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Impact of material properties of intervertebral disc on dynamic response of the human lumbar spine to vertical vibration: a finite element sensitivity study

  • Li-Xin Guo
  • Wei Fan
Original Article
  • 26 Downloads

Abstract

This study aimed to determine the effect of variations in material properties of the intervertebral disc on dynamic response of the human lumbar spine to vertical vibration using a finite element model of the lumbar L1–S1 motion segment. The present material sensitivity study was conducted by varying elastic moduli for annulus ground substance (AGS), annulus fibers (AF), and nucleus pulposus (NP) in the disc. Transient dynamic analysis was performed initially on the model with basic material property under a sinusoidal vertical vibration load. Subsequently, the same analysis was done for each of the three disc components corresponding to high and low material property cases. The computed results were plotted as a function of time and compared. The AGS property displayed a larger impact on vertebral axial displacement and von Mises stress in AGS, and the AF property displayed a larger impact on disc bulge. In contrast, the NP property had little effect on all the response parameters. Additionally, the intradiscal pressure was found to be not sensitive to any of the disc properties. These findings may be helpful in adoption of appropriate material parameters for the intervertebral disc in finite element model of the lumbar spine used for vibration analysis.

Graphical abstract

Material property sensitivity analysis on vibration characteristics of the human lumbar spine.

Keywords

Lumbar spine Finite element model Vibration Material sensitivity Dynamic response 

1 Introduction

The finite element (FE) method is a very powerful tool for investigating the biomechanical behavior of the human spine. The method is frequently employed to quantify internal responses (e.g., strain and stress) of the spine, which is difficult or even impossible to achieve by mathematical models and experiments [12]. Geometry and material properties are the two most important parameters for FE modeling of the spine [19]. The accurate geometry of the spine can be procured from some well-known techniques such as computed tomography scans and digitizer. However, material properties of the spinal components used in many published FE models vary in a wide range due to variations in specimens [9]. The scatter of material properties is inherent, and it is impossible and not acceptable to completely eliminate [18]. Therefore, the material sensitivity analyses are needed as indicated by the literature [1, 35].

Numerous studies have reported the impact of the material properties on the spinal biomechanics [10, 13, 14, 22, 34]. For example, Kumaresan et al. [13] studied the effect of variations in material properties on output responses (angular rotation and stress) of the C4–C6 cervical spine under physiologic load vectors. The results indicated that material properties of the soft tissue components were more dominant in the output responses than the hard tissue components. Jebaseelan et al. [10] developed a FE model of the juvenile lumbar spine (L1–S1) to conduct a material sensitivity study by varying the elastic modulus for various spinal components and found that intervertebral disc material properties displayed the most impact on the output responses under compressive loading, whereas those of bone components impacted least. There is no doubt that these previous studies have provided a valuable insight into the material property sensitivity of the human spine under static loading modes (compression, bending, and torsion moments). However, very few sensitivity studies of the material properties have been conducted under whole-body vibration (WBV) loading that is typically present when driving vehicles [15, 17].

Biodynamic experiments have demonstrated that WBV may cause disruption of the intervertebral disc and thus play an important role in low-back pain and degenerative diseases of the human spine [2, 31, 33]. The intervertebral disc is the most critical component in spine FE models as indeed it is in the natural spine, and therefore, its representation in the models is of crucial importance [4]. Accordingly, the present study was designed to investigate effect of the variations in material properties of the disc components on output dynamic responses of the lumbar spine under vibration loading using a 3D FE model of the L1–S1 motion segment.

2 Materials and methods

2.1 FE model

A previously developed and validated FE model of the human whole lumbar spine (L1–S1) subjected to a compressive follower preload was employed to serve as the basis for the present material sensitivity study. Material properties used in this basic model are listed in Table 1. Details for development and validation of the model were reported elsewhere [5]. Six vertebrae, five intervertebral discs, and seven ligaments were included in the basic model, as shown in Fig. 1. Each vertebra consisted of a cancellous bone surround by a cortical shell (including endplate). Each intervertebral disc consisted of annulus ground substance (AGS), annulus fibers (AF), and nucleus pulposus (NP). The spinal ligaments including anterior longitudinal, posterior longitudinal, supraspinous, interspinous, flavum, intertransverse, and capsular ligaments were modeled as tension-only truss elements.
Table 1

Material properties used in the basic FE model

Component

Elastic modulus (MPa)

Poisson’s ratio

Cross-sectional area (mm2)

Density (kg/mm3)

References

Bone

 Cortical bone

12,000

0.3

 

1.7e−6

[30]

 Cancellous bone

100

0.2

 

1.1e−6

[30]

 Posterior bony elements

3500

0.25

 

1.4e−6

[30]

