiffness reached a

maximum initial strain of approximately 18% while those which did not show a

decreased stiffness had reached a maximum strain of 13%. This suggested that the

derangement strain when measured radially under biaxial compression was between

13 and 18% which was the same as the estimated derangement strain range for the

circumferential orientation.

4.10 Discussion of Edge Effects

When loaded the cartilaginous/bony endplates on the specimens of anulus ground

substance caused some restriction to the deformation of the specimen. This behaviour

is characteristic of edge effects and is shown under biaxial compression loading in

Figure 4-31 although edge effects were present under all loading modes. Edge effects

resulted in increased shear stresses near the endplates and caused an increase in the

overall stiffness of the tissue.

A B C

Figure 4-31 Deformation under biaxial compression loading. A. Undeformed specimen; B. Specimen without end constraint; C. Specimen with end constraint

Ideally, the experimental testing would have been carried out on anulus specimens

which did not include any cartilaginous/bony endplate and with an attachment to the

testing fixtures that did not cause any restriction to the deformation of the tissue. To

avoid edge effects experimentation should be carried out on specimens with an aspect

ratio between approximately 1:10 and 1:20 (Figure 4-32).

Biaxial compression force, F

F F

F Endplate

Increased shear stress near endplates

F

F

F

F

F

F

F

F



A B

Figure 4-32 Aspect ratio. A. Aspect ratio on the test specimens; B. Aspect ratio to avoid edge effects

However, it was not possible to obtain specimens with a suitably large aspect ratio

(Figure 4-32 A) since the average height of the sheep discs was between 1.5-2mm and

the available specimen sectioning techniques would not accurately provide specimens

of anulus fibrosus with a cross-sectional width less than 2mm. If this cross-sectional

dimension was below 3mm manufacture of a piston with a small enough cross-

sectional area to use for the biaxial compression tests was not possible. The testing

methods required the attachment of two opposite faces of the test specimens to the

fixtures on the Hounsfield and on the components of the biaxial compression device.

This attachment would have caused restriction to the deformation of the specimen if

the endplates were not present; therefore, the endplates were preserved on the

specimens for ease of experimental setup.

It should be noted that the constraint at the ends of the specimens and the non-uniform

deformation of the specimen depicted in Figure 4-31C would have resulted in a slight

axial force. This axial force would have caused an increase in the deformation of the

specimen along the other two orthogonal axes (i.e. radial and circumferential

directions). However, the edge effects caused a constraint to the deformation of the

specimen along these axes. Since these conditions resulted in opposing effects on the

deformation of the specimen, the overall errors in the measured deformations were

deemed to be negligible.

3

2 10-20

1



4.11 Potential Sources of Error in the Results

Table 4-5 Potential sources of error in the experimental data

Source of Error Accuracy Percentage error Specimen

dimensions 3 ± 0.2mm ± 6.67%

Hounsfield (Section 4.4.6.1)

- 2.2%

Sigmoscope (Section 4.4.2.4)

± 0.05%

Edge Effects Unknown Druck pressure

calibrator ± 0.1%

Norgren pressure regulator

0.435psi, with maximum pressure

of 30psi applied ± 1.45%

Piston friction (Section 4.4.6.1)

- 1.07%

Table 4-5 details possible sources of error in the experimental results. The cumulative

effect of these potential sources of error was + 8.27% and – 11.54% variation in the

final experimental data presented. These figures did not take into account the possible

increase in stiffness resulting from the edge effects because a numerical value for this

error could not be determined.

4.12 Conclusion

Experimental data for the response of the anulus fibrosus ground substance was

obtained under uniaxial compression, biaxial compression and simple shear loading.

These data allowed for an improved hyperelastic equation for the anulus ground

substance to be developed and this is the subject of Chapter 5.

There was a significant difference between the response of the tissue to the initial

loading and to repeated loading for all load types except biaxial compression in the

anterior anulus in the radial direction. This difference was attributed to the



derangement of the anulus fibrosus ground substance which occurred at strains

between 20 and 35% during uniaxial compression and simple shear and between 12

and 18% during biaxial compression. Assessment of the effect of strain rate on the

tissue response indicated that the rate of 1%/sec which was employed for all testing

was a suitable choice. The results from this testing provided the necessary data for

determination of a set of hyperelastic parameters for the anulus fibrosus ground

substance.

It was hypothesised that a possible mechanism for degeneration in the anulus fibrosus

related to the derangement of the anulus fibrosus ground substance during daily

activities. Possibly, in younger discs there are biological or biochemical processes

active which permit the recovery of the anulus. With age these processes may cease

to function effectively and subsequently, signs of degeneration become evident.


Chapter 5: Determining Hyperelastic Parameters for the Anulus Fibrosus Ground Substance 203

CChhaapptteerr

55

DDeetteerrmmiinniinngg HHyyppeerreellaassttiicc

PPaarraammeetteerrss ffoorr tthhee AAnnuulluuss

FFiibbrroossuuss GGrroouunndd SSuubbssttaannccee

Analyses of the preliminary FEM in chapter 3 suggested that while the Mooney-

Rivlin strain energy equation captured the material response of the anulus ground

substance in compression, it oversimplified the shear response such that the shear

stresses may have been higher/lower than in the real material for shear dominated

loading conditions. A significant limitation of this equation was the inherent

assumption of linearity during shear loading. The results of simple shear loading

presented in chapter 4 indicated that the anulus ground substance behaved nonlinearly

under simple shear loading. This suggested that FEMs of the intervertebral disc and

other biological tissues which had been developed by previous researchers did not

accurately represent the nonlinearity of the material in shear dominated loading states.

This chapter presents the determination of improved hyperelastic properties for the

anulus ground substance by fitting hyperelastic strain energy equations which permit

nonlinear behaviour under shear loading.

5.1 Chapter Overview

Hyperelastic constants for the improved material model were determined using the

Abaqus/Standard software by the input of experimental stress-strain data for uniaxial

compression and biaxial compression and calculated data for the pure shear response.



The simple shear data required manipulation to obtain representative data for the pure

shear response of the tissue.

Several hyperelastic strain energy equations were considered, however, the Ogden

and the polynomial equations provided the best fit to the experimental data collected

in Chapter 4. It was demonstrated in Chapter 4 that the mechanical response of the

anulus varied with anulus region. Two approaches to modelling the anulus ground

substance were used:

1. A simplified homogeneous model was developed in which the entire anulus

ground substance was assigned the same hyperelastic constants; and

2. An inhomogeneous model was developed in which hyperelastic constants

varied with location in the anulus, according to the regional variations

identified in Chapter 4.

The circumferential inhomogeneity of the anulus was represented by determining a

set of hyperelastic constants for the 3 anulus regions. However, implementation of

these inhomogeneous material properties was preceded by the implementation of a set

of homogeneous hyperelastic constants. These homogeneous hyperelastic constants

were determined because it was desirable to introduce the inhomogeneous material

properties progressively. It was first ascertained whether the improved properties of

the anulus fibrosus ground substance were compatible with the other FEM

components and whether they were mechanically capable of carrying the loads

applied to the intervertebral disc.

The hyperelastic constants could be calculated manually using the least squared error

approach; however, the constants determined using Abaqus were found to provide a

good fit for the experimental data. Therefore, both the homogeneous and

inhomogeneous hyperelastic constants were calculated by Abaqus. These constants

were determined for both the intact and the deranged response of the tissue. The

homogeneous hyperelastic constants were fitted using a 2nd order polynomial strain

energy equation and the inhomogeneous constants were fitted using a 3rd order Ogden

equation. These strain energy equations demonstrated a similar response to the

experimental data.



5.2 Manipulation of Experimental Regression Lines to Obtain Input for the

Strain Energy Equations

In order to determine hyperelastic parameters for a material Abaqus required input of

experimental data for specific testing modes. Uniaxial compression and biaxial

compression experimental data was obtained for the anulus ground substance.

Experimental results for simple shear loading were obtained, but the necessary

experimental input for Abaqus was pure shear data. This section details the algorithm

that was developed to convert simple shear data to pure shear data.

5.2.1 Simple shear compared to pure shear (Treloar, 1975)

Simple shear loading involves the sliding of parallel planes in a material. This creates

a constant volume deformation of the specimen. The lateral faces of the specimen are

transformed into parallelograms due to the deformation and the shear deformation is

measured as the tangent of the angle θ (Figure 5-1 A). The height of the specimen is

maintained during the deformation. Simple shear deformation creates no strain in the

direction perpendicular to the skewed faces and is characterised by a rotation of the

principal directions of strain in relation to the orientation of the shear loading applied.

A B

Figure 5-1 Shear deformation detailing the stretch ratios. A. Simple shear; B. Pure shear. The shear stress is denoted as τxy, the principal stresses are σ1 and

σ2 and λ is the extension/stretch ratio (Section 2.6)

This is in contrast to pure shear deformation (Figure 5-1 B). This loading creates a

constant volume deformation but the axes of stress are orientated parallel to the

τxy

y y

x x τxy

1

λ

1/λ

σ2

σ1

σ1

σ2

1

θ



principal directions of stress and strain. Pure shear deformation is obtained by

applying principal stresses, σ1 and σ2, with σ3=0.

The extension/stretch ratios, λi=1,2,3, associated with both simple and pure shear are the

same. This is due to the zero volume change associated with both these deformations

which creates a third strain invariant of unity.

13211

1;1; λλλλλ ===

Eqn 5-1

Pure shear is comparable to simple shear without the rotation. All stretch in the

sheared material is defined using only the first stretch ratio. The maximum principal

extension ratio, 1λ is defined as the major axis of the strain ellipsoid in a shear

deformed cube (Figure 5-4).

Treloar (1975) defined simple shear strain using Eqn 5-2.

11

1)tan( λλφγ −==

Eqn 5-2 Simple shear strain.

5.2.2 Manipulating simple shear data to obtain pure shear data

Treloar (1975) stated that it is possible to use the strain energy per unit volume, U, for

a deformed specimen to determine the equivalent simple shear stress, τ, for a pure

shear loading case. Therefore, it is possible to determine the pure shear stress/strain

state associated with a simple shear loading case.

Under simple shear the work done on the material is due to the simple shear stress

creating the shear angle θ (Figure 5-2).



where, τxy is the simple shear stress

A B

Figure 5-2 Simple shear loading on a cubic specimen. A. undeformed; B. deformed

Using Treloar’s expression for simple shear strain (Eqn 5-2) an expression for work

on a material resulting from the simple shear stress is stated in Eqn 5-3.

dydDdzdxFDFW xy .;..; γτ ==×=

therefore,

dyddzdxW xy .... γτ=

where, W = work; F = force; D = displacement

Eqn 5-3

The strain energy density, U, or work per unit volume is expressed in Eqn 5-4.

γτ dU xy .=

Eqn 5-4

The work done on a specimen subjected to pure shear loading is a result of the stress

σ1. The second principal stress σ2 does not perform any work since the extension

ratio in this direction remains at 1 during the deformation (Figure 5-3).

θ

dz

dy

dx Displ.,D

dy

τxy



Figure 5-3 Pure shear loading on a cubic specimen. A. undeformed; B. deformed

Define f as the force which generates the pure shear stress, σ1. An expression for

the work on the specimen is stated in Eqn 5-5.

dyddzdxWdydD

dzdxf

.....

..

11

1

1

λσλ

σ

=∴==

Eqn 5-5

From this, the strain energy per unit volume is expressed in Eqn 5-6.

1. λdfU =

Eqn 5-6

When the strain energy density for both the simple shear and pure shear are equated,

an expression for the pure shear stress in terms of the simple shear stress is


11 .

λγ

τσdd xy

xy=

Eqn 5-7

Using Eqn 5-2 for simple shear strain, the derivative of the simple shear strain with

respect to the maximum principal extension ratio was determined. Thus an

expression for the pure shear stress in terms of the simple shear stress and extension

ratio is stated in Eqn 5-8.

dz

dy

dx

λy = λ1

λz = 1/λ1

λx = 1

Where, σ1=force per unit unstrained area

σ1

σ2



)1.( 211−+= λτσ xy

where, 1λ = the maximum principal simple shear extension ratio

Eqn 5-8 Pure shear stress expressed in terms of simple shear stress and the maximum principal simple shear extension ratio

5.2.3 Principal extension ratios for Simple Shear deformation

The directions and magnitudes of principal extension in a specimen are described

using a strain ellipsoid (Treloar, 1975). A circle is drawn within the confines of an

unstrained cube. After the application of either simple or pure shear the deformed

shape of this cube is redrawn and the strained shape of the circle is an ellipse (Figure

5-5). In the case of pure shear the major and minor axes of the strained ellipsoid are

parallel to the direction of the applied stress (Figure 5-4).

Figure 5-4 Unstrained circle and strain ellipse for pure shear loading

The major and minor axes of the strain ellipse define the maximum and minimum

extension ratios, respectively (Figure 5-5). Simple shear loading results in a strain

ellipse with a major axis which is rotated from the horizontal direction. Each point on

the locus of the unstrained circle is displaced by dx due to the shear deformation. The

resulting shape is an ellipse (Figure 5-5). In Figure 5-5, λ1 is the maximum principal

extension ratio and λ2 is the minimum principal extension ratio and they are

perpendicular to one another.

Strain ellipse

Unstrained circle

Pure strain on a cube σ1



A B The maximum displacement of any point on the unstrained circle was dx

Figure 5-5 Simple shear deformation A. Unstrained cube and strained parallelogram; B. Unstrained circle and the strain ellipsoid (Schematic B

obtained from Reish and Girty, 2001)

In the case of the experimental testing carried out on the anulus specimens, the

unstrained shape of the specimen was a rectangle. Therefore, an unstrained ellipse

was deformed into a skewed ellipse. An algorithm was developed to numerically

determine the magnitude and orientation of the principal extension ratios in this

strained ellipse. This algorithm was based on the equation for an ellipse that was

orientated with the major axis in the y direction (Eqn 5-9). When the ellipse was

strained the y values remained constant and the x values varied by a value between

zero and dx (Figure 5-6).

( )( )

( )( )

1

2

22

2

2

22

.2

=−

+−−

Y

Yy

X

XyYdxx

Where, the centre was

2,

2YX

Eqn 5-9 Equation for strained ellipse

dx

Original x of a point on the new ellipse

halla




Figure 5-6 Schematic of the deformation of the test specimen during simple shear loading

This equation was parameterised in terms of cosine and sine and expressed in vector

form as )(tr′ . The curvature for the strained ellipse was determined using Eqn 5-10.

)(

)()(3

tr

trtrK

′

′′×′=

Eqn 5-10 Equation for curvature

The maximum and minimum of the curvature, K, were the turning points of the

strained ellipse and therefore, were the locations of the intercept of the major and

minor axes with the strained ellipse. Once these points were found the equation for

the lines representing the major and minor axes were plotted and their lengths

determined.

This procedure was automated using Matlab executable files (Appendix A). The

input for the code was a matrix containing the simple shear deformation dx. These

deformations were obtained from the regression lines fit to the experimental data for

simple shear. A matrix of associated maximum and minimum principal stretch ratios

was output from the Matlab code. The stretch ratios were converted to engineering

strain which was the form required by Abaqus.

Using values for the maximum principal stretch ratios Eqn 5-8 was utilised to

determine associated pure shear stresses for these stretch ratios. These data for pure

shear stress were input into Abaqus.

Origin

Unstrained ellipse Simple shear strain ellipse

dx

x

y



5.2.4 Average Biaxial Compression Data

The experimental results for biaxial compression indicated that there was a significant

difference between the results for radial and circumferential measurement of

specimen deformation. It was considered that use of the experimental results for

either of these deformation orientations would either overstate or understate the

stiffness of the anulus. Therefore, the stress-strain data input to Abaqus for biaxial

compression was the average of the response for the radial and circumferential

measurement directions. In retrospect, given the accuracy of the fit between the

experimental data and the hyperelastic strain equations outlined in section 5.3 this was

not a significant source for error.

5.3 Approach to Choosing Hyperelastic Models for the Anulus Fibrosus

Ground Substance

The response of the anulus fibrosus ground substance during initial loading was

determined to be significantly different to the response under repeated loading

(Section 4.5). This difference was modelled by developing a set of hyperelastic

parameters for both initial and repeated loading. Separate analyses were carried out

for each loading circumstance.

Statistical analysis of the regression lines fit to the experimental results demonstrated

that the anulus ground substance in the intervertebral disc was a circumferentially

inhomogeneous structure. This was in keeping with the findings of previous

researchers (Acaroglu et al., 1995, Skaggs et al., 1994, Galante, 1967). It was

desirable to model the varied mechanical response of the anterior, lateral and posterior

anulus fibrosus ground substance. In order to do this, separate hyperelastic

parameters were obtained for each of these regions and the 3D continuum elements in

the anulus of the model were divided into 3 regions. However, the FEM of the

intervertebral disc was a comparatively complex model when only one material

description was provided for the entire anulus fibrosus. Introduction of

inhomogeneity in the anulus fibrosus would increase this level of complexity

considerably.



Before the inhomogeneous material properties were implemented a set of average

homogeneous hyperelastic parameters were developed for the entire anulus fibrosus

ground substance. This model was based on the results from the compression and

shear testing over all the regions of the disc. The use of this homogeneous

hyperelastic model allowed assessment of how effectively a representative sample of

the experimental results would perform in the model. Analysis of the FEM using the

homogeneous parameters ascertained whether the experimental properties were

numerically compatible with the other materials in the model and mechanically

capable of carrying the loads applied to the intervertebral discs in vivo.

5.3.1 Possible Strain Energy Equations for the Anulus Fibrosus Ground

Substance

The primary requirements for the hyperelastic constitutive model being developed in

this chapter included the following:

• The strain energy formula used to describe the anulus fibrosus ground substance

needed to reproduce the materials nonlinearity in shear loading;

• The strain energy equation implemented was a well tested and robust equation.

The intention of this requirement was to avoid the use of a hyperelastic strain

energy model which had been developed for a specific purpose or system. Such

equations may not have been robust for a wide range of strains or for the range

of stress states likely to be encountered when modelling the entire disc during

flexion/extension, lateral bending and axial rotation loading conditions; and

• That the equation incorporated the assumptions of incompressibility

• That the derivation of the equation was strain-rate-independent and could

simulate isotropic material behaviour.

There were a considerable number of strain energy equations which had been

developed and could potentially be applied to the ground substance. A critique of the

models which were considered for use in the FEM is provided in the following

sections.



5.3.1.1 Veronda and Westmann

[ ] )3.(2.

1. 221))3(

11(2 −−−= − I

CCeCU IC

where, C1 and C2 are material constants; and I1 and I2 are the first and second strain invariants

Eqn 5-11 Veronda and Westmann strain energy equation

Veronda and Westmann (1970) proposed a strain energy equation which incorporated

nonlinear behaviour, was strain-rate-independent and isotropic. This equation was

validated using experimental results from feline skin.

An attempt to implement the Veronda and Westmann (1970) strain energy equation

was not successful. The fit between the experimental and theoretical data was

unacceptable. It was later noted that Veronda and Westmann (1970) only validated

the equation under uniaxial loading and additionally, Crisp (in Fung, 1972) mentioned

that this particular formula was only suited to uniaxial tension.

5.3.1.2 Ogden

The Ogden strain energy equation (Eqn 5-12) was developed using the assumption of

nonlinear shear behaviour. This equation had been widely applied to various

engineering applications for large strain rubber elasticity and compressible or

incompressible materials. The use of the Ogden hyperelastic equation to represent

biological tissues is well documented. Assumptions inherent in the Ogden strain

energy equation also included isotropy and strain-rate independence. Therefore it was

considered to be a candidate strain energy equation for use in the FEM of the

intervertebral disc.

)3.(.2

3211

2 −++= −−−

=∑ iii

N

i i

iU ααα λλλαµ

where, αi and µi are experimentally determined material parameters

Eqn 5-12 Ogden strain energy equation



5.3.1.3 Extended Mooney equation

Mooney further developed the classic Mooney-Rivlin equation to account for the

nonlinearity of rubbers when subjected to large shear deformations (Eqn 5-13). 23

11

23

11

11 ∑∑∑∑=

∞

==

∞

=

−+

+=

ini

ni

nini

ni

n

BAUλ

λλ

λ

where, A, B = material constants

Eqn 5-13

This equation was later stated to be inaccurate (Rivlin, 1984).

5.3.1.4 Polynomial

Tschoegl (1971) found that the use of higher order combinations of the Mooney-

Rivlin strain energy equation provided for closer agreement between experimental

and theoretical results, especially at high strains. Consequent to this finding, various

strain energy equations were proposed which extended the Mooney-Rivlin strain

energy equation to include polynomial combinations of the expressions, (I1-3) and

(I2-3). These polynomial strain energy equations were nonlinear for shear loading.

For incompressible materials, the polynomial equation as stated in the Abaqus Theory

Manual (§ 4.6.1) is defined in Eqn 5-14.

jiN

jiij IICU )3()3( 21

1

−−= ∑=+

where, Cij are material constants

Eqn 5-14

The maximum value of N which can be defined using the Abaqus software is N=2

(Eqn 5-15).

2

2022

1202111201110 )3.()3.()3).(3.()3.()3.( −+−+−−+−+−= ICICIICICICU

Eqn 5-15

The use of the polynomial equation for mechanical applications had been

demonstrated by several previous researchers (Pearson and Pickering, 2001; MARC

White paper, 1996; Juming et al., 1997). However, there was a lack of previous



publications detailing the use of the polynomial strain energy equation for modelling

biological tissues. Despite this, the polynomial equation, with an N order of 2 was

thought to be a good representation of the material behaviour of the anulus fibrosus

ground substance on the basis of the criteria stated in Section 5.3.1.

5.3.2 Verification of the Abaqus Algorithm used to Determine Hyperelastic

Parameters

As defined in §2.6 an expression for the uniaxial compression, biaxial compression or

simple shear nominal stresses in a material was derived from the particular strain

energy density function assigned to the material. These expressions for nominal

stress were expressed in terms of strain invariants and a series of constants.

Generally, the constants in hyperelastic constitutive models have no intuitive

relationship to the stiffness of the material in a specific loading mode which is in

contrast with classic linear elastic materials.

The parameters which define hyperelastic material behaviour are commonly

calculated using a least squares approach (Twizell and Ogden, 1983, Weiss et al.,

2001, Vossoughi, 1995). The Abaqus pre-processor provided a function whereby raw

experimental data from uniaxial compression, biaxial compression and/or pure shear

tests could be input and hyperelastic parameters for the particular hyperelastic

formula calculated. A least squared error algorithm is implemented in the Abaqus

pre-processor to determine these constants and output an assessment of the stability of

this particular hyperelastic function when the calculated parameters were used.

The Abaqus documentation provided little information on specifically how Abaqus

obtained the hyperelastic parameters. Because these constants were an integral part of

the FEM, it was considered necessary that the algorithm employed by Abaqus was

verified.

To achieve this, a Matlab code was written which implemented a least squared error

algorithm to determine parameters for a 2nd order polynomial equation (Appendix A)

under an unconfined uniaxial compression loading condition.



The 2nd order polynomial strain energy equation for an incompressible material is

defined in Eqn 5-16.

22022111

2120201110 )3.()3).(3.()3.()3.()3.( −+−−+−+−+−= ICIICICICICU

Eqn 5-16 Polynomial strain energy equation

Using the principal of virtual work for incompressible materials (Eqn 5-17) an

expression for the nominal stress vs. strain invariant relationship in a specific loading

mode can be determined.

22

11

.. IIUI

IUU δδδ

∂∂

+∂∂

=

Eqn 5-17 Principal of virtual work

The extension ratios for unconfined uniaxial compression loading are stated in Eqn

5-18.

;1 Uλλ = 21

32−== Uλλλ

Eqn 5-18 Stretch ratios for uniaxial compression

Expressions for the strain invariants are shown in Eqn 5-19.

3213

23

22

21

2

23

22

211

..

111

λλλλλλ

λλλ

=

++=

++=

I

I

I

Eqn 5-19 Strain invariants for an incompressible material

In accordance with the relationship for incompressible materials as stated in Eqn 5-20,

third strain invariant, I3 is equal to 1 and the first and second strain invariant, I1 and

I2, are stated in Eqn 5-21.

1.. 321 =λλλ

Eqn 5-20 Relationship between principal extension ratios in an incompressible

material



UU

UU

I

I

λλ

λλ

.2

.22

2

121

+=

+=−

−

Eqn 5-21 Strain invariants for uniaxial compression

Using the equation for virtual work (Eqn 5-16) the virtual strain energy potential in

uniaxial compression is expressed in Eqn 5-22.

UU

UU

IIUI

IUU δλ

λδλ

λδ .... 2

2

1

1 ∂∂

∂∂

+∂∂

∂∂

=

Eqn 5-22 Equation for virtual work

The relationship between strain energy, force and displacement can be expressed in

terms of virtual quantities Eqn 5-23.

UUTU δλδ =

where, δU = virtual strain energy potential

TU = nominal uniaxial stress

δλU = virtual extension ratio

δ denotes a small change

Eqn 5-23

Using Eqn 5-22 and Eqn 5-23 an expression for the nominal uniaxial stress, TU, is


UUU

IIUI

IUT

λλ ∂∂

∂∂

+∂∂

∂∂

= 2

2

1

1

..

Eqn 5-24

Using the expressions for the polynomial strain energy equation (Eqn 5-16) and for

the strain invariants in an incompressible material (Eqn 5-21) an expanded

relationship for TU is determined (Eqn 5-25).

−+−+−

+−++−= −

)3.(.2))3.(3.()3.(..2.

).1(22022111

12001103

ICIICICCC

TU

UUUU λ

λλλ

Eqn 5-25 Nominal stress based on the virtual work equation



The constants, C10, C01, C20, C11 and C02 in Eqn 5-25 were determined using the least

squared error method.

The least squared error equation uses experimental data to fit constants for any

predefined equation (Eqn 5-26). This equation minimised the error, E, between the

experimental curve and the theoretical curve defined by the equation. In this case, the

equation to be fit was that for TU (Eqn 5-25).

21

0))((∑ −

=−=

n

i ii yxFE where, F(xi) = theoretical function yi = experimental values

Eqn 5-26 Least squared error

The least squared error equation used by Abaqus was based on Eqn 5-26 but was

normalized for the experimental result, TUTheoretical (Eqn 5-27). This was obtained

from the Abaqus theory manual, §4.6.2.

21

)1(∑=−=

n

i alExperimentU

lTheoreticaU

TTE where, TU

Theoretical = TU

Eqn 5-27 Least squared error equation normalized for the experimental result

Substituting Eqn 5-24 into Eqn 5-26 gave an expression for the error in the uniaxial

compression nominal stress.

2

1

02

02212

11

122001103

)3.2.(.2))3.2.(3.2.(

)3.2.(..2.).1(2

∑ −

=−−−

−−

−

−++−++−+

+−+++−

=

n

ialExperiment

U

UUUUUUU

UUUUU

T

CC

CCC

E

λλλλλλλ

λλλλλ

Eqn 5-28 Least squared error expression for uniaxial compression

The least squared error procedure involved determining the derivative of the error

equation with respect to the constants, C10, C01, C20, C11 and C02. These derivative

equations were set to zero and solved simultaneously for the constants. Setting the

derivative equations to zero ensured that the constants were values which minimised

the error, E.



This Matlab algorithm was verified using a set of contrived experimental uniaxial

compressive stress data. A comparison between the experimental stress response and

the stress determined using the hyperelastic constants obtained from the least squared

error algorithm showed good agreement (Figure 5-7).

-0.0395-0.0345-0.0295-0.0245-0.0195-0.0145-0.0095-0.00450.0005

0 0.2 0.4 0.6 0.8 1

Uniaxial Extension Ratio

Uni

axia

l Nom

inal

Str

ess

(MPa

)

T-ExperimentalT-Theoretical (determined using least squares)

Figure 5-7 A comparison between the experimental data for uniaxial compression and the theoretical stress calculated using hyperelastic constants

obtained from the least squared error algorithm.

In order to assess the accuracy of the least squared error algorithm used by Abaqus, a

set of experimental data for the nominal uniaxial compression stress and strain were

obtained from a specimen. These were input into both the least squared error

algorithm and into an Abaqus input file for pre-processor analysis. The constants

calculated from both sources are presented in Table 5-1.



Table 5-1 Comparison of hyperelastic parameters determined by Abaqus and

determined using the Matlab algorithm for the polynomial, N=2 hyperelastic

equation

Abaqus Pre-Processor Matlab Algorithm

C10 = 0.01879 C10 = -0.0239

C01 = -0.01202 C01 = 0.025

C20 = 0.01242 C20 = -0.0073

C11 = -0.005312 C11 = 0.000286

C02 = 0.0008385 C02 = -0.0000346

A comparison of these constants showed that, while they were generally of the same

order of magnitude, the numerical values were different. The experimental response

was compared with the theoretical response calculated with the two sets of constants

(Figure 5-8). This comparison showed a good fit. The maximum percentage error of

7% between the theoretical results and the experimental results occurred at an

extension ratio of 0.25. A maximum error of 6% between the results from Abaqus

and the experimental data occurred at an extension ratio of 0.30.

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

00 0.2 0.4 0.6 0.8 1 1.2

Uniaxial Extension Ratio

Uni

axia

l Nom

inal

Str

ess

(MPa

)

Experimental Theoretical - AbaqusTheoretical - Matlab algorithm

Figure 5-8 Comparison of the theoretical response calculated using Abaqus constants and the theoretical response calculated using the Matlab algorithm

with the experimental data



On the basis of these results it was thought that the least squared error algorithm

employed in the Abaqus pre-processor produced accurate results. Further Matlab

code could have been developed to implement the additional loading states and to use

other hyperelastic strain energy expressions such as the Ogden equation. However, it

was considered that there was a high potential for human error involved in

determining and coding the lengthy differential equations. Additionally, there would

have been considerable time spent coding the equations which was unnecessary given

the demonstrated accuracy of the least squared error algorithm employed by Abaqus.

5.4 Strain Energy Equations Used for the Anulus Fibrosus Ground Substance

The choice of the final strain energy equation for the disc regions was based on:

• the root-mean-squared error associated with the model;

• the observed correlation between experimental and finite element analyses

stress-strain response for uniaxial compression, biaxial compression, simple shear

and planar tension;

• the order of the model and therefore the computational time associated with

solution of analyses; and

• to a lesser degree on the level of stability of the model as stated in the results

of pre-processing carried out by Abaqus.

With reference to the four criteria outlined in Section 5.3.1 it was decided that the

polynomial and Ogden equations were well suited for application to the anulus ground

substance.

5.4.1 Inhomogeneous hyperelastic model for the ground substance

For the purpose of the following section, 5.4.1, the term ‘model’ will refer to the

algorithm associated with the constitutive equations for hyperelasticity.

Twizell and Ogden (1983) note that in fitting parameters to the Ogden strain energy

equation, increasing orders of the equation would provide improvements in the fit

between the experimental and theoretical results. These researchers also state that the



material constants which were found individually from the three experimental tests –

uniaxial compression, biaxial compression and simple shear – should be similar

values. Any variation in these values would be due to experimental error. However,

use of higher orders of the Ogden equation would reduce this variation.

An attempt was made to fit both the polynomial and the Ogden strain energy

equations to the experimental data from the disc regions. The same strain energy

theory was applied to each region. On the basis of the criterion in section 5.4 the

experimental data from the majority of the disc regions were best fit with the Ogden

equation.

