Abstract
Introduction
Animal models are often used to make the transition from scientific concepts to clinical applications. The sheep model has emerged as an important model in spine biomechanics. Although there are several experimental biomechanical studies of the sheep cervical spine, only a limited number of computational models have been developed. Therefore, the objective of this study was to develop and validate a C2-C7 sheep cervical spine finite element (FE) model to study the biomechanics of the normal sheep cervical spine.
Methods
The model was based on anatomy defined using medical images and included nonlinear material properties to capture the high flexibility and large neutral zone of the sheep cervical spine. The model was validated using comprehensive experimental flexibility testing. Ten adult sheep cervical spines, from C2-C7, were used to experimentally ascertain overall and segmental flexibility to ±2 Nm in flexion-extension, lateral bending, and axial rotation.
Results
The ranges of motion predicted by the computational model were within one standard deviation of the respective experimental motions throughout the load cycle, with the exception of extension and lateral bending. The model over- and under predicted the peak motions in extension and lateral bending, respectively. Nevertheless, the model closely represents the range of motion and flexibility of the sheep cervical spine.
Discussion
This is the first multilevel model of the sheep cervical spine. The validated model affords additional biomechanical insight into the intact sheep cervical spine that cannot be easily determined experimentally. The model can be used to study various surgical techniques, instrumentation, and device placement, providing researchers and clinicians insight that is difficult, if not impossible, to gain experimentally.
Introduction
Animal models are essential for making the transition from scientific concepts to clinical applications1,2. Such models have proven valuable for spinal research3,4, providing insight into fusion techniques. The cervical spine of sheep, for example, compare favorably with that of human, exhibiting similarities in both vertebral geometry and lordotic curvature5. Although anatomic similarities are important, biomechanical correspondence is imperative to understand the effects of disorders, surgical techniques, and implant designs.
Several studies have focused on the biomechanics of the sheep cervical spine, utilizing both functional spinal units3,4,6 and multilevel specimen5,7. Such studies highlight the flexibility of the sheep cervical spine, especially in lateral bending (approximately ±65°). The sheep cervical spine also exhibits a large neutral zone, accounting for 50% to 75% of the total motion7. These studies provided details regarding the external biomechanics (i.e. motion, stiffness), however, oftentimes internal biomechanics (i.e. stresses, strains) are desired as well.
Finite element (FE) models afford the ability to study internal biomechanics in response to a given external stimulus. Consequently, to better understand spinal biomechanics, FE analyses are often performed. Several studies have focused on the human cervical spine8-11. To our knowledge there is only one study using FE analysis to study the sheep cervical spine, and it was limited to the C3-C4 level12. Since the sheep is often used for in vivo studies13-15, it is important to have a comprehensive understanding of both external and internal biomechanical parameters. Additionally, it is essential to model the multilevel spine as opposed to single levels because oftentimes in vivo studies focus on multiple levels. Therefore, the objective of this study was to develop and validate a C2-C7 sheep cervical spine FE model to study the biomechanics of the normal sheep cervical spine. Since the sheep spine is highly flexible, the focus of the validation study was on the large range of motion and neutral zone (external biomechanics).
Materials and Methods
Model Development
A detailed, geometrically accurate FE model of the C2- C7 sheep spine was created using IA-FEMesh16 coupled with custom-written tools devoted to modeling the spine17 (Figure 1). The vertebral surfaces were defined using CT data (Siemens Sensation 64 CT scanner, slice thickness 0.6mm, 0.5mm in-plane resolution) by manually segmenting the regions of interest, similar to the methods described previously by DeVries and colleagues18. The vertebrae were modeled using eight-noded hexahedral elements with an average element length of 1 mm, comparable to the mesh density reported by Kallemeyn19, resulting in 8640 elements for each vertebral body and 6,956 (C2) to 14,630 (C6) elements in the posterior region.
Figure 1. The C2-C7 FE model incorporating the detailed intervertebral disc and the five major ligaments (anterior and posterior longitudinal ligaments, ligamentum flavum, interspinous ligaments, and capsular ligaments).
