A finite element model of the human lower thorax to pelvis spinal segment: Validation and modal analysis

Abstract

BACKGROUND:

Several finite element (FE) models have been developed to study the effects of vibration on human lumbar spine. However, the authors know of no published results so far that have proposed computed tomography-based FE models of whole lumbar spine including the pelvis to conduct dynamic analysis.

OBJECTIVE:

To create and validate a three-dimensional ligamentous FE model of the human lower thorax to pelvis spinal segment (T12–Pelvis) and provide a detailed simulation environment to investigate the dynamic characteristics of the lumbar spine under whole body vibration (WBV).

METHODS:

The T12–Pelvis model was generated based on volume reconstruction from computed tomography scans and validated against the published experimental data. FE modal analysis was implemented to predict dynamic characteristics associated with the first-order vertical resonant frequency and vibration mode of the model with upper body mass of 40 kg under WBV.

RESULTS:

It was found that the current FE model was validated and corresponded closely with the published data. The obtained results from the modal analysis indicated that the first-order vertical resonant frequency of the T12–Pelvis model was 6.702 Hz, and the lumbar spine mainly performed vertical motion with a small anteroposterior motion. It was also found that shifting the upper body mass centroid onwards or rearwards from the normal upright sitting posture reduced the vertical resonant frequency.

CONCLUSIONS:

These findings may be helpful to better understand vibration response of the human spine, and provide important information to minimize injury and discomfort for these WBV-exposed occupational groups.

Keywords

Finite element modeling human spine lower thorax to pelvis segment modal analysis validation

1. Introduction

Long-term exposure to whole body vibration (WBV) is associated with an increased risk of low-back pain and degenerative diseases of the spine among vehicle drivers and other vibration machine operators [1,2]. It has been reported that WBV exposure can result in degenerative and geometric changes of vertebral bodies [3]. Some experimental studies have also found that long-term WBV exposure may cause proteoglycan loss and intervertebral disc degeneration [4,5]. Finite element (FE) analysis is a powerful tool routinely used to study complex biological systems, recent years have seen an increasing interest in adopting three-dimensional FE method to understand dynamic mechanical behavior of the human spine under WBV, such as strain and stress distributions in the intervertebral discs, contact force transmission of facet joints and so on, which are difficult to be studied through experimental protocols.

Several FE models have been developed to look into dynamic characteristics of the spine. In the early times, Kasra et al. [6] developed a three-dimensional FE model of the L2–L3 unit to predict the free and force vibration responses, and the results showed a good agreement with the experimental data. Soon afterwards, Goel et al. [7] created an FE model of two motion segments L4–S1 and analyzed its dynamic response without damping. However, it is difficult to reflect the force transmission and stress distribution of the whole lumbar spine for the short motion segments. Therefore, Kong and Goel [8] established a more complete FE model of the upper body from the head to sacrum and performed modal analysis to extract resonant frequencies of the whole spine. There is no doubt that these previous FE studies have offered valuable insights into dynamic characteristics of the human spine. Nonetheless, to the best of our knowledge, most of the existing FE models employed to perform the dynamic analysis lack an intact pelvis structure, which is one of the most important components within the human body, and facilitates the transfer of the weight of the upper body to the hip [9,10]. Moreover, it has been reported that the sacrum-pelvis joints may influence dynamic response of the spine [11–13]. Therefore, developing a lumbar spinal model with the intact pelvis is helpful to understand biomechanical parameters of the spine under vibration.

The aim of this study was to investigate holistic mechanical characteristics of the lumbar spine system under WBV. For this purpose, a three-dimensional FE model of the T12–Pelvis segment was created by the computed tomography scanning protocols. The model was firstly validated through comparison with the published results. After that, modal analysis was applied to predict the resonant frequency and vibration mode of the model. Finally, we attempted to investigate how the anteroposterior (A-P) shifting of the human upper body affected dynamic characteristics of the spine.

2. Materials and methods

2.1. FE modeling

The T12–pelvis ligamentous FE model was generated in ABAQUS software, as shown in Fig. 1. The geometrical information of vertebrae and pelvis from an adult female with no current physical abnormalities was obtained via computed tomography scans having in-plane resolution 512 × 512 pixels and 0.6 mm slice thickness. The present study was approved by the Science and Ethics Committee of Northeastern University, and informed consent was obtained from the volunteer.

Fig. 1.

Ligamentous FE model of the T12–Pelvis segment in front view and lateral view.