Consider the results for the anterior anulus ground substance during the initial loading

(Figure 5-9).



A -3

-2.5

-2

-1.5

-1

-0.5

0-0.7 -0.5 -0.3 -0.1

Uniaxial Compressive Strain

Uni

axia

l Com

pres

sive

Stre

ss (M

Pa)

B -0.5

-0.4

-0.3

-0.2

-0.1

0-0.3 -0.2 -0.1 0

Biaxial Compressive Strain

Bia

xial

Com

pres

sive

Stre

ss (M

Pa)

C

0

0.05

0.1

0.15

0.2

0 0.1 0.2 0.3 0.4Planar Shear Strain

Plan

ar S

hear

Str

ess

(MPa

)

Figure 5-9 Comparison of the theoretical results from the Ogden, N=2, N=3, N=4 and Polynomial, N=2 hyperelastic strain energy equations with the experimental

results for A. Uniaxial compression, B. Biaxial compression, C. Planar shear. Experimental; Ogden, N=2; Ogden, N=3;

Ogden, N=4; Polynomial



Abaqus output for the stability of these models was:

The polynomial, N=2 hyperelastic equation was unstable for strains over

approximately 0.5.

the Ogden, N=2 equation was stable, root-mean-squared error of 43.07%.

the Ogden, N=3 equation was stable, root-mean-squared error of 40.83%; and

the Ogden, N=4 equation was unstable for strains over 0.54 under all loading

conditions, root-mean-squared error of 38.94%.

The polynomial, N=2 equation was not a suitable fit for the experimental data, as

shown in Figure 5-9 B and C. A comparison between the Ogden equations and the

experimental results showed good agreement and the decreasing root-mean-squared

error for these models indicated that the increasing order of N on the Ogden equation

improved the fit with experimental data. Under uniaxial compression and planar

shear loading, the Ogden, N=3 and N=4 equations showed a response which was

closer to the experimental data than the Ogden, N=2 equation. The response of the

Ogden, N=4 model demonstrated an instability in the uniaxial compression stress-

strain response at 0.54 strain. Owing to this it was decided that the N=3 model was

best suited to the anterior anulus fibrosus for initial loading.

A similar procedure was undertaken to determine the strain energy equation which

best suited each anulus region for both the initial and repeated loading. The

experimental data for the three loading cases was compared to the theoretical results

for the 2nd order polynomial strain energy equation and various orders of the Ogden

strain energy equation.

5.4.1.1 Explanation of the criterion used to select the hyperelastic strain energy

equation for the anterior, lateral and posterior anulus during initial and

repeated loading

The remaining regions of the anulus demonstrated a similar trend to that of the

anterior anulus during initial loading. The Ogden, N=3 equation consistently

provided a superior fit for the experimental data. This assessment was based on a

detailed analysis of the selection criteria outlined in the following section.



Agreement Between Experimental and FEM Results for Simple Loading

The correlation between the experimental and theoretical hyperelastic response was

the principal selection criteria. A single element FEM was either assigned Ogden or

polynomial hyperelastic material parameters that were determined from the

experimental data. This model was analysed with loading conditions of uniaxial

compression to 60% strain, simple shear to 50% shear strain, planar tension to ≅ 25%

strain and biaxial compression to 24% biaxial compressive strain. The FEM results

were compared to the experimental data used to find the hyperelastic constants. The

3rd order Ogden model consistently showed the best results for the agreement of the

experimental data and the FEM response.

All the hyperelastic models demonstrated excellent agreement with the experimental

data for the uniaxial compression response (Figure 5-10 B). The models either

demonstrated an acceptable fit at all biaxial compressive strains (Figure 5-10 A) or

overestimated the biaxial compression experimental results at higher strains and

demonstrated a reasonable fit at low strains.



A

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3Biaxiaxial Compressive Strain

Bia

xial

Com

pres

sive

Stre

ss (M

Pa)

FEM Response Experimental Data

B

-2.5

-2

-1.5

-1

-0.5

0-0.8 -0.6 -0.4 -0.2 0


Uni

axia

l Com

pres

sive

Stre

ss (M

Pa)

FEM Response Experimental Response

C

0

0.02

0.04

0.06

0.08

0 0.1 0.2 0.3 0.4Planar Shear Strain

Plan

ar S

hear

Str

ess

(MPa

)

FEM Response Experimental Response

Figure 5-10 Comparison of the experimental response and the theoretical hyperelastic response for A. Biaxial compression loading – anterior anulus, initial loading; B. Uniaxial compression loading – anterior anulus, repeated

loading; C. Planar shear loading – lateral anulus, repeated loading.



The least squared error equation (Eqn 5-27) can be solved using either uniaxial

compression, biaxial compression or planar shear experimental data. If a set of

hyperelastic constants for a specific anulus region were determined using

experimental data for the three loading types it was necessary to carry out three

separate least squared error procedures. The results of these calculations showed that

a different set of constants were determined for the biaxial and uniaxial compression

data. An assessment of Eqn 5-27 using planar shear experimental data indicated that

this equation was indeterminant for this loading condition. Therefore, hyperelastic

constants could not be determined solely on the basis of planar shear experimental

data.

These results suggested that Abaqus employed an algorithm to combine the constants

obtained from the three types of experimental data input. The Abaqus documentation

provided no information on the methods employed to combine the hyperelastic

parameters from the different loading conditions. It seemed likely that the algorithm

placed most emphasis on the uniaxial compression data, less importance on the

biaxial compression data and used the planar shear data to interpolate between these

two. This explained why Abaqus did not require planar shear data in order to

determined hyperelastic constants – this data was possibly used only as a smoothing

tool. Furthermore, under planar shear loading the hyperelastic models rarely showed

close agreement with the experimental data (Figure 5-10 C). It was thought that this

was a due to the limited influence of this data in determining the hyperelastic

parameters. The fit between the experimental data and the hyperelastic response for

the three loading types was generally slightly improved when planar shear data was

included.

When calculating the parameters for the posterior anulus during initial loading,

inclusion of the planar shear data in the experimental input resulted in the inability of

the software to converge on a set of constants which fit the biaxial compression data

acceptably. The constants which were obtained for initial loading on this region

(Table 5-3) were based only on uniaxial and biaxial compressive experimental data.



Root-Mean-Squared Error

The root-mean-squared error was output by Abaqus when the hyperelastic constants

were determined. For the inhomogeneous models the errors ranged from 25.28% to

71.70% with an average of 43.20%.

The order of the Ogden model chosen affected these errors. The N=1 Ogden models

returned the highest error percentage and this error decreased as the order of the

model increased. This decrease in error followed a pattern similar to an inverse

exponential – a comparison of N=1 and N=2 models showed a larger decrease in the

error compared to the difference between the error observed in the N=2 and N=3

models. The N=3 models consistently exhibited an error value which was a “plateau”

value. Increasing the order of the Ogden model to N=4 did not show a notable

reduction in the root-mean-squared error and this observation was consistent for most

regions of the disc.

Model Stability Reported by Abaqus

The final criterion for determining the accuracy of the Ogden, N=3 hyperelastic

parameters was based on the Drucker stability analysis carried out by Abaqus. In

addition to the hyperelastic parameters, the Abaqus software generated a root-mean-

squared error value and an assessment of the stability of the parameters when

implemented in the hyperelastic energy equation. The instability was calculated for

the six primary loading modes.

The Drucker stability criterion assessed the material stability using Eqn 5-29 (Abaqus

Users Manual §10.5.1).

0: >εσ dd

Eqn 5-29 Drucker Stability Criterion

This criterion required that the material obey classic laws of physics and did not

create energy. That is, the energy of the material when strained was a positive value.

This stability was assessed in uniaxial tension/compression, biaxial

tension/compression and planar tension/compression with a range of nominal strains



between 0.1 and 10.0. The strain at which the material relation in Eqn 5-29 was not

fulfilled was reported.

Of the 6 hyperelastic models developed, Abaqus reported that 3 of these were stable

for all strains between 0.1 and 10.0 (Table 5-2).

Table 5-2 Summary of inhomogeneous hyperelastic material parameters

Region Stable/Unstable Material

Unstable Loading Conditions

Anterior, Initial Loading

Stable for all strains

Anterior, Repeated Loading

Unstable for Uniaxial tension > 0.100 Uniaxial compression < -0.1266 Biaxial tension > 0.0700 Biaxial compression < -0.0465 Planar tension > 0.0900 Planar compression < -0.0826

Lateral, Initial Loading Stable for all strains Lateral, Repeated Loading

Unstable for Uniaxial tension > 0.1000 Biaxial compression < -0.0465 Planar tension > 0.1000 Planar compression < -0.0909

Posterior, Initial Loading

Stable for all strains

Posterior, Repeated Loading

Unstable for Uniaxial tension > 0.1000 Biaxial compression < -0.0465 Planar tension > 0.1300 Planar compression < -0.1150

Despite these reported instabilities, the stress-strain response from the single element

FEM showed acceptable agreement in comparison to the experimental results. For

example, consider the results for the anterior anulus under repeated loading (Figure

5-10 B). According to the Drucker analysis of instability this hyperelastic model was



unstable for uniaxial compressive strains greater than -0.1266. However, the

correlation between the hyperelastic model and the experimental data was assessed up

to a uniaxial compressive strain of -0.6000.

It had been expected that the model instability would manifest as a discontinuity or

spike in the stress-strain response of the FEM. However, given that there was no

observed discontinuity in the stress-strain response of any of the unit FEM at strains

similar to those reported to be points of instability, further investigations of the model

parameters were undertaken. It was postulated that the material instability was a

result of inconsistencies in the stresses or strains in the planes perpendicular to the

plane of principal strain. For example, under uniaxial compression, the axial strain

and stress were acceptable but the stress/strain in the directions perpendicular to the

uniaxial loading direction were abnormal. The parameters investigated were out-of-

plane strain vs. axial strain, out-of-plane stress vs. axial strain and strain energy in the

model vs. axial strain. These parameters were investigated for the single element

FEM loaded to strains similar to that which was reportedly the instability strain and

there were no inconsistencies observed.

Further assessment of the material stability was carried out by determining the 3rd

strain invariant for an extensive range of strains. An incompressible material would

have no net volume change during loading therefore, I3=λ1λ2λ3 = 1. It was postulated

that an inconsistency in the strain of the material would result in an inconsistency in

the strain invariants. The 3rd strain invariant for the singe element FEM was

determined for the duration of the loading and plotted against the axial strain. This

produced a straight line at a value of 1 indicating the hyperelastic material was

behaving as an incompressible material.

On the basis of this assessment of the hyperelastic models fit to the regions for initial

and repeated loading it was considered that they were an acceptable fit and the

erroneous instability evaluation stated by Abaqus was neglected.



5.4.1.2 Inhomogeneous Hyperelastic Constants for Initial and Repeated

Loading

It was considered that the Ogden, N=3 strain energy equation was best suited to the

inhomogeneous experimental data (Table 5-3).

Table 5-3 Specifications for the Ogden, N=3 hyperelastic parameters for the

three disc regions during initial and repeated loading

Region Loading Type Constants Anterior Initial µ1 = -76.5566 α1 = 0.444505

µ2 = 38.0248 α2 = 0.658874 µ3 = 38.6030 α3 = 0.232102

Repeated µ1 = -94.7200 α1 = -0.589100 µ2 = 49.0900 α2 = -0.392400 µ3 = 45.6600 α3 = -0.783500

Lateral Initial µ1 = -96.435 α1 = -1.2012x10-2 µ2 = 48.3595 α2 = 0.1670090 µ3 = 48.1117 α3 = -0.189028

Repeated µ1 = -32.5100 α1 = -0.271300 µ2 = 16.9300 α2 = -0.042160 µ3 = 15.5900 α3 = -0.497100

Posterior Initial µ1 = -335.299 α1 = 1.56578 µ2 = 166.922 α2 = 1.74099 µ3 = 168.422 α3 = 1.39183

Repeated µ1 = -0.67480 α1 = -1.87300 µ2 = 0.30980 α2 = 0.95790 µ3 = 0.37900 α3 = -3.20500

5.4.2 Homogeneous Hyperelastic Model for the Ground Substance

For the purpose of the following section, 5.4.2, the term ‘model’ will refer to the

particular relationship and algorithms associated with the various constitutive

equations for hyperelasticity.

To determine a set of homogeneous hyperelastic parameters for the anulus ground

substance, the experimental results for an individual test type – uniaxial compression,

biaxial compression or simple shear – were compared for the three disc regions. Only

data for the initial loading was used. The region which showed the ‘mid-stiffness’



response was chosen as representative of the disc response for that particular test type.

For example, the anterior anulus demonstrated the mid-stiffness response under

uniaxial compression (Figure 5-11).

For uniaxial compression, the anterior response was used; for biaxial compression, the

anterior response was used; and for simple shear, the lateral response was used.

0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6Uniaxial Compressive Strain

Uni

axia

l Com

pres

sive

Stre

ss(M

Pa)

Anterior Lateral Posterior

Figure 5-11 Uniaxial compression stress vs. strain for the anterior, lateral and posterior anulus fibrosus ground substance. The anterior disc shows the 'mid-

stiffness' response used for the homogeneous hyperelastic parameters.

Using similar criteria to that employed to determine a set of inhomogeneous

hyperelastic parameters (section 5.4.1.1), a Polynomial, N=2 hyperelastic model was

chosen for the homogeneous hyperelastic FEM (Table 5-4).



Table 5-4 Polynomial, N=2 hyperelastic strain energy parameters for the

homogeneous anulus under initial loading

Hyperelastic Parameters

C10 = 0.02121 C01 = 0.02971 C20 = -0.05379 C11 = 0.1218 C02 = -0.04291

Uniaxial tension, nominal strain > 0.9500 Uniaxial compression, nominal strain < -0.5891 Biaxial tension, nominal strain > 0.5600 Biaxial compression, nominal strain <-0.2839 Planar tension, nominal strain > 0.7600

Unstable material

model for loading

conditions Planar compression, nominal strain < -0.4318

The instability of the hyperelastic parameters was not considered to be detrimental to

the results. As detailed in Section 5.4.1.1 the warnings of instability in the material

behaviour were misleading and additionally, the strains at which the hyperelastic

parameters were unstable under biaxial loading and planar tension were very high in

comparison to the expected nominal strains in the FEM.

It was not believed that the use of different strain energy equations in the

inhomogeneous and the homogeneous FEM would not cause inaccuracies in the

comparison of results. The choice of the strain energy equation was based on how

well the equation predicted the experimental data. The polynomial strain energy

equation provided the most accurate correlation with the experimental data for the

homogenous material properties while the Ogden equation provided a better

correlation for the anulus regions in the inhomogeneous FEM. Each of these models

acceptably predicted the stress in the material due to the various loading conditions

applied and it was considered that the ability of the models to predict the stress-strain

response of the material was of most importance.

The hyperelastic parameters based on the initial loading results were used to assess

the validity of the experimental results in the FEM.



5.5 Conclusion

With the use of representative experimental results from uniaxial compression, biaxial

compression and simple shear loading on the anulus fibrosus ground substance, a set

of inhomogeneous and homogeneous hyperelastic parameters were determined. With

the use of extensive criteria to choose the model which best suited the material

response a reliable representation of the behaviour of the anulus ground substance was

obtained. Details of implementation of these hyperelastic constants in the FEM are

provided in Chapter 6. It was thought that knowledge of the strain to initiate

derangement in the FEM anulus fibrosus ground substance and hyperelastic constants

which represented the response of the tissue both before and subsequent to this

derangement would permit the FEM to better simulate the in vivo condition.

The hyperelastic constitutive models developed in this chapter provided a good fit for

the experimental data up to maximum strains of 60% under uniaxial compressive,

60% under simple shear and 25% under biaxial compression. These were the

maximum strains achieved during the experimental testing under the three loading

conditions and therefore, the constitutive models fit to this data could only reasonably

provide information on the mechanical response of the tissue when loaded in this

range.

It was envisaged that the use of strain energy equations which were based on an

assumption of nonlinear shear behaviour would provide a mechanical response of the

FEM anulus fibrosus which was superior to those observed in previous hyperelastic

models which had used a Mooney-Rivlin strain energy equation.


Chapter 6: Implementation of the Improved Anulus Fibrosus Material Propeties 236

CChhaapptteerr

66

IImmpplleemmeennttaattiioonn ooff tthhee

IImmpprroovveedd AAnnuulluuss FFiibbrroossuuss

MMaatteerriiaall PPrrooppeerrttiieess This chapter presents the implementation of the hyperelastic parameters determined in

Chapter 5 and the subsequent modification of the model to obtain a converged

solution.

6.1 Chapter Overview

Initially the homogeneous material parameters for the anulus ground substance were

implemented in the FEM developed in Chapter 3 to establish the compatibility of

these parameters with the other disc components. This analysis was followed by

simulation using inhomogeneous material parameters. It was not possible to obtain a

solution for the model using the inhomogeneous material parameters under the

comparatively simple loading case of torso compression.

This difficulty was addressed by improving the configuration of the hydrostatic fluid

elements defining the nucleus pulposus and recalculating the material properties and

geometry of the collagen fibres in an attempt to obtain a converged solution in the

FEM. These changes allowed successful analysis of the disc model using both

homogeneous and inhomogeneous hyperelastic material parameters. The material

parameters developed in Chapter 5 to represent the anulus ground substance are

notably more compliant compared to the parameters employed in the preliminary



FEM in Chapter 3. As a result, the nucleus pulposus pressures in these models were

reasonably high in comparison to the experimental evidence of the in vivo nucleus

pressure presented by Nachemson (1960). The intimate relationship between the

anterior and posterior longitudinal ligaments and the intervertebral disc, suggests that

inclusion of these ligaments in the FEM would result in a model which more closely

represented the in vivo loading condition of the disc. The inclusion of the longitudinal

ligaments is the subject of Chapter 7.

6.2 Implementation of the Homogeneous Anulus Ground Substance into the

FEM

The homogeneous hyperelastic constants that were determined for the initial loading

on the anulus ground substance were implemented in the FEM developed in Chapter

3. Other than the material properties of the anulus ground substance, all other features

of this model remained the same. This new analysis will be referred to as the

Homogeneous FEM. The loading applied to the Homogeneous FEM was a torso

compressive load of 500N. The homogeneous hyperelastic parameters defined in

Section 5.2.4 were implemented in the FEM.

It was not possible to obtain a completed solution for the homogeneous model. The

attempted solution failed on the first increment of the first time step in the nonlinear,

static analysis. Abaqus warned that the solution appeared to be diverging. The first

step of this analysis was attempted with an initial increment time of 0.1 and a

minimum time increment of 1 x 10-7 which were similar values to those used in the

previous analyses of the preliminary FEM. This time increment was reduced to 3.906

x 10-4 by the end of the attempted increment. Inability to achieve convergence on the

first increment of the first step indicated that the analysis had significant complexity

and inconstancies in the finite element algorithms.

The analysis was re-attempted with a reduction in the initial time increment to 1x10-7.

The solution completed 0.63 of the 500N compressive torso loading step which was

comparable to a compressive load of 315N. Abaqus was not able to obtain

convergence for the displacements on the inner anterior anulus surface, at the



boundary of the hydrostatic fluid. Excessive displacement corrections were observed

in this location.

It had been observed during the experimental procedures that the anulus ground

substance was very compliant. It was predicted that the model would not be

adversely affected by this apparent compliance since the rebar elements representing

the collagen fibres were comparatively stiff and these elements had demonstrated

extremely low strains in the analysis of the preliminary FEM. However, subsequent

to the difficulties encountered in obtaining convergence in the homogeneous model, it

was a reasonable assumption that this high compliance of the ground substance may

have been the cause for the convergence problems.

In order to determine how compliant the experimental results were in comparison to

the Mooney-Rivlin hyperelastic parameters which had been used previously, a single

element model was analysed. The model was a solid 3D element of unit edge length.

This model was first analysed using the homogeneous 3rd order Ogden hyperelastic

model as the material property for the element. The element was separately analysed

under uniaxial compression and simple shear and the stress-strain response for these

loadings recorded. The material definition for the element was then changed to the

Mooney-Rivlin hyperelastic parameters of C10=0.7, C01=0.2. Figure 6-1 and Figure

6-2 show a comparison of the results of analyses using these material parameters.

-30

-25

-20

-15

-10

-5

0-0.8 -0.6 -0.4 -0.2 0


Uni

axia

l Com

pres

sive

Str

ess

(MP

a)

Polynomial, N=2 Mooney-Rivlin

Figure 6-1 Comparison of uniaxial compression response for the Polynomial, N=2 and Mooney-Rivlin hyperelastic models



0

0.5

1

1.5

2

2.5

0 0.2 0.4 0.6 0.8Simple Shear Strain

Sim

ple

Shea

r St

ress

(MP

a)

Polynomial, N=2 Mooney-Rivlin

Figure 6-2 Comparison of simple shear response for the Polynomial, N=2 and Mooney-Rivlin hyperelastic models

It may be seen from Figure 6-1 and Figure 6-2 that the response of the FEM that

incorporated hyperelastic parameters which were derived from experimental data was

considerably more compliant than the response of the Mooney-Rivlin material.

Therefore, the analyses carried out in Chapter 3, whilst validating according to the

criterion described, were based on material properties for the anulus ground substance

with an overestimated stiffness.

There were two possible scenarios which may have resulted in the lack of

convergence of the Homogeneous FEM. This could have been due to the compliance

of the anulus elements and the inability of this material to carry a sufficient portion of

the applied load, resulting in overload of the collagen fibre rebar elements. However,

this situation would not be expected to result in a lack of convergence in the solution,

rather it would have become evident once the analysis had solved and the stress in the

rebar elements was determined to be too large in relation to the failure stresses of

collagen fibres. Another potential cause for the convergence problems could have

been the incompatible material properties of the collagen fibres and the anulus ground

substance. The collagen fibres had a linear elastic modulus of 500MPa and the

comparable elastic modulus of the ground substance was 2-3MPa. It was possible

that this intimate contact between two materials with such a significant difference in

stiffness may have caused the stiffness matrix to become “ill-conditioned”. An ill-



conditioned matrix has a large difference between the largest and the smallest value in

the stiffness matrix. This ill-conditioning would cause complications in the matrix

solution to determine the stress/strain.

6.3 Compatibility of the Material Stiffness of the Collagen Fibres and the

Anulus Fibrosus Ground Substance

To determine whether the material properties of the anulus ground substance and the

collagen fibres were incompatible, several pilot analyses were carried out:

1. The collagen fibre rebar elements in the Homogeneous FEM were removed

and the analysis was re-solved.

2. A single 3D hyperelastic continuum element model with embedded collagen

fibre rebar elements was solved.

3. A hyperelastic model with a hollow cylinder geometry was solved. Embedded

collagen fibre rebar elements were simulated and a surface pressure that was

normal to the inner cylinder face simulated the nucleus pressure.

4. A hyperelastic model with simplified disc geometry was solved. This model

included the endplates. Embedded collagen fibre rebar elements were

simulated and a nucleus pressure was applied using a surface pressure normal

to the inner face.

The results of these analyses indicated that the Ogden hyperelastic material was

capable of generating a solution. However, the stiffness of this material was not

sufficient to withstand the loads to which the disc was subjected in vivo without the

added reinforcement of the rebar elements. Analysis of a model with a simplified disc

geometry and elements representing the collagen fibres resulted in a converged

solution. This suggested that the inability of the Homogeneous FEM to obtain a

converged solution was not a result of the incompatibility of the material properties

defined for the anulus ground substance and the collagen fibres.

The reason for the failure of the Homogeneous FEM was not clarified after these pilot

studies. Further investigation of the model mesh was necessary.



6.4 Improved Element Configuration for the Hydrostatic Fluid Elements on

the Inner Anulus Fibrosus

The Abaqus 3D element library offered only 3 or 4 node fluid elements. The 3D

continuum elements used to model the anulus fibrosus ground substance were 20 node

elements therefore, on any one face there were 8 nodes to constrain to the fluid. It

was not possible to attach one 4 node fluid element to the face of the continuum

element as this would have resulted in the 4 midside nodes being unconstrained.

In order to constrain all the nodes on the faces of each continuum element, it was

necessary to use five hydrostatic elements for each continuum element – four 3 node

fluid elements defined at each corner and one 4 node fluid element defined diagonally

in the centre of the continuum element (Figure 6-3).

Figure 6-3 Attachment of 3 and 4 node fluid elements to the face of the continuum elements on the inner anulus surface

Inspection of the deformed geometry in the simplified disc FEM analysed in Section

6.3 showed an irregular, saw-tooth appearance on the inner anulus surface, where the

hydrostatic fluid elements were defined (Figure 6-4). This was similar to that of

deformed meshes which demonstrate hourglassing.

Hourglassing is a phenomenon which commonly occurs in near incompressible or

incompressible materials. It takes place when the mathematical relations for a

particular set of elements result in a zero energy change but the actual deformed shape

of those elements is such that they have “turned inside out” (Figure 6-4). For

example, one surface of a 3D element may have “pushed through” the surface on the

opposite side of the element, which is not generally a physical possibility. However,

3 node hydrostatic fluid element

4 node hydrostatic fluid element

Boundary of one face of the 3D

continuum element

Node



because there has been no apparent net change in the volume of the element the

mathematical relations governing the incompressibility criterion reach a solution.

Figure 6-4 The undeformed and deformed shape of one element on the inner anulus surface, at the boundary of the anulus and nucleus. Note the nodes A and

B on the undeformed element have moved “through” the element causing the midside nodes to create a jagged, saw-tooth profile for the element edge. (The ’

denotes the deformed location of the node)

While the appearance of the inner anulus elements was similar to that observed during

hourglassing, this was not believed to be the cause for the unusual deformed shape.

Abaqus did not provide for any hourglass control to be prescribed for the 3D

continuum elements used to model the anulus ground substance. It was stated that

this element type was a 2nd order element with midside nodes and hourglassing would

not occur in this element type.

Upon higher magnification of the deformed shape of the preliminary FEM analysed in

Chapter 3 it was observed that a similar saw-tooth profile was present on the nucleus.

However, it was not as pronounced as in the FEM which used the experimental

hyperelastic material properties. Possibly this was because the Mooney-Rivlin

Cranial Direction

Radially Outward Direction

Left Lateral Direction

Deformed Shape of one element

Undeformed Shape of one element

9

11’3’

9’2’

5’6’

B’

A’

8’

13

8

5

2

6

B

A

20

18

20’

18’ 17’

17

3

11

7’

13’

19

15

15’

19’

7 16

10

12

14

14’

10’

12’

16’



hyperelastic material incorporated in the preliminary FEM was stiffer than the

experimental hyperelastic response and therefore, the relative deformations of the

nodes were not as large.

Whilst this saw-tooth deformation pattern had not caused convergence problems for

the comparatively stiffer anulus ground substance in the preliminary FEM, it was

considered that the abnormal deformation of the inner anulus surface may have been a

cause for the difficulties in solving the Homogeneous FEM. The error messages

returned from the failed analysis included warnings of large displacements at nodes

and statements on the inability of the solver to obtain convergence for these

displacements. The large displacements were partly due to the excessive

displacement of the nodes on the inner anulus surface.

To improve the deformed shape of the anulus wall in the Homogeneous FEM, the 3

node hydrostatic fluid elements were removed. The midside nodes on the 2nd order

continuum elements of the inner anulus surface were constrained to the adjacent

corner nodes. This effectively created a 1st order element surface on the inner anulus,

as the computational advantage created by having midside nodes was removed. The

constraint restricted all the degrees of freedom at the midside nodes to be an

extrapolation of the degrees of freedom of the corner nodes. With the constraint of

the midside nodes, there were effectively 4 nodes on the face of each continuum

element at the boundary of the nucleus and anulus. Therefore, only 4 node

hydrostatic fluid elements were required (Figure 6-5, compare with Figure 3-18).

Figure 6-5 Improved hydrostatic fluid elements on the anulus wall



The preliminary FEM was reanalysed using these improved hydrostatic elements – a

500N uniaxial compression load was applied. The deformed shape of the inner anulus

wall of the mesh was improved (Figure 6-6).

Figure 6-6 The undeformed and deformed shape of one element on the inner anulus surface after a single 4 node hydrostatic element was attached to the continuum element face. (The ’ denotes the deformed location of the node)

A comparison of the results of this analysis with the results presented in Chapter 3 for

the compressive load of 500N applied to the preliminary FEM showed:

• An increase in the nuclear pressure in the FEM with the 3 node hydrostatic

elements removed – the increased pressure was 0.74MPa compared to the

previous value of 0.66MPa.

• The region of high stress in the anulus fibrosus of the FEM with only 4 node

hydrostatic fluid elements was on the superior, posterior surface of the anulus

fibrosus and the von Mises stress ranged from 1.67-2.24MPa. The peak von

Mises stress in the FEM with 3 node fluid elements occurred at the inner

posterior anulus surface and ranged from 2.91-3.18MPa. On the superior,

posterior anulus of this FEM the von Mises stress was 1.07-1.33MPa.

Therefore, the stress state of the FEM when subjected to a 500N compressive

5

Cranial Direction

Radially Outward Direction

Left Lateral Direction

Deformed Shape of one element Undeformed shape of one element

1

2 4

3

6

7

8

9 10

12

11

13 14

15 16

17 18

19 20

10’

2’

1’

3’

4’ 9’

11’12’

18’17’

19’ 20’

5’

6’ 14’

15’

13’

8’

7’ 16’



load was reduced due to the improved method of modelling the inner anulus

wall.

It was apparent that the use of 3 node hydrostatic fluid elements in the preliminary

FEM had resulted in erroneous results for the stress and strain on the inner anulus

fibrosus surface. When only 4 node fluid elements were used at this surface, the

location of maximum stress and logarithmic principal strain was on the superior,

posterior anulus. The magnitude of this strain was similar to that observed in the

same location on the FEM with 3 node fluid elements.

With the 3 node fluid elements removed the pressure in the nucleus was 1.77 times

the applied stress which was slightly higher than the ratio found in the preliminary

FEM. Previously the pressure was 1.57 times the applied stress which was closer to

the results of Nachemson (1960). This was an excellent result in comparison to

Nachemson’s (1960) findings of a ratio of 1.5.

6.4.1 Results of analysis of the Homogeneous FEM with improved hydrostatic

fluid element configuration

Rather than defining a single stress value and location for the maximum stress

observed in the model, regions of high stress were observed in the analysis of the

FEM. These regions were delineated such that the high stress contour regions

represented as red/orange in the von Mises stress contours, extended no more than the

distance between the corner node/s at which the individual maximum was observed

and the next closest midside node. The results for von Mises stress in Chapter 6 and

the Chapters that follow will present values for this range. Also, the highest stress

contour band corresponded to approximately 8% of the total stress range. A range of

stresses was used in preference to a specific maximum stress since the over-riding aim

of the FEM analyses was to provide information on the change in mechanics of the

disc rather than to provide information on the maximum stress in the model for the

purpose of analysing potential failure initiation. As such, the use of a range of stress

observed in regions of high stress was thought to provide more useful information

than data for the specific maximum stress.



A reanalysis of the Homogeneous FEM with the revised 4 node fluid elements

resulted in a converged solution for the 500N compressive loading step.