The mesh definitions for the intervertebral discs were defined by interpolating between the nodes of the adjacent vertebral bodies; the discs were divided into the nucleus and the surrounding fiber-reinforced annulus. Due to its nearly incompressible nature, the nucleus was modeled using fluid elements. The annulus grounds were modeled using eight-noded hexahedral elements with embedded rebar elements to mimic the annulus fibers. The rebar elements were angled at ±25° with alternating directions for each layer. The annulus was further divided into the anterior, posterior, and lateral regions to account for region-dependent variability in the material properties of the disc.
The five major ligaments of the cervical spine (anterior longitudinal ligament, posterior longitudinal ligament, ligamentum flavum, interspinous ligament, and capsular ligaments) were modeled using two-noded truss elements in tension only. The anterior and posterior longitudinal ligaments were defined to span from mid-body to adjacent mid-body. The ligamentum flavum was defined between adjacent lamina; the interspinous ligaments spanned between adjacent spinous processes. The capsular ligaments were defined between corresponding facets, angled to mimic the physiological orientation. Each ligament element was assigned a cross-sectional area so that the total ligament area represented the physiological cross section reported by DeVries20 (Table 1).
Table 1.
The Young's Modulus and crosssectional areas for each spinal ligament
| Young's Modulus (MPa) | Cross-Sectional Area (mm2) | ||
|---|---|---|---|
| ALL | C23 | 20.6 (<11%), 25.06 (>11%) | 12.78 |
| C34 | 5.77 (<11%), 12.84 (>11%) | 14.42 | |
| C45 | 8.95 (<11%), 12.94 (>11%) | 12.49 | |
| C56 | 17.428 (<11%), 20.62 (>11%) | 9.19 | |
| C67 | 0.86 (<11%), 16.496 (>11%) | 11.61 | |
| PLL | C23 | 16.73 (<11%), 59.94 (>11%) | 6.92 |
| C34 | 1.62 (<27%), 28.264 (>27%) | 7.46 | |
| C45 | 0.33 (<11%), 50.61 (>11%) | 12.49 | |
| C56 | 1.65 (<27%), 11.10 (>27%) | 5.86 | |
| C67 | 0.68 (<27%), 7.21 (>27%) | 4.48 | |
| CL | C23 | 4.08 (<5%), 56.88 (>5%) | 84.97 |
| C34 | 2.87 (<32%), 33.19 (>32%) | 109.06 | |
| C45 | 2.55 (<37%), 34.91 (>37%) | 92.55 | |
| C56 | 1.88 (<27%), 50.61 (>27%) | 77.35 | |
| C67 | 1.24 (<27%), 23.51 (>27%) | 72.73 | |
| IS | C23 | 3.04 (<11%), 68.03 (>11%) | 32.74 |
| C34 | 1.22 (<15%), 58.74(>15%) | 28.98 | |
| C45 | 0.82 (<27%), 33.23 (>27%) | 23.86 | |
| C56 | 1.37 (<27%), 52.05 (>27%) | 15.08 | |
| C67 | 2.76 (<37%), 41.08 (>37%) | 16.53 | |
| LF | C23 | >52.64 (<5%), 120.43 (>5%) | 17.22 |
| C34 | 1.89 (<27%), 40.78 (>27%) | 24.15 | |
| C45 | 2.65 (<37%), 52.23 (>37%) | 22.76 | |
| C56 | 1.03 (<27%), 100.83 (>27%) | 9.95 | |
| C67 | 2.22 (<37%), 70.90 (>37%) | 14.13 | |
Additionally, the facet joints were defined as finite-sliding surface interactions. The cartilage was represented by an exponential pressure-overclosure relationship, mimicking a uniform cartilage layer8,21,22. The relationship was defined such that as the distance between the facets decreases the contact pressure increases, eventually reaching the modulus of the posterior bone.