The vertebral bodies were modeled by accounting for the separation of the cancellous bone and cortical shell (including bony endplates) using 4-nodes solid and 3-nodes shell elements, respectively (Fig. 2(a)). All the shell elements have a constant thickness of 0.7 mm [14]. The intervetebral discs were created by replicating the outer geometry of the actual discs from the tested woman as closely as possible via CAD modeling and Boolean interaction operation [15]. As shown in Fig. 2(b), the newly created disc was partitioned into annulus fibrous and the nucleus pulpous (about 40%_nucleus, 60%_annulus). The annulus was constructed as a homogenous ground substance using 8-nodes hexahedral elements and reinforced in the radial direction by 6 crisscross fiber layers using tension-only truss elements. In each layer, the fibers angulations were closed to ±30° [16] and their total volume was assumed to be 19% of the annulus volume [17]. In addition, the fluid-like behavior of the nucleus pulpous is simulated to be nearly incompressible [18]. A complete model of the pelvis structure was presented including the innominate bones, sacrum and soft tissues connection (Fig. 2(c)). Using the same method for intervetebral disc modeling, the soft tissues (sacroiliac joints and the pubic symphysis) were created, and then the pelvis domain was discretized with 4-nodes solid elements. The spinal ligaments were modeled with tension-only truss elements, and the contact between the facet joints was simulated by surface to surface contact elements without friction [19]. The material properties used in the FE model were taken based on the literature [7,20–22] and are listed in Table 1.

Fig. 2.

Modeling description for the spinal motion segment. (a) Vertebral bodies, (b) intervetebral disc and (c) intact pelvis.

Table 1

Material properties implemented to various components of the T12–Pelvis FE model

Component	Element type	Young’s modulus (MP)	Poisson’s ratio	Density (Kg/mm³)	Cross-sectional area (mm²)
Cortical bone	Shell	12000	0.3	1.7E-06
Bony endplate		23.8	0.4	1.2E-06
Cancellous bone	Solid	100	0.2	1.1E-06
Posterior bone element		3500	0.25	1.4E-06
Pelvis		6000	0.4	1.2E-06
Pubic symphsis		27.22	0.4	1.4E-06
Sacroiliac joints		24.3	0.4	1.4E-06
Annulus ground substance		4.2	0.45	1.02E-06
Nucleus pulpous		1	0.499	1.02E-06
Annulus fiber	Truss	500	0.3	1.0E-06
Anterior longitudinal ligament		7.8	0.3	1.0E-06	63.7
Posterior longitudinal ligament		10	0.3	1.0E-06	20
Intertransverse ligament		10	0.3	1.0E-06	3.6
Ligamentum flavum		15	0.3	1.0E-06	40
Capsular ligament		7.5	0.3	1.0E-06	30
Interspinous ligament		10	0.3	1.0E-06	40
Supraspionous ligament		8	0.3	1.0E-06	30
Sacrospinous ligament		12.5	0.3	1.0E-06	30
Sacrotuberous ligament		50	0.3	1.0E-06	50
lliolumbar ligament		30	0.3	1.0E-06	30
Anterior sacroiliac ligament		20.8	0.2	1.0E-06	100
Posterior sacroiliac ligament		13.3	0.2	1.0E-06	100
Pubic ligament		10	0.3	1.0E-06	30

2.2. Boundary and loading conditions

FE modeling requires several simplifications and assumptions with impact on the predicted results. Therefore, validation of the FE model regarding its kinematical behavior (i.e. static validation) needs to be conducted. There have been very few investigations for the whole T12–Pelvis model reported, so the model was dissected to form three segmental models (T12–L1, L1–S1 and S1–Pelvis), and they were separately validated by comparison with the available results in the literature. For the T12–L1 model, according to tests by Oxland et al. [23] and the FE study by Qiu [24], the inferior surface of the L1 vertebra was fixed in all directions. Then, the values of the rotation angle at T12–L1 were calculated using FE method under 7.5 Nm pure moment loading in flexion, extension, lateral bending and axial rotation. For the L1–S1 model, following the tests by Guan et al. [25] and Panjiabi et al. [26] and the FE study by Moramarco et al. [27], the lower face of the vertebra S1 was fully constrained and then an incremental 3 Nm pure flexion moment and an incremental 7.5 Nm extension moment with 100 N axial compressive pre-loading were applied respectively on the vertebra L1 to conduct the validation with respect to the spinal range of motion (ROM) predictions. For the S1–Pelvis model, according to the FE study by Mei et al. [28], the model was constrained at the ischial tuberosity of the pelvis and FE method was used to predict the vertical displacement of the pelvis under vertical load.