Figure 6-7 Deformed shape of Homogeneous FEM – wireframe shows undeformed shape and arrows define translation and rotation

The deformed shape of the Homogeneous FEM demonstrated a significant anterior

translation and rotation of the superior surface (Figure 6-7). The posterolateral anulus

demonstrated an inward radial bulge of approximately 0.5mm while the mid posterior

anulus bulged outward (Figure 6-8).

The nucleus pressure in the Homogeneous FEM after the 500N compression load was

applied reached 0.96MPa which was 2.31 times the applied compressive stress. This

value was significantly higher than the expected ratio between the nucleus pressure

and the applied pressure of 1.5 (Nachemson, 1960).

The maximum von Mises stress in the FEM occurred in the posterior region of the

endplates. This was at the junction between the anulus fibrosus, the nucleus pulposus

and the cartilaginous endplates and the stress ranged from 1.50 to 2.39MPa (Figure 6-

9). Given the stiffness of the cartilaginous endplates, this stress was very low.

This maximum stress location on the endplates was reasonable in relation to the

posterior bulge of 2.25mm of the inner posterior anulus fibrosus surface in relation to

the posterior junction of this surface and the superior endplate (Figure 6-9). The


1.06mm

1.36mm

2.08mm

0.88mm

5.97o



significant deformation of the anulus fibrosus in response to the applied compressive

load was resisted by the comparatively stiff cartilaginous endplate.



Figure 6-8 Posterior FEM demonstrating outward bulge of posterior anulus and inward bulge of posterolateral anulus

Figure 6-9 The inferior surface of the intervertebral disc FEM viewed from an anterior direction. The inner posterior surface of the anulus fibrosus is shown.

Outward mid-posterior bulge

Junction of anulus fibrosus, nucleus pulposus and superior cartilaginous endplate – peak stress in the endplates ranged from 1.50 to

2.39MPa

Bulge of the posterior anulus in relation to

junction of the posterior anulus and the superior endplate

was 2.25mm




Figure 6-10 Von Mises stress contour in the Homogeneous FEM demonstrating maximum stress on superior, posterior anulus and high stress on inferior,

posterolateral anulus

The anulus fibrosus demonstrated comparatively small von Mises stresses (Figure

6-10) in relation to the stresses observed in the endplates. The maximum stress in the

anulus fibrosus was on the superior, posterior surface and ranged from 0.68 to

0.82MPa. There was a region of increased stress on the inferior, posterolateral

margins of the anulus which ranged from 0.48 to 0.61MPa (Figure 6-10). This region

of high stress was a result of the significant deformation of the posterolateral anulus

fibrosus and the rigid boundary conditions on the inferior surface of the FEM. A high

maximum principal logarithmic strain was observed in a similar location to the high

von Mises stress on the inferior, posterolateral anulus fibrosus. This strain ranged

from 0.63 to 0.99.

The maximum principal logarithmic strain on the inner anulus surface varied from -

0.81 to -1.17.

Peak stress on superior, posterior anulus 0.68 to 0.82MPa

High stress on inferior, posterolateral anulus 0.48 to 0.61MPa



6.4.2 Discussion

Examination of the results of the Homogeneous FEM indicated that the material

properties for the anulus fibrosus were extremely compliant compared to those for this

structure in the preliminary FEM. The lower stiffness of this material resulted in

large anterior translations and rotations of the superior surface of the intervertebral

disc and a high nucleus pulposus pressure. The high pressure was likely a result of

the significant deformation of the anulus fibrosus which enclosed the hydrostatic

fluid. This deformation was evident from an observation of the high compressive

maximum principal logarithmic strains in this region.

It should be noted that the undeformed geometry of the Homogeneous FEM was not

manipulated to reflect the accurate sagittal dimensions as determined by Tibrewal and

Pearcy (1985). The iterative procedure required to obtain the correct anterior and

posterior heights for the L4/5 intervertebral disc during relaxed standing was outlined

in Section 3.5.6. These accurate dimensions were not incorporated in the FEM at this

stage because the analyses in this chapter were performed to determine whether the

hyperelastic material properties generated acceptable results in the FEM. Once this

was established the accurate sagittal geometry was incorporated (Chapter 7, Section

7.2). Any observed anterior translation and rotation of the superior surface would be

removed in the final FEM geometry by iterating until the anterior height of the

intervertebral disc FEM was 14mm and the posterior height was 5.5mm.

A discussion of the deformed sagittal geometry of the FEM with particular reference

to the inward posterolateral bulge of the anulus fibrosus is provided in Section 6.6.2.

6.4.3 Summary

The improved method of representing the hydrostatic nucleus pulposus using 4 node

fluid elements on the inner anulus boundary significantly improved the results of the

FEM. Use of both 3 and 4 node fluid elements had resulted in the inability of the

FEM to achieve a completed solution due to the excessive deformation of the midside

nodes on the inner anulus fibrosus surface.



The increased compliance of the anulus fibrosus in comparison to the preliminary

FEM was a result of the compliance of the anulus fibrosus ground substance. Data for

the mechanical response of the ground substance was obtained from experimentation

and the procedure and analysis techniques employed to obtain material parameters

from this experimentation were considered to be accurate. Therefore, no

improvements could be made to the material behaviour of the ground substance in the

Homogeneous FEM. However, further investigations were undertaken on the

geometry and material properties of the collagen fibres which performed the

reinforcing role in the anulus fibrosus in vivo.

It was also noted that the analysis of the Homogeneous FEM was intended to confirm

the compatibility of the anulus ground substance with the remaining FEM materials

and to assess the ability of the ground substance to bear the loads applied to the

intervertebral disc in vivo. Subsequent models were developed which incorporated

the inhomogeneous material properties for the anulus ground substance. Further

conclusions relating to the compliance of the anulus fibrosus ground substance in the

FEM were made once the inhomogeneous ground substance material properties were

incorporated (Section 6.6.2).

6.5 Improved Properties for the Collagen Fibres in the Anulus Fibrosus

The collagen fibres in the anulus fibrosus are responsible for carrying tensile force in

the loaded anulus fibrosus. This tensile force is generated by hoop stress in the anulus

fibrosus due to the pressure in the nucleus pulposus, as well as the axial fibre stress

due to bending, torsion and shearing motions of the disc. These fibres provide

reinforcement for the comparatively compliant anulus fibrosus ground substance.

Given the integral role of the collagen fibres during loading of the intervertebral disc,

the accuracy of the material properties assigned to the rebar elements in the

Homogeneous FEM was crucial. Further investigation of the geometry, placement

and material properties of the collagen fibre rebar elements was undertaken to ensure

the accuracy of the properties selected for the rebar elements in the Homogeneous

FEM. It was thought that an increase in the stiffness of the rebar elements



representing the collagen fibres in the Homogeneous FEM would provide additional

stiffness to the modelled anulus fibrosus during loading.

Cassidy et al (1989) observed that the tilt angle of the collagen fibres in the anulus

fibrosus varied radially. The fibres in the outer anulus lamellae were inclined at a

larger angle to the cranio-caudal direction than the fibres in the inner anulus lamellae.

The angle of inclination of the rebar elements representing the collagen fibres in the

disc FEM alternated between adjacent lamellae and had a constant magnitude of 70o.

This had not been considered to detrimentally affect the response of the preliminary

FEM since several previous finite element researchers had made a similar assumption

(Kumaresan et al., 1999; Belytschko, 1974; Shirazi-Adl et al., 1986). Additionally, as

a result of the higher stiffness of the Mooney-Rivlin hyperelastic properties employed

for the anulus ground substance in the preliminary FEM, a slight change in the

geometry or material properties of the collagen fibres did not significantly affect the

response of this FEM. However, the use of more compliant material properties for the

anulus ground substance in the Homogeneous FEM resulted in a higher contribution

to load bearing from the rebar elements. It was believed that increased reinforcement

in the outer anulus would result in a reduction in the lateral bulge and axial

deformation of the FEM.

6.5.1 Collagen fibre inclination

As outlined in section 2.1.2 there was considerable variation in the quoted angles for

the inclination of the collagen fibres to the caudo-cranial direction. These values

varied between 45 and 70o. Cassidy et al. (1989) reported that the inclination of the

fibres varied radially with the collagen fibres in the inner anulus inclined at angles as

low as 45o and the fibres in the outer anulus inclined at 62o. It was thought that the

considerable range of variation in the experimentally observed inclination angle of the

collagen fibres (45-70º) may have been a result of experimental techniques which

considered the fibre inclination did not vary with position in the anulus. Therefore,

the inclination of the rebar elements representing the collagen fibres in the FEM was

varied radially.



The average inclination angle based on the observations of previous researchers was

approximately 60o (Section 2.1.2, Table 1). Cassidy et al. (1989) stated a difference

in fibre inclination between the inner and outer anulus of 17o. On the basis of this

value and on the range of inclination angles observed in previous experimental

studies, a conservative estimate of fibre inclination in the inner anulus was 55o and in

the outer anulus was 65o. In the disc FEM the inclination of the collagen fibres in the

element layers was varied linearly between these values.

6.5.2 Collagen fibre stiffness

Morgan (1960) defined a stress-strain curve for a single collagen fibre. The

mechanical response of the material was nonlinear and demonstrated an increase in

instantaneous stiffness with increasing strain up to 20% strain. A straight line of best

fit drawn between zero strain and the collagen failure strain of 15% (Viidik, 1973)

had a gradient of 630MPa. Shirazi-Adl et al. (1986) summarized the experimental

results of previous researchers to define a stress-strain curve for collagen fibres up to

25% strain. The gradient of a straight line of best fit drawn between zero strain and

the collagen fibre failure strain was 680MPa which was similar to the results of

Morgan (1960). However, the curvature of the stress-strain response showed an

increase in the instantaneous stiffness of the tissue up to a strain of approximately 3%

then a gradual decrease in this stiffness up to 25% strain.

The obvious difference in the stress-strain responses stated by Morgan (1960) and

Shirazi-Adl et al. (1986) provided contradictory evidence for the nonlinear behaviour

of collagen fibres. However, the nonlinear behaviour of the fibres was not

represented in the FEM of the intervertebral disc. Therefore, the agreement between

the linear elastic moduli determined from these studies was of primary importance

and these values were used to determine an improved elastic modulus for the rebar

elements in the Homogeneous FEM.

It was considered that the use of an elastic modulus for the collagen fibres that was

based on the nonlinear stress-strain curves of the material would improve the material

properties for the rebar elements. The elastic modulus of 500MPa which was used in



the preliminary FEM and the Homogeneous FEM was based on the stiffness

employed in previous finite element studies rather than data obtained for the

mechanical response of the fibres. Therefore, the average stiffness for the collagen

fibres was calculated to be 655MPa based on the results of Morgan (1960) and

Shirazi-Adl et al. (1986). The average stiffness of the fibres was 655MPa.

The outermost lamellae of the anulus fibrosus contained primarily type I collagen

fibres which were generally found in materials that experienced tensile loading. The

inner lamellae contained primarily type II collagen fibres. To incorporate this radial

variation in the collagen content of the anulus fibrosus, Shirazi-Adl et al. (1986)

radially varied the elastic stiffness of the elements representing the collagen fibres.

The FEM developed by Shirazi-Adl et al. (1986) contained 8 circumferential element

layers and the distribution of collagen fibre stiffness is detailed in Table 6-1.

Table 6-1 Radial variation of fibre stiffness (Shirazi-Adl et al., 1986)

The collagen fibres in the outermost lamellae were the stiffest and were modelled as

nonlinear materials. Given that the stress-strain curve for the nonlinear collagen

fibres in the FEM developed by Shirazi-Adl et al. (1986) was used to determine the

average linear elastic stiffness of the fibres stated above – 655MPa – this stiffness was

assumed to be the elastic modulus of the rebar elements in the outermost 2 layers of

the Homogeneous FEM and the other layers were assigned values in the same

proportion as Shirazi-Adl et al. (1986) (Table 6-2).

halla

This table is not available online. Please consult the hardcopy thesis available from the QUT Library



Table 6-2 Radially varying elastic modulus of the rebar elements representing

the collagen fibres

Circumferential Element Layer (Layer 1 was

adjacent to the nucleus)

Elastic Modulus (MPa)

Proportion of the Stiffness of the Outermost Rebar

Element 1 426 0.65 2 426 0.65 3 491 0.75 4 491 0.75 5 590 0.90 6 590 0.90 7 655 1.00 8 655 1.00

These improved values for the rebar element stiffness in the intervertebral disc were

implemented in all subsequent models analysed. It was believed that these properties

provided a more realistic representation of the collagen fibres within the intervertebral

disc in vivo. The Homogeneous FEM was then reanalysed using these radially

varying values for the elastic modulus and inclination of the rebar elements

representing the collagen fibres.

6.5.3 Results of the analysis of the Homogeneous FEM using improved collagen

fibre geometry and material properties

A compressive torso load of 500N was applied to the FEM. Anterior translation and

rotation of the superior surface was observed; however, the magnitude of these

deformations was reduced in comparison to the results of the Homogeneous FEM.

The displacement and rotation of the superior FEM surface were as follows:

• The anterior margin of the superior surface was displaced 0.54mm in the

anterior direction and 1.30mm in the caudal direction;

• The posterolateral margin of this surface displaced 0.60mm anteriorly and

0.57mm in a cephalic direction; and

• A rotation of the superior surface of 3.31o was observed.



The maximum von Mises stress of 3.86-4.50MPa (compared with previous values of

1.49-2.39MPa) occurred in the posterior endplates near the junction of the

cartilaginous endplates, the anulus fibrosus and the nucleus pulposus.

The von Mises stress distribution in the anulus fibrosus of the FEM (Figure 6-11) was

similar to that observed in Section 6.4.1, Figure 6-10; however, the magnitude of the

stresses was increased.

Figure 6-11 Von Mises stress distribution for the Homogeneous FEM with improved collagen fibre properties

While both the right and left inferior posterolateral anulus demonstrated higher

stresses, the stress in the left anulus was approximately double the stress in the right

anulus. It was considered that the difference in these stresses was due to the

orientation of the collagen fibres in this outer circumferential layer of elements.

When viewing the frontal plane of the disc from the posterior aspect, the collagen

fibres were orientated toward the left of the disc at 65o to the axial direction through

the disc. This orientation of the fibres was such that the anterior rotation of the disc

resulted in fibres in the left posterolateral anulus experiencing a higher stress than

those in the right posterolateral anulus. This higher fibre stress was evident from the

rebar element stresses. Rebar elements in the right posterolateral anulus demonstrated

nominal axial stresses in the order of 0.83MPa while those in the left posterolateral

Maximum stress in anulus – 0.48MPa – 0.52MPa

High stress on left inferior posterolateral anulus – 0.39MPa – 0.48MPa

Stress on the right posterolateral anulus – 0.26MPa – 0.35MPa



anulus demonstrated stresses as high as 7.45MPa. The maximum rebar element stress

in the FEM of 13.1MPa occurred in the circumferential element layer second in from

the right lateral peripheral anulus.

A hydrostatic pressure of 0.83MPa was present after the 500N compressive load.

This was comparable to a ratio between the nucleus pressure and the applied pressure

of 2. Although this ratio was higher than 1.5 it is an improvement on the ratio of 2.31

times observed in the Homogeneous FEM which incorporated a constant stiffness and

inclination of the rebar elements.

6.5.4 Discussion and conclusions

An elastic modulus of 500MPa was employed for the stiffness of all collagen fibres in

the Homogeneous FEM. This value was based on the finite element studies carried

out by Kumaresan et al. (1999) and Ueno and Liu (1987). However, this stiffness was

improved to reflect the radial variation of the collagen fibre stiffness in the anulus

fibrosus in accordance with the work of Shirazi-Adl et al. (1986). Additionally the

radial variation in the inclination of the collagen fibres within the lamellae was

incorporated into the FEM. The fibre inclination from the axial direction in the disc

ranged from 55o in the inner lamellae to 65o in the outer lamellae.

The results of analysis of the Homogeneous FEM incorporating the improved

geometry and material properties for the rebar elements indicated that the response of

the anulus fibrosus was stiffer. The stresses observed in the anulus fibrosus were

increased compared to the results presented in Section 6.4.1 for the uniform rebar

properties. The anterior translation and rotation was reduced – the angle of rotation of

the superior surface was reduced from 5.97o to 3.31o. This reduced deformation in the

FEM resulted in a reduction in the pressure in the nucleus pulposus. It was concluded

that the simulation of a radially varying stiffness and inclination angle for the rebar

elements representing the collagen fibres resulted in an FEM which more closely

simulated the intervertebral disc both morphologically and mechanically.



It was noted that the sagittal geometry of this FEM did not incorporate the correct

anterior and posterior height of the disc in accordance with the findings of Tibrewal

and Pearcy (1985). As stated in Section 6.4.2, these dimensions were incorporated

into the FEM once the accuracy of the hyperelastic material parameters was verified

and once it was established that the structures represented in the FEM were both

mechanically and morphologically accurate.

6.6 Implementation of the Inhomogeneous Anulus Ground Substance into the

FEM

Inhomogeneous material properties for the anulus fibrosus ground substance were

implemented in the FEM using the 4 node hydrostatic fluid elements and improved

mechanical properties and geometry for the collagen fibre rebar elements as described

previously. This FEM is referred to as the Inhomogeneous FEM. These

inhomogeneous hyperelastic parameters were defined in Chapter 5 and are based on

the initial loading data. Material properties for the repeated loading were

implemented in subsequent models once the robustness of these experimentally

determined hyperelastic parameters was established.

The inhomogeneous hyperelastic parameters for the Ogden, N=3 strain energy

equation are stated in Table 6-3.

Table 6-3 Inhomogeneous hyperelastic material parameters for the Ogden, N=3

strain energy equation

Anterior anulus ground substance

µ1 = -76.5566 α1 = 0.444505 µ2 = 38.0248 α2 = 0.658874 µ3 = 38.6030 α3 = 0.232102

Lateral anulus ground substance

µ1 = -96.435 α1 = -1.2012x10-2 µ2 = 48.3595 α2 = 0.1670090 µ3 = 48.1117 α3 = -0.189028

Posterior anulus ground substance

µ1 = -335.299 α1 = 1.56578 µ2 = 166.922 α2 = 1.74099 µ3 = 168.422 α3 = 1.39183



The anulus fibrosus in the FEM was divided into 3 regions (Figure 6-12). The

Inhomogeneous FEM was analysed under 500N compressive torso loading.

Figure 6-12 Anulus regions in the Inhomogeneous FEM mesh

6.6.1 Results of the Inhomogeneous FEM

The analysis of the Inhomogeneous FEM solved successfully. The deformed shape of

the Inhomogeneous FEM and the anterior translation and rotation is shown in Figure

6-13.

Figure 6-13 Deformed shape of Inhomogeneous FEM (Wireframe shows undeformed mesh)

The posterior anulus fibrosus bulged outward (Figure 6-14). When the deformed

shape of the mesh was viewed sagittaly, the posterolateral regions of the anulus

Posterior

Lateral

Anterior

0.62mm 0.69mm 3.78o

Inward posterior bulge



fibrosus bulged radially inward by 0.34mm (Figure 6-14). This was similar to the

observed deformation of the preliminary FEM when analysed under full flexion

loading (Section 3.6.3). However, the magnitude of this outward posterior and inward

posterolateral bulge was not as high as for the flexed preliminary FEM.

Figure 6-14 Posterior anulus bulges outward, posterolateral anulus bulges inward

The maximum von Mises stress in the anulus fibrosus (0.90 to 1.35MPa) was

observed on the inferior surface at the interface between the posterior and lateral

regions (Figure 6-15). This stress ranged from 0.90 to 1.35MPa. A higher stress on

the inferior posterolateral edge of the disc was reasonable given the forward rotation

and translation of the superior surface of the disc. However, it was considered that

this location of the maximum stress was partly an artefact of the discontinuity created

by the use of different material parameters for these regions. A region of high stress

(0.83 to 0.90MPa) was observed on the superior, posterior anulus ground substance.

This stress range was higher than the maximum von Mises stress observed in the

anulus fibrosus of Homogeneous FEM but in a similar location.

Outward mid-posterior bulge

Inward postero-lateral bulge of 0.34mm with respect to point X

X



A

B C

Figure 6-15 Von Mises stress contours for the anulus fibrosus. A. View of the superior disc; B. View of the inferior disc; C. Posterior anulus, viewing from the

superior disc into the nucleus pulposus

Displacements observed in the Inhomogeneous FEM were compared with the average

displacements observed in the literature (Markolf and Morris, 1974; Brown et al.,

1957; Shirazi-Adl et al., 1984; Virgin, 1951; Nachemson, 1992) (Figure 6-16).

Results for the anterior, lateral and posterior bulge of the peripheral anulus fibrosus

and the axial displacement of superior surface of the disc under a 500N compressive

load were obtained from the experimental work of previous researchers. These results

were averaged and the standard deviation of these data was determined (Figure 6-16).

The anterior and posterior bulges in the FEM were determined at mid height. Axial

displacement in the FEM varied from 0.95mm in a cephalic direction at the posterior

margin of the superior anulus to 1.42mm in the caudal direction on the anterior

margin of this surface. This significant variation was a result of the forward rotation

High stress on superior, posterior anulus – 0.83-0.90MPa

Maximum von Mises stress at junction between posterior and lateral anulus – 0.90-1.35MPa



of the superior surface of the FEM. The axial displacement that was compared to

experimental data in Figure 6-16 was obtained at a mid anterior-posterior point on the

superior disc.

A

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

AB LB PB AD

Disp

lace

men

t (m

m)

Experimental FEA

B

0

0.5

1

1.5

2

2.5

500N Compression

Nuc

lear

Pre

ssur

e Ra

tio

Nachemson (1960) FEA

Figure 6-16 Comparison of FEA and experimental results. A. Displacements (the maximum lateral bulge in the FEM was on the right lateral anulus fibrosus);

B. Nucleus pulposus pressure (AB=anterior bulge, LB=lateral bulge, PB=posterior bulge, AD=axial displacement)

The anterior bulge of the anulus was within the first standard deviation from the

average based on the experimental results of previous researchers. A comparison of

the average experimental data with the results from the FEM showed that the lateral

and posterior bulge in the FEM overestimated the average experimental bulge (Figure

6-16). The axial displacement at a mid anterior-posterior location on the superior disc

surface was within the first standard deviation of the experimental data; however, this



value was significantly lower than the displacements observed at the anterior and

posterior margins of the disc – 1.421 and 0.95mm, respectively. These displacements

were outside the first standard deviation of the experimental results.

Pressure in the nucleus pulposus of the Inhomogeneous FEM was 2.06 times the

applied pressure (Figure 6-16). This was considerably higher than the experimentally

determined ratio of 1.5 (Nachemson, 1960).

6.6.2 Discussion and conclusions for the Inhomogeneous FEM

This section discusses the posterior and posterolateral bulge of the anulus fibrosus, the

anterior translation and rotation of the superior surface of the FEM and the

compliance of the inhomogeneous ground substance.

6.6.2.1 Posterior and posterolateral bulge of the anulus fibrosus

The observed forward rotation of the superior surface in the Inhomogeneous FEM

was 30% of the full flexion rotation applied to the preliminary FEM. This was very

high considering the loading was simulating torso compression. The inward

posterolateral bulge of the Inhomogeneous FEM was likely due to this high forward

rotation.

It was likely that the inward bulge of the posterolateral anulus was a result of the axial

extension in this region. In order to maintain the incompressibility of the anulus

fibrosus ground substance, there was a contraction in the posterolateral direction in

response to the axial extension. This resulted in an inward bulge. A similar cause

was postulated for the deformed shape of the preliminary FEM when analysed under

full flexion. However, it was noted that this deformed geometry had not been

reported in vivo.

In Chapter 3 it was suggested that this inward deformation may have been previously

unobserved due to the inability of current imaging techniques to capture images of the

disc when in a flexed posture. Alternatively, it was suggested that this inward



deformation may have been a result of the inaccuracy of the hyperelastic strain energy

equation employed to represent the anulus fibrosus ground substance. The

observation of a similar deformed geometry in the FEM which incorporated an

improved hyperelastic strain energy equation and material parameters for the anulus

fibrosus ground substance determined from experimentation, suggested that this

deformed geometry may be accurate but would require confirmation from

experimental studies.

6.6.2.2 Anterior translation and rotation of the superior surface of the

Inhomogeneous FEM

The analysis of the preliminary FEM in Chapter 3 demonstrated an anterior

translation and rotation of the superior surface of 0.127mm and 0.68o, respectively.

This translation and rotation was significantly increased in the Inhomogeneous FEM

due to the increased compliance of the anulus fibrosus ground substance in this FEM.

It was thought that the high values of anterior translation and the observed anterior

flexion of the FEM under torso compression was a result of a low shear stiffness of

the anulus fibrosus ground substance. This low shear stiffness of the ground

substance was apparent from the experimental results presented in Chapter 4. The

simple shear stiffness of the anulus fibrosus was approximately 0.1-0.5MPa. This

range was an order of magnitude lower than the average stiffness under uniaxial and

biaxial compression loading.

6.6.2.3 Compliance of the inhomogeneous anulus fibrosus ground substance

The high compliance of the ground substance was apparent from comparison of the

average experimental data obtained by previous researchers and the FEM results for

the peripheral anulus bulge and axial displacement of the superior disc surface.

Displacements in the FEM were up to 2 times the average experimental values. The

pressure in the nucleus pulposus was 40% higher than the expected value of 625kPa

according to the nucleus pressure ratio determined by Nachemson (1960). Excessive



deformations of the nucleus pulposus in the FEM would have resulted in an elevated

nucleus pressure in comparison to the in vivo condition.

Since the material properties of the anulus fibrosus ground substance were obtained

from experimentation on specimens of sheep anulus fibrosus, at this stage these

parameters were considered to provide a reasonable representation of the human disc

(Further discussion of the similarity between the human and the sheep anulus fibrosus

ground substance is given in Chapter 7). The hyperelastic parameters were fit to this

experimental data using specific criteria and were believed to predict the experimental

behaviour of the tissue with a reasonable level of accuracy. Therefore, while the

compliance of anulus fibrosus ground substance was one likely cause for the

inconsistencies between the experimental behaviour of the intervertebral disc and the

results of the FEM, these material properties were believed to provide similar

mechanical behaviour to that of the human anulus fibrosus ground substance.

It was noted that other potential causes for the discrepancies between the experimental

behaviour of the intervertebral disc and the results of the FEM were the assumption of

linearity for the collagen fibre material properties and the method used to apply the

compressive torso loading condition (Section 6.6.2.4).

6.6.2.4 Method for applying compressive torso load

The method for applying the compressive torso loading condition involved applying a

rigid beam constraint between all the nodes on the superior disc surface and a node

located at the centroid of this surface in the transverse plane. The compressive load

was then applied to the centroid node. This method ensured that all the nodes on the

superior disc were subjected to the same compressive force and additionally, that

these nodes were constrained to deform with respect to one another. That is, these

nodes deformed as a plate, rather than permitting inhomogeneous force distributions

across the superior disc due to the differing stiffness of the disc components.

This method of load application incorporated three assumptions which included:

• The superior vertebra was a rigid body;



• The in vivo compressive torso load acted through a centroid; and

• The centre of motion during torso compression in the spine was located at the

centroid of the superior disc surface in the transverse plane.

The first of these assumptions was justified in Chapter 3. With respect to the second

assumption, it seemed reasonable to assume that as a result of the considerable sagittal

wedge shape of the L4/5 intervertebral disc and also, due to its location toward the

caudal spine, a compressive load could result in some forward rotation of the disc.

However, this rotation would be difficult to quantify in vivo and no evidence was

found for this in the literature. Furthermore, the point or axis about which this

rotation occurred was not known. In the absence of more accurate data, it seemed

reasonable to assume that the location of the point about which the compressive

loading was applied in vivo would be the centroid of the disc in the transverse plane.

If however, this location was not accurately specified then the application of

compressive loading through the centroid of the disc would result in large rotations

and translations of the superior surface.

6.7 Discussion and Conclusions on Implementation of the Homogeneous and

Inhomogeneous Material Parameters for the Anulus Fibrosus Ground

Substance

The improvement of the material properties and geometry for the rebar elements

representing the collagen fibres in the Homogeneous FEM resulted in a reduction in

the displacements, rotations and nucleus pressure. However, the magnitude of these

parameters remained higher than expected. The Inhomogeneous FEM was analysed

using the improved material properties and geometry for the rebar elements in the

anulus fibrosus. The results of this analysis demonstrated translations and rotations of

the superior surface of the FEM which were high in comparison to in vivo

observations of the intervertebral disc. Additionally, due to the significant

deformation of the nucleus pulposus – in terms of anterior translation, axial

displacement and outward radial bulge – the nucleus pressure overestimated the

values observed in vivo.



A likely explanation for the high rotation and translation of the superior disc surface

and the large nucleus pulposus pressure was the lack of resistance to deformation

offered by the compliant anulus ground substance. Another possible cause was the

location for the point through which the compressive loading was applied to the

superior disc surface.

To limit the incorporation of unnecessary complexity into the analyses of the model

geometries analysed in this and previous chapters, these meshes did not include any

anatomical structures outside the intervertebral disc. However, subsequent to the

analyses of the Homogeneous and Inhomogeneous FEMs it was concluded that the

representation of the disc structures alone oversimplified the anatomy of the disc.

Owing to the close physical relationship between the anterior and posterior

longitudinal ligaments, these structures were included in the geometries of subsequent

models.

It was thought that the inclusion of the anterior and posterior longitudinal ligaments

would not significantly increase the complexity of the model and would permit a

closer representation of the in vivo anatomy. It seemed likely that the inclusion of the

anterior and posterior longitudinal ligaments into the FEM would provide shear

stiffness to the loaded structure. These ligaments would provide additional resistance

to the anterior translation and rotation of the superior disc and thereby reduce the axial

deformation and lateral bulge in the anulus and the excessive nucleus pressures. In

the FEMs analysed in the previous Chapters, the control of trunk movement provided

by the ligaments was simulated through the loading and boundary conditions

prescribed (Section 3.5). However, the resistance to excessive anterior and posterior

bulge of the anulus fibrosus and therefore, the bracing stiffness provided by the

ligaments could not be simulated without the inclusion of elements to define the

geometry and material properties of these structures.

Chapter 7 details the modelling techniques employed to represent the anterior and

posterior longitudinal ligaments. Results are presented for the Homogeneous and

Inhomogeneous FEMs.


Chapter 7: Modelling Anterior and Posterior Longitudinal Ligaments 268

CChhaapptteerr

77

MMooddeelllliinngg AAnntteerriioorr aanndd

PPoosstteerriioorr LLoonnggiittuuddiinnaall

LLiiggaammeennttss

The results of the compressive analysis carried out on the FEM in Chapter 6

suggested that both the inhomogeneous and the homogeneous material properties for

the anulus fibrosus ground substance produced a disc that was too compliant. Large

anterior displacements were observed in the deformed mesh for a torso uniaxial

compressive load of 500N and excessively high nucleus pulposus pressures were

obtained.

In vivo, the anterior longitudinal ligament (ALL) and the posterior longitudinal

ligament (PLL) span the peripheral surfaces of the anterior and posterior

intervertebral disc, thus connecting the adjacent vertebrae. The longitudinal ligaments

were not included in either the preliminary FEM or the FEM analysed in Chapter 6.