 Endplate

23.8

0.4

 

1.2e−6

[16]

Intervertebral disc

 Annulus ground substance

Hyperelastic, Mooney-Rivlin C10 = 0.18, C01 = 0.045

  

1.05e−6

[25]

 Nucleus pulpous

Hyperelastic, Mooney-Rivlin C10 = 0.12, C01 = 0.03

  

1.02e−6

[25]

Annulus fibers

    

[21]

 Outermost

550

0.3

0.7

1.0e−6

 

 Second

495

0.3

0.63

1.0e−6

 

 Third

440

0.3

0.55

1.0e−6

 

 Fourth

420

0.3

0.49

1.0e−6

 

 Fifth

385

0.3

0.41

1.0e−6

 

 Innermost

360

0.3

0.30

1.0e−6

 

Ligaments

    

[11]

 Anterior longitudinal

7.8 (< 12.0%) 20.0 (> 12.0%)

 

63.7

1.0e−6

 

 Posterior longitudinal

10.0 (< 11.0%) 20.0 (> 11.0%)

 

20

1.0e−6

 

 Ligamentum flavum

15.0 (< 6.2%) 19.5 (> 6.2%)

 

40

1.0e−6

 

 Supraspinous

8.0 (< 20.0%) 15 (> 20.0%)

 

30

1.0e−6

 

 Interspinous

10.0 (< 14.0%) 11.6 (> 14.0%)

 

40

1.0e−6

 

 Intertransverse

10.0 (< 18.0%) 58.7 (> 18.0%)

 

1.8

1.0e−6

 

 Capsular

7.5 (< 25.0%) 32.9 (> 25.0%)

 

30

1.0e−6

 
Fig. 1

FE model of the human whole lumbar spine L1–S1

2.2 Sensitivity analyses

The material sensitivity analyses were conducted by varying elastic modulus for the AGS, AF, and NP in the intervertebral disc. The modulus for each disc component of the basic model as shown in Table 1 was increased and decreased by 25% to create high and low material property cases, respectively. In the basic model, the AGS and NP were modeled using the hyperelastic Mooney-Rivlin formulation, and the material coefficients C10 and C01 were defined using the following approximate equation for the elastic modulus E [25]:
$$ E\cong 6\left({C}_{10}+{C}_{01}\right),\kern0.36em \mathrm{with}\kern0.6em {C}_{01}\cong 0.25{C}_{10} $$
(1)

Therefore, the corresponding C10 and C01 coefficients for high and low material property cases can be determined by Eq. (1). In addition, it should be noted that for any particular disc component, it was decided to conduct the sensitivity analysis by varying elastic modulus of that component, maintaining the material properties of the other disc components at the basic values.

2.3 Vibration simulations

To mimic vibration loading of the human body, a sinusoidal vertical load of ± 40 N at a frequency of 5 Hz was imposed on superior surface of the L1 vertebra according to some previous studies [6, 8, 32]. To mimic the physiologic compressive load induced by muscle activities, a compressive preload of 400 N was applied to the model using the follower load technique (Fig. 1) [20, 23]. A mass point of 40 kg was added on the top of the model to include the effect of human upper body mass as suggested by Goel et al. [6], and the point was located on the superior surface of the L1 vertebra by 1 cm anterior to the L3–L4 vertebral centroid [27]. The caudal part of the sacrum was fully constrained.

Then, transient dynamic analyses were performed on the model using Abaqus/Standard 6.10 (Dassault Systèmes Simulia Corp., Providence, RI, USA). The first simulation was conducted on the basic model. Two additional simulations were done for each of the disc components corresponding to high and low material property cases. The output responses of the dynamic analyses, including vertebral axial displacement, disc bulge, intradiscal pressure, and von Mises stress in AGS, were considered. In this study, disc bulge was defined as deformation of the annulus in lateral direction, intradiscal pressure was defined as the average value of the pressures in the elements used to model the nucleus, and von Mises stress in AGS was defined as the average value of the stresses in the elements used to model the AGS.

3 Results

Figure 2 shows the output time domain dynamic response of the basic model using the material properties listed in Table 1. It was observed that plots of these output responses revealed a cyclic characteristic with time, and the response values in the five lumbar levels were different. During the vertical vibration, the vertebral axial displacement was decreased from L1 to L5, the disc bulge at L4–L5 and L5–S1 was larger than those at other levels, the intradiscal pressure differed moderately for different levels, and the highest von Mises stress in AGS appeared at L5–S1. Figures 3, 4, and 5 show the corresponding dynamic responses after varying elastic modulus values of the AGS, AF, and NP, respectively. It was found that variations in material properties of the disc components induced changes in the response values. The maximum response values for the basic, high, and low material property cases were provided in Table 2. For example, maximum axial displacement of the L1 vertebra was changed from − 5.53 mm (basic) to − 5.15 (high) and − 6.36 mm (low) due to variation of AGS material property; maximum disc bulge of the L2–L3 disc was changed from 0.740 mm (basic) to 0.666 mm (high) and 0.852 mm (low) due to variation of AF material property; maximum von Mises stress in AGS of the L3–L4 disc was changed from 0.192 MPa (basic) to 0.181 MPa (high) and 0.203 MPa (low) due to variation of NP material property; and maximum intradiscal pressure of the L4–L5 disc was changed from 0.576 MPa (basic) to 0.546 MPa (high) and 0.611 MPa (low) due to variation of AGS material property (Table 2).
Fig. 2