For the consideration of dynamic validation and comparison with published results, the T12–Plevis model was altered to generate segmental models, and the inferior-most surface of each model is fixed in all directions. A mass point of 40 kg was imposed on these models to simulate the effects of human upper body mass. For one motion segment (T12–L1, L2–L3) and two motion segments (L3–L5, L4–S1), the mass point was assigned to the center of the corresponding upper vertebral body. For long motion segments (T12–Pelvis, L1–L5, L1–S1), the location of the mass point was derived from Schirazi-Adl et al. [29]. Finally, modal analysis was performed in all above segmental models to extract their resonant frequencies. In current study, only the first-order vertical resonant frequency and vibration mode will be specifically described.

Fig. 3.

(a) Comparison of rotation angle between the current study and the literature for the T12–L1 segment under 7.5 Nm pure-moment loading in four directions and (b) comparison of vertical displacement between the current study and the literature for the S1–Pelvis segment under axial compressive loading.

Fig. 4.

Comparison of the rotation angle between the current study and the literature for the L1–S1 segment under pure flexion moment: (a) 1 Nm, (b) 2 Nm and (c) 3 Nm.

3. Results

3.1. Static validation of the FE model

In Fig. 3(a), the rotation angle predictions by the current T12–L1 model under various pure-moment loadings were compared with the values from the literature. It was observed that the predicted ROMs of 2.8°, 3.5°, 3.4° and 1.4° fell within the range of the experimental deviation [23] of 2.9 ± 1.4°, 3.9 ± 1.4°, 3.7 ± 1.1° and 1.2 ± 0.7° for flexion, extension, lateral bending and axial rotation, respectively. Also, the present predictions showed close agreement with those from the previous FE study by Qiu [24]. Figure 3(b) compares the current vertical displacement predictions of the S1–Pelvis model with the results of Mei et al. [28] under different axial compressive loadings, and they agreed well with each other. Figure 4 shows the response of the L1–S1 model to the incremental pure flexion moment at different spinal segments, and the present results displayed a good agreement with the experimental data of Guan et al. [25]. The maximum ROM under the flexion moment was observed in L5–S1 segment and the minimum in L1–L2, which were in concordance with the findings in the literature [25,27]. In Fig. 5, the L1–S1 model responses to the incremental extension moment are illustrated, and in general the predictions also remained in the range of the standard deviation interval measured by Panjiabi et al. [26].

Fig. 5.

Comparison of the rotation angle between the current study and the literature for the L1–S1 segment under extension moment (with 100 N axial compressive pre-loading): (a) 2.5 Nm, (b) 5 Nm and (c) 7.5 Nm.

3.2. Reports on resonant frequencies and vibration modes

Table 2 lists first-order vertical resonant frequencies of the different spinal motion segments obtained from modal analysis, as well as other published results [6–8]. The results showed that the resonant frequency of the T12–Pelvis model was 6.702 Hz with consideration of 40 kg upper body mass. For one motion segment, the resonant frequencies of T12–L1 and L2–L3 were 21.338 Hz and 25.128 Hz, respectively, and they were near the experimental values of Kasra et al. [6]. For two motion segments L3–L5 and L4–S1, the resonant frequencies were 17.860 Hz and 13.694 HZ, respectively, and both of them were slightly smaller than the results in the literature [7,8]. For the entire lumbar spine (L1–L5) and lumbosacral spine (L1–S1), the resonant frequencies were 12.475 Hz and 8.391 HZ, respectively, and they were also close to the results reported by Kong and Goel [8]. Overall, it can be seen that our predictions were in close agreement with the published results, which further validated the current FE model. Additionally, it can be seen from Table 2 that increasing the number of motion segment decreased the resonant frequency. Figure 6 describes the amplitude of different spinal motion segments for the T12–Pelvis model in the vertical direction and A-P direction under WBV, and indicates that the spine mainly performed vertical motion with a small A-P motion. It was observed in Fig. 6(a) that the vertebra T12 had the largest vertical amplitude. From Fig. 6(b) it can be seen that the relative amplitude of vertebra L5 against the sacrum was largest. The maximum A-P amplitude was found in the intervertbral disc between vertebra L3 and L4, as shown in Fig. 6(c).