In order to improve the level of anatomical detail in the FEM, the ALL and PLL were

simulated (7.1). These ligaments were represented as tension-only structures. They

provided resistance to the peripheral bulge of the anulus fibrosus and limited the

anterior translation of the disc.

The pressure in the nucleus pulposus is dependent on the deformation of the nucleus

pulposus in the FEM – excessive deformation of the nucleus resulted in an increase in

the nucleus pressure. Therefore, the high pressures observed in the nucleus were



possibly due to the significant deformation of the anulus fibrosus. It was believed that

the inclusion of the anterior and posterior longitudinal ligaments would have the

effect of stiffening the anulus fibrosus, reducing the deformation of the nucleus and

therefore, reduce the nucleus pressure.

Both the Inhomogeneous FEM and the Homogeneous FEM were analysed with the

anterior and posterior longitudinal ligaments included in the model geometries.

Initially these models included a simplified sagittal geometry which had not been

manipulated to ensure the anterior and posterior disc heights were comparable to the

in vivo condition. Section 7.2 and 7.4 detail the results of these models.

Once the technique for modelling the ALL and PLL was established, the sagittal

dimensions of the Inhomogeneous and Homogeneous FEM were manipulated using

the technique outline in Chapter 3, Section 3.5.6. The results of these analyses are

presented in Section 7.3 and 7.6. The Homogeneous model analysed in these sections

are employed for the analysis of anular lesions in Chapter 8.

All stresses stated in this chapter are expressed in MPa.

7.1 Method of Representing the Longitudinal Ligaments in the FEM

The anterior longitudinal ligament (ALL) and the posterior longitudinal ligament

(PLL) were represented using spring elements. Details of the material properties and

geometry of these ligaments were obtained from the literature.

7.1.1 Spring elements

The anterior and posterior longitudinal ligaments do not carry compressive loading.

Therefore, the elements used to model these structures were tension-only members.

Spring elements were considered to best represent the mechanical behaviour of the

longitudinal ligaments. The spring elements were linear elements joining two nodes

and the line of action of the spring element was between these nodes. As such, this

line of action would displace in relation to the bounding nodes during large-



displacement analyses such as the intervertebral disc FEM. The spring elements

possess no rotational degrees of freedom and the element output is the spring force

and the nominal strain in the line of action of the element.

The anterior longitudinal ligament (ALL) is “loosely” connected to the anterior anulus

fibrosus (White and Panjabi, 1990; Bogduk, 1991). Bogduk (1991) stated that the

anterior longitudinal ligament was connected to the anterior anulus with areolar tissue.

However, specific details of the connection between the ligament and the peripheral

anulus fibrosus were not available in the literature. Details of the tensile strength of

the connection, the mechanical nature of the connection or specifics of the physical

nature of the connection were not clear. The posterior longitudinal ligament, PLL, is

“intimately” connected to the posterior anulus (White and Panjabi, 1990) and fibres

from this ligament insert into the anulus fibrosus (Bogduk, 1991). Details of the

connection between the PLL and the peripheral anulus fibrosus were not evident from

the literature.

The close connection between the PLL and the posterior anulus fibrosus was

simulated by axially linking all the vertex nodes on the continuum elements at this

interface with spring elements (Figure 7-1 A). In an attempt to represent the limited

connection between the anterior anulus fibrosus and the ALL, selected vertex nodes

on the continuum elements at the interface between the anterior anulus and the ALL

were linked using spring elements (Figure 7-1 B). However, this was not successful.

Since the spring elements were essentially a link which tied the nodes, they did not

provide resistance to the bulge of the anulus at the nodes that were not connected to

the springs. Therefore, there were large anterior displacements of some nodes on the

anterior anulus and the analysis was not successful. An improved result was obtained

when the spring elements linked all the vertex nodes on the anterior anulus (Figure

7-1 C).



A

B

C

Figure 7-1 Spring elements connected to corner nodes. A. Spring elements on the posterior anulus fibrosus – connected to all corner nodes; B. Spring elements on the anterior anulus fibrosus – connected to selected corner nodes to represent

“loose” connection; C. Spring elements on the anterior anulus fibrosus – connected to all corner nodes.

Given the peripheral bulge of the anterior anulus fibrosus in response to the majority

of the loading conditions applied to the FEM, the altered method of connecting the

anterior anulus fibrosus elements and the ALL spring elements was not considered to

compromise the results of the analyses on the FEM. The manner of connection

between the anterior anulus and the ALL was considered to be realistic for all loading

conditions except when the anterior anulus bulged radially inward, thereby separating

the two structures and causing the connection between them to be tensioned. The

only loading mode which resulted in the inward bulge of the anterior anulus fibrosus

was extension. While extension of the lumber spine resulted in a reduction of the

anterior bulge of the anulus, this reduction was not sufficient to separate the ALL and

the anterior anulus fibrosus in the FEM. Therefore, the method employed to represent

Corner nodes on anulus fibrosus continuum elements

Spring elements axially connecting corner nodes on anulus fibrosus continuum elements



the connection of the ALL to the anterior anulus fibrosus was considered to be

acceptable.

It was not possible to connect the spring elements to the midside nodes on the anterior

and posterior peripheral anulus fibrosus. Linking these elements to the midside nodes

resulted in inconsistent loading in the spring elements and erroneous results. This was

similar to the difficulties encountered when the hydrostatic fluid elements were

connected to the midside nodes on the inner anulus surface (Section 6.2). Therefore,

it was necessary to constrain all the midside nodes on the anterior and posterior anulus

fibrosus to possess degrees of freedom that were calculated from the adjacent corner

nodes. This method was similar to that employed to constrain the midside nodes on

the inner anulus surface.

7.1.2 Anterior and posterior longitudinal ligament geometry

On the basis of the literature reviewed in Section 2.1.4.2 the width of the ALL over

the L4/5 intervertebral disc was modelled as 20mm. Corner nodes on the continuum

elements representing the anulus fibrosus ground substance were selected such that

the width of the ALL in the FEM was 20mm. A similar approach was used to define

the lateral boundary of the PLL. A width of 15.75mm was used to model the lateral

dimension of the PLL (Section 2.1.4.2).

The cross-sectional profile of the PLL was an ellipse (Tkaczuk, 1968) and it was

assumed that the ALL cross-section was also an ellipse. As detailed in Section

2.1.4.1, the average cross-sectional area of the ALL and PLL were 43.2mm2 and

25.2mm2, respectively.

7.1.3 Crimp and pre-tension in the anterior and posterior longitudinal

ligaments

The mechanical response of connective tissue such as ligaments is a result of the

mechanical response of the collagen fibres, elastin fibres and the ground substance in



which these fibres are embedded (Behrsin and Briggs 1988). The ground substance

consists of cells, proteoglycans, water and other non-collagenous proteins.

A typical load-deformation curve for a ligament could be divided into 3 distinct

regions (Betsch and Baer, 1986, White and Panjabi, 1990, Bogduk, 1991). Initially

there is a neutral zone or toe phase, where a small increase in stress results in a large

increase in strain. This is followed by the elastic zone or linear phase where the

stiffness of the material is increased and the tissue stress-strain response is near linear.

These two regions constitute the physiological loading range of the ligament. The

final region is the plastic zone or macro-failure phase where permanent damage and

failure occur.

The neutral zone represents the loading range during which the ‘crimp’ in the

collagen fibres is removed (Betsch and Baer, 1986, Bogduk, 1991). Crimp is the

buckling in the collagen fibres which is present when they are at rest (Betsch and

Baer, 1986, Bogduk, 1991). The increase in stiffness of the ligament with increasing

strain is a result of their crimped structure (Shah et al., 1979).

As such, the mechanical response of ligaments is complex and is intrinsically

dependant on the behaviour of the collagen fibres. The ligament representation

employed in the FEM was a simplification of the in vivo behaviour.

Kirby et al. (1989) reported that the crimp in the collagen fibres of both longitudinal

ligaments was removed once a strain of 12% was achieved. Neumann et al. (1992)

stated that the strain response of the ALL exhibited delineation between the neutral

zone and the elastic zone at a strain of 10%. It was noted by Hukins et al. (1990) that

the waviness in the collagen fibres of the ALL disappeared at a strain of 10% ± 1%

and in the PLL at a strain of 8% ± 1%.

On the basis of these values, it was considered that crimp in the ALL and PLL would

be removed once a strain of 10% was reached.

Nachemson and Evans (1968) stated that the ligamentum flavum was in tension when

the spine was in the neutral position. This tension varied linearly between 1.8N in



subjects under 20 years of age and 0.5N in subjects over 70 years of age. Assuming a

similar loading condition was present in the longitudinal ligaments and given the age

of the disc specimen used for the FEM was 49 years, the pre-tension in the ALL and

PLL would be approximately 1N. Using the average cross-sectional areas of the ALL

and the PLL, the stress associated with this pre-tension was calculated. A pre-tension

of 1N in the ALL would create a stress of approximately 0.025MPa. A pre-tension of

1N in the PLL would create a stress of 0.042MPa.

The mathematical model of the spinal ligaments proposed by McGill (1988) found

that for a strain of 10.1%, the stress in the PLL was 0.2MPa. According to the stress-

strain response of the longitudinal ligaments reported by Chazal et al. (1985) a strain

of 10% corresponded to a stress of between approximately 0.4 and 1 MPa in the ALL

and over 1 MPa in the PLL. Therefore, the pre-tension stress associated with the 1N

load would not be sufficient to remove the crimp in the ALL or in the PLL.

Even so, for the purpose of the current model it was assumed that the state of tension

in the ALL and PLL due to relaxed standing was sufficient to cause any crimp in the

ligaments to be removed and the mechanical response of the ligament to be in the

‘elastic zone’. In this way, the ALL and PLL were modelled as linear elastic

materials. This avoided the introduction of more complexity into the FEM by

defining a nonlinear elastic or hyperelastic material description for these tissues.

7.1.4 Stiffness of the anterior and posterior longitudinal ligaments

The mechanical properties of the spring elements were defined in terms of force per

relative displacement or spring stiffness. As was outlined in Section 2.5.4, the

average elastic modulus of the ALL and PLL was 32.7MPa and 42MPa, respectively.

Using the stiffness and the average cross-sectional area of the ligaments as well as

details of the anterior and posterior height of the anulus fibrosus in vivo (Tibrewal and

Pearcy, 1985) the stiffness of the ALL and PLL were calculated (Section 2.5.4). The

stiffness of the ALL and PLL were 103.4N/mm and 192.44N/mm, respectively.



The spring elements did not carry compressive loading and only possessed tensile

stiffness.

7.2 Analysis of the Homogeneous FEM with Longitudinal Ligaments

A similar finite element model to that analysed in Chapter 6 was solved to investigate

the effects of modelling the ALL and PLL. This FEM was based on the preliminary

FEM analysed in Chapter 3. Several features of this homogeneous model were

improved in Chapter 6 and these improvements included:

• A polynomial, N=2 hyperelastic material represented the anulus fibrosus

ground substance

• 4 node hydrostatic fluid elements modelled the nucleus pulposus

• The collagen fibres in the anulus fibrosus included a radially varying stiffness

of 655-426MPa and a radial variation in the collagen fibre inclination. This

angle varied between 55 and 65o to the axial direction through the disc.

The ALL and PLL were included in this FEM. A pressure of 70kPa was introduced

into the nucleus pulposus during the first loading step (section 3.5.2) and a

compressive torso load of 500N was analysed during the second loading step.

7.2.1 Results

When the 70kPa nucleus pressure was introduced, the superior surface of the FEM

displaced in a cephalic direction by between 5.12 x 10-2 and 2.18 x 10-2mm. The mid-

posterior anulus bulged outward by 0.33mm and the posterolateral anulus bulged

inward by up to 9.24 x 10-2mm. This deformation pattern was similar to that observed

in the preliminary FEM (Section 3.6.2).

Displacements and rotation of the superior surface of the FEM in response to the

500N compression are detailed in Table 7-1. These are compared to the results from

the analysis of the homogeneous FEM which did not incorporate the longitudinal

ligaments (Section 6.4.3).



Table 7-1 Displacements and rotation of the superior surface of the FEM due to

the 500N load

Displacements Rotation

Anterior displacement

of anterior edge

Caudal displacement

of anterior edge

Anterior displacement of postero-lateral edge

Cephalic displacement of postero-lateral edge

Anterior Rotation

FEM without

ALL or PLL (Chapter 6,

Section 6.4.3)

0.54mm 1.30mm 0.60mm 0.57mm 3.31o

FEM with ALL and

PLL 0.45mm 1.07mm 0.48mm 0.29mm 2.44o

Sagitally, the deformed shape of the posterolateral anulus fibrosus in the FEM did not

demonstrate a significant inward radial bulge as was observed in the previous

analyses of the FEM. It was believed that the stiffening mechanism provided by the

ALL and PLL reduced the forward translation and rotation of the FEM and therefore,

did not encourage the inward posterolateral anulus bulge (Figure 7-2).

Figure 7-2 Deformed shape of the inhomogeneous FEM with the ALL and PLL modelled – there was no inward posterolateral bulge of the peripheral anulus

fibrosus

Details of the stresses observed in the FEM are listed in Table 7-2 and compared with

the results of the FEM that did not include the longitudinal ligaments.



Table 7-2 Comparison of von Mises stress in the anulus ground substance of the

FEMs with and without ALL and PLL

Location Maximum in the FEM

Superior, posterior anulus

Inferior, left posterolateral

anulus

Lateral margin of PLL

FEM without ALL or PLL (Chapter 6,

Section 6.5.3)

In posterior endplates:

3.86- 4.50MPa 0.48-0.52MPa 0.39-0.48MPa

FEM with ALL and PLL


2.95-3.93MPa 0.18-0.20MPa 0.22-0.27MPa 0.16-0.18MPa

The maximum von Mises stress in the model occurred in the posterior region of the

endplates. This stress was at the junction of the anulus fibrosus, the nucleus pulposus

and the cartilaginous endplates and ranged from 2.95 to 3.93MPa. Peak stresses in

the anulus ground substance (Figure 7-3) were in a similar location to those observed

in the FEM without the longitudinal ligaments present; however, there was an

additional region of high stress at the lateral margin of the PLL.



Figure 7-3 Von Mises stress distribution in the anulus fibrosus ground substance of the homogeneous FEM with longitudinal ligaments modelled.

Higher stresses were observed in the left posterolateral anulus fibrosus compared to

the right anulus due to the orientation of the collagen fibres in these elements. The

fibre orientation was such that the rebar elements in the left anulus carried a greater

axial load than those in the right anulus. This was confirmed by observation of the

nominal axial stress in the rebar elements in the right and left posterolateral anulus

fibrosus. The rebar elements in the left anulus demonstrated stresses as high as

8.81MPa and in the right anulus the stresses were approximately 0.18MPa. A similar

observation was made in the stress distribution in the homogeneous FEM without the

ALL and PLL present (Section 6.4.3). The maximum rebar element stress occurred in

the right lateral anulus fibrosus. This stress was in the circumferential element layer,

second in from the outermost anulus layer and was 13.05MPa.

Stresses in the PLL ranged from 3.17MPa at the mid-lateral location to 1.38MPa at

the lateral margins. Due to the forward translation and rotation of the disc and the

tension-only nature of the ligaments, the ALL did not experience any stress. The

nominal axial strain in the PLL ranged from 11.80% in the mid lateral spring elements

to 5.13% in the lateral-most elements. The ultimate tensile strength of the PLL is

High stress at lateral margin of posterior

longitudinal ligament –

0.16-0.18MPa

Peak stress on the inferior left posterolateral

anulus due to the collagen fibre inclination in

this circumferential element layer – 0.16-0.20MPa



between 2.9-20MPa and the ultimate tensile strain is between 8-44% (Chazal et al.,

1985; Dvorak et al., 1988; Goel et al., 1986; Nachemson and Evans, 1968; Tkaczuk,

1968). This indicated that the PLL had not reached failure stress/strain.

Pressure in the nucleus pulposus reached 0.88MPa due to the 500N compression.

This was 2.1 times the applied pressure and was again higher than the value of 1.5

stated by Nachemson (1960).

7.2.2 Discussion

The presence of the ALL and PLL in the FEM resulted in a reduction in the anterior

displacement and axial movements of the superior surface of the FEM. Anterior

rotation of this surface was reduced from 3.31o to 2.44o. The observed reductions in

the anterior displacement and rotation of the superior surface of the disc FEM

suggested that the ligaments were providing additional shear stiffness to the disc.

The general state of stress in the FEM demonstrated lower stresses compared to the

FEM analysed in Section 6.4. This was attributed to the additional portion of the

applied load which was carried by the longitudinal ligaments.

The initial anterior and posterior disc heights used in this model were 14mm and

5.5mm, respectively. These were the average in vivo disc heights reported by

Tibrewal and Pearcy (1985). However, due to the rotation of the superior surface of

the FEM the disc heights after the 500N load was applied did not reflect the in vivo

values. Using the iterative procedure outlined in Section 3.11.6 the disc heights of the

FEM were manipulated in the model analysed in Section 7.3 to reflect the correct in

vivo anterior and posterior disc heights.

It was considered that the excessive pressure in the nucleus pulposus was indicative of

excess deformation of the anulus fibrosus. Conclusions in relation to the inaccuracy

of the nucleus pressure were reserved until further analysis of the FEM with accurate

disc heights was modelled in the homogeneous FEM and until inhomogeneous

material properties for the anulus fibrosus were included.



7.3 Analysis of the Homogeneous FEM with Longitudinal Ligaments – Correct

Disc Heights

The displacements of the anterior and posterior margin of the superior surface in the

FEM analysed in Section 7.2 (Table 7-1) were used to obtain accurate disc heights.

Using the iterative process detailed in Section 3.11.6 these displacements were

applied as offsets for the position of the superior surface of the homogeneous FEM in

the unloaded state. Tibrewal and Pearcy (1985) determined that the in vivo sagittal

height of the anterior disc ranged from 11 to 16mm with an average height of 14mm

and the posterior height ranged from 3 to 8mm with an average height of 5.5mm. The

axial dimensions in the sagittal plane of the FEM were monitored to achieve a final

deformed anterior height of approximately 14mm and a posterior height of

approximately 5.5mm.

7.3.1 Results

After a sensible number of iterations the anterior and posterior heights of the FEM

with a 500N compressive torso load applied were 14.5mm and 5.27mm, respectively.

These dimensions were well within the in vivo range determined by Tibrewal and

Pearcy (1985). Observation of the deformed geometry of the FEM showed that the

mid posterior anulus bulged outward. The right lateral posterior anulus was near

vertical (Figure 7-4 A); however, the left, inferior, posterolateral anulus fibrosus

bulged inward (Figure 7-4 B).



A

B

Figure 7-4 Deformed sagittal geometry of the homogeneous FEM with the correct disc heights. A. Viewed from right lateral direction; B. Viewed from left

lateral direction

It was unusual that the left posterolateral anulus bulged inward while the right

posterolateral anulus was near vertical. The inward bulge in this region was due to

the forward rotation of the superior disc during the 500N loading step. This rotation

was outlined in Section 3.5.6 and was used to generate final sagittal dimensions of the

FEM that were comparable to in vivo observations. Higher shear stresses were

present in the left, inferior, posterolateral anulus as a result of this rotation (Figure

7-5) and these stresses ranged from 0.13-0.15MPa. The presence of a higher stress in

the left anulus rather than the right anulus was due to the orientation of the rebar

elements in this circumferential element layer. These elements were orientated such

that they connected the inferior disc surface to the superior disc surface in a clockwise

direction when the disc was viewed superiorly. This orientation predisposed the

elements in the left posterolateral anulus to bear a higher portion of tensile load during

Inward bulge of the posterolateral anulus



forward motion of the disc compared to the right posterolateral anulus. The nominal

axial stress in the rebar elements in this region was between 5.9 and 11.1MPa

Figure 7-5 Shear stress in the anulus fibrosus due to the anterior translation of

the superior surface with respect to the inferior surface

The nucleus pressure in the homogeneous FEM reached 0.85MPa after the 500N load

was simulated. This pressure was 2 times the pressure applied to the superior surface

of the FEM and therefore, exceeded the expected ratio of 1.5 as stated by Nachemson

(1960). The nucleus pressure in the FEM was dependent on the deformation of the

nucleus pulposus volume – large deformations of this volume resulted in higher

nucleus pressures. Figure 7-6 shows the deformed shape of the nucleus. The anterior

bulge of the anterior nucleus wall was 0.62mm and the posterior bulge of the mid-

posterior nucleus wall was 0.72mm. To obtain an accurate posterior and anterior

sagittal height in the FEM the anterior edge of the nucleus displaced by 0.72mm and

the posterior edge displaced by 0.23mm.



Figure 7-6 Sagittal view of the deformed nucleus pulposus. (Wireframe lines

denote the undeformed mesh)

Peak von Mises stresses of 3.79-4.14MPa were found at the posterior junction of the

inferior endplate, the anulus fibrosus and nucleus pulposus. Due to the inward

deformation of the left, inferior, posterolateral anulus fibrosus, this region

demonstrated high von Mises stress in the anulus fibrosus ground substance (0.28-

0.34MPa). The anulus ground substance at the lateral margins of the PLL

demonstrated a higher region of stress as did the superior, posterior surface of the

anulus (Figure 7-7). These stresses were 0.17-0.20MPa.



Figure 7-7 Von Mises stress distribution in the anulus fibrosus ground substance

of the homogeneous FEM with corrected sagittal dimensions

7.3.2 Discussion

The state of stress in the FEM with the corrected sagittal dimensions was similar to

the stress state in the uncorrected FEM. These stresses were low which was

reasonable since the loading condition was torso compression.

It was necessary to incorporate the unusual deformed shape of the left inferior,

posterolateral anulus in the homogeneous FEM in order to accurately represent the

sagittal dimensions of the intervertebral disc in vivo. It was thought that this

deformation was caused by the compliance of the anulus fibrosus ground substance.

Future analyses of the homogeneous FEM which incorporated more complex loading

conditions were interpreted with consideration of the region of higher stress/strain

which occurred in the left, inferior, posterolateral anulus fibrosus.

It was noted that while the magnitude of the inward radial displacement of the inferior

posterolateral anulus was questionable, the mechanism by which it occurred could

potentially manifest in vivo. If the inclination of the collagen fibres in the anulus

fibrosus was such that there was an imbalance in the strain/stress in the fibres in either

Peak von Mises stress in the anulus 0.28-0.34MPa

Region of high stress at the lateral margins of the PLL and on the superior, posterior surface of the anulus 0.17-0.20MPa



the left or right posterolateral anulus during relaxed standing, this may predispose this

region of the disc to damage.

As was observed in Section 7.2.2, the nucleus pressure was high in comparison to the

experimental observations of Nachemson (1960). It was believed that this was a

result of the compliance of the anulus fibrosus ground substance and in particular, the

deformation of the nucleus walls that were in contact with the anulus fibrosus. Radial

displacements as high as 0.72mm were observed on these walls.

7.4 Analysis of the Inhomogeneous FEM with Longitudinal Ligaments

Spring elements representing the ALL and PLL were incorporated into the

inhomogeneous FEM that was analysed in Chapter 6, section 6.5. The FEM was

reanalysed using a 70kPa nucleus pulposus pressure in the first loading step and a

500N compressive torso load in the second loading step.

7.4.1 Results and discussion of unsuccessful analyses of the Inhomogeneous

FEM

A converged solution for this FEM was not obtained.

The partial solution completed the step which introduced the nucleus pulposus

pressure and completed 333 increments while attempting to apply the 500N

compression. The final results for these increments in the 500N compression step

were for a completed time of 3.87 x 10-6 which corresponded to an applied

compressive load of 0.002N. This extremely high number of increments for such a

small completed time indicated that the software had encountered significant

difficulties in obtaining convergence.

It was apparent from the analysis output that it was the displacement algorithms that

encountered difficulty in converging. Abaqus requires that the largest correction to

the displacement at a node be less than 1% of the largest increment of displacement in

the entire model. This convergence requirement was achieved on the completed



increments; however, it was not achieved with sufficient ease to permit the time

length on each increment to increase significantly. In the previous analyses which

had returned a completed solution, the Abaqus software reduced the time length of the

increments if the algorithms for force, displacement and nucleus pulposus volume in

this increment converged both successfully and without the need for a time cut-back

in the increment. In the case of the inhomogeneous FEM with the ALL and PLL

present, the time length of the increment remained in the order of 1 x 10-9 for the

entire analysis.

The nodes in the FEM which commonly caused difficulties in convergence of the

displacement algorithms were on the left and right lateral anulus fibrosus (Figure 7-8).

It was noted that these nodes were not common to either the ALL or the PLL. This

suggested that while the inclusion of these structures into the FEM had resulted in an

analysis which could not solve completely, the direct attachment of these ligaments

was not the cause of the displacement convergence problems.

A B

Figure 7-8 Nodes in the anulus fibrosus where difficulties were encountered in the displacement algorithms

There were no nodes in the cartilaginous endplates that caused difficulties in

displacement convergence. There was no apparent explanation for the high

displacement corrections in the lateral anulus ground substance. The rebar elements

which demonstrated the highest nominal axial stress were in the posterior anulus

fibrosus, therefore, the high displacements were not a result of excessive stresses in

the reinforcing elements in the lateral anulus. The maximum displacements in the

Nodes on left lateral anulus with convergence

difficulties

Nodes on right anterolateral anulus with convergence difficulties



three translational degrees of freedom were not observed on the nodes in the lateral

regions where the large displacement corrections were found. However, these nodes

did correspond to regions of variation in the peripheral bulge of the anulus (Figure

7-8).

A B

Figure 7-9 Deformed geometry of the circumferential element layer in the anulus fibrosus where the nodes with the largest displacement correction were

located. The scale on the deformation is 10:1. (Wireframe lines show the undeformed shape) A. Region 1 – the left lateral anulus; B. Region 2 – the right

lateral anulus

Figure 7-9 shows a circumferential element layer in the anulus fibrosus – this layer is

the second layer in from the peripheral anulus. The scale of the deformed geometry is

10:1. Region 1 and 2 in Figure 7-9 contained several nodes which repeatedly

demonstrated high displacement corrections. Observation of the deformed element

layer geometry in comparison to the undeformed geometry showed that the left

posterolateral anulus and the left anterolateral anulus both bulged inward radially

(Figure 7-9 A). However, the anulus nodes in region 1 bulged radially outward. The

magnitude of this outward radial bulge was smaller compared to the inward bulge of

the surrounding regions. It was thought that this variation in the radial bulge of the

anulus was related to the transverse geometry of the disc and in particular, the

curvature of the anulus in the lateral regions. It was unlikely that the inclination of

the rebar elements in this circumferential element layer caused the increased inward

Region 1 – outward radial bulge

Region 2 – outward radial bulge

Inward bulge



bulge of the anterolateral and posterolateral anulus compared to the lateral region 1.

The stress in the rebar elements in this region was in the order of 1 x 10-3N. This

rebar force was low compared to the maximum rebar force in the model at the end of

the partial solution of 0.11N.

Similarly the deformed geometry of the anulus elements in and around region 2

(Figure 7-9 B) showed limited outward radial bulge in this anterolateral region

compared to an inward radial bulge in the right lateral anulus and an inward radial

bulge in the anterior anulus.

In an attempt to obtain a converged solution the analysis was run with only the 500N

compression step. This analysis completed one increment with a time length of 1 x

10-9. The analysis failed on the next increment. Further analysis was not attempted

with a smaller time increment since this minimum value was already extremely low.

In a second attempt to improve the results of the inhomogeneous FEM the inclination

of the rebar elements in the circumferential element layers was altered. The rebar

elements in the outermost element layer linked the inferior anulus surface to the

superior anulus surface in a clockwise direction when viewing the transverse plane

from a superior aspect (Figure 7-10 A). These fibres were inclined at 65o to the axial

direction through the disc. This inclination was changed by 90o such that the rebar

elements in the outermost element layer connected the inferior and superior anulus in

an anti-clockwise direction (Figure 7-10 B). The angle between the rebar elements in

the outer element layer and the axial direction through the disc was 65o. Rebar

element inclination in the successive circumferential element layers was alternated to

create a criss-cross pattern.

The analysis involved a 70kPa pressure introduced to the nucleus in the first step and

a 500N compressive load applied to the superior surface in the second step.

Unfortunately, this did not improve the response of the inhomogeneous FEM and

displacement convergence difficulties were encountered in similar regions of the

anulus to those depicted in Figure 7-8. This confirmed the conclusion that the cause



for the unusual deformation pattern observed in Figure 7-9 was not a result of the

stress in the rebar elements and was likely due to the geometry of the anulus fibrosus.

Figure 7-10 Orientation of rebar elements in outermost circumferential element layer of anulus fibrosus. A. Orientation in the partially completed analysis; B.

Orientation in the second analysis

The FEM with the altered fibre orientation was re-run without the initial step

introducing the 70kPa nucleus pulposus pressure. The analysis completed

successfully. While this completed analysis was an achievement, the lack of the

70kPa loading step in the analysis limited the ability of the FEM to simulate the in

vivo condition of the intervertebral disc.

7.4.1.1 Effects of removing the 70kPa loading condition

To determine the percentage error introduced into the results of the analysis if the

70kPa loading step was removed a sensitivity analysis was carried out on the

homogeneous FEM with the corrected sagittal geometry analysed in section 7.3. This

model was re-analysed with only one loading step to simulate a 500N compressive

load. The results for this analysis were compared with the results presented in Section

7.3.1 and are detailed in Table 7-3.

A

B



Table 7-3 Comparison of the results for the homogeneous FEM loaded with both

a 70kPa nucleus pressure and a 500N compression load and loaded with only a

500N compression load

Data at the end of the 500N compression step

70kPa nucleus pressure and 500N

compression 500N compression

Nucleus pressure 0.85MPa 0.79MPa

Maximum von Mises stress in FEM

Posterior endplate 3.79-4.14MPa

Posterior endplate 3.40-3.71MPa

Maximum von Mises stress in inferior posterolateral anulus ground substance

0.28-0.34MPa 0.28-0.34MPa

Anterior 0.62mm 0.68mm Nucleus

Deformation Posterior 0.72mm 0.60mm

Anterior 1.10mm 1.18mm Axial

displacement Posterior 0.37mm 0.34mm

The results presented in Table 7-3 indicated that the axial displacement and maximum

stress in the anulus fibrosus were the same in both models. However, the nucleus

pressure and deformation were approximately 10% lower in the FEM without a 70kPa

loading condition. The maximum von Mises stress observed in the FEM was located

in similar regions of the endplates; however, the magnitude of this stress was

approximately 10% lower in the FEM without the 70kPa nucleus pressure.

This data indicated that the removal of the 70kPa loading condition would result in a

lower stress state in the FEM and the error in the results would be approximately

10%. Therefore, the results of analyses on the inhomogeneous FEM that did not

incorporate the 70kPa nucleus pressure were interpreted with consideration of this

potential error in the nucleus pressure and the peak von Mises stress observed in the

endplates.



7.4.2 Results of the successful analysis of the Inhomogeneous FEM using a

single loading condition of 500N compression

Further to the results presented in Section 7.4.1 the inhomogeneous FEM was

successfully analysed by removing the 70kPa loading condition and altering the

orientation of the rebar elements in the circumferential anulus element layers by ±90o.

This analysis completed with 65 increments and no convergence difficulties due to

displacement corrections were observed in these increments.