Material Properties
Since bone is significantly stiffer than soft tissues and the main focus of this study was to capture the overall motion, the bone material properties were simplified as homogenous isotropic material. Note, if bone stress and strains are of particular importance, specimen-specific material properties could be easily incorporated using the relationship between the CT Hounsfield value and bone density. For this study, the vertebral body was subdivided into cancellous and cortical regions, where the cortical bone was defined as the outermost layer of elements. Linear elastic material properties were incorporated for all the bony anatomy. The material properties are summarized in Table 2.
Table 2.
The material properties used to model the bone and soft tissues. The bony material properties are based on human material properties taken from literature.
| Material | Young's Modulus (MPa) | Poisson's Ratio | Reference |
|---|---|---|---|
| Cortical Bone | 10000 | 0.30 | Kallemeyn et al.8 |
| Cancellous Bone | 450 | 0.25 | Kallemeyn et al.8 |
| Posterior Bone | 3500 | 0.25 | Kallemeyn et al.8 |
| Ligaments | Nonlinear (Hypoelastic) | 0.30 | DeVries20 |
| Nucleus | Fluid | – | Kallemeyn et al.8 |
| Annulus Grounds | Nonlinear (Yeoh Hyperelastic) | 0.45 | Fujita el al.25, Ambrosetti-Giudici et al. 29 |
| Annulus Fibers | Nonlinear (Hypoelastic) | 0.30 | Kallemeyn et al.8 |
As previously mentioned, the nucleus was modeled as fluid elements to capture the nearly incompressible nature. The nonlinear nature of the annulus fibers was represented using hypoelastic material properties. The nonlinearity of the annulus grounds was captured using the Yeoh hyperelastic function, similar to the technique used to model the human lumbar disc23. The Yeoh model defines the strain energy function as follows:
where C10, C20, C30, and D1 are the material coefficients, Ī1 is the first invariant of the deviatoric component of the Cauchy-Green strain tensor, and Jel is the elastic volume ratio. The elastic volume ratio accounts for the compressibility of the material and is determined based on the Poisson's ratio.24
The annulus material properties were based on the stress-strain curve reported by Fujita and colleagues25 for the human lumbar intervertebral disc. This curve was used as a baseline model that was then altered to depict varying levels of stiffness. The material properties for each disc were determined using the experimental flexibility data described in a later section. Thus, for more flexible levels (i.e., C5-C6 and C6-C7), the stiffness was decreased as compared to the baseline properties, and vice versa for levels with minimal motion (i.e., C2-C3). Yeoh material coefficients for the various curves were determined using Abaqus CAE (SIMULIA, Providence, RI). Table 3 summarizes the material coefficients for the annulus.
Table 3.
The material coefficients for the disc material properties at each level
| Disc | Yeoh Coefficients | ||||
|---|---|---|---|---|---|
| Region | Level | C10 | C20 | C30 | D1 |
| Anterior | C23 | 5.508E-2 | −8.880E-1 | 24.804 | 1.878 |
| C34 | 2.911E-2 | −1.351E-1 | 1.014 | 3.554 | |
| C45 | 1.318E-2 | −1.271E-2 | 1.822E-2 | 7.849 | |
| C56 | 8.978E-3 | −3.877E-3 | 2.483E-3 | 11.523 | |
| C67 | 9.997E-3 | −5.437E-3 | 4.364E-3 | 10.348 | |
| Posterior | C23 | 5.059E-2 | −4.025 | 169.767 | 1.749 |
| C34 | 1.785E-2 | −4.770E-1 | 13.175 | 5.796 | |
| C45 | 1.785E-2 | −4.770E-1 | 13.175 | 5.796 | |
| C56 | 8.430E-3 | −7.532E-3 | 6.257E-3 | 12.271 | |
| C67 | 9.997E-3 | −5.437E-3 | 4.364E-3 | 10.348 | |
| Lateral | C23 | 2.911E-2 | −1.351E-1 | 1.014 | 3.554 |
| C34 | 1.134E-2 | −8.020E-3 | 8.384E-3 | 9.124 | |
| C45 | 8.978E-3 | −3.879E-3 | 2.483E-3 | 11.523 | |
| C56 | 7.513E-3 | −2.198E-3 | 9.681E-4 | 13.770 | |
| C67 | 8.170E-3 | −2.876E-3 | 1.510E-3 | 12.662 | |
The nonlinearity of the ligaments was captured using hypoelastic material properties8,17,26, based on the sheep cervical spine ligament stress-strain curves reported by DeVries20. Table 1 summarizes the ligament properties.