Dynamic response of the basic model under the sinusoidal vertical load and the compressive follower preload. a Vertebral axial displacement. b Disc bulge. c Intradiscal pressure. d von Mises stress in AGS

Fig. 3

Effect of the variation of AGS material property on these output dynamic responses. a, c, e, g High material property case. b, d, f, h Low material property case

Fig. 4

Effect of the variation of AF material property on these output dynamic responses. a, c, e, g High material property case. b, d, f, h Low material property case

Fig. 5

Effect of the variation of NP material property on these output dynamic responses. a, c, e, g High material property case. b, d, f, h Low material property case

Table 2

Maximum response values for the basic, high and low material property cases

   

AGS

AF

NP

Dynamic response

Vertebra/disc

Basic

High

Low

High

Low

High

Low

Vertebral axial displacement (mm)

L1

− 5.53

− 5.15

− 6.36

− 5.48

− 5.90

− 5.40

− 5.83

L2

− 4.51

− 4.18

− 5.21

− 4.47

− 4.81

− 4.40

− 4.75

L3

− 3.56

− 3.18

− 4.07

− 3.46

− 3.72

− 3.42

− 3.73

L4

− 2.60

− 2.32

− 2.98

− 2.52

− 2.71

− 2.51

− 2.71

L5

− 1.43

− 1.27

− 1.64

− 1.38

− 1.50

− 1.39

− 1.49

Disc bulge (mm)

L1–L2

0.889

0.838

0.941

0.818

0.995

0.857

0.924

L2–L3

0.740

0.710

0.763

0.666

0.852

0.709

0.774

L3–L4

0.680

0.653

0.701

0.611

0.784

0.650

0.713

L4–L5

1.084

1.006

1.162

1.008

1.199

1.040

1.132

L5–S1

1.144

1.091

1.211

1.058

1.294

1.110

1.198

Intradiscal pressure (MPa)

L1–L2

0.721

0.681

0.767

0.723

0.720

0.721

0.718

L2–L3

0.668

0.633

0.708

0.672

0.666

0.669

0.666

L3–L4

0.586

0.557

0.618

0.590

0.583

0.586

0.583

L4–L5

0.576

0.546

0.611

0.581

0.571

0.577

0.575

L5–S1

0.655

0.632

0.704

0.668

0.654

0.655

0.653

Von Mises stress in AGS (MPa)

L1–L2

0.220

0.240

0.194

0.213

0.230

0.209

0.231

L2–L3

0.211

0.232

0.185

0.205

0.221

0.200

0.223

L3–L4

0.192

0.211

0.166

0.185

0.201

0.181

0.203

L4–L5

0.220

0.242

0.191

0.214

0.229

0.208

0.232

L5–S1

0.254

0.287

0.223

0.249

0.268

0.244

0.267

Figure 6 illustrates the percent changes in maximum values of the output dynamic responses associated with variations of the respective material properties of each disc components. The results indicated that variations in material properties of the different disc components produced dissimilar changes in the response results. The AGS property displayed a larger impact on the vertebral axial displacement (− 11.2 to 15.5%) and von Mises stress in AGS (− 13.5 to 13.0%). The disc bulge was found to be most sensitive to the AF property (− 10.1 to 15.1%), and this was followed by variations of AGS (− 8.0 to 7.2%). In contrast, variations in NP property had little effect on all these response parameters (− 3.9 to 5.4% for the vertebral axial displacement, − 4.4 to 4.9% for the disc bulge, − 0.51 to 0.17% for the intradiscal pressure, − 5.7 to 5.7% for the von Mises stress in AGS). In addition, it was found that the intradiscal pressure was not sensitive to any of the disc properties (− 5.5 to 5.9% in the AGS study; − 0.87 to 2.0% in the AF study; − 0.51 to 0.17% in the NP study).
Fig. 6

Percent changes in maximum values of the output dynamic responses due to variations of the material properties of AGS, AF, and NP. a Vertebral axial displacement. b Disc bulge. c Intradiscal pressure. d von Mises stress in AGS

4 Discussion

There is a lack of studies investigating material property sensitivity of the human spine under WBV condition. This study evaluated effect of the variations in material properties of the intervertebral disc components (AGS, AF, and NP) on time domain dynamic response of the human lumbar spine to the vertical vibration using the FE method. A previous FE model of the whole lumbar spine (L1–S1) [5] was employed, and the model had been validated against the results of previously published studies [3, 7, 23, 29] under static and cyclic loading conditions.