Data for the displacement and von Mises stresses observed in the FEM were

compared with the results of the Inhomogeneous FEM analysed without the ALL and

PLL present (Section 6.5.1). These data are presented in Table 7-4 and Table 7-5 and

the contour plot for the von Mises stress is presented in Figure 7-12.

Table 7-4 Comparison of the displacements observed in the inhomogeneous FEM

with and without the ALL and PLL present.

Displacements Rotation

Anterior displacement

of anterior edge

Caudal displacement

of anterior edge

Anterior displacement of postero-lateral edge

Cephalic displacement of postero-lateral edge

Anterior Rotation

FEM without

ALL or PLL (Chapter 6,

Section 6.6.1)

0.62mm 1.43mm 0.69mm 0.71mm 3.78o

FEM with ALL and

PLL 0.51mm 1.14mm 0.54mm 0.33mm 2.52o



Table 7-5 Comparison of the von Mises stress observed in the Inhomogeneous

FEM with and without the ALL and PLL present

Maximum in

the FEM

Superior, posterior

anulus ground substance

Inferior posterolateral anulus ground

substance – interface

between regions

Anulus ground

substance at the lateral margin of

PLL FEM without ALL or PLL (Chapter 6,

Section 6.6.1)


4.68-6.01MPa 0.83-0.90MPa 0.90-1.35MPa

FEM with ALL and PLL

In posterior endplates*:

3.51-4.21MPa 0.20-0.23MPa 0.32-0.39MPa 0.32-0.39MPa

The displacement and rotation of the superior surface of the inhomogeneous FEM was

reduced when the ALL and PLL were simulated. It was thought that as with the

homogeneous FEM, the longitudinal ligaments provided additional shear stiffness to

the loaded model. Negligible inward bulge (reduced from 0.34mm to 0.05mm) was

observed on the posterolateral surface of the anulus (Figure 7-11).

Figure 7-11 Deformed geometry of the inhomogeneous FEM with the ALL and

PLL present. (Wireframe lines are the undeformed geometry)

Very slight inward posterolateral bulge



Von Mises stresses in the model were decreased when the longitudinal ligaments were

present. This was attributed to the additional portion of the applied load which was

resisted by these structures. The location of the regions of high von Mises in the

anulus fibrosus ground substance are shown in Figure 7-12.

The region of high stress on the inferior anulus ground substance at the junction of the

posterior and lateral anulus was a result of the discontinuity in the materials.

The maximum von Mises stress in the PLL was in the middle of the ligament in a

lateral direction (3.00MPa). This corresponded to an axial strain of 11.12%. When

these values were compared with the ultimate tensile strain of the PLL, 8-44%, it was

apparent that the ligament was at the lower end of this range and failure was unlikely

(Chazal et al., 1985; Dvorak et al., 1988; Goel et al., 1986; Nachemson and Evans,

1968; Tkaczuk, 1968).

Similarly, the maximum stress and strain in the rebar elements of 12.95MPa and 2%

strain were below the tensile failure stress/strain of the collagen fibres.

A nucleus pulposus pressure of 0.79MPa was observed due to the 500N compressive

load. However, there was a possible error of 10% in this value as a result of the

removal of the loading condition that applied a 70kPa nucleus pressure into the

unloaded disc. The nucleus pressure could be as high as 0.87Pa. Both these values

exceeded the expected pressure of 0.63MPa (Nachemson, 1960).



A

B

C

Figure 7-12 Von Mises stress distribution in the anulus fibrosus ground substance of the inhomogeneous FEM with the ALL and PLL simulated

High stress on superior and inferior, posterior anulus – 0.20-0.23MPa

Peak stress in anulus –0.32-0.39MPa High stress at

interface of posterior and lateral anulus – 0.32-0.39MPa



7.4.3 Discussion

Simulation of the ALL and PLL in the FEM resulted in a similar response of the FEM

to that observed in the homogeneous FEM (Section 7.3). The inclusion of the ALL

and PLL in the FEM successfully reduced the high anterior translation and the

forward rotation of the superior surface of the FEM. This reduced rotation resulted in

a lessening of the inward posterolateral bulge of the anulus. It was thought that this

geometry of the loaded FEM more closely represented the loaded intervertebral disc

and data from Table 7-4 were used to manipulate the position of this surface to

generate accurate sagittal dimensions in the FEM (Section 7.6).

Modelling the longitudinal ligaments resulted in a decrease in the general state of

stress in the model and the reduction in the anterior motion of the FEM suggested that

these structures were providing additional shear stiffness. The high nucleus pressure

in comparison to the experimental results of Nachemson (1960) suggested that the

anulus ground substance was too compliant and the deformation of the nucleus

pulposus volume was causing excessive pressures.

7.5 Discussion of the Displacement Convergence Problems in the Unsuccessful

Analyses of the Inhomogeneous FEM

While it was possible to observe behaviour in the displacements of the nodes in the

deformed anulus fibrosus which could be related to the difficulties in displacement

convergence that were encountered in the analysis, this did not sufficiently explain

why these convergence problems occurred in the model. Similar difficulties had not

been observed in convergence of the degrees of freedom in previous analyses of the

preliminary, homogeneous or inhomogeneous FEM.

Problems in the loading conditions, boundary conditions or prescribed conditions at

nodes in the FEM could reasonably be expected to result in the algorithms for either

displacement or force encountering convergence difficulties. However, there were no

conditions defined on the nodes in the lateral anulus fibrosus nor were there any

inconsistencies in the mesh in this region of the anulus. There was no apparent

explanation for the slow displacement convergence on these nodes and the inability to



achieve a completed solution until the rebar element inclination was altered and the

70kPa pressure removed from the analysis.

Considering there were no apparent problems with the prescribed conditions on the

nodes in the anulus fibrosus, the extremely small time increments employed in the

analysis and the necessity for several attempts to be made for convergence of an

increment when these small time steps were used was questionable. These facts were

indications that the constitutive equations governing the anulus fibrosus ground

substance were not stable. It was questioned whether incorporation of the material

parameters for this material into the FEM could permit a sound analysis to be carried

out.

Additionally, in order to obtain a converged solution it was necessary to remove the

physiological loading condition of a 70kPa pressure in the nucleus pulposus of the

unloaded disc. The removal of this loading condition limited the physiological

similarity between the inhomogeneous FEM and the in vivo intervertebral disc and

introduced an error into the results of approximately 10%.

7.6 Analysis of the Inhomogeneous FEM with Longitudinal Ligaments –

Correct Disc Heights

Data for the axial and anterior displacement of the anterior and posterior edge of the

FEM superior surface were used to obtain a deformed mesh with anterior and

posterior dimensions similar to in vivo values. These displacement data are listed in

Table 7-4. The average anterior and posterior height of the L4/5 intervertebral disc

are 14mm and 5.5mm, respectively (Tibrewal and Pearcy, 1985).

7.6.1 Results for the Inhomogeneous FEM with longitudinal ligaments, correct

sagittal geometry and a single 500N compression loading condition

The deformed sagittal geometry and dimensions for the FEM are shown in Figure

7-13. These dimensions were similar to the in vivo values.



A

B

Figure 7-13 Deformed sagittal geometry of the inhomogeneous FEM with the ALL and PLL present and a single 500N compression loading condition. A.

Viewed from the right lateral direction; B. Viewed from the left lateral direction

Details of the von Mises stress and nominal axial stress in the FEM are listed in Table

7-6 and the von Mises stress contours are shown in Figure 7-14 for the anulus fibrosus

ground substance. The FEM demonstrated low stresses in comparison to the potential

failure stress of the anulus fibrosus.

14.57mm

5.30mm



Table 7-6 Stress in the FEM

Maximum von Mises stress in

FEM

Maximum von Mises

stress in the anulus ground

substance

Other regions of high von

Mises stress in the anulus ground

substance

Maximum rebar

element nominal

axial stress

Maximum PLL axial

stress

Stress * 3.04-

3.64MPa 0.38-

0.45MPa 0.23-

0.30MPa 13.41MPa 12.04MPa

Location Posterior, inferior endplate

1. Right inferior, postero-lateral

2. Right lateral

boundary of PLL

1. Superior, posterior

2. Inferior, posterior 3. Left inferior, postero-lateral

Left lateral anulus, second circum-ferential element

layer

Mid lateral PLL

A nucleus pulposus pressure of 0.78MPa was generated due to the 500N compression

which was 2.05 times the applied pressure. Similar to the results presented for the

Homogeneous FEM, this high nucleus pressure was attributed to the deformation of

the nucleus. The anterior bulge of the anterior wall of the nucleus was 0.63mm and

the posterior bulge of the posterior wall was 0.67mm. These values were reasonably

high considering the bulges in similar regions of the nucleus in the preliminary FEM

with the improved 4 node hydrostatic fluid elements were 0.24mm and 0.55mm, on

the anterior and posterior nucleus walls, respectively. The nucleus pressure in this

preliminary FEM was 0.74MPa.



A

B

C

Figure 7-14 Von Mises stress distribution in anulus ground substance of the Inhomogeneous FEM. A. Superior surface viewed from the posterior direction;

B. Maximum stress in anulus; C. Inferior surface viewed from the posterior direction.

Maximum stress in anulus – 0.340-0.415MPa

High stress in anulus on superior, posterior surface – 0.227-0.265MPa

Regions of high stress in the inferior anulus – 0.265-0.302MPa



7.6.2 Conclusions with respect to the Inhomogeneous FEM

In light of the potential instability of the anulus ground substance (Section 7.5) and

due to the limited physiological similarity between the loading conditions of the FEM

and the human disc given the nucleus pulposus pressure in the unloaded disc could

not be simulated, the Inhomogeneous FEM was not used to analyse the biomechanical

effect of anular lesions. Difficulties were encountered in obtaining a converged

solution for a torso loading condition and it was questioned whether the model would

provide accurate, converged results for more complex loading conditions or when

anular lesions were simulated. It was concluded that the Homogeneous FEM would

be used to analyse the effects of anular lesions.

7.7 Discussion of the Mechanical Properties of the Anulus Fibrosus Ground

Substance

Since the position of the superior surface was manipulated to reflect the in vivo

dimensions of the L4/5 intervertebral disc, it was no longer possible to use data for the

in vitro bulge of the peripheral anulus fibrosus or the axial displacement of the

superior surface of the disc. Therefore, the main validation criterion for the FEM with

the corrected sagittal geometry was the nucleus pulposus pressure.

The nucleus pressure in both the inhomogeneous and the homogeneous FEM was

high in comparison to the experimental observations of Nachemson (1960). This high

pressure was attributed to the compliance of the anulus fibrosus ground substance and

the resulting deformation of the anulus walls bounding the nucleus pulposus.

The material parameters used to represent the anulus fibrosus ground substance were

determined from the experimental results presented in Chapter 4. It was questioned

why these material parameters generated a mechanical response for the anulus ground

substance that was too compliant. Three possible causes were highlighted for the

inaccuracy of the experimental results:

• Incorrect choice of strain rate;

• Inappropriate methods of maintaining the testing environment; and



• The appropriateness of using sheep anulus fibrosus ground substance to

represent human anulus fibrosus ground substance.

7.7.1 Strain rate

A strain rate of 0.01 sec-1 was employed in the experimental testing. It was intended

that this loading rate would not permit fluid loss from the tissue and therefore, the

results of testing would provide data on the incompressible mechanical response of

anulus fibrosus ground substance. However, if this loading rate was too slow then

fluid would have been lost from the tissue during testing and due to the lack of pore

fluid pressure a more compliant response would have been recorded. However, the

strain rate sensitivity tests detailed in Chapter 4 suggested that the strain rate of 0.01

sec-1 was an appropriate choice. Higher strain rates resulted in micro-damage of the

tissue at strains observed during relaxed standing, and lower strain rates provided

results that were consistent with fluid loss from the tissue.

The choice of strain rate in the experimental testing of the anulus fibrosus was not

considered to be the cause for the unexpectedly high compliance of the tissue.

7.7.2 Testing environment

The hydration of the test specimens during the uniaxial compression and simple shear

tests was maintained using Ringers soaked muslin and by direct application of

Ringers solution to the tissue. It was suggested that this method of hydration did not

maintain the fluid content of the tissue effectively and a better means of hydrating the

specimens would have involved the use of an environmental chamber.

While the use of more sophisticated methods to hydrate the specimen would have

undoubtedly ensured the fluid content remained constant, the use of the soaked muslin

and direct application of Ringers solution to the specimen were successful. The test

specimens did not appear to dry out during the testing procedures. Additionally, the

repeatability of the results for both the mechanical tests and the derangement strain

tests suggested that condition of the specimen was not degrading during the recovery



periods between the load applications. Therefore, it was unlikely that the test

specimens dried during the testing procedure. This was not considered to be the cause

for the compliance of the tissue tested.

7.7.3 Compatibility of sheep and human anulus fibrosus ground substance

Wilke et al. (1997) found that sheep were an acceptable model for the human spine in

terms of the range of motion, neutral zone and general stiffness of the spinal

components. Additionally, the morphology of the intervertebral disc and vertebrae

were similar. Sheep spines have been successfully used in previous studies of the

biomechanics of the spine (Latham et al., 1994; Thompson et al., 2003). Reid et al.

(2002) demonstrated that the collagen content, collagen inclination and water content

of the sheep intervertebral disc was similar to that of the human intervertebral disc. It

was on the basis of these studies that sheep specimens were used to quantify the

mechanical response of the anulus fibrosus ground substance in the FEM of the

human disc.

However, closer examination of the results of Reid et al. (2002) suggested that the

compatibility of the sheep and human anulus ground substance was due to similarity

in trends of the variables tested but not necessarily similarity in the magnitudes of the

variables. Of the parameters measured by Reid et al. (2002) the water content of the

sheep anulus was the most relevant to the mechanical behaviour of the anulus ground

substance. These data and data for the human intervertebral disc are listed in Table

7-7.

Table 7-7 Water content (by total mass) in the anulus fibrosus of human and

sheep intervertebral discs.

Inner anulus

fibrosus Outer anulus

fibrosus Reference

Human 82% 66% Lyons and

Eisenstein (1981)

Sheep 82% 74% Reid et al. (2002)



The decrease in water content of the anulus fibrosus from the inner to the outer anulus

was demonstrated in both the human and the sheep anulus. The magnitude of this

decrease was similar but not the same (Table 7-7).

Reid et al. (2002) stated that the purpose of their study was not to determine any

differences in the types of proteoglycans present in the sheep anulus compared to the

human anulus fibrosus. The purpose was only to identify the variation in water

content. The primary proteoglycan type in the human anulus fibrosus ground

substance is chondroitin sulphate. However, there is little information available on

the primary proteoglycan type in the sheep ground substance.

It was believed that further investigations of the biochemical composition of the

human and sheep discs would provide beneficial data on the compatibility of the

human and sheep anulus fibrosus ground substance. In the absence of such

investigations it was concluded that the similarity between the kinematics,

biomechanics, morphology and general biochemical nature of the anulus fibrosus in

the sheep and human discs would result in a similarity in the mechanical behaviour of

this tissue between the species. However, the stiffness of these materials would be

intrinsically linked to the water content and proteoglycan type present. Therefore, it

was possible that the compliance of the ground substance in the FEM which resulted

in excessive nucleus pulposus pressures may have been due to the higher compliance

of the sheep tissue in comparison to the human tissue.

7.7.4 Justification for continued use of the overly compliant anulus ground

substance

On the basis of the nucleus pulposus pressure observed in the FEM it appeared that

the anulus ground substance in the FEM may have been too compliant. The similarity

between the biomechanics and the general biochemical nature of the sheep and human

anulus ground substance established in Section 7.7.3 suggested that the mechanical

behaviour of the modelled ground substance would effectively simulate the human

intervertebral disc. Quantitatively, the anulus ground substance in the FEM did not

accurately represent the stiffness of the human anulus fibrosus. However, the FEM



was believed to be an effective analysis tool to qualitatively investigate the

biomechanical effects of anular lesions on the behaviour of the intervertebral disc.

7.8 Conclusions

Based on the higher nucleus pulposus pressure in the FEM in comparison to the in

vivo nucleus pulposus it was apparent that the material properties for both the

homogeneous and the inhomogeneous anulus ground substance were overly

compliant in comparison to the human intervertebral disc. This compliance was

attributed to possible differences in the water content and specifically the

proteoglycan type present in the human and sheep anulus fibrosus. However, the

similarity between the biomechanics, morphology and general biochemical nature of

the anulus ground substance in the human and sheep anulus fibrosus had been

extensively demonstrated in the literature. Therefore, it was thought that while the

stiffness of these two materials may vary, their mechanical behaviour would be

similar.

The material parameters for the polynomial and Ogden strain energy equations which

were determined from experimentation on sheep discs were incorporated in

subsequent analyses of the model to investigate the effects of anular lesions on the

biomechanics of the intervertebral disc. Owing to the possibly higher compliance of

the anulus ground substance in the sheep discs compared to the human tissue, the

numerical value of the observed stresses and strains in these analyses would not be

compared. However, the relative increase and decrease in these parameters would

provide valuable information on the biomechanical effects of anular lesions on the

intervertebral disc.

Difficulties were encountered in obtaining a converged solution for the

inhomogeneous anulus ground substance due to excessive displacement corrections in

the FEM ground substance. It was thought that these inhomogeneous parameters

were unstable when implemented in the FEM. Given these difficulties in completing

an analysis with the comparatively simple load of torso compression, it was

questioned whether the inhomogeneous FEM would be able to obtain a completed



solution under more complex loading conditions. Additionally, introducing lesions

into the anulus of the FEM would increase the complexity of the analyses

significantly.

Analysis of the effects of anular lesions was carried out using the comparatively

robust Homogeneous FEM. Due to the difficulties encountered in obtaining a

converged solution for the Inhomogeneous FEM and the time constraints of the

project, this FEM was not used to analyse the biomechanical effects of anular lesions.

Chapter 8 details the method of modelling the anular lesions and the loading and

boundary constraints applied to simulate the rotation of the intervertebral disc about

the three axes of motion. The results of the analyses of anular lesions in the

Homogeneous FEM are presented in this chapter.


Chapter 8: Simulation and Analysis of Anular Lesions in the FEM 306

CChhaapptteerr

88

SSiimmuullaattiioonn aanndd AAnnaallyyssiiss ooff

AAnnuullaarr LLeessiioonnss iinn tthhee FFEEMM This chapter presents further analysis of the Homogeneous FEM presented in Chapter

7 to simulate the physiological loading and degeneration of the intervertebral disc.

Degeneration was simulated by removing the hydrostatic nucleus pulposus and

modifying the mesh to simulate anular lesions. The results of analyses of the

degenerate model are compared with the experimental work of Thompson (2002).

(The Homogeneous FEM without anular lesions present will be referred to as the

Healthy FEM. The Homogeneous FEM with anular lesions simulated and the

hydrostatic nucleus pulposus removed will be referred to as the Degenerate FEM.)

8.1 Physiological Loading Simulated in the FEM

The loading conditions applied to the model attempted to simulate the physiological

loading on the L4/5 intervertebral disc. To achieve this, rotations were defined about

an instantaneous centre of rotation (ICR). As stated in Section 2.1.3, the ICR is the

point about which pure rotation occurs. The locations of the ICRs during rotation in

the 3 planes of motion are defined in Section 3.6. All rotation loading on the FEM

was applied after the loading step simulating uniaxial compressive torso loading.

The rotations implemented in the FEM were based on Pearcy (1985). A comparison

of the angular motions observed by Pearcy (1985) and the rotations used in the FEM

is shown in Table 8-1.



Table 8-1 Angles of rotation for maximum physiological movements expressed in

degrees (SD – standard deviation)

Angle of rotation

(Pearcy, 1985) Mean (SD)

Angle of rotation used in

the FEM

Flexion 13 (4) 13

Extension 2 (1) 2

Right lateral bending 3 (2) 3

Left lateral bending 2 (3) 3

Right axial rotation 1 (1) 2

Left axial rotation 2 (1) 2

The rotations outlined in Table 8-1 were analysed in the FEM and the moment about

the axis of rotation was measured.

8.2 Representing the degenerate disc

In order to represent the altered mechanical state of the degenerate disc, anular lesions

were represented in a FEM without the nucleus pulposus hydrostatic pressure. The

following sections detail the mechanical properties employed for the anulus ground

substance and the rationale for removing the hydrostatic nucleus pulposus in the

Degenerate FEM.

8.2.1 Use of initial loading parameters for the anulus ground substance

The Homogeneous FEM with material parameters determined from the initial loading

of the anulus ground substance specimens was used to analyse the effects of anular

lesions on the disc biomechanics. Originally it was intended that the model would be

analysed using the initial loading parameters and if the strains in the anulus ground



substance were higher than the derangement strain (Chapter 4), then the material

parameters for the repeated loading of the material would be used. However,

subsequent to the discussion of the high compliance of the anulus ground substance in

the FEM which was presented in Section 7.7 it was thought that the anulus ground

substance in the FEM was already very compliant in comparison to the mechanics of

the human anulus fibrosus and further reductions in the stiffness of this material may

not benefit the results of the analysis.

8.2.2 Removal of the nucleus pulposus pressure

As stated in Chapter 2, the degeneration of the intervertebral disc is marked by a

reduction in fluid content, a less distinguishable boundary between the nucleus and

anulus and a more granular texture (Eyre, 1976). Therefore, in order to represent the

degeneration of the intervertebral disc, it was necessary for the models to simulate the

degeneration of the nucleus pulposus.

Degeneration of the nucleus may be characterised by a loss of the hydrostatic nucleus

pulposus pressure and a loss of hydration. This may occur subsequent to prolapse of

the nucleus pulposus through lesions in the anulus or possibly through fractures in the

endplates (Adams et al., 2000, Holm et al., 2004). Sato et al. (1999) and Panjabi et

al. (1988) measured nucleus pulposus pressures in vivo and in vitro and observed a

significant reduction in nucleus pressure in degenerate discs. Therefore, the use of a

hydrostatic nucleus pulposus in the FEM when analysing the effects of anular lesions

was not accurate. All hydrostatic fluid elements on the inner surface of the anulus

fibrosus and the inner surface of the superior and inferior cartilaginous endplates were

removed in the Degenerate FEM. Previous researchers had used a similar approach to

represent the degenerate condition of the intervertebral disc in a finite element model

(Goto et al., 2002; Shirazi-Adl et al., 1986; Goel et al., 1995).

Consequent to these changes in the FEM, the initial unloaded condition of the

intervertebral disc FEM was comparable to a closed hollow cylinder, void of material

in the inner cavity.



8.3 Simulating Anular Lesions

Rim, radial and circumferential lesions were simulated in the FEM. The locations of

the anular lesions in the FEM were similar to the location of the lesions introduced

into the sheep discs in the experimental study carried out by Thompson (2002). This

study was used for validation of the results of the FEM, therefore, it was necessary

that the location of the lesions be comparable.

In order to obtain a broader understanding of the biomechanical effect of anular

lesions on the disc, several Degenerate FEMs were developed with various lesions

simulated. These models are outlined in Table 8-2.

Table 8-2 Lesions present in the degenerate finite element models

Model 1 Rim lesion only

Model 2 Radial lesion only

Model 3 Circumferential lesion only

Model 4 Rim and radial lesion

Model 5 Rim and circumferential

lesion

Model 6 All lesions

In particular the simulation of rim lesions was considered to be of importance since

Thompson (2002) had observed that this type of lesion most frequently resulted in a

variation in maximum moments resisted by the joint. Conversely, it was observed

that radial and circumferential lesions did not significantly affect the peak moments

resisted.

Each of the degenerate models defined in Table 8-2 was analysed under six separate

rotational loading conditions. These were outlined in Table 8-1.



8.3.1 Rim lesions

In keeping with the observations of Hilton et al. (1976) and Osti et al. (1990) the rim

lesion was simulated in the anterior anulus fibrosus of the FEM (Figure 8-1).

Thompson produced rim lesions in the right anterolateral anulus using a scalpel blade.

The rim lesion in the FEM extended from the outermost anulus surface through the 6

circumferential element layers. It did not extend as far as the inner anulus wall.

Figure 8-1 Position of rim lesion in FEM viewed from right anterolateral direction (Rim lesion surface in blue). The lesion extended through 6

circumferential element layers.

8.3.2 Radial lesion

Hirsch and Schajowicz (1953) and Osti et al (1990) reported that radial lesions were

commonly observed in the posterolateral anulus. Thompson (2002) produced radial

lesions in the left posterolateral anulus using a scalpel orientated in the axial disc

direction. The scalpel was inserted until there was no significant resistance to motion

and it was assumed that the nucleus had been penetrated. The radial lesion in the

FEM was positioned in a similar location to that used experimentally and the lesion

extended the full radial width of the anulus into the nucleus (Figure 8-2).

The elements on the superior surface above the rim lesion have been removed for visualisation of the lesion



Figure 8-2 Position of the radial lesion (Radial lesion surface shown in blue)

8.3.3 Circumferential lesions

Osti et al. (1990) observed that circumferential lesions were most prevalent in the

anterior anulus. The circumferential lesions created using injections of fluid between

the lamellae in sheep discs by Thompson (2002) extended between 25% and 50% of

the perimeter of the disc anulus and were inserted in the outer radial half of the

anulus, from the right anterolateral to the left posterolateral anulus. This location of

the circumferential lesions was reproduced in the FEM (Figure 8-3).

A Thin green lines are spring

elements modelling anterior longitudinal

ligament

Elements have been removed so the lesion can be visualised



B

Figure 8-3 Position of circumferential lesion in the FEM A. Viewed from right superior anterolateral direction; B. Viewed from left superior posterolateral

direction (Circumferential lesion surface in blue)

8.3.4 Contact relationships

At the locations where the lesions were to be modelled, nodes on the existing element

faces were duplicated in order to create a discontinuity in the mesh (Figure 8-4). The

interactions between the faces of the lesion were defined using contact definitions.

Figure 8-4 Schematic of contact simulation for the radial lesion. A similar method was used for the rim and circumferential lesions

It was noted that the creation of the contact surfaces caused a stress concentration at

the corner of the lesion (Figure 8-4). However, the stresses in these regions were not

an accurate indication of the state of stress in the anulus fibrosus ground substance in

the immediate area around the lesion. In order to provide correct data for the stress at

the lesion edge, a fracture mechanics analysis would need to be carried out in order to

The outer two element layers have been removed so the lesion can be visualised

Duplicate nodes at lesion face

Circled regions may exhibit high stresses; however, this is an artefact of the discontinuity in the anulus.



accurately simulate the stresses in this region. This was beyond the capabilities of the

FEMs developed in this thesis and beyond the scope of the project.

Contact definitions are required in any analysis where surfaces may interact with one

another. If contact definitions are not provided, the surfaces which are in physical

contact have no computational knowledge of one another and can technically

penetrate into each other without any resulting variation in the stress state at the site.

An extensive selection of contact definitions is available in the Abaqus software;

however, little is known about the specifics of the surface interaction at the faces of

lesions in the anulus fibrosus. Frictionless behaviour (µ=0) was assumed for the

lesion contact faces based on the hydrated, lubricated nature of the tissue.

When two surfaces in a contact pair come into contact, a normal stress/pressure will

be present between these surfaces. Abaqus requires data for the magnitude of this

pressure when the surfaces are in contact. It is possible to define a contact

relationship such that the surfaces experience no contact pressure until they are in

complete contact and the distance between the surfaces is zero – this contact

definition is referred to as ‘hard’ contact and is graphically represented in Figure 8-5

A. This type of contact definition is such that as the faces come into contact there is

no pressure acting across the physical space between them. However, when they

contact there is a sudden increase in the pressure acting on the elements in this region.

This may cause difficulties in convergence of the algorithms for force equilibrium.

Instead a ‘softened’ contact relationship was defined for the interaction between the

lesion faces. This relationship is illustrated in Figure 8-5B. A soft contact definition

allows for the contact pressure between contact surfaces to gradually increase from

zero when they are not in contact to the full contact pressure when the distance

between the surfaces is zero. This type of contact definition was a reasonable choice

for the anulus fibrosus because the material was highly deformable and it was

reasonable to expect that in vivo the faces of a lesion would gradually come into

contact and transmit pressure.



A

0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8 1 1.2

Clearance, h

Con

tact

pre

ssur

e, P

B

02468

1012141618

-0.15 0.35 0.85 1.35

Clearance, h

Con

tact

pre

ssur

e, P

Figure 8-5 Two types of contact definitions offered by Abaqus. A. Hard contact; B. Soft contact

The parameters defined for this contact are the contact pressure when the clearance is

zero and the clearance when the contact pressure is zero – the x and y intercept on

Figure 8-5 B. These values are difficult to define since they are not experimentally

determined. The convergence of the analysis at the beginning of the load application

is largely dependant upon the appropriateness of the parameters selected since it is at

this time that the surfaces initially come into contact. If the parameters are not

appropriate for the material and geometry of the model the state of contact at the

lesion faces cannot be determined and the solution will fail before any load has been

applied.



The contact parameters selected for the three lesions were determined iteratively.

Various combinations of contact pressure and clearance were tested in the Degenerate

FEMs with either, a rim lesion, a radial lesion or a circumferential lesion simulated.

If the analysis successfully completed the torso compression loading step, the

parameters chosen were used for further analysis of the mechanical effects of lesions.

It was apparent at this stage that the contact definitions in the Degenerate FEMs

resulted in significant difficulties in obtaining converged solutions for the analysis of

the model under all loading conditions. If a converged solution could be obtained for

the torso compressive loading condition, it was thought that the model would be

robust enough to at least complete the validation analyses (Section 8.4). Further

analysis of the Degenerate FEMs under additional loading conditions would be

carried out subsequent to the validation analyses. A discussion of the difficulties in

obtaining a converged solution for the FEM with a circumferential lesion simulated

due to the contact interaction at the lesion faces is provided in Section 8.4.3.1.

The contact definition provided for the rim and radial lesion required the definition of

contact surfaces. These surfaces were on the adjacent faces of the elements at the

interface of the lesion. The Abaqus software recognised these surfaces as contact

surfaces if they were defined as contact pairs. The contact between the faces of the

circumferential lesion was defined using gap elements. These elements are used to

define contact by connecting the adjacent nodes on the element faces that are in

contact. The contact parameters for gap elements are defined and used in a similar

way to those for contact defined using contact pairs. The results of contact

simulations using these methods produce similar results for the mechanical response

of the underlying material. However, the output from gap elements does not permit

the normal pressure between the contacting surfaces to be determined. It should be

noted that there is some potential for minor variations in the peak rotational moment

as a result of the different methods for defining the contact relationship in the models

with the circumferential lesions.



8.4 Validation of the Degenerate Disc Model

In order to validate the finite element model with anular lesions simulated, the loading

conditions used by Thompson (2002) were reproduced. Thompson (2002) tested

intact lumbar sheep joints under rotational loading to simulate physiological ranges of

motion. There was no compressive load applied to these joints to simulate

compressive torso loading, therefore, the compressive loading condition in the FEM

was removed from the validation analyses.

The joint rotations detailed in Table 8-1 were used in these validation analyses and the

response of the FEM with lesions present was compared to the response of FEMs

without lesions simulated. The loading steps applied closely reproduced the loading

conditions employed by Thompson (2002) who compared the results of rotational

loading applied to both intact sheep discs and discs with lesions present.