Boundary Conditions
To mimic the experimental testing conditions described below, the nodes of the inferior endplate of C7 were fixed in all directions. A nondestructive, physiologic moment of 2.5 Nm was applied at the superior endplate of C2 using a rigid surface. Moments were applied in flexion (+), extension (−), right (+) and left (−) lateral bending, and right (−) and left (+) axial rotation. At C2, the model was unconstrained in the remaining uncontrolled degrees of freedom. The finite element software Abaqus 6.11 (Dassault Systèmes Simulia, Providence, RI) was used to perform the analyses. The spinal motions throughout the entire loading curves were analyzed.
Experimental Biomechanics for FE Validation
Experimental flexibility tests were performed on ten adult Suffolk sheep cervical (C2-C7) specimens to determine the average moment-rotation curves for flexion-extension, lateral bending, and axial rotation. The specimens were tested to ±2.5 Nm at a rate of 5.0 Nm/minute3. Specimens underwent three loading and unloading cycles (the first two served as specimen preconditioning) with data captured on the third cycle. Refer to the study by DeVries et al.7 for detailed information regarding these methods. For model validation, the predicted FE motions were compared to the average experimental motions for the complete loading curve, focusing on the entire spinal segment (C2-C7) as well as each individual spinal level.
Results
The finite element model successfully simulated the nonlinear moment-rotation behaviors observed experimentally in each of the loading directions. In general, the range of motion predictions were within one standard deviation of the sample mean throughout the load cycle (Figure 2). For all loading directions, the motion was within one standard deviation of the loading curve up to 1.0 Nm. During extension, after 1.8 Nm the model overpredicted the motion; at −2.5 Nm the model-predicted motion was 6.6° greater than the average experimental motion (−39.82° ± 4.08°). The FE model under predicted lateral bending after 1.0 Nm. At ±2.5 Nm, the model-predicted motion was about 16° less than the mean experimental motion (62.94° ± 9.61°) for right lateral bending; for left lateral bending, the model-predicted motion was approximately 20° less than the experimental motion (−67.04° ± 9.62°). For axial rotation (±30°) and flexion (37°), the motion matched the experimental results well, including at moments greater than 1.0 Nm.
Figure 2. The C2-C7 finite element predicted motion compared to the averaged experimental moment-rotation curves for flexion (+) and extension (−), right (+) and left (−) lateral bending, and right (−) and left (+) axial rotation.
The FE ranges of motion at ±2.5 Nm for each vertebral level are summarized in Figure 3. The model matches well in flexion, and was within one stand deviation of the experimental motion at each level. The model overpredicted extension at level C2-C3 and C5-C6. Similar to the C2-C7 trend, the model underpredicted the motion at C2-C3, C4-C5 and C5-C6 during right lateral bending and at C4-C5, C5-C6 and C6-C7 during left lateral bending. For axial rotation, the model underpredicted at the C5-C6 and C6-C7 levels and overpredicted at the C2-C3 level.
Figure 3. The finite element model predicted motion compared to the experimental range of motion at 2.5 Nm for flexion (+) and extension (−), right (+) and left (−) lateral bending, and right (−) and left (+) axial rotation.