Table 2
Comparison of the first-order vertical resonant frequencies between the current study and the literature for different spinal motion segments

Spinal motion segment Resonant frequency (Hz) Literature

One motion segment T12–L1 23.5 Experimental study by Kasra et al. [6]

T12–L1 24.055 Current study

L2–L3 26.0 Experimental study by Kasra et al. [6]

L2–L3 23.217 Current study

Two motion segments L3–L5 18.0 FE study by Kong and Goel [8]

L3–L5 16.860 Current study

L4–S1 17.5 FE study by Goel et al. [7]

L4–S1 15.178 Current study

Lumbar spine L1–L5 12.2 FE study by Kong and Goel [8]

L1–L5 11.384 Current study

Lumbosacrum L1–S1 10.6 FE study by Kong and Goel [8]

L1–S1 9.59 Current study

Lower thorax to pelvis T12–Pelvis 7.304 Current study

Spinal motion segment		Resonant frequency (Hz)	Literature
One motion segment	T12–L1	23.5	Experimental study by Kasra et al. [6]
	T12–L1	24.055	Current study
	L2–L3	26.0	Experimental study by Kasra et al. [6]
	L2–L3	23.217	Current study
Two motion segments	L3–L5	18.0	FE study by Kong and Goel [8]
	L3–L5	16.860	Current study
	L4–S1	17.5	FE study by Goel et al. [7]
	L4–S1	15.178	Current study
Lumbar spine	L1–L5	12.2	FE study by Kong and Goel [8]
	L1–L5	11.384	Current study
Lumbosacrum	L1–S1	10.6	FE study by Kong and Goel [8]
	L1–S1	9.59	Current study
Lower thorax to pelvis	T12–Pelvis	7.304	Current study

Fig. 6.

Amplitude at the centers of different vertebrae or intervertebral discs of the T12–Pelvis FE model for the first-order vertical vibration mode: (a) absolute vertical amplitude, (b) relative vertical amplitude (every vertebra against the next level vertebra) and (c) A-P amplitude.

3.3. Effect of A-P shifting of upper body mass

Herein the mass of human upper body was accounted for by imposing a mass point of 40 kg on the T12–Pelvis FE model, and when the mass point was located on the top of vertebra T12 by 1 cm anterior to the L3–L4 vertebral centroid, the model was assumed to be a normal posture for an upright sitting posture [29]. Figure 7 shows the effect of the A-P upper body mass shifting on the first-order vertical resonant frequency of the T12–Pelvis model. The results demonstrated that the human spine had the maximum vertical resonant frequency at the normal posture, and the frequency was decreased with shifting the mass point onwards or rearwards from the normal posture. Figure 8 compares the absolute and relative vertical amplitudes of the T12–Pelvis model in the first-order vertical vibration modes under different A-P upper body mass shifting postures. The results indicated that shifting the mass point onwards decreased the vertical amplitude for every vertebra, and the vertical amplitude was increased with shifting the mass point rearwards.

Fig. 7.

Effect of A-P shifting of human upper body mass on the first-order vertical resonant frequency of the T12–Pelvis FE model.

Fig. 8.

Comparison of vertical amplitude at the different vertebral centers of the T12–Pelvis FE model in the first-order vertical vibration mode for the different A-P upper body mass shifting: (a) absolute amplitude and (b) relative amplitude (every vertebra against the next level vertebra).

4. Discussion

Long-term WBV exposure is known to change the biomechanical response of the lumbar spine and thus may contribute to spinal discords. Research has found that dynamic loads with frequencies close to that of the human spine resonant frequencies have a destructive effect, which may expose the vertebrae to large forces and high risk of injury in seat vibration [30]. Experimental measurement is the most widely used method of identifying the intrinsic relationship between WBV and spinal health risk at present. However, compared with the FE method, the experimental approach is difficult or even impractical to quantify effect of the WBV exposure on the geometric and structural changes in the spine. Therefore, this study attempted to develop a spinal FE model to investigate the dynamic characteristics of human lumbar spine under WBV. Considering that it was difficult for short spinal motion segments to reflect holistic mechanical characteristics of the lumbar spine, a detailed ligamentous FE model of T12–Pelvis human spine segment, including intact pelvis structure, was created.

Static validation of the current FE model was firstly conducted by comparing the predicted values with the published results. As stated earlier, the T12–Pelvis model was dissected to generate three segmental sub-models (T12–L1, L1–S1 and S1–Pelvis), and they were loaded and constrained by the same conditions as those in the literature. These sub-models were validated separately, and the predicted values were in general agreement with the published results. As shown in Fig. 3(a), all our ROM predictions were almost perfect for the T12–L1 segment under different loading conditions, and the maximum found deviation from the experimental data was smaller than 0.4°. But as shown in Figs 4 and 5, both of the predicted ROMs, from current and previous FE studies, for L5–S1 segment under flexion and extension loadings were not in good accordance with mean value of the experimental results. This deviation was assessed in relation to the fix boundary applied in this segment, and in reality the vertebra was not constrained from moving in space.