Two sets of models without lesions simulated were developed. In the first model a

hydrostatic nucleus represented the nucleus pulposus – thus this model represented a

healthy intervertebral disc and was referred to as the Healthy FEM. In the second set

of models the hydrostatic nucleus was removed. This model simulated a disc with a

healthy anulus fibrosus but without a nucleus present and was referred to as the

Healthy Anulus FEM. The development of both these models allowed for the

separate investigation of the biomechanical effects of anular lesions and the effect of

removing the nucleus pulposus in order to simulate a degenerate disc.

Differences in the moments generated in the Healthy, the Healthy Anulus and the

Degenerate FEMs in response to rotational loading were compared with the results of

Thompson (2002). She found that rim lesions resulted in a reduction in the moment

resisted by the joint during extension, left lateral bending and right axial rotation.

There was no difference in the joint moments during flexion, right lateral bending or

left axial rotation and circumferential and radial lesions had no notable effect on the

moments resisted by the joint.

It should be noted that two main differences existed between the disc model tested

experimentally and the disc model analysed with the FEM. Firstly, the disc FEM



simulated a human L4/5 intervertebral disc with geometry obtained from cadaveric

human specimens. The dimensions of this model and the sheep discs tested by

Thompson (2002) were significantly different – the lateral width of the sheep disc is

approximately 30mm while the lateral width of the disc FEM was 45.9mm and the

anterior-posterior width of the sheep and the disc FEM were 15mm (approximately)

and 33mm, respectively. Secondly, the geometry of these discs may have differed in

terms of the severity of the posterior curvature, the skewed location of the nucleus in

the transverse plane and the overall curvature of the outer anulus profile.

8.4.1 Results

A comparison of the results for the rotational stiffness in the Healthy FEM, the

Healthy Anulus FEM and the Degenerate FEMs when the three lesion types were

simulated is shown in Figure 8-6 (A-F). To determine the percentage change in peak

moment in the Degenerate FEMs the peak moments in these models were normalized

with the peak moment in the Healthy Anulus FEM (Figure 8-7). In this way, the

effects of simulating the anular lesions were separated from the effects of removing

the nucleus pressure. The comparison presented in Figure 8-7 shows the effect of

simulating rim lesions in a disc without a nucleus present. Data presented in Table

8-3 provides information on the variation in peak moment due to the removal of the

nucleus pressure from a healthy disc.

It was not possible to obtain a converged solution for the Degenerate FEMs with a

circumferential lesion simulated when lateral bending or axial rotation loading

conditions were applied. These analyses failed on the first increment of the first step.

Reasons for this are discussed in Sections 8.4.3.1 and 8.4.3.2. Consequently, the

Degenerate FEMs simulating rim and circumferential lesions and simulating all three

lesions (Table 8-2) were not analysed under the validation loading or under

compressive and rotational loading conditions.

It should be noted that a completed flexion loading condition could not be achieved in

the FEMs and the results were compared for the maximum flexion rotation that was

achieved by all Degenerate FEMs (6.9o).



A

2o Extension2074

1454 1471 1451 1427 1467

0

500

1000

1500

2000

2500

Healthydisc

Healthyanulus

Rim Radial Circ'l Rim/Rad

Model Type

Mom

ent (

Nmm

)

B

6.9o Flexion5303

40333647 3932 3838 3972

0

1000

2000

3000

4000

5000

6000

Healthydisc

Healthyanulus

Rim Radial Circ'l Rim/Rad

Model Type

Mom

ent (

Nm

m)

C

3o Left Lateral Bending

21744

3461 3543 3314 3393

0

5000

10000

15000

20000

25000

Healthydisc

Healthyanulus

Rim Radial Rim/Rad

Model Type

Mom

ent (

Nm

m)



D

3o Right Lateral Bending

21004

2864 3281 2859 3277

0

5000

10000

15000

20000

25000

Healthydisc

Healthyanulus

Rim Radial Rim/Rad

Model Type

Mom

ent (

Nm

m)

E

2o Left Axial Rotation3137

909 899 906 896

0

500

1000

15002000

2500

3000

3500

Healthydisc

Healthyanulus

Rim Radial Rim/Rad

Model Type

Mom

ent (

Nm

m)

F

2o Right Axial Rotation

2960

990 955 985 947

0

500

1000

1500

2000

2500

3000

3500

Healthydisc

Healthyanulus

Rim Radial Rim/Rad

Model Type

Mom

ent (

Nm

m)

Figure 8-6 Comparison of peak moments. A. Extension; B. Flexion; C. Left lateral bending; D. Right lateral bending; E. Left axial rotation; F. Right axial

rotation



A

2° Extension

0102030405060708090

100110120

Rim Radial Circ'l Rim/RadModel Type

Perc

enta

ge o

f Hea

lthy

Anu

lus

Mom

ent

B

7.4° Flexion

0102030405060708090

100110120

Rim Radial Circ'l Rim/RadModel Type

Perc

enta

ge o

f Hea

lthy

Anu

lus

Mom

ent

C

3o Left Lateral Bending

0102030405060708090

100110120

Rim Radial Rim/RadModel Type

Perc

enta

ge o

f Hea

lthy

Anu

lus

Mom

ent

6.9o Flexion



D

3o Right Lateral Bending

0102030405060708090

100110120


Perc

enta

ge o

f Hea

lthy

Anu

lus

Mom

ent

E

2o Left Axial Rotation

0102030405060708090

100110120


Perc

enta

ge o

f Hea

lthy

Anu

lus

Mom

ent

F

2o Right Axial Rotation

0102030405060708090

100110120


Perc

enta

ge o

f Hea

lthy

Dis

c M

omen

t

Figure 8-7 Comparison of peak moments in Degenerate FEMs with the peak moment in the Healthy Anulus FEM



Figure 8-8 shows the variation in extension moment with rotation for the Healthy

FEM, the Healthy Anulus FEM and the Degenerate FEM. This plot was similar for

all the loading conditions applied to the healthy models and the degenerate models.

The moment in the Healthy FEM was higher than the moment in both the Degenerate

and the Healthy Anulus FEMs from the beginning of the rotation.

0

5000

10000

15000

20000

25000

0 0.01 0.02 0.03 0.04 0.05 0.06

Rotation (degrees)

Mom

ent (

Nm

m)

Healthy FEM Healthy Anulus FEM Degenerate FEM

Figure 8-8 Right lateral bending moment for the Healthy FEM, the Healthy Anulus FEM and the Degenerate FEM with a rim lesion present

Figure 8-6 shows a substantial reduction in moment between the Healthy and Healthy

Anulus FEMs. In particular, removing the nucleus pressure greatly reduced the

intervertebral disc’s resistance to rotation during lateral bending and axial rotation.

Table 8-3 details the percentage reduction in peak moment between these models for

the six rotational loading conditions. The largest reduction in rotational stiffness was

observed during right lateral rotation (86%) and the smallest reduction occurred

during flexion loading (24%). While this loading condition resulted in the lowest

reduction in peak moment between the Healthy and Healthy Anulus FEMs it was only

simulating 53% of the average full range of motion in the human spine. The results

for the other rotational loading conditions were for the full range of rotational motion.



Table 8-3 Percentage reduction in peak moment of the Healthy Anulus FEM

compared with the Healthy FEM

Loading Condition Percentage reduction in

moment Extension 30% Flexion 24%

Left lateral bending 84% Right lateral bending 86%

Left axial rotation 71% Right axial rotation 67%

The results presented in Figure 8-6 and Figure 8-7 showed minimal reduction in the

moment after lesions were simulated in the FEM. Thompson (2002) stated that rim

lesions resulted in the most significant change to the peak moments observed in the

disc and radial and circumferential lesions did not cause a notable change. The

variation in moment when the rim lesions were present was apparent under extension,

left lateral bending and right axial rotation while the other motions did not cause a

significant change.

The rim lesion created a discontinuity between the outer six of the eight

circumferential element layers in the anulus (Section 8.3.1). When the rotational

loads were applied to this FEM, the elements at the lesion face that were in the outer

circumferential element layer were separated. These elements on the outer edge of

the lesion and in the lower face bulged outward (Figure 8-9). The remaining elements

in the lesion face were separated under extension and left lateral bending. Under

flexion, right lateral bending, left axial rotation and right axial rotation the remaining

elements on the lesion face were in contact.

The radial lesion extended the full radial depth of the left posterolateral anulus

(Section 8.3.2). The elements at the lesion interface between the two outermost

element layers separated during the rotational loading (Figure 8-10). All other

elements on this interface remained in contact when flexion, left lateral bending and

left axial rotation were applied. Under extension, right lateral bending and right axial

rotation the remaining elements on the lesion face were separated.



Figure 8-9 Deformed geometry of the anulus fibrosus in the Degenerate FEM with a rim lesion simulated and with a 200N compressive load applied – viewed

from the right lateral direction.

Figure 8-10 Deformed geometry of the Degenerate FEM with a radial lesion simulated and with a 200N compressive load applied – viewed from the left

posterolateral direction (Wireframe shows undeformed shape)

It may be seen from Table 8-4 that the results from the Degenerate FEMs do not agree

with the experimental results presented by Thompson (2002). With reference to the

effects of rim lesions, under extension, left lateral rotation and right axial rotation

loading conditions there was minimal change in the peak moment. The results of the

FEM indicated that flexion and right lateral bending in the presence of a rim lesion

caused the most notable change in stiffness, with a 10% decrease and a 15% increase,

Lower face of lesion bulged outward

Radial lesion in left posterolateral anulus. Outer face of lesion is open under compression



respectively. However, for these loading conditions Thompson (2002) observed a

12% increase under flexion and only a 7% increase under right lateral bending.

Table 8-4 Comparison of the change in peak moments in the Degenerate FEMs

and in the results of Thompson (2002) (The experimental values from

Thompson, 2002 were average data)

Rim lesion Radial LesionCircumferential

Lesion

FEM 1% 0% -2% Extension

Experimental -20% 20% -20%

FEM -10% -3% -5% Flexion

Experimental 12% -3% 0%

FEM 2% -4% - Left lateral bending Experimental -8% -9% -2%

FEM 15% 0% - Right lateral

bending Experimental 7% 2% 3%

FEM -1% 0% - Left axial rotation Experimental -20% -3% -2%

FEM -4% -1% - Right axial rotation Experimental -1% 19% -3%

With the exceptions of flexion and right lateral bending in the presence of a rim

lesion, the variation between peak moments observed in the Healthy Anulus FEM and

the Degenerate FEM ranged from +2% to -4%. The results from Thompson (2002)

demonstrated a variation for similar loading conditions between +20 and -20%. It

was noted that variations of such a magnitude as were observed in the Degenerate



FEM were very low and while they highlighted trends in the mechanical behaviour of

the disc, they were not quantitatively significant.

The percentage changes observed under all the loading conditions except flexion were

thought to be too low to be of any significance. Thompson (2002) observed

percentage changes of approximately 20% in the discs that were deemed to be

significantly affected by the presence of the rim lesion.

The deformed geometry of all the Degenerate FEMs showed an inward bulge of the

inner surface of the anulus (Figure 8-11 A, B) and an outward bulge of the outer

anulus surface. The inward bulge of the inner anulus in the absence of the nucleus

pulposus pressure was observed experimentally by Meakin and Hukins (2000) and

Meakin et al. (2001).

When the von Mises stress distribution in a radial direction through the anulus was

investigated, regions of higher stress were observed both at the inner anulus boundary

and through the depth of the Degenerate FEM anulus (Figure 8-11 C). Meakin et al.

(2001) observed an inward bulge of the inner anulus and an outward bulge of the

outer anulus during experimentation on human discs subjected to compression after

partial removal of the nucleus pulposus. They suggested that this may provide a

mechanism for the development of anular lesions whereby the differing directions of

radial bulge in the anulus may result in higher stresses in the anulus and cause the

lamellae to separate to form a circumferential lesion. The observation of higher

stresses within the anulus ground substance of the Degenerate FEM was in keeping

with these deductions.



A

B

C

Figure 8-11 Deformed geometry of the anulus ground substance. The disc is viewed from the posterior direction and has been sectioned in the mid-frontal

plane to view the deformation through the anulus (Wireframe shows the undeformed mesh). A. Healthy FEM; B. Degenerate FEM – rim lesion; C. Von Mises stress distribution through ground substance in the Degenerate FEM –

rim lesion



8.4.1.1 Rim and radial lesion simultaneously represented in the Degenerate

FEM

When both the radial and the rim lesion were simulated and either extension, right

lateral bending, left axial rotation or right axial rotation were applied, the final

moment was the same as that for the Degenerate FEM with a rim lesion present.

Under flexion and left lateral bending the peak moment in the model with both lesions

was similar to the results of the Degenerate FEM with only the radial lesion present.

It was possible that the similarity between the results of the Degenerate FEM with

both the lesions simulated and the results of the Degenerate FEMs with the individual

lesion present suggested that specific lesions were of most importance during certain

rotations. For example, the similarity between the extension peak moment in the

FEM with both radial and rim lesions and the FEM with a rim lesion simulated may

have suggested that during this rotational motion, the rim lesion was of most

importance in the generation of the moment.

In this case, the results of the simulations with both the rim and radial lesion present

did reproduce the observation made by Thompson (2002) that rim lesions caused the

most significant change to peak moments under extension and right axial rotation.

However, it was noted that the variation in the peak moments was small (ranged 2%

to -4%).

8.4.2 Simulation of a rim lesion in a disc FEM with a hydrostatic nucleus

pulposus

Subsequent to the validation analyses it was observed that the Degenerate FEMs did

not simulate the exact condition of the sheep discs that were tested in vitro with anular

lesions inserted. Thompson (2002) inserted lesions into an otherwise healthy disc and

then carried out mechanical testing on the joint. The nucleus in this disc could no

longer be considered a hydrostatic structure once the radial lesion was inserted since

this lesion penetrated the nucleus. However, for the duration of the tests the nucleus

in these discs would have remained hydrated and would not have exhibited the

granular structure characteristic of degenerate discs. The nucleus in the discs with a



rim or circumferential lesion inserted were initially hydrostatic structures since these

lesions did not penetrate the nucleus.

The results of experimental testing on these discs with a partially degenerate nucleus

were compared with the results of an FEM without any nucleus present. The latter

was an extreme case and simulated a very degenerate disc. It was thought that a

simulation of the nucleus pulposus that captured the slight compressibility and

increased shear stiffness of this structure in the Degenerate FEMs with a radial lesion

present may have more closely reproduced the results presented by Thompson (2002).

Due to time constraints it was not possible to carry out further investigations into the

material characteristics which could adequately describe the slightly degenerate

nucleus or to modify the intervertebral disc mesh to include the nucleus pulposus.

However, since the nucleus in the sheep discs tested with a rim lesion present was

initially hydrostatic, this could be simulated by modifying the Healthy FEM to

incorporate a rim lesion. Furthermore, Thompson (2002) found that rim lesions

resulted in the most significant effect on the disc mechanics. This model will be

referred to as the Rim Lesion FEM.

The Rim Lesion FEM was analysed under the rotational loading conditions presented

in Table 8-1. A converged analysis was not obtained for the flexion, left lateral

bending or right lateral bending loading conditions. These analyses failed due to

displacement convergence problems which may have been related to the high nucleus

pressures exhibited under these loading conditions (this is discussed in Section 8.4.3.3

and Figure 8-14). The maximum rotation reached in the flexion analysis was 4.69o, in

the left lateral bending analysis it was 2.72o and in the right lateral bending it was

0.58o. The results for these analyses were compared with the results for the Healthy,

the Healthy Anulus FEM and the Degenerate FEM with a rim lesion simulated

(Figure 8-12).

The results for the peak moment in the Rim Lesion FEM, the Healthy Anulus FEM

and the Degenerate FEM with a rim lesion present were compared with the peak

moment in the Healthy FEM (Figure 8-13). This allowed for the separate

determination of the effect on peak moments due to the simulation of a rim lesion, due

to the removal of the nucleus pressure and as a result of both a rim lesion and the

removal of the nucleus pressure.



A

2° Extension

2074 2081

1454 1471

0

500

1000

1500

2000

2500

Healthy disc Rim +Hydrostatic

nucleus

Healthyanulus

Rim

Peak

Mom

ent (

Nm

m)

B

4.69° Flexion3174 3167

2565 2605

0

500

1000

15002000

2500

3000

3500


nucleus

Healthyanulus

Rim

Peak

Mom

ent (

Nm

m)

C

2.72° Left lateral bending18535 17918

2899 2999

02000400060008000

100001200014000160001800020000


nucleus

Healthyanulus

Rim

Peak

Mom

ent (

Nm

m)



D

0.58° Right lateral bending

1975 1987

327 386

0

500

1000

1500

2000

2500


nucleus

Healthy anulus Rim

Peak

Mom

ent (

Nm

m)

E

2° Left axial rotation3137 3029

909 899

0

500

1000

1500

2000

2500

3000

3500


nucleus

Healthy anulus Rim

Peak

Mom

ent (

Nm

m)

F

2° Right axial rotation

2960 2827

990 955

0

500

1000

1500

2000

2500

3000

3500


nucleus

Healthy anulus Rim

Peak

Mom

ent (

Nm

m)

Figure 8-12 Comparison of peak moments A. Extension; B. Flexion; C. Left lateral bending; D. Right lateral bending; E. Left axial rotation; F. Right axial rotation (Rim+Hydrostatic nucleus = rim lesion simulated in the Healthy FEM)



A

2 ° Extension

100

70 71

0

20

40

60

80

100

120

Rim + Hydrostaticnucleus

Healthy anulus Rim

Perc

enta

ge o

f Hea

lthy

Dis

c

B

4.69° Flexion

100

81 82

0

20

40

60

80

100

120


Healthy anulus Rim

Perc

enta

ge o

f Hea

lthy

Dis

c

C

2.72° Left lateral bending

97

16 16

0

20

40

60

80

100

120


Healthy anulus Rim

Perc

enta

ge o

f Hea

lthy

Dis

c



D

0.58° Right lateral bending

101

17 20

0

20

40

60

80

100

120


Healthy anulus Rim

Perc

enta

ge o

f Hea

lthy

Dis

c

E

2° Left axial rotation

97

29 29

0

20

40

60

80

100

120


Healthy anulus Rim

Perc

enta

ge o

f Hea

lthy

Dis

c

F

2° Right axial rotation

96

33 32

0

20

40

60

80

100

120


Healthy anulus Rim

Perc

enta

ge o

f Hea

lthy

Dis

c

Figure 8-13 The peak moments in the Healthy Anulus FEM, the Degenerate FEM with a rim lesion and the Healthy FEM with a rim lesion simulated are

compared with the peak moment in the Healthy FEM (Rim+Hydrostatic nucleus = rim lesion simulated in the Healthy FEM)



These results suggest that the presence of the rim lesion did not affect the resistance

of the disc to rotation when loaded under extension, flexion and right lateral bending.

However, left lateral bending resulted in a 3% decrease in the peak moment and left

and right axial rotation resulted in a 3% and 4% reduction in peak moment,

respectively. These values for axial rotation were of a similar magnitude to the

variations in peak moment that were observed in Figure 8-7. The loading conditions

that resulted in a change in the peak moments of the Rim Lesion FEM were different

to the Degenerate FEMs with the rim and radial lesions simulated either individually

or simultaneously (Section 8.4.1). It was noted that these percentage variations in

peak moment were very small and while they provided information on trends in the

mechanical behaviour of the degenerate disc, they were not quantitatively significant.

8.4.3 Discussion of validation analyses

The following sections present potential causes for the lack of convergence of the

contact algorithms when the circumferential lesions were simulated, provides an

explanation for the large decrease in peak moment between the Healthy FEM and the

Healthy Anulus FEM, discusses the comparison of results between the Degenerate

FEM and the results of Thompson (2002) and suggests possible reasons for the lack of

agreement between these results.

8.4.3.1 Discussion of the Degenerate FEMs with circumferential lesions

Since the Degenerate FEM in which the circumferential lesion was simulated failed

during the first increment, this suggested that the contact parameters defined for the

lesion contact faces were not appropriate. However, the successful solution of the

mesh when an extension load was applied and the completion of 6.9o of the flexion

loading indicated that these parameters were capable of establishing the initial contact

state between the lesion faces under these loading conditions. Several attempts were

made to obtain a set of contact parameters for the circumferential lesion that were

successful for all the validation analyses, but this was not possible.

The difficulty in determining a suitable set of parameters was likely related to the

significant deformation of the circumferential lesion faces. Of the three types of



anular lesions, the contact parameters defined are of most relevance to the

circumferential lesions since all loading conditions on the FEM resulted in the anulus

bulging outward and the faces of the circumferential lesion contacting. Analyses of

the Degenerate FEM with rim and radial lesions demonstrated that the faces of these

lesions at the completion of several of the rotational loading conditions were partially

opened. Owing to the radial bulge of the anulus it seemed likely that the faces of the

circumferential lesion would always be closed at the beginning and for the duration of

all loading conditions. This was evident in the analyses of extension and flexion

loading. The faces of the circumferential lesion were closing from the beginning of

the step.

8.4.3.2 Difficulties in obtaining a converged solution for the validation analyses

When contact relationships are defined in the FEM, the procedure that is applied in

the Abaqus software involves iterations in order to obtain convergence for the contact

surfaces. Once the algorithms relating to the contact state in the model have

converged, equilibrium iterations are carried out to obtain a solution for the force and

displacements in the model.

The iterations which are performed to determine the contact state between a pair of

contacting surfaces are called severe discontinuity iterations. These iterations involve

determining any changes in the contact pressure, P or the clearance, h. If these values

do not change then there are no severe discontinuity iterations performed. Severe

discontinuity iterations will always be performed at the beginning of a contact

analysis and if the contact state of the surfaces changes during the loading step,

further severe discontinuity iterations will be performed.

The two causes identified for the uncompleted validation analyses were:

• Lack of convergence of the algorithms defining the contact between the lesion

faces.

• Lack of convergence of the equilibrium iterations due to excessive displacement

corrections and forces residuals at nodes on the lesion faces.



In an attempt to obtain completed solutions for the analyses that failed due to lack of

convergence of the contact algorithms, several remedies were attempted, including:

• reducing the mesh density in the region of the anulus fibrosus where the lesion

was simulated;

• employing different methods to define the contact surfaces – either contact

pairs or gap elements; and

• using different contact parameters to define the contact relationship.

These remedies were not successful.

The failure of some analyses due to lack of convergence of the equilibrium iterations

was also related to the contact at the lesion faces. The displacements and forces at

some nodes on the lesion faces were so high that the convergence algorithms

employed in Abaqus could not converge.

A possible cause for the failure of the analyses was the choice of user-defined contact

parameters (Section 8.3.4). Generally, in the successful analyses of the Degenerate

FEM once the severe discontinuity iterations were performed during the initial

increment of the analysis, the contact relationship was established and few subsequent

increments required further iterations of this kind. This indicated that the parameters

were an appropriate choice in order to determine the initial contact state at the lesion

interface. Any change in the contact state at the lesion interface during subsequent

increments (i.e. if the contact surfaces opened or closed) resulted in a severe

discontinuity iteration. The user-defined clearance at zero pressure and the user-

defined pressure at zero clearance were not used in this iterative process. They were

only relevant in establishing the initial contact relationship between the surfaces. So,

it was not likely that the inability to determine appropriate contact parameters for the

circumferential lesion was the cause for the lack of convergence in the validation

analyses.

Difficulties in obtaining convergence in contact simulations may be overcome with

the careful choice of the contact definitions to obtain values that are physically

realistic and that result in a converged solution. However, significant effort was



expended to achieve convergence of the Degenerate FEM and a completely successful

combination of contact definitions and parameters could not be determined. In order

to achieve the overall objective of the research which was to investigate the

biomechanical effects of anular lesions on the disc behaviour the contact definitions

that had been employed in the partially successful validation analyses were used.

8.4.3.3 Discussion of the decrease in peak moment between the Healthy FEM

and the Healthy Anulus FEM

In comparison to the results for the Healthy FEM there was a large decrease in the

moment observed when the Healthy Anulus and Degenerate FEMs were loaded under

the six rotations (Figure 8-6 ). The peak moment observed in the Healthy Anulus

models was as little as 14% of the moment observed in the Healthy FEM. This

decrease was much higher than the decrease observed between the Healthy Anulus

FEM and the Degenerate FEM (maximum of 10%). Also, the moment observed in

the models loaded under lateral bending was extremely high in comparison to the

moment under axial rotation or sagittal rotations. Further investigation of the cause

for this high moment when the Healthy FEM was subjected to lateral bending

provided an explanation for the large decrease in moment when the nucleus pulposus

was removed.

It was found that the nominal axial stress in the rebar elements when the Healthy

FEM was loaded in lateral bending was 34 and 40MPa for right and left lateral

bending, respectively. These rebar stresses were between two and six times the rebar

stress observed in the Healthy FEM when subjected to the other loading conditions.

The nominal axial stress in the rebar elements in the Healthy Anulus and the

Degenerate FEMs were:

• In the Healthy Anulus FEM the maximum rebar stress reduced to 7.22 and

11.68MPa for right and left lateral bending, respectively;

• In the Degenerate FEM with the rim lesions simulated the maximum rebar

stress was 15.76 and 11.67MPa, for right and left lateral bending, respectively;

and



• When the radial lesions were simulated the maximum stress was 7.22 and

16.64MPa for right and left lateral bending, respectively.

A reduction in the rebar stress was also observed when the lesions were simulated in

models subjected to the other rotation motions. The increase in rebar stress between

the Healthy Anulus FEM and the Degenerate FEM under lateral bending indicated

that the presence of a rim or radial lesion resulted in a greater portion of the applied

loading being carried by the rebar elements simulating the collagen fibres.

Also, when rotations were applied to the Healthy and the Healthy Anulus FEM there

were associated off axis rotations in the sagittal and transverse planes. However, in

the Healthy Anulus FEM these rotations were larger than for the Healthy FEM.

These results suggested that there was less resistance to rotation when the nucleus

pressure was removed.

The observation that the rebar stress was higher in the Healthy FEM subjected to

lateral bending suggested that there was greater resistance offered by the collagen

fibres when the disc was rotated laterally. The increased pressure within the nucleus

during lateral rotation may have resulted in this increased fibre stress (Figure 8-14).

When the degenerate nucleus was simulated – that is, when there was no hydrostatic

fluid elements used to define the nucleus pulposus – the resistance to inward bulge of

the anulus was decreased and this in turn reduced the resistance to lateral rotation.



00.20.40.60.8

11.21.41.61.8

2

Flexion Extension L lateralbend

R lateralbend

L axial rot R axial rot

Loading Conditions

Nuc

leus

Pre

ssur

e (M

Pa)

Figure 8-14 Nucleus pressure in the healthy disc FEM during rotational loading

The finding that the rebar stress reduced when the nucleus pulposus was removed was

evidence that a significant portion of the stress resisted by the collagen fibres in the

anulus fibrosus was a result of the pressurisation of the nucleus pulposus. This was in

keeping with the observations of Hickey and Hukins, 1980. With the removal of this

pressure, the fibres were no longer as highly stressed and in turn did not provide as

much resistance to the rotation. Therefore, the peak moment in the disc due to the

rotations in the three planes of motion was significantly reduced when the nucleus

pressure was removed.

This conclusion was confirmed by comparing the nucleus pulposus pressure during

rotation in the three planes of motion (Figure 8-14). Lateral bending simulated in the

healthy disc FEM resulted in the highest nucleus pressure. Hence, this motion

resulted in the highest rotational moment and the most significant reduction in

rotational stiffness when the nucleus pulposus pressure was removed and anular

lesions simulated.

8.4.3.4 Discussion of the validation results

From the results of the Degenerate FEM with two lesions simulated it was apparent

that rim lesions were of most importance to the disc mechanics during extension, right



lateral bending, left axial rotation and right axial rotation. This was similar to the

findings of Thompson (2002) in that she found extension and right axial rotation

resulted in a significant change in the peak moments when the rim lesions were

present. However, she also observed that left lateral bending affected these moments.

The presence of the lesion will have the greatest effect on the disc mechanics when

the loading conditions applied to the disc result in the faces of the lesion separating.

In this case, there is no resistance offered by the anulus fibrosus to the rotation of the

disc and the peak moments will be reduced. Conversely, if the loading applied results

in compression of the faces of the lesion this should not affect the peak moments

since the presence of the lesion does not hinder the ability of the anulus to carry

compressive loads. This discussion is similar to the discussion presented by

Thompson (2002) in relation to the observed variation in peak moments when the rim

lesion was inserted in the sheep discs.

The observation that the rim lesions were of most importance during extension and

axial rotation in the FEM was in keeping with this discussion. Extension resulted in

the rim lesion faces being separated in a direction normal to the face and axial rotation

caused the lesion faces to translate in the plane of the face. Hence the peak moments

in the Degenerate FEMs with the rim lesion present were reduced when these loading

conditions were applied.

While Thompson (2002) found that rim lesions affected the peak moments during left

lateral bending, the results of the Degenerate FEM suggested that radial lesions were

of most importance during this loading condition. This disagreement may have been

a result of the differing geometry of the sheep discs tested experimentally and the

human disc simulated in the model. The reduction in peak moment during left lateral

bending may be a result of the separation of the lesion faces when the anulus bulges

outward. This separation was described in Section 8.4.1.

Similarities were observed between the deformation of the anulus fibrosus in the

Degenerate FEMs and the deformation of human anulus fibrosus subsequent to partial

removal of the nucleus (Meakin et al., 2001). Furthermore, the observation of regions

of higher stress within the anulus fibrosus in a radial direction were in keeping with



the conclusions drawn by Meakin et al. (2001). These researchers suggested that

degeneration and the development of circumferential lesions may be a result the

higher stresses in the anulus caused by the bulge of the inner and outer anulus in

opposite directions.

8.4.3.5 Discussion of the Rim Lesion FEM

The results presented in Table 8-5 show the reduction in peak moment as a result of

either removing the nucleus or simulating a rim lesion.

Table 8-5 Percentage variation in the peak moment in the Degenerate FEM with a rim lesion and in the Rim Lesion FEM. The values in brackets are the magnitude of the increase or decrease in the peak moment.

Loading

Degenerate FEM – Rim lesion, no nucleus pressure

– compared to Healthy Anulus FEM

Rim Lesion FEM – Rim lesion, hydrostatic nucleus

- compared to Healthy FEM

Extension 1%

(17Nmm) 0%

Flexion -10%

(-386Nmm) 0%

Left lateral bending 2%

(82Nmm) (radial higher, -4%)

-3% (-617Nmm)

Right lateral bending

15% (417Nmm)

1% (12Nmm)

Left axial rotation -1%

(-10Nmm) -3%

(-108Nmm)

Right axial rotation -4%

(-35Nmm) -4%

(-133Nmm)

The reductions in moment observed in the Rim Lesion and the Degenerate FEMs are

of a similar magnitude under left and right axial rotation. However, the removal of

the hydrostatic nucleus resulted in a change in the mechanics of the disc when the rim

lesion was simulated. The variations in peak moment when the nucleus was removed

were not comparable to the results of the models with the hydrostatic nucleus present.