Discussion
To date, there has only been one finite element study focusing on the sheep cervical spine12; the study was limited to a functional spinal unit (C3-C4). However, it is important to study the multilevel spine in order to have a more detailed comparison between the sheep and human cervical spine. Since sheep are often used for in vivo studies13-15, it is important to have a comprehensive understanding of both external (i.e., motion) and internal (i.e., bone stress and strains, disc pressures, facet contact, etc.) responses for the sheep. To address this, we developed and validated the first multilevel model of the sheep cervical spine. Overall, the model compared well with experimental studies, thus providing a method for researchers and clinicians to study the biomechanical effects of various surgical procedures and device placement. Details of these findings are further discussed.
There are several benefits of FE analyses. First, it allows researchers to study several scenarios using one model. Due to the cost of cadavers, researchers are looking to other methods to obtain relevant information. Animal models are one way of doing this; however, animal testing can be expensive as well, depending on the species and the number of specimen needed. Finite element analysis allows researchers to study the effects of different scenarios by modifying the computational model as opposed to testing several specimens. Another advantage of finite element analyses is that computer simulations can be done without the expense of fabricating and testing multiple prototypes, saving significant time and money. Using an FE model, design changes can be made and a new analysis rerun within a short period of time. In addition, FE analyses provide thorough information that physical testing cannot provide. For example, stress and stain can be obtained at any location in the finite element model; it could be prohibitively expensive or impossible to collect all of this data in an experimental test. As an example, Figure 4 shows the von Mises stress distribution in the sheep cervical spine at 2.5 Nm of flexion. That being said, an FE model relies on experimental studies for validation. Once validated experimentally, FE models provide a powerful design tool.
Figure 4. The von Mises stress distribution at 2.5 Nm of flexion.
Overall, the model corresponds well with the experimental data, capturing the high flexibility. Although the model over- and underpredicted some of the peak ranges of motion, the model compared well throughout the majority of the loading curve. The model predicted the C2-C7 motion during flexion and axial rotation very well (within one standard deviation). The model overpredicted extension and underpredicted lateral bending; however, due to the highly nonlinear behavior seen experimentally it is difficult to determine material properties that can account for the large neutral zone while still capturing the elastic zone. Additionally, the material properties must accommodate the motions in all six directions, thus accounting for different levels of stiffness (i.e., axial rotation vs lateral bending).
The experimental data was shifted such that the neutral position was calculated as the mid-point of the neutral zone and centered about zero. The neutral position (i.e., starting position) is different for each loading direction (i.e., flexion-extension, lateral bending, and axial rotation). However, the FE model assumes the same neutral position for each loading direction, which could account for the discrepancies between the model-predicted motions and experimental motions.
To more accurately predict all motions, future studies should focus on determining the intervertebral disc material properties at the each spinal level in addition to any regional variations within the levels. The annulus properties of the current model were based on variations of the stress-strain curves for the human lumbar spine25. Although this curve was adjusted to capture the more flexible nature of the sheep cervical spine, experimental testing of sheep intervertebral discs would provide species-specific properties to better define the annulus grounds and fibers.
Additionally, determining the annulus fiber orientation and material properties on a regional basis would be beneficial. Currently, the model incorporates the same fiber angle throughout the entire annulus. Previous studies of the human lumbar spine27, 28 have reported that the fiber orientation and material properties vary between annular layers (i.e., inner versus outer) as well as annulus regions (i.e., posterior versus anterior). This may be true for the sheep intervertebral disc as well. The regional differences were taken into consideration with the region dependent annulus ground material properties, but in the future this should be extended to the annular layers as well.
Overall, this is the first multilevel finite element model of the sheep cervical spine. The FE model predicted the large neutral zone and captured the high flexibility shown experimentally. The model affords additional biomechanical insight into the intact sheep cervical spine that cannot be easily determined experimentally. The model can provide stress distributions for the given loading conditions and can be used to predict regions of high stress concentration in the bone, facets, and intervertebral discs. Additionally, this validated model can be used to study changes in disc pressures and facet contact, as well as the effects of various surgical techniques and material properties of new implant designs.