Many previous studies have indicated that the first-order vertical vibration mode may make the mainly contribution to the human lumbar spine under WBV. Therefore, after validating the current FE model regarding its kinematical behavior, modal analysis was employed to predict the first-order vertical resonant frequency of the model and its corresponding vibration mode in order to better understand dynamic characteristics of the lumbar spine under WBV. The predicted resonant frequency for the T12–Pelvis model with a 40 kg mass was 6.702 Hz. To verify our predictions, resonant frequencies for shorter motion segments of T12–Pelvis were also extracted and compared with the results in the literature, as shown in Table 2. It can be seen that our results were broadly in agreement with the published data. In addition, Kong and Goel [8] stated that the resonant frequency in the vertical direction decreased with the increase in the number of spinal motion segments. The same conclusion was also drawn in this study. Vertical resonant frequency measured at the human upper body have been reported to occur between 4 and 6 Hz [31,32] and our result (6.702 Hz) was very close to the experimental data. The difference may be attributed to the fact that the current FE model did not included the rib cage and muscles. Moreover, Kong and Goel [8] also found that the inclusion of muscles with self-weight might decrease the first-order resonant frequency degeneration under WBV. This is in line with experimental observations of Sandover and Dupuis [32], who found that the largest motion occurred in the upper lumbar and lower thoracic region of the spine under vibration. As mentioned above, in the first-order vertical vibration mode, the lumbar spine performs not only vertical motion, but A-P motion. The larger A-P amplitudes were found at vertebra L3 and L4 and the intervertbral disc between them (Fig. 6(c)), which was close to the reported results by Kong and Goel [8]. This implies that larger shear stress or shear force might appear at these regions. The above-mentioned findings may provide important information to minimize injury and discomfort for the WBV-exposed occupational groups of their H–S1 FE model (from 8.91 to 6.82 Hz). Therefore, it can be imagined that an improved FE model based on the current T12–Pelvis model, including muscles, rib cage, etc., may give more accurate resonant frequency prediction results. In future studies, the authors will try to provide this improvement. As shown in Fig. 6(b), the larger relative vertical amplitudes for the first-order vertical vibration mode were found at T12–L1 and L5–S1 levels. This implies that larger strains and stress might appear at these joints and thus accelerate their degeneration under WBV. This is also in line with experimental observations of Sandover and Dupuis [32], who found that the largest motion occurred in the upper lumbar and lower thoracic region of the spine under vibration. As mentioned above, in the first-order vertical vibration mode, the lumbar spine performs not only vertical motion, but A-P motion. The larger A-P amplitudes were found at vertebra L3 and L4 and the intervertbral disc between them (Fig. 6(c)), which was close to the reported results by Kong and Goel [8]. This implies that larger shear stress or shear force might appear at these regions. The above-mentioned findings may provide important information to minimize injury and discomfort for the WBV-exposed occupational groups.

Some earlier studies [33,34] have demonstrated the influence of decreasing (or increasing) human upper body mass on resonant frequency of the human spine. However, very few studies have been conducted to analyze influence of the mass locations shifting onwards or rewards on dynamics characteristics of the lumbar spine under WBV. In this study, the authors used the T12–Pelvis FE model to determine the effect of A-P shifting of upper body mass from the upright sitting posture on the resonant frequency and vertical amplitude of the lumbar spine. As shown in Figs 7 and 8, the first-order vertical resonant frequencies and amplitudes were presented under different postures. These findings could also assist researchers design safer and more comfortable environments for seated operators under WBV. For example, a new car seat may be developed to help drivers maintain a correct driving posture in order to prevent WBV-related spinal discords.

5. Conclusions

This study created and validated a detailed three-dimensional ligamentous FE model of the human T12–Pelvis segment to investigate holistic mechanical characteristics of the lumbar spine under WBV. The obtained results from the modal analysis indicated that the first-order vertical resonant frequency of the T12–Pelvis model was 6.702 Hz, and the lumbar spine mainly performed vertical motion with a small A-P motion. It was also found that shifting the upper body mass centroid onwards or rearwards from the normal upright sitting posture reduced the vertical resonant frequency. These findings may be helpful to better understand vibration response of the human spine, and provide important information to minimize injury and discomfort for these WBV-exposed occupational groups.

Footnotes

Acknowledgements

This project was supported by the National Natural Science Foundation of China (Grant nos 52005089 and 51875096) and Fundamental Research Funds for the Central Universities (Grant no. N2103010).

Conflict of interest

None declared.

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