The results presented in Section 8.4.2 and Table 8-5 suggested that the removal of the

nucleus altered the affect that the lesions had on the disc mechanics. For example,

when a hydrostatic nucleus was present, the rim lesion did not effect the peak moment

under flexion; however, when the hydrostatic nucleus was removed the simulation of

the rim lesion caused a 10% reduction in peak moment under flexion. When flexion

loading was applied to the Rim Lesion FEM both the inner and outer anulus bulged

outward and the faces of the lesion were compressed. Possibly the simulation of a rim

lesion in the Degenerate FEM caused a reduction in the peak flexion moment because

the discontinuity of the anulus at the lesion face resulted in less resistance to the

inward bulge of the inner anulus. This was evidenced by an increase in the

displacement of nodes on the lower face of the lesion (in the plane of the lesion) in the

Degenerate FEM compared to the Rim Lesion FEM.

The results presented in Table 8-5 showed that left lateral bending and right and left

axial rotation resulted in a reduction in the peak moment. Conversely, Thompson

(2002) found that extension, left lateral bending and right axial rotation affected the

peak moments when a rim lesion was inserted in the sheep discs. The finding that

both left and right axial rotation affected the disc mechanics in the Rim Lesion FEM

was reasonable since both these motions could result in translation of the lesion faces

in the plane of the lesion and therefore, reduce the resistance to rotation. However, it

was not clear why extension did not result in a reduction in moment in the Rim Lesion

FEM since this motion would cause a separation of the lesion faces and thereby,

should pose less resistance to rotation. Possibly the disagreement between these

results was due to the differing geometry and dimensions of the sheep discs and the

human disc FEM.

In summary, the comparison of results from the Rim Lesion FEM and the Degenerate

FEM indicated that the removal of the hydrostatic nucleus affected the rotational

stiffness of the disc and also affected the response of the disc when a rim lesion was

present. These results suggested that the reduction in peak moments observed by

Thompson (2002) may have been a result of both the presence of the lesion and of the

loss of a hydrostatic nucleus and the resulting compressibility of the nucleus pulposus

in the sheep discs.



8.4.3.6 Discussion of the approach to simulating anular lesions

The presence of rim and radial anular lesions in vivo creates a discontinuity within the

anulus fibrosus ground substance, but also results in the degradation and severing of

the collagen fibres within the lamellae. Possibly the dissimilarity between the

experimental results for anular lesions in sheep and the results of the FEM was due to

the inability of the FEM to simulate the disruption to both the anulus ground

substance and the collagen fibres.

Two analyses were carried out on a Degenerate FEM to separately simulate radial and

rim lesions under extension loading. These analyses investigated the effects of

varying the stiffness in the elements adjacent to the lesion face. Rebar element

definitions were provided for the anulus ground substance elements in only one of the

faces adjacent to the lesion. The results of these analyses provided the same results as

those presented in Figure 8-6 and Figure 8-7. This indicated that the stiffness

provided by the rebar elements in the anulus fibrosus immediately around the lesion

did not affect the peak moment resisted by the disc.

However, in vivo the severing of collagen fibres due to the presence of a lesion results

in the reduction in stiffness of regions of the anulus fibrosus over the entire length of

the collagen fibres. Therefore, depending on the radial location of the collagen fibres

which are compromised a significant portion of the anulus fibrosus would be affected.

When a rim lesion is present a greater portion of the anulus would be affected due to

the transverse orientation of the discontinuity in the anulus fibrosus. Owing to the

method employed to simulate the collagen fibres, it was not possible to effectively

simulate the laxity in the collagen fibres that would have been severed when the

lesion was inserted in the sheep discs.

This was a potential cause for the dissimilar peak moments observed in the Healthy

Anulus FEM and the Degenerate FEMs. The collagen fibres are responsible for

bearing a large portion of the load applied to the disc due to the hoop stresses in the

anulus under loading. If the rebar elements simulating the collagen fibres in the

anulus fibrosus of the Degenerate FEM were still providing stiffness in regions which

should not have contained continuous collagen fibres then this FEM may not have



accurately simulated the compromised mechanics of the anulus fibrosus when anular

lesions were present.

8.5 Analysis of the Healthy and Degenerate FEM using Compressive and

Rotational Loading Conditions

Initial analyses of the Degenerate FEM with either radial or rim lesions present

indicated that the use of a 500N compressive loading step in the degenerate model

resulted in difficulties in convergence of the displacement corrections. The deformed

geometry of the partially completed solutions demonstrated very high displacements

in the anulus ground substance elements near these lesions. The displacement

convergence difficulties were likely related to the extreme deformation these

elements.

Further analyses were carried out using loading conditions that included a reduced

compressive load of 200N (40% of the full torso compressive load) and rotations in

the three planes of motion. The models used for these analyses were Degenerate

FEMs with either a rim or radial lesion simulated. Given the difficulties in obtaining

results for the circumferential lesion during the validation analyses this lesion type

was not simulated in these analyses.

A converged solution was not obtained for any of these analyses. The analysis of

flexion loading in the Degenerate FEM with a radial lesion simulated completed

0.020o of the full flexion rotation. When this model was analysed under right lateral

rotation the maximum rotation applied was 0.045o. A maximum rotation of 0.0012o

was achieved in this FEM under right axial rotation. The final rotations under the

other loading conditions when either the rim or radial lesions were simulated were

slightly higher than these values. Since the maximum rotations were so low, the

comparison of the peak moments in the Degenerate FEM with either the rim or radial

lesion present did not offer useful information and therefore, were not included in the

thesis.



The cause for the failure of the solutions was the lack of convergence of the

algorithms defining either, the force equilibrium, the displacement equilibrium or the

contact state. However, in general the over-riding cause for the failure of these

analyses before the complete rotation load had been applied was the contact

simulation between the lesion faces. If the displacement or force algorithms could not

converge, the large displacement corrections or force residuals became evident after

severe discontinuity iterations were performed. A discussion of the causes for lack of

convergence in contact simulations is given in Section 8.4.3.2.

The failed analyses exhibited difficulties in the convergence of the force and

displacement equilibrium iterations. High force residuals and displacement

corrections were generally only observed subsequent to severe discontinuity

iterations. This suggested that the lack of convergence of the force and displacement

algorithms was also related to the contact simulations for the lesions. It was thought

that the failure of the analyses due to non-convergence of the contact algorithms was

partly attributable to the inherent numerical complexity of finite element analyses that

include contact definitions. Additionally, the FEM incorporated contact definitions

between surfaces of a material with a relatively complex nonlinear constitutive

equation.

The FEM involved contact definitions between extremely deformable contact surfaces

in the anulus fibrosus ground substance. The failure of validation analyses due to lack

of convergence of the displacement and force algorithms may have resulted from the

excessive deformation of these contact surfaces. When elements in the finite element

mesh become distorted under the applied load, the accuracy of the results for these

elements is compromised and it is difficult to obtain convergence for the equilibrium

calculations. The elements at the face of the lesions showed high deformations.

The algorithms which the Abaqus software employed to determine the contact state

between contact surfaces were extremely complicated and a variety of numerical

difficulties may have been encountered. Examples of these difficulties were

‘chattering’ which was a phenomenon where the contact state changed from open to

closed during subsequent iterations and the severe discontinuity iterations could not

converge on a final contact state for the interfaces. Difficulties may also occur if



contact surfaces did not have a fine enough mesh density. These were two common

problems encountered in contact simulations; however, neither of them was the cause

for the convergence difficulties of the Degenerate FEM. Reducing the mesh density

did not improve the analysis results and the contact surfaces of the circumferential

lesion did not appear to be ‘chattering’.

A trial analysis of the Degenerate FEM was carried out using the stiffer Mooney-

Rivlin constants from the preliminary FEM presented in Chapter 3 to define the

anulus fibrosus ground substance. This model was loaded with a 200N compressive

load and an extension rotation. The Mooney-Rivlin constitutive equation was linear

under shear loading and the FEM resulted in a completed solution for the analysis of

the Degenerate FEM with the radial lesion present. Possibly, the more linear

stress/strain response of the Mooney-Rivlin constitutive equation under shear loading

generated stress-strain equations with less complexity that could more readily

converge on a solution. The rotational loading on the disc in any of the three planes

of motion would have resulted in shear stress in the anulus ground substance and

possibly the nonlinear shear behaviour added complexity to the force and

displacement algorithms.

The successful solution of the FEM using the Mooney-Rivlin constants may also have

been related to the higher stiffness of the anulus fibrosus ground substance compared

to the models using the parameters determined from experimentation (Chapter 4, 5).

Perhaps the lower deformation of the stiffer anulus ground substance when the

Mooney-Rivlin parameters were used resulted in less mesh distortion and therefore

the force and displacement equilibrium algorithms converged more readily.

8.6 Conclusions

The results of the analyses carried out on the Degenerate FEM with both compressive

and rotational loading applied suggested that the model was not capable of

representing all the loading cases applied to the intervertebral disc in vivo. It was not

possible to simulate full flexion loading of either the healthy discs or the degenerate

discs and circumferential lesions could not be modelled when lateral bending or axial



rotation was applied to the Degenerate FEM. However, useful information on the

mechanics of the intervertebral disc could be obtained for other loading cases and

lesion types.

Furthermore, the results of the validation analyses did not completely reproduce the

findings of Thompson (2002) in a sheep study. However, it should be noted that the

research carried out by Thompson (2002) was a novel body of work and prior to this

research there was little information available in the literature for the change in the

peak moments of a lumbar joint due to the insertion of the three types of anular

lesions. Also, both the geometry and the dimensions of the sheep discs tested by

Thompson (2002) were different to that of the human L4/5 intervertebral disc

simulated in the FEM. These differences may have resulted in variations in the

mechanics of the discs and therefore, been related to the discrepancy between the

findings of the two studies.

It was concluded that a possible cause for the disagreement between the results of

Thompson (2002) and the results of the validation analyses on the FEM was the

removal of the hydrostatic nucleus pulposus. It was possible that the change in disc

mechanics consequent to the simulation of anular lesions was overshadowed by the

significant changes in mechanics due to the simulation of a completely degenerate

nucleus pulposus. The changes in moment subsequent to the removal of the nucleus

pressure were as high as 86%, while the maximum reduction in peak moment

observed by Thompson (2002) was 20%.

Simulation of rim lesions in a disc with a hydrostatic nucleus pulposus allowed the

effects of the presence of the lesions to be assessed separately to the effects of

removing the nucleus pulposus pressure. These analyses of the Rim Lesion FEM

indicated the removal of the nucleus pressure changed the affects of the rim lesions on

the disc mechanics. Therefore, the results presented by Thompson (2002) may have

shown variations in the peak moments as a consequence of both the presence of the

lesions and due to the loss of the hydrostatic state of the nucleus pulposus in the sheep

discs.



Possibly the dissimilarity between the validation analyses and the experimental results

for sheep anulus was because the FEM did not simulate the discontinuity in the

collagen fibres of the anulus fibrosus when rim and radial lesions were present.

Simulation of the collagen fibres using rebar elements did not permit the

representation of severing of these fibres. Rebar elements are used in the finite

element algorithm applied by Abaqus to increase the stiffness of the material in which

they are embedded. While they are referred to as elements they are not individually

connected to nodes and therefore, cannot be disconnected to produce the effects of a

lesion that is severing collagen fibres.

It was thought that the inability to obtain convergence when the circumferential

lesions were present was due to the contact parameters selected. While an extensive

selection of parameters were trialled, it was not possible to obtain constants which

were suitable to describe the contact between the circumferential lesion faces.

Trial analyses of the Degenerate FEM using the Mooney-Rivlin material parameters

determined in Chapter 3 for the anulus ground substance were successful. This

provided insight into the cause for the lack of convergence of the models after the

compressive load was applied. This may have been related to the nonlinearity under

shear loading of the Polynomial strain energy equation compared to the linear

Mooney-Rivlin equation. Alternatively, the large deformations of the inner anulus

surface and of the lesion faces (Figure 8-11, Figure 8-9, Figure 8-10) in the

comparatively compliant Polynomial anulus ground substance may have lead to

excessive mesh distortion. Excessive mesh distortion can result in an unconverged

solution.

While the quantitative results for analyses of the FEM did not reproduce values

observed in vitro, it was important to note that the deformed shape of the inner anulus

surface in the Degenerate FEM without the nucleus pulposus pressure was similar to

in vitro observations of bovine discs (Hickey and Hukins, 1980). This indicated that

the FEM was capable of reproducing similar results for the overall deformed shape of

the anulus fibrosus. The observation of higher von Mises stresses in the anulus

ground substance due to the inward bulge of the inner anulus was in keeping with the



conclusions drawn by Meakin et al. (2001) with respect to a possible mechanism for

the development of circumferential lesions.

Additionally, the considerable reduction in peak moment when the nucleus pulposus

was removed and lesions were simulated was supported by the findings of Goel et al.

(1995). These researchers simulated gradual development of a radial lesion and the

final removal of the nucleus pulposus. Finite element results for this analysis showed

that the extension rotation subsequent to the removal of the nucleus was five times the

rotation of the healthy disc. This indicated that the rotational resistance of the disc

was reduced. These findings supported the observation of a considerable reduction in

peak moment when the nucleus pressure was removed in the current study. The

reduced moment was indicative of a reduction in stiffness of the healthy disc.

The reduction in the peak moments as a result of the removal of the nucleus pulposus

pressure showed that there was a significant change in the mechanics of the

intervertebral disc. This finding suggested that if the nucleus pulposus is extremely

degraded, it will offer little resistance to the inward bulge of the anulus fibrosus.

Furthermore, it is reasonable to assume that with the removal of the nucleus pulposus,

the stresses and strains in the anulus ground substance would be increased. These

higher strains would likely exceed the derangement strain of the tissue. When the

Healthy Anulus FEM was analysed with left and right lateral bending, the nominal

strains in the ground substance were 153.3% and 133%, respectively. These values

were higher than the derangement strain for uniaxial compression of 27% that was

stated in Chapter 4. The strains in the anulus ground substance during the other

rotational loading conditions also exceeded the derangement strain.

From this it may be hypothesised that with the degradation of the nucleus pulposus,

the strains in the anulus ground substance are increased and may exceed the

derangement strain for the tissue. Furthermore, it is postulated that a possible

mechanism for the development of anular lesions involves the degradation of the

nucleus pulposus and the consequent straining of the anulus ground substance to

strains above that at which permanent damage is initiated. The initiation of

permanent damage may result in the development of anular lesions.



The similarity between the results of the FEM and those of Goel et al. (1995) and

Hickey and Hukins (1980) provided support for the conclusion that the FEM

developed in this study was capable of reproducing the overall intervertebral disc

mechanics under certain loading and boundary constraints. The overall deformed

shape of the model was comparable to that observed in vivo for compressive and

rotational loading. However, further work was necessary to obtain a model that could

adequately simulate all physiological loading conditions applied to the intervertebral

disc.


Chapter 9: Conclusions and Recommendations 351

CChhaapptteerr

99

CCoonncclluussiioonnss aanndd

RReeccoommmmeennddaattiioonnss

The goal of this research was to develop a finite element model of an L4/5 lumbar

intervertebral disc. This model would be used to investigate the biomechanical effects

of anular lesions on the disc behaviour. The attainment of this goal required the

achievement of four main objectives.

Firstly, a preliminary FEM was developed in order to ensure the modelling techniques

employed were capable of producing a geometrically accurate simulation of the

intervertebral disc. This preliminary FEM highlighted features of the model which

required improvement if the results of model analyses were to provide meaningful

information on the disc mechanics. Subsequent to analysis of the preliminary FEM it

was concluded that a constitutive equation which better described the nonlinear shear

behaviour of the anulus ground substance under all shear loading modes would be

employed.

The second objective that was addressed was the acquisition of hyperelastic

parameters that accurately represented the mechanical behaviour of the anulus

fibrosus ground substance. In order to determine these parameters experimental

testing was carried out on specimens of sheep anulus fibrosus under uniaxial

compression, simple shear and biaxial compression loading modes. The

experimental data from these tests was used to determine representative responses for

the tissue. There was a significant difference between the mechanical response of the



anterior, lateral and posterior anulus ground substance and a third order Ogden

hyperelastic equation was fit to this data in order to define the inhomogeneous

mechanical response of the anulus. A second order polynomial hyperelastic equation

was fit to the overall response of the anulus ground substance to define a

homogeneous mechanical response.

The anulus specimens were tested under repeated loading. It was found that loading

the anulus fibrosus ground substance to strains above 27% in uniaxial compression

and above 35% in simple shear loading resulted in permanent damage to the tissue.

The strain at which permanent damage was initiated was referred to as the

derangement strain. The derangement of the tissue was evident since the stiffness of

the specimens reduced when they were loaded to strains above the derangement strain

and this reduced stiffness was not regained after the specimens were allowed to

recover for 1 hour in a hydrated environment. This suggested that the loss of stiffness

was not attributable to lack of viscoelastic recovery or pore fluid. This reduced

stiffness characteristic was reproducible upon repeated loading which indicated that

the tissue had been damaged but was not incapable of bearing a load during

subsequent testing.

The derangement strains of the tissue were strains that could be experienced during

common physiological motions, such as full flexion. It was hypothesised that a

possible mechanism for degeneration of the intervertebral disc may be that the anulus

fibrosus experiences derangement as a result of daily activities. With increasing age

the regenerative capabilities of the disc materials may cease to function effectively

and consequently signs of degeneration manifest in the intervertebral disc.

The third objective of this research was to implement the improved mechanical

properties for the anulus fibrosus ground substance into the preliminary FEM. The

successful fulfilment of this objective was achieved with the development of a finite

element model that closely represented the geometry of the intervertebral disc and

incorporated the mechanical behaviour determined experimentally.

The homogeneous material parameters and the inhomogeneous material parameters

were implemented in separate models. Further improvements were made to the



geometry and material representation for the collagen fibres and nucleus pulposus in

these models in order to more closely represent the materials in vivo. Owing to the

close relationship between the intervertebral disc and the anterior and posterior

longitudinal ligaments in the spine, these structures were simulated in the model to

more closely represent the in vivo geometry immediately around the intervertebral

disc.

Subsequent to analyses of the Homogeneous and the Inhomogeneous FEMs it was

observed that once implemented, the material parameters describing the anulus

fibrosus ground substance resulted in high deformations and excessive pressures in

the nucleus pulposus in comparison to experimental evidence. It was concluded that

the material parameters determined using experimental data from sheep anulus

fibrosus ground substance may have been overly compliant in comparison to the

human tissue. Possibly this was a result of differences in the fluid content or

proteoglycan type in the human and sheep disc. However, numerous prior

experimental studies had demonstrated the analogous behaviour between human and

sheep intervertebral discs. It was believed that the results of the FEM which

incorporated data from sheep anulus fibrosus would provide valuable qualitative

information on the mechanical response of the human intervertebral disc.

It was concluded that the hyperelastic parameters describing the inhomogeneous

anulus ground substance were not stable for the range of loading conditions employed

in the FEM. This lack of stability was apparent since the Inhomogeneous FEM was

not capable of converging on a solution due to excessive displacement corrections at

nodes in the anulus ground substance. However, a converged solution for the

Inhomogeneous FEM was obtained when the initial loading condition of 70kPa

pressure was removed. It was questioned whether the Inhomogeneous FEM would be

capable of converging on a completed solution for more complex loading conditions

such as physiological rotations. Also, it was unclear whether the results of simulation

of lesions would be accurate. As such, the Inhomogeneous FEM was not employed to

analyse the effects of anular lesions.

The final objective was to simulate both a healthy disc and a degenerate disc using the

hyperelastic parameters determined from experimentation on sheep anulus ground



substance. The Homogeneous FEM was used for these analyses. Degeneration of the

intervertebral disc was simulated by removing the nucleus pulposus pressure and

using contact simulations to represent the faces of rim, radial and circumferential

lesions in the anulus fibrosus. Analyses were carried out using physiological rotation

loading conditions. However, a converged solution was not obtained for some

analyses which simulated a circumferential lesion.

A large reduction in peak moment was observed when the nucleus pulposus was

removed from the FEM that was simulating the healthy disc condition. This reduced

moment was attributed to the reduction in stress in the rebar elements representing the

collagen fibres and the subsequent reduction in resistance to rotation offered by these

elements. These findings indicated that when the nucleus pulposus is extremely

degenerate and offers little resistance to inward bulge of the inner anulus, the

mechanics of the disc are significantly changed. Specifically, the resistance to

rotation of the intervertebral joints that is provided by the intervertebral disc is greatly

reduced and consequently the stability of the affected joint may be compromised. If

there is less resistance to rotation offered by the intervertebral disc, then the rotational

loading applied to the spine must be resisted by other spinal structures such as the

ligaments and zygapophysial joints. This may lead to the overload and subsequent

damage of these structures in spines with degenerate discs.

The ligaments and joint capsules of the lumbar spine are innervated (See Bogduk,

1997, for a comprehensive review). From the results of the current study it is

hypothesised that the potential overloading of the spinal ligaments in degenerate discs

may result in pain if these structures are subjected to deformations outside their range

of motion in the healthy spine.

Furthermore, Malinsky (1959), Yoshizawa et al. (1980) and Bogduk (1983)

demonstrated the innervation of the peripheral regions of the anulus fibrosus.

Assuming that normal disc deformations in a healthy intervertebral disc do not result

in back pain, then it was postulated that the abnormal local disc deformations present

in the anulus fibrosus as a result of the anular lesions may give rise to back pain that

originates from the sites of innervation in the outer anulus.



The results also indicated that the removal of the nucleus pulposus pressure altered the

response of the disc to the presence of lesions. The loading conditions that resulted in

a reduction in peak moments when a rim lesion was simulated in a model with a

hydrostatic nucleus were different to those that resulted in a decreased peak moment

when hydrostatic nucleus was removed.

When anular lesions were simulated in the FEM without the nucleus pressure,

minimal changes in rotational stiffness were observed. This suggested that the loss of

nucleus pressure had a much greater effect on the mechanics of the intervertebral disc

than the presence of lesions. The reduction in peak moments observed in the FEM

were as high as 86% while a maximum reduction of 20% was observed during

experimentation on sheep discs with lesions (Thompson, 2002).

Degeneration of the intervertebral disc may be characterised by a loss of hydration, a

dry granular texture and the presence of anular lesions. The removal of the nucleus

pulposus in the Degenerate FEMs simulated a very granular and dry nucleus with no

compressive resistance. The results presented in this thesis suggest that the

development of lesions in the anulus prior to the degradation of the nucleus pulposus

would result in less variation in the mechanics of the disc than if the nucleus pulposus

degraded first. However, the overall response of the entirely degenerate disc would

show a reduced resistance to rotation. It should be noted that the precise link between

the presence of anular lesions and the biochemical changes in the degenerate disc that

cause the loss of hydration is not known.

However, from the observations of the significant reduction in peak moments when

the nucleus pressure was removed, a possible link between the biochemical changes

evident in degenerate discs and the presence of lesions is postulated. If the nucleus

pulposus is degraded, possibly as a result of biochemical changes in the disc, there

will be a loss of mechanical integrity of the nucleus and consequently the anulus

fibrosus may be subjected to increased loads. These increased loads would result in

increased strains in the anulus ground substance which may exceed the derangement

strain of this tissue causing some permanent damage to be initiated. It is hypothesised

that this initiation of damage in the ground substance may result in the development

of anular lesions.



Overall, the finite element model that has been developed has provided an improved

understanding of the mechanics of the intervertebral disc and of the mechanical

behaviour of the materials in the intervertebral disc. The experimentation on sheep

anulus ground substance provided useful information on the strains at which damage

is initiated in this material and permitted deductions to be made regarding possible

mechanisms for disc degeneration.

9.1 Recommendations for Further Work

The results of the model did not reproduce in vivo observations of the pressure in the

nucleus pulposus under compressive loading. Furthermore, the peak moments

observed in the Degenerate FEM with lesions present did not show similar trends to

the peak moments observed in experimentation on sheep discs with lesions

(Thompson, 2002).

Three main causes were attributed to the disagreement between the results of the FEM

and the in vivo and in vitro results of previous researchers.

• The methods employed to obtain material parameters for the disc components;

• The methods used to simulate the anular lesions in the Degenerate FEMs; and

• The assumptions made with respect to the compressive loading on the disc.

These three factors highlighted areas for future work that would improve the results of

the FEM. Furthermore, it was thought that the simulation of a sheep intervertebral

disc may provide a useful validation tool.

9.1.1 Parameters for the disc components

The material parameters used to describe the anulus ground substance were

determined from experimentation on sheep anulus fibrosus. The high compliance of

the anulus ground substance in the FEM was thought to be the cause for the high

nucleus pulposus pressures observed under torso compression loading. It was

suggested that the compliance of the anulus ground substance in the FEM may have

been due to the dissimilarity between the human and sheep anulus ground substance.



Experimentation on human specimens to determine the mechanical response and

derangement strain of the anulus ground substance using the loading conditions

outlined in Chapter 4 would provide useful information on the similarity between the

human and sheep ground substance; however, this was not possible as part of this

project. Determination of hyperelastic parameters using this data would ensure the

material parameters used in the FEM accurately simulated the human tissue.

Alternatively, biochemical analysis of the sheep and human anulus ground substance

in terms of the types and concentration of proteoglycans present could confirm the

compatibility of the two tissues and in particular, may highlight any differences in

their propensity to absorb fluid.

Finally, it was thought that further investigations to obtain meshing parameters to

describe the anulus fibrosus in relation to the contact relationship between the lesion

faces were necessary. This would involve further investigation of the contact

parameters and exploration of the effects of varying the location of the lesion and the

extent of the discontinuity created in the mesh. Perhaps the simulation of a smaller

lesion would result in a converged solution or highlight the cause for the lack of

convergence in the contact simulations already carried out.

9.1.2 Simulation of anular lesions

An improved representation of the collagen fibres in the anulus fibrosus would

involve the use of tension-only spring elements. Individual elements could then be

disconnected when either rim or radial lesions were simulated. It was thought that the

simulation of these lesions by both creating a discontinuity in the anulus ground

substance and disabling the action of the collagen fibres that pass through the lesion

face would more closely simulate the effects of anular lesions.

Finally, it was thought that future analyses of the biomechanical effects of anular

lesions that used a fracture mechanics approach would provide useful information on

the development and progression of the lesions. Static analyses such as those

undertaken in this study cannot provide information on the stresses in the immediate

vicinity of the lesion and cannot predict the development and progression of lesions.



9.1.3 Compressive loading conditions

The assumptions inherent in the application of compressive loading to the FEM were

highlighted in Chapter 6 (Section 6.6.2.4). These included the application of

compressive load through the centroid of the disc in the transverse plane. It is

suggested that experimentation on cadaveric spines to determine the centre of pressure

at which pure compression occurs when torso compressive loading is applied to the

disc should be carried out. Possibly the methods employed by Tibrewal and Pearcy

(1985) to determine the centre of rotation during flexion could be employed on

compressively loaded cadaveric lumbar spines to determine this location. These

experimental techniques would involve determining the point in the disc at which

pure compression occurs without any associated rotation. If this centre of pressure

was not at the centroid of the intervertebral disc, then the use of a correct centre may

result in a reduction in the anterior rotation of the Homogeneous FEM under torso

compression. This would in turn reduce the deformation of the nucleus pulposus

volume and decrease the nucleus pressure.

9.1.4 Simulation of a sheep intervertebral disc

At the outset of this research, the lack of information regarding the biomechanical

effect of anular lesions on the disc mechanics was highlighted as an important topic,

with clinical relevance to the understanding of back pain. Therefore, the finite

element analysis of anular lesions was carried out on a model that geometrically

represented a human disc.

Human intervertebral discs were not available for the experimental determination of

the anulus ground substance mechanical properties. Due to the extensively published

similarity between the mechanics of the human and sheep discs, the sheep anulus

ground substance was assumed to provide an acceptable representation of the human

tissue. However, the results presented in this thesis suggested there may be some

biochemical differences between these tissues. Even so, the experimental testing of

sheep anulus ground substance provided useful information for qualitative

comparative analysis of the biomechanics of the human disc.



It is suggested that analysis of an FEM using the geometry of a sheep disc and the

material properties presented in Chapter 4 and 5 would provide useful results for

validation. The geometry of the sheep discs would be obtained using the

mathematical algorithm presented in Chapter 3.

In summary, the finite element model of the L4/5 intervertebral disc developed in this

study has demonstrated similar behaviour to that observed for the disc in vivo and in

vitro. The deformation of the model components in terms of anulus bulge and axial

displacement did reproduce the behaviour of the intervertebral disc. However,

improvements could be made to the material parameters describing the anulus

fibrosus ground substance, the method for applying the compressive torso loading, the

representation of the collagen fibres in the anulus fibrosus and finally the techniques

employed to simulate the degenerate disc. With these improvements it is expected

that the model would provide a very powerful analysis tool for the simulation of

various loading conditions and for the investigation of the biomechanical effects of

degeneration and anular lesions on the disc.


Appendices: A, B, C 360

AAppppeennddiixx Appendix A: Matlab and Fortran executables for generating mesh geometry, to determine maximum and minimum principal stretch ratios and to carry out the least-squared-error algorithm Appendix B: All engineering drawings for the testing devices that were designed and manufactured to carry out the experimental testing detailed in Chapter 4. Appendix C: This contains raw data, regression lines of best fit and data for the envelopes of measurement. These data are referenced in Chapter 4 and 5. All appendices are contained on the CD.


Bibliography 361

BBiibblliiooggrraapphhyy

Acaroglu, E. R., Iatridis, J. C., Setton, L. A., Foster, R. J., Mow, V. C. and

Weidenbaum, M. (1995). "Degeneration and Aging Affect the Tensile Behaviour of

Human Lumbar Annulus Fibrosus." Spine 20(24): 2690-2701.

Adams, M. A., Freeman, B. J., Morrison, H. P., Nelson, I. W. and Dolan, P. (2000).

"Mechanical initiation of intervertebral disc degeneration." Spine 25(13): 1625-36.

Adams, M. and Hutton, W. C. (1981). "The Relevance of Torsion to the Mechanical

Derangement of the Lumbar Spine." Spine 6(3): 241-8.

Adams, M. A., McNally, D. S., Chinn, H. and Dolan, P. (1994). "Posture and the

Compressive Strength of the Lumbar Spine." Clinical Biomechanics 9: 5-14.

Adams, P., Eyre, D. and Muir, H. (1977). "Biochemical Aspects of Development and

Ageing of Human Lumbar Intervertebral Discs." Rheumatologic Rehabilitation 16:

22-29.

Ahmed, M., Bjurholm, A., Kreicbergs, A. and Schultzberg, M. (1993). "Sensory and

autonomic innervation of the facet joint in the rat lumbar spine." Spine 18(14): 2121-

6.

Andra, H., Warby, M. K. and Whiteman, J. R. (2000). "Contact problems of

hyperelastic membranes: existence theory." Mathematical Methods in the Applied

Sciences 23(10): 865-95.

Armstrong, J. R. (1958). Lumbar Disc Lesions - Pathogenesis of Low Back Pain and

Sciatica. Edinburgh, E & S Livingstone Ltd.

Australian Bureau of Statistics (2002). "4364.0 National Health Survey - Summary of

Results, Australia." [Web document] Available:


Bibliography 362

http://www.abs.gov.au/ausstats/[email protected]/0/CAC1A34167E36BE3CA2568A9001393

64?Open [23rd Dec., 2003]

Behrsin, J. F. and Briggs, C. A. (1988). "Ligaments of the lumbar spine: a review."

Surg Radiol Anat 10(3): 211-9.