References
- 1.Singh K, Masuda K, An H. Animal models for human disc degeneration. The Spine Journal. 2005;5:267S–279S. doi: 10.1016/j.spinee.2005.02.016. [DOI] [PubMed] [Google Scholar]
- 2.Sheng S, Wang X, Xu H, et al. Anatomy of large animal spines and its comparison to the human spine: a systematic review. European Spine Journal. 2010;19:46–56. doi: 10.1007/s00586-009-1192-5. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 3.Wilke H, Kettler A, Claes L. Are sheep spines a valid biomechanical model for human spines? Spine. 1997;22:2365–2374. doi: 10.1097/00007632-199710150-00009. [DOI] [PubMed] [Google Scholar]
- 4.Kandziora F, Pflugmacher R, Scholz M, et al. Comparison between sheep and human cervical spines. Spine. 2001;26:1028–1037. doi: 10.1097/00007632-200105010-00008. [DOI] [PubMed] [Google Scholar]
- 5.Szotek S, Szust A, Pezowicz C, et al. Animal models in biomechanical spine investigations. Bulletin of the Veterinary Institute in Pulawy. 2004;48:163–168. [Google Scholar]
- 6.Clarke E, Appleyard R, Bilston L. Immature sheep spines are more flexible than mature spines: an in vitro biomechanical study. Spine. 2007;32:2970–2979. doi: 10.1097/BRS.0b013e31815cde16. [DOI] [PubMed] [Google Scholar]
- 7.DeVries N, Gandhi A, Fredericks D, et al. Biomechanical analysis of the intact and destabilized sheep cervical spine. Spine. 2012;37:E957–E963. doi: 10.1097/BRS.0b013e3182512425. [DOI] [PubMed] [Google Scholar]
- 8.Kallemeyn N, Gandhi A, Kode S, et al. Validation of a C2-C7 cervical spine finite element model using specimen-specific flexibility data. Medical Engineering & Physics. 2010;32:482–489. doi: 10.1016/j.medengphy.2010.03.001. [DOI] [PubMed] [Google Scholar]
- 9.Yoganandan N, Kumaresan S, Pintar F. Biomechanics of the cervical spine part 2. Cervical spine soft tissue responses and biomechanical modeling. Clinical Biomechanics. 2001;16:1–27. doi: 10.1016/s0268-0033(00)00074-7. [DOI] [PubMed] [Google Scholar]
- 10.del Palomar A, Calvo B, Doblare M. An accurate finite element model of the cervical spine under quasi-static loading. Journal of Biomechanics. 2008;41:523–531. doi: 10.1016/j.jbiomech.2007.10.012. [DOI] [PubMed] [Google Scholar]
- 11.Wheeldon J, Stemper B, Yoganandan N, et al. Validation of a finite element model of the young normal lower cervical spine. Annals of Biomedical Engineering. 2008;36:1458–1469. doi: 10.1007/s10439-008-9534-8. [DOI] [PubMed] [Google Scholar]
- 12.Kiapour A, Khere A, Dennis M, et al. San Francisco, CA: Orthopaedic Research Society; 2008. Biomechanics of C3-C4 sheep cervical spine compared to human: an experimentally validated FE study. [Google Scholar]
- 13.Thompson R, Barker T, Pearcy M. Defining the neutral zone of sheep intervertebral joints during dynamic motions: an in vitro study. Clinical Biomechanics. 2003;18:89–98. doi: 10.1016/s0268-0033(02)00180-8. [DOI] [PubMed] [Google Scholar]
- 14.Slivka M, Spenciner D, Seim H, et al. High rate of fusion in sheep cervical spines following anterior interbody surgery with absorbable and nonabsorbable implant devices. Spine. 2006;31:2772–2777. doi: 10.1097/01.brs.0000245935.69927.a1. [DOI] [PubMed] [Google Scholar]
- 15.Kandziora F, Pflugmacher R, Scholz M, et al. Bioabsorbable interbody cages in a sheep cervical spine fusion model. Spine. 2004;29:1845–1855. doi: 10.1097/01.brs.0000137060.79732.78. [DOI] [PubMed] [Google Scholar]