The sinusoidal vertical load of ± 40 N at 5 Hz used to simulate vibration loading of the human body was applied to the top vertebra due to the fact that a study by Wilder [32] showed that when an person in a sitting posture was exposed to vertical vibration at 5 Hz, a cyclic axial load of about 40 N might be imposed on the top of the lumbar spine. Moreover, Goel et al. [6] and Guo et al. [8] had employed this vibration loading mode in their FE studies for predicting time domain dynamic response of the lumbar spine to the vertical vibration. The frequency of 5 Hz was adopted due to its correlation with human body vibration and many transport vehicles. A follower preload of 400 N was used to represent the physiologic compressive loading on the spine model due to the fact that for the whole lumbar spine, the follower load technique allows for the application of compressive loads of high physiologic magnitude (up to 2700 N) without generating instability [26].

Previous studies [10, 13, 22] have indicated that material properties of the soft tissue structures (e.g., intervertebral discs) have a preponderant effect on biomechanical responses of the spine compared with material properties of the hard tissue structures (e.g., cancellous core and cortical shell). Therefore, this study focused on analyzing material property sensitivity of the disc components under vibration. By comparison of these response plots obtained from the transient dynamic analysis (Figs. 2, 3, 4, and 5), it was found that variations in material properties of the AGS, AF, and NP had different effects on the output responses. In general, maximum response values of the vertebral axial displacement, intradiscal pressure, and von Mises stress in AGS were more sensitive to the AGS property than the AF and NP properties (Table 2; Fig. 6). This implies the AGS is more dominant than other disc components under the vertical vibration. The AGS also has the highest influence on the vertical resonant frequency and the static biomechanical responses of the spine as reported by Guo et al. [9] and Ng et al. [18], respectively. Maximum response value of the disc bulge was most sensitive to the AF property, and it implies that under the vertical vibration, variations in material property of the AF might have a more significant influence on horizontal stiffness of the intervertebral disc than the AGS and NP properties. In contrast, a FE study by Fegan et al. [4] found that the disc bulge was not sensitive to any of the material properties of the intervertebral disc under static loading conditions. The NP property had little effect on these predicted dynamic responses under the vertical vibration. This might be due to the fact that the NP was defined as incompressible (Poisson’s ratio nearly equal to 0.5) in the FE model, and therefore, the NP was relatively insensitive to its elastic modulus [4]. In addition, results from the present study also suggested that the intradiscal pressure was not sensitive to the disc properties under the vertical vibration. Spilker et al. [28] and Rao and Dumas [22] also reported a similar conclusion under static loading conditions.

The present study has several limitations. Firstly, only one FE model of the lumbar spine with specific geometry was employed, which means the computed absolute values may be unrepresentative of an average person as demonstrated by Rohlmann et al. [24]. However, the principal effect of changes in material properties of the intervertebral disc on vibration response of the lumbar spine was not affected due to the fact that the same model was used for the material sensitivity study of each disc components (AGS, AF, and NP). Secondly, the viscoelastic and poroelastic effects of the disc on its vibration response were not included in the current model, and the possible time-varying changes in disc properties were not accounted for. Thirdly, variations in material properties of the disc in this study were limited to only changes in elastic modulus, and changes in other material parameters (e.g., Poisson’s ratio) were not considered. In addition, to study the effect of specific disc component, variations of the material property for each disc component were investigated separately (i.e., one changed, the others kept constant) in this study.

5 Conclusions

The sensitivity of dynamic response of the human spine to variations in material property of the intervertebral disc (AGS, AF, and NP) was investigated using a FE model of the L1–S1 lumbar segment under the vertical vibration. In summary, the AGS property was more dominant than other disc properties. The AF property displayed a larger impact on the disc bulge. In contrast, the predicted response parameters were insensitive to the NP property. The findings may be helpful in adoption of appropriate material parameters for the disc in FE model of the lumbar spine used for vibration analysis.

Notes

Funding information

We are grateful for the grants from the Fundamental Research Funds for the Central Universities (02090022118032) and the National Natural Science Foundation of China (51275082, 11272273).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© International Federation for Medical and Biological Engineering 2018

Authors and Affiliations

  1. 1.School of Mechanical Engineering and AutomationNortheastern UniversityShenyangChina

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