Belytschko, T., Kulak, R. F., Schultz, A. B. and Galante, J. O. (1974). "Finite Element

Stress Analysis of an Intervertebral Disc." Journal of Biomechanics 7: 277 - 285.

Benzel, E. C. (1995). Biomechanics of Spine Stabilization. New York, McGraw-Hill

Inc. , Health Professions Division.

Best, B. A., Guilack, F., Setton, L. A., Zhu, W., Saed-Nejad, F., Ratcliffe, A.,

Weidenbaum, M. and Mow, V. (1994). "Compressive Mechanical Properties of the

Human Anulus Fibrosus and Their Relationship to Biochemical Composition." Spine

19(2): 212-221.

Betsch, D. F. and Baer, E. (1986). "Structure and Mechanical Properties of Rat Tail

Tendon." Spine 11(9): 914-27.

Bilston, L. E., Liu, Z. and Phan-Thien, N. (2001). "Large Strain Behaviour of Brain

Tissue in Shear: some experimental data and differential constitutive model."

Biorheology 38(4): 335-45.

Bischoff, J. E., Arruda, E. M. and Grosh, K. (2002). "Finite element simulations of

orthotropic hyperelasticity." Finite Elements in Analysis and Design 38(10): 983-98.

Bogduk, N. (1983). "The Innervation of the Lumbar Spine." Spine 8(3): 286-93

Bogduk, N. (1997). Clinical Anatomy of the Lumbar Spine and Sacrum. New York,

Churchill Livingstone.

Brown, T., Hansen, R. J. and Yorra, A. J. (1957). "Some Mechanical Tests on the

Lumbosacral Spine with Particular Reference to the Intervertebral Discs." The

Journal of Bone and Joint Surgery 39A(5).

Burns, M. L., Kaleps, I. and Kazarian, L. E. (1984). "Analysis of Compressive Creep

Behaviour of the Intervertebral Unit Subjected to a Uniform Axial Loading Using


Bibliography 363

Exact Parametric Solution Equations of Kelvin-sold models: Part I, Human

intervertebral joints." Journal of Biomechanics 17: 113-30.

Cassidy, J. J., Hiltner, A. and Baer, E. (1989). "Hierarchical structure of the

intervertebral disc." Connect Tissue Res 23(1): 75-88.

Chazal, J., Tanguy, A., Bourges, M., Gaurel, G., Escande, G., Guillot, M. and

Vanneuville, G. (1985). "Biomechanical properties of spinal ligaments and a

histological study of the supraspinal ligament in traction." J Biomech 18(3): 167-76.

Cossette, J. W., Farfan, H. F., Robertson, G. H. and Wells, R. V. (1971). "The

Instantaneous Centre of Rotation of the Third Lumbar Intervertebral Joint." Journal of

Biomechanics 4: 149-153.

Criscione, J. C., Humphrey, J. D., Douglas, A. S. and Hunter, W. C. (2000). "An

Invariant Basis for Natural Strain which Yields Orthogonal Stress Response Terms in

Isotropic Hyperelasticity." Journal of Mechanics and Physics of Solids 48: 2445-

2465.

Crisp, J. D. C. (1972). Properties of Tendon and Skin. Biomechanics: Its foundations

and objectives. Y. C. Fung, N. Perrone and M. Anliker. Englewood Cliffs, New

Jersey, Prentice-Hall Inc.

Dickson, I., Happey, F., Pearson, C., Naylor, A. and Turner, R. (1967). "Variations in

the Protein Components of Human Intervertebral Disk with Age." Nature (London)

215: 52.

Duncan, N. A., Ashford, F. A. and Lotz, J. C. (1996). "The Effect of Strain Rate on

the Axial Stress-Strain Response of the Human Anulus Fibrosus in Tension and

Compression: Experiment and Poroelastic Finite Element Predictions." Advances in

Bioengineering 33: 401-2.

Dvorak, J., Schneider, E., Saldinger, P. and Rahn, B. (1988). "Biomechanics of the

Craniocervical Region: The alar and transverse ligaments." Journal of Orthopaedic

Research 6(3): 452.


Bibliography 364

Ebara, S., Iatridis, J. C., Setton, L. A., Foster, R. J., Mow, V. C. and Weidenbaum, M.

(1996). "Tensile Properties of Nondegenerate Human Lumbar Annulus Fibrosus."

Spine 21(4): 452-461.

Eyre, D. and Muir, H. (1976). "Types I and II Collagen in Intervertebral Discs.

Interchanging Radial Distributions in Annulus Fibrosus."Biochemistry Journal 157:

267.

Eyre, D. and Muir, H. (1977). "Quantitative Analysis of Types I and II Collagens in

Human Intervertebral Discs at Various Ages." Acta Biochem Biophys 492: 29.

Eyre, D. R. (1976). "Biochemistry of the Intervertebral Disc." International Reviews

of Connective Tissue Research 8: 227-287.

Farfan, H. F. (1973). Mechanical Disorders of the Low Back. Philadelphia, Lea and

Febiger.

Fujita, Y., Duncan, N. A. and Lotz, J. C. (1997). "Radial Tensile Properties of the

Lumbar Annulus Fibrosus are Site and Degeneration Dependent." Journal of

Orthopaedic Research 15: 814-9.

Fujita, Y., Wagner, D. R., Biviji, A. A., Duncan, N. A. and Lotz, J. C. (2000).

"Anisotropic Shear Behaviour of the Annulus Fibrosus: effect of harvest site and

tissue prestrain." Medical Engineering and Physics 22: 349-57.

Fung, Y. C. (1972). Stress-Strain-History Relations of Soft Tissues in Simple

Elongation. Biomechanics: Its foundations and objectives. Y. C. Fung, N. Perrone and

M. Anliker. Englewood Cliffs, New Jersey, Prentice-Hall Inc.

Galante, J. O. (1967). "Tensile Properties of Human Lumbar Annulus Fibrosis." Acta

Orthopaedica Scandinavica Supplementum 100.

George, E. D., Haduch, G. A. and Jordan, S. (1988). "The Integration of Analysis and

Testing for the Simulation of the Response of Hyperelastic Materials." Finite

Elements in Analysis and Design 4: 19-42.


Bibliography 365

Goel, V. K., Monroe, B. T., Gilbertson, L. G., Brinckmann, P. and Nat, R. (1995).

"Interlaminar Shear Stresses and Laminae Separation in a Disc: Finite Element

Analysis of the L3-L4 Motion Segment Subjected to Axial Compressive Load." Spine

20(6): 689-98.

Goel, V. K. and Njus, G. O. (1986). Stress-strain Characteristic of Spinal Ligaments.

32nd Trans. Orthop. Res. Soc., New Orleans.

Goel, V. K. and Weinstein, J. N., Eds. (1990). Biomechanics of the Spine: Clinical

and Surgical Perspective. United States, CRC Press.

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." Journal of Orthopaedic Science 7: 243-46.

Haughton, V. M., Syvertsen, A. and Williams, A. L. (1980). "Soft-tissue anatomy

within the spinal canal as seen on computed tomography." Radiology 134(3): 649-55.

Hibbit, Karlsson and Sorenson (2002). Abaqus Theory Manual, HKS Abaqus.

Hickey, D. S. and Hukins, D. W. L. (1980). "Relation Between the Structure of the

Annulus Fibrosus and the Function and Failure of the Intervertebral Disc." Spine 5:

106-16.

Higginson, G. R., Litchfield, M. R. and Snaith, J. (1976). "Load-displacement

Characteristics of Articular Cartilage." International Journal of Mechanical Sciences

18(9-10): 481-86.

Hilton, R. C., Ball, J. and Benn, R. (1976). "Vertebral Endplate Lesions (Schmorl's

Nodes) in the Dorsolumbar Spine." Annals of the Rheumatic Diseases 35: 127-32.

Hirsch, C. (1955). "The Reaction of the Intervertebral Disc to Compression Forces."

Journal of Bone & Joint Surgery 37-A(6): 1188-1196.

Hirsch, C. and Schajowiz (1953). "Studies on Structural Changes in the Lumbar

Anulus Fibrosus." Acta Orthop Scand 22: 184-231.


Bibliography 366

Holm, S., Holm, A. K., Ekstrom, L., Karladani, A. and Hansson, T. (2004).

"Experimental disc degeneration due to endplate injury." J Spinal Disord Tech 17(1):

64-71.

Horton, W. G. (1958). "Further Observations on the Elastic Mechanism of the

Intervertebral Disc." Journal of Bone and Joint Surgery 40B(3): 552-7.

Hukins, D. W., Kirby, M. C., Sikoryn, T. A., Aspden, R. M. and Cox, A. J. (1990).

"Comparison of Structure, Mechanical properties and Functions of Lumbar Spinal

Ligaments." Spine 15(8): 787-95.

Hukins, D. W. L., Davies, K. E. and Hilton, R. C. (1989). Constancy of collagen fibre

orientations in the annulus fibrosus during ageing of the intervertebral disc. The

Ageing Spine. D. W. L. Hukins and M. A. Nelson. Manchester, Manchester

University Press: 150-6.

Iatridis, J. C., Kumar, S., Foster, R. J., Weidenbaum, M. and Mow, V. C. (1999).

"Shear Mechanical Properties of Human Lumbar Annulus Fibrosus." Journal of

Orthopaedic Research 17: 732-37.

Inoue, H. (1981). "Three-Dimensional Architecture of Lumbar Intervertebral Discs."

Spine 6(2): 139-146.

Jemiolo, S. and Telega, J. J. (2001). "Transversely Isotropic Materials Undergoing

Large Deformations and Application to Modelling of Soft Tissues." Mechanics

Research Communications 28(4): 397-404.

Jemiolo, S. and Turteltaub, S. (2000). "Parametric model for a class of foam-like

isotropic hyperelastic materials." Journal of Applied Mechanics, Transactions ASME

67(2): 248-54.

Juming, T., Tung, M. A., Lelievre, J. and Yanyin, Z. (1997). "Stress-Strain

Relationships for Gellan Gels in Tension, Compression and Torsion." Journal of Food

Engineering 31(4): 511-529.

Kaleps, I., Kazarian, L. E. and Burns, M. L. (1984). "Analysis of compressive Creep

Behaviour of the Vertebral Unit Subjected to a Uniform Axial Loading using Exact


Bibliography 367

Parametric Solution Equations of Kelvin-solid models: Part III Rhesus monkey

intervertebral joints." Journal of Biomechanics 17: 131-36.

Kelley, B. S., Lafferty, J. F., Bowman, D. A. and Clark, P. A. (1983). "Rhesus

Monkey Intervertebral Disk Viscoelastic Response to Shear Stress." Journal of

Biomechanical Engineering 105, Transactions of the ASME: 51-54.

Kirby, M. C., Sikoryn, T. A., Hukins, D. W. and Aspden, R. M. (1989). "Structure

and mechanical properties of the longitudinal ligaments and ligamentum flavum of the

spine." J Biomed Eng 11(3): 192-6.

Klein, J. A. and Hukins, D. W. L. (1983). "Functional Differentiation in the Spinal

Column." Engineering in Medicine 12(2): 83-85.

Klisch, S., M., and Lotz, J. C. (1999). "Material Constants for Human Anulus

Fibrosus Depend on Water Content and Reference Configuration." Advances in

Bioengineering 43: 223-4.

Kumaresan, S., Yoganandan, N., Pintar, F. A. and Maiman, D. J. (1999). "Finite

Element Modelling of the Cervical Spine: Role of Intervertebral disc under Axial and

Eccentric Loads." Medical Engineering and Physics 21: 689 - 700.

Kurowski, P. and Kubo, A. (1986). "The Relationship of Degeneration of the

Intervertebral Disc to Mechanical Loading Conditions on Lumbar Vertebrae." Spine

11(7): 726-731.

Laible, J. P., Pflaster, D., Simon, B. R., Krag, M. H., Pope, M. and Haugh, L. D.

(1994). "A Dynamic Material Parameter Estimation Procedure for Soft Tissue Using a

Poroelastic Finite Element Model." Journal of Biomechanical Engineering 116(1):

19-29.

Laible, J. P., Pflaster, D. S., Krag, M. H., Simon, B. R. and Haugh, L. D. (1993). "A

Poroelastic-Swelling Finite Element Model with Application to the Intervertebral

Disc." Spine 18(5): 659-670.


Bibliography 368

Latham, J. M., Pearcy, M. J., Costi, J. J., Moore, R., Fraser, R. D. and Vernon-

Roberts, B. (1994). "Mechanical Consequences of Annular Tears and Subsequent

Intervertebral Disc Degeneration." Clinical Biomechanics 9(4): 211-219.

Lieu, C., Nguyen, T. and Payant, L. (2001). "In Vitro Comparison of Peak

Polymerisation Temperatures of 5 Provisional Restoration Resins." Journal of

Canadian Dental Association 67(1): 36-39.

Malinsky, J. (1959). "The Ontogenetic Development of Nerve Terminations in the

Intervertebral Discs of Man." Acta Anatomica 38: 96-113.

MARC (1996). White Paper on: Nonlinear Finite Element Analysis of Elastomers.

Palo Alto, US, MARC Analysis Research Corp.

Marchand, F. and Ahmed, A. (1990). "Investigation of the Laminate Structure of

Lumbar Disc Annulus Fibrosus." Spine 15(5): 402-410.

Marieb, E. N. (1998). Human Anatomy and Physiology. California,

Benjamin/Cummings Publishing Company, Inc. an imprint of Addison Wesley

Longman, Inc.

Markolf, K. L. and Morris, J. M. (1974). "The Structural Components of the

Intervertebral Disc: a study of their contributions to the ability of the disc to

withstand compressive force." Journal of Bone and Joint Surgery 56A(4): 675-87.

McGill, S. M. (1988). "Estimation of force and extensor moment contributions of the

disc and ligaments at L4-L5." Spine 13(12): 1395-402.

McGill, S. M. and Norman, R. W. (1986). "Partitioning of the Dynamic L4/5 Moment

into Disc, Ligamentous and Muscular Components During Lifting." Spine 11: 666-78.

McNally, D. S. and Arridge, R. G. C. (1995). "An Analytical Model of Intervertebral

Disc Mechanics." Journal of Biomechanics 28(1): 53-68.

Meakin, J. R. and Hukins, D. W. L. (2000). "Effect of removing the nucleus pulposus

on the deformation of the annulus fibrosus during compression of the intervertebral

disc." Journal of Biomechanics 33: 575-580.


Bibliography 369

Meakin, J. R., Redpath, T. W. and Hukins, D. W. (2001). "The Effect of Partial

Removal of the Nucleus Pulposus from the Intervertebral Disc on the Response of the

Human Annulus Fibrosus to Compression." Clinical Biomechanics 16(2): 121-8.

Miller, J. A. A., Schultz, A. B., Warwick, D. N. and Spencer, D. L. (1986).

"Mechanical Properties of Lumbar Spine Motion Segments Under Large Loads."

Journal of Biomechanics 19: 79-84.

Miller, K. and Chinzei, K. (2002). "Mechanical properties of brain tissue in tension."

Journal of Biomechanics 35(4): 483-90.

Minns, R. J., Soden, P. D. and Jackson, D. S. (1973). "The Role of the Fibrous

Components and Ground Substance in the Mechanical Properties of Biological

Tissues: a preliminary investigation." Journal of Biomechanics 6(2): 153-165.

Mooney, M. (1940). "A Theory of Large Elastic Deformation." Journal of Applied

Physics 11: 582-592.

Morgan, F. R. (1960). "The Mechanical Properties of Collagen Fibres: Stress-strain

curves." Journal of the Society of Leather Trades' Chemists. 44: 170-.

Myklebust, J. B., Rauschning, W., Sances, A. J., Pintar, K. and Larson, S. J. (1984).

Failure Levels and Dimensinos of Lumbar Spinal Ligaments. Tenth Meeting of the

International Society for the Study of the Lumbar Spine, Montreal.

Nachemson, A. (1960). "Lumbar Intradiscal Pressure: experimental studies on post-

mortem material." Acta Orthopaedica Scandinavica 43: 9-104.

Nachemson, A. (1992). Lumbar mechanics as revealed by lumbar intradiscal pressure

measurements. The Lumbar Spine and Back Pain. M. Jayson. Edinburgh, Churchill

Livingstone: 157-71.

Nachemson, A. and Evans, J. H. (1968). "Some Mechanical Properties of the 3rd

Lumbar Interlaminar Ligament (ligamentum flavum)." Journal of Biomechanics 1:

211-220.


Bibliography 370

Nachemson, A. and Morris, J. M. (1964). "In Vivo Measurements of Intradiscal

Pressure." Journal of Bone and Joint Surgery 46A(5): 1077-1092.

Nachemson, A. L. (1963). "Influence of spinal movement on the lumbar intradiscal

pressure and on the tensile stresses in the annulus fibrosus." Acta Orthopaedica

Scandinavica 33(183): 183-207.

Nachemson, A. L. (1966). "The Load on Lumbar Disks in Different Posistions of the

Body." Clinical Orthopaedics 45: 107-122.

Nachemson, A. L. (1981). "Disc Pressure Measurements." Spine 6: 93-97.

Natali, A. and Meroi, E. (1990). "Nonlinear Analyis of Intervertebral Disc Under

Dynamic Load." Transactions of the ASME Journal 112(Aug.): 358-362.

Natali, A. N. (1991). "Hyperelastic and almost Incompressible Material Model as an

Approach to Intervertebral Disc Analysis." Journal of Biomedical Engineering 13(2):

163-168.

Natarajan, R. N., Ke, J. H. and Andersson, G. B. J. (1994). "A Model to Study the

Disc Degeneration Process." Spine 19(3): 259-265.

National Health and Medical Research Council (2000). "Guidelines for consultation

on management of low back pain." [Web document] Available:

http://www.nhmrc.gov.au/media/2000rel/pain.htm [23rd Dec., 2003]

Naylor, A., Happey, F. and Macrae, T. (1954). "The Collagenous Changes in the

Intervertebral Disc with Age and their Effects on Elasticity: an X-ray Crystallographic

Study." Br Med J 2: 570-573.

Neumann, P., Ekstrom, L. A., Keller, T. S., Perry, L. and Hansson, T. H. (1994).

"Aging, vertebral density, and disc degeneration alter the tensile stress-strain

characteristics of the human anterior longitudinal ligament." J Orthop Res 12(1): 103-

12.


Bibliography 371

Neumann, P., Keller, T., Ekstrom, L., Hult, E. and Hansson, T. (1993). "Structural

properties of the anterior longitudinal ligament. Correlation with lumbar bone mineral

content." Spine 18(5): 637-45.

Neumann, P., Keller, T. S., Ekstrom, L. and Hansson, T. (1994). "Effect of strain rate

and bone mineral on the structural properties of the human anterior longitudinal

ligament." Spine 19(2): 205-11.

Neumann, P., Keller, T. S., Ekstrom, L., Perry, L., Hansson, T. H. and Spengler, D.

M. (1992). "Mechanical properties of the human lumbar anterior longitudinal

ligament." J Biomech 25(10): 1185-94.

Ogden, R. W. (1972). "Large Deformation Isotropic Elasticity - on the correlation of

theory and experiment for incompressible rubberlike solids." Proceedings of the

Royal Society of London, Series A Mathematical and Physical Sciences 326: 565-584.

Ohshima, H., Hirano, N., Osada, R., Matsui, H. and Tsuji, H. (1993). "Morphologic

variation of lumbar posterior longitudinal ligament and the modality of disc

herniation." Spine 18(16): 2408-11.

Osti, O. L., Vernon-Roberts, B. and Fraser, R. D. (1990). "Anulus Tears and

Intervertebral Disc Degeneration: An experimental study using an animal model."

Spine 15(8): 762-767.

Panagiotacopulos, N. D., Pope, M. H., Krag, M. H. and Bloch, R. (1987). "A

Mechanical Model for the Human Intervertebral Disc." Journal of Biomechanics

20(9): 839-850.

Panjabi, M., Brown, M., Lindahl, S., Irstam, L. and Hermens, M. (1988). "Intrinsic

disc pressure as a measure of integrity of the lumbar spine." Spine 13(8): 913-7.

Panjabi, M. M., Goel, V., Oxland, T., Takata, K., Duranceau, J., Krag, M. and Price,

M. (1992). "Human Lumbar Vertebrae: Quantitative three-dimensional anatomy."

Spine 17(3): 299-306.


Bibliography 372

Pearce, R. H., Grimmer, B. J. and Adams, M. E. (1987). "Degeneration and the

Chemical Composition of the Human Intervertebral Disc." Journal of Orthopaedic

Research 5: 198-205.

Pearcy, M. J. (1985). "Stereo Radiography of Lumbar Spine Motion." Acta

Orthopaedica Scandinavica Supplementum 56(212).

Pearcy, M. J. and Bogduk, N. (1988). "Instantaneous Axes of Rotation of the Lumbar

Intervertebral Joints." Spine 13(9): 1033-1041.

Pearcy, M. J. and Hindle, R. J. (1991). "Axial Rotation of Lumbar Intervertebral

Joints in Forward Flexion." Proceedings of Institution of Mechanical Engineers: Part

H Journal of Engineering in Medicine 205: 205-209.

Pearson, I. and Pickering, M. (2001). "The determination of a highly elastic adhesive's

material properties and their representation in finite element analysis." Finite

Elements in Analysis and Design 37: 221-232.

Pedrini, V., Ponseti, I. V. and Dohrman, S. C. (1973). Journal of Laboratory and

Clinical Medicine 82: 938.

Pintar, F. A., Yoganandan, N., Myers, T., Elhagediab, A. and Sances, A., Jr. (1992).

"Biomechanical properties of human lumbar spine ligaments." J Biomech 25(11):

1351-6.

Rao, A. A. and Dumas, G. A. (1991). "Influence of material properties on the

mechanical behaviour of the L5-S1 intervertebral disc in compression: a nonlinear

finite element study." J Biomed Eng 13(2): 139-51.

Reid, J. E., Meakin, J. R., Robins, S. P., Skakle, J. M. and Hukins, D. W. (2002).

"Sheep lumbar intervertebral discs as models for human discs." Clin Biomech (Bristol,

Avon) 17(4): 312-4.

Reish, N. E. and Girty, G. H. (2001). "Visualizing Stress and Strain: Using Visual

Basic 6.0 to Provide a Dynamic Learning Environment." [Web document] Available:

http://www.geology.sdsu.edu/visualstructure/vss/htm_hlp/intro.htm [12th Aug., 2002]


Bibliography 373

Rissanen, P. M. (1960). "The Surgical Anatomy and Pathology of the Supraspinous

and Interspinous Ligaments of the Lumbar Spine with Special Reference to Ligament

Ruptures." Acta Orthop Scand (Suppl) 46.

Rivlin, R. S. (1984). Forty Years of Non-Linear Continuum Mechanics. IX

International Congress on Rheology, Acapulco, Mexico, Universidad Nacional

Autonoma De Mexico.

Roberts, C. S., Voor, M. J., Rose, S. M. and Glassman, S. D. (1998). "Spinal ligament

loading during axial distraction: a biomechanical model." Am J Orthop 27(6): 434-40.

Rolander, S. D. (1966). "Motion of the Lumbar Spine with Special Reference to the

Stabilizing Effect of Posterior Fusion." Acta Orthop Scand 90(S): 1-144.

Rubin, M. B. and Bodner, S. R. (2002). "A Three-Dimensional Nonlinear Model for

Dissipative Response of Soft Tissue." International Journal of Solids and Structures

(Article in press, uncorrected proof used).

Salomon, O., Oller, S. and Barbat, A. (1999). "Finite element analysis of base isolated

buildings subjected to earthquake loads." International Journal for Numerical

Methods in Engineering 46(10): 1741-61.

Sato, K., Kikuchi, S. and Yonezawa, T. (1999). "In vivo intradiscal pressure

measurement in healthy individuals and in patients with ongoing back problems."

Spine 24(23): 2468-74.

Schendel, M. J., Wood, K. B., Buttermann, G. R., Lewis, J. L. and Ogilvie, J. W.

(1993). "Experimental measurement of ligament force, facet force, and segment

motion in the human lumbar spine." J Biomech 26(4-5): 427-38.

Schmidt, T. A., An, H. S., Lim, T, Nowicki, B. H., Haughton, V. M. (1998). "The

Stiffness of Lumbar Spinal Motion Segments with a High-Intensity Zone in the

Anulus Fibrosus." Spine 23(20): 2167-2173.

Schultz, A. B., Warwick, D. N., Berkson, M. H. and Nachemson, A. L. (1979).

"Mechanical Properties of Human Lumbar Spine Motion Segments - Part I:


Bibliography 374

Responses in Flexion, Extension, Lateral Bending and Torsion." Transactions of the

ASME 101(Feb.): 46-52.

Shah, J. S., Jayson, M. I. V. and Hampson, W. G. J. (1979). "Mechanical Implications

of Crimping in Collagen Fibres of the Human Spinal Ligaments." Engineering in

Medicine 8: 95-102.

Shirazi-Adl, A., Ahmed, A. M. and Shrivastava, S. C. (1986). "A Finite Element

Study of a Lumbar Motion Segment Subjected to Pure Sagittal Plane Moments."

Journal of Biomechanics 19(4): 331-350.

Shirazi-Adl, A., Ahmed, A. M. and Shrivastava, S. C. (1986). "Mechanical Response

of a Lumbar Motion Segment in Axial Torque Alone and Combined with

Compression." Spine 11(9): 914-927.

Shirazi-Adl, A., Ahmed, A. M. and Shrivastava, S. C. (1986). "Mechanical response

of a lumbar motion segment in axial torque alone and combined with compression."

Spine 11(9): 914-27.

Shirazi-Adl, A. and Drouin, G. (1987). "Load-Bearing Role of Facets in a Lumbar

Segment Under Sagittal Plane Loadings." Journal of Biomechanics 20(6): 601-613.

Shirazi-Adl, A., Shrivastava, S. C. and Ahmed, A. M. (1984). "Stress Analysis of the

Lumbar Disc-Body Unit in Compression - A three dimensional nonlinear finite

element study." Spine 9(2): 120-134.

Simon, B. R., Wu, J. S. S., Carlton, M. W., Evans, J. H. and Kazarian, L. E. (1985).

"Structural Models for Human Spinal Motion Segments Based on a Poroelastic View

of the Intervertebral Disk." Journal of Biomechanical Engineering 107(Nov.): 327 -

335.

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): 1310-1319.


Bibliography 375

Tang, D., Yang, C., Huang, Y. and Ku, D. N. (1999). "Wall stress and strain analysis

using a three-dimensional thick-wall model with fluid-structure interactions for blood

flow in carotid arteries with stenoses." Computers and Structures 72(1): 341-56.

Thompson, R. E. (2002). Mechanical Effects of Degeneration in Lumbar

Intervertebral Discs. School of Mechanical, Manufacturing and Medical Engineering.

Brisbane, Queensland University of Technology.

Thompson, R. E., Barker, T., M., and J., P. M. (2003). "Defining the Neutral Zone of

Sheep Intervertebral Joints During Dynamic Motions: an in vitro study." Clinical

Biomechanics 18: 89-98.

Thompson, R. E., Pearcy M. J., Downing K. J., Manthey B. A., Parkinson I. H. and

N.L., F. (2000). "Disc lesions and the mechanics of the intervertebral joint complex."

Spine. 25(23): 3026-3035.

Tibrewal, S. B. and Pearcy, M. J. (1985). "Lumbar Intervertebral Disc Heights in

Normal Subjects and Patients with Disc Herniation." Spine 10(5): 452-454.

Tkaczuk, H. (1968). "Tensile Properties of Human Lumbar Longitudinal Ligaments."

Acta Orthop Scand (Suppl) 115.

Treloar, L. R. G. (1975). The Physics of Rubber Elasticity. London, Oxford

University Press.

Tschoegl, N. W. (1971). "Constitutive Equations for Elastomers." Journal of Polymer

Science, A1: 1959-1970.

Twizell, E. H. and Ogden, R. W. (1983). "Non-Linear Optimization of the Material

Constants in Ogden's Stress-Deformation Function for Incompressible Isotropic

Elastic Materials." Journal of Australian Mathematical Society. Series B 24: 424-434.

Ueno, K. and Liu, Y. K. (1987). "A Three-Dimensional Nonlinear Finite Element

Model of Lumbar Intervertebral Joint in Torsion." Transactions of the ASME 109:

200-209.


Bibliography 376

Ugural, A. C. and Fenster, S. K. (1995). Advanced Strength and Applied Elasticity.

New Jersey, Prentice Hall PTR.

Vernon-Roberts, B. (1988). Disc Pathology and Disease States. The Biology of the

Intervertebral Disc. P. Ghosh. Florida, USA, CRC Press, Inc. 2: 73-120.

Vernon-Roberts, B., Fazzalari, N. and Manthey, B. A. (1997). "Pathogenesis of tears

of the anulus investigatedby multiple-level transaxial analysis of the T12-L1 disc."

Spine 22(22): 2641-6.

Veronda, D. R. and Westmann, R. A. (1970). "Mechanical Characterisation of Skin -

finite deformations." Journal of Biomechanics 3: 111-124.

Viidik, A. (1973). "Functional Properties of Collagenous Tissues." International

Reviews of Connective Tissue Research 6: 127-215.

Virgin, W. J. (1951). "Experimental Investigations into the Physical Properties of the

Intervertebral Disc." Journal of Bone and Joint Surgery 33B(4): 607-611.

Vossoughi, J. (1995). "Determination of Mooney Material Constants for Highly

Noninear Isotropic INcompressibly Materials Under Large Elastic Deformations."

Experimental Techniques 19(2): 24-27.

Weiss, J. A., Bonifasi-Lista, C. and Gardiner, J. C. (2001). Determination of Ligament

Shear Properties Using a Finite Element Parameter Estimation Technique. BED-

Bioengineering Conference, ASME, ASME.

White, A. A. and Panjabi, M. M. (1978). Clinical Biomechanics of the Spine.

Philadelphia, Lippincott.

Wilke, H.-J., Neef, P., Caimi, M., Hoogland, T. and Claes, L. E. (1999). "New In Vivo

Measurements of Pressures in the Intervertebral Disc in Daily Life." Spine 24(8): 755-

762.

Williams, J. G. (1973). Stress analysis of polymers. London, Longman group.

Wu, H. and Yao, R. (1976). "Mechanical Behavior of the Human Annulus Fibrosus."

Journal of Biomechanics 9: 1-7.


Bibliography 377

Wu, J. and Chen, J. (1996). "Clarification of the mechanical behaviour of spinal

motion segments through a three-dimensional poroelastic mixed finite element

model." Medical Engineering and Physics 18(3): 215-24.

Yamada, H. (1970). Strength of Biological Materials. USA, The Williams and

Wilkins Company.

Yoshizawa, H., O'Brien, J. P., Thomas-Smith, W. and Trumper, M. (1980). "The

Neuropathology of Intervertebral Discs Removed for Low-Back Pain." Journal of

Pathology 132: 95-104.

Yu, J., Peter, C., Roberts, S. and Urban, J. P. (2002). "Elastic fibre organization in the

intervertebral discs of the bovine tail." J Anat 201(6): 465-75.

Zienkiewicz, O. C. (1980). The Finite Element Method, McGraw-Hill.

Zobitz, M. E., Luo, Z. and An, K. (2001). "Determination of the Compressive

Material Properties of the Supraspinatus Tendon." Journal of Biomechanical

Engineering 123(1): 47-51.

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