- 16.Grosland N, Shivanna K, Magnotta V, et al. IAFEMesh: An open-source, interactive, multiblock approach to anatomic finite element model development. Computer Methods and Programs in Biomedicine. 2009;94:96–107. doi: 10.1016/j.cmpb.2008.12.003. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 17.Kallemeyn N, Tadepalli S, Shivanna K, et al. An interactive multiblock approach to meshing the spine. Computer Methods and Programs in Biomedicine. 2009;95:227–235. doi: 10.1016/j.cmpb.2009.03.005. [DOI] [PubMed] [Google Scholar]
- 18.DeVries N, Gassman E, Kallemeyn N, et al. Validation of phalanx bone three-dimensional surface segmentation from computed tomography images using laser scanning. Skeletal Radiology. 2008;37:35–42. doi: 10.1007/s00256-007-0386-3. [DOI] [PubMed] [Google Scholar]
- 19.Kallemeyn NA. The University of Iowa: Doctoral Dissertation; 2009. Toward patient-specific finite element mesh development and validation of the cervical spine. [Google Scholar]
- 20.DeVries N. The University of Iowa: Doctoral Dissertation; 2011. The biomechanics of the sheep cervical spine: an experimental and finite element analysis. [Google Scholar]
- 21.Faizan A, Goel V, Garfin S, et al. Do design variations in the artificial disc influence cervical spine biomechanics? A finite element investigation. European Spine Journal. 2009 doi: 10.1007/s00586-009-1211-6. [DOI] [PMC free article] [PubMed] [Google Scholar]
- 22.Sasa T, Yoshizumi Y, Imada K, et al. Cervical spondylolysis in a judo player: A case report and biomechanical analysis. Archives of orthopaedic and trauma surgery. 2009;129:559–567. doi: 10.1007/s00402-008-0609-7. [DOI] [PubMed] [Google Scholar]
- 23.Ayturk U, Garcia J, Puttlitz C. The micromechanical role of the annulus fibrosus components under physiological loading of the lumbar spine. Journal of Biomechanical Engineering. 2010;132:61007–1. doi: 10.1115/1.4001032. [DOI] [PubMed] [Google Scholar]
- 24. Abaqus Version 6.5. Analysis user's manual: Materials - hyperelastic behavior. In: Vol 3. 2004. [Google Scholar]
- 25.Fujita Y, Duncan N, Lotz J. Radial tensile properties of the lumbar annulus fibrosus are site and degeneration dependent. Journal of Orthopaedic Research. 1997;15:814–819. doi: 10.1002/jor.1100150605. [DOI] [PubMed] [Google Scholar]
- 26.Goel V, Clausen J. Prediction of load sharing among spinal components of a C5-C6 motion segment using the finite element approach. Spine. 1998;23:684–691. doi: 10.1097/00007632-199803150-00008. [DOI] [PubMed] [Google Scholar]
- 27.Schmidt H, Heuer F, Simon U, et al. Application of a new calibration method for a three-dimensional finite element model of a human lumbar annulus fibrosus. Clinical Biomechanics. 2006;21:337–344. doi: 10.1016/j.clinbiomech.2005.12.001. [DOI] [PubMed] [Google Scholar]
- 28.Chen S, Zhong Z, Chen C, et al. Biomechanical comparison between lumbar disc arthroplasty and fusion. Medical Engineering & Physics. 2009;31:244253. doi: 10.1016/j.medengphy.2008.07.007. [DOI] [PubMed] [Google Scholar]
- 29.Ambrosetti-Giudici S, Gedet P, Ferguson S, et al. Viscoelastic properties of the ovine posterior spinal ligaments are strain dependent. Clinical Biomechanics. 2010;25:97–102. doi: 10.1016/j.clinbiomech.2009.10.017. [DOI] [PubMed] [Google Scholar]