Introduction

The lumbar spine, as a key part of the spine, supports the weight of the upper body and provides the necessary movement and flexibility for daily activities. Long-term occupational exposure to whole-body vibration has been identified as an important factor leading to spinal degenerative diseases and back pain, particularly for truck drivers and helicopter pilots1,2,3. Whole-body vibration can harm the lumbar spine, resulting in tissue damage and functional disorders. Studies4 have shown that vibration loads significantly increase the stress and strain between lumbar tissues. According to Wolff’s law, these large forces and strain energies can induce tissue regeneration and lead to spinal disorders, such as degeneration of intervertebral discs, osteophyte formation, and osteoarthritis.

To this end, many scholars have attempted to study the biomechanical relationship between vibration and lower back pain. Due to the limitations of experimental methods and mathematical models, the finite element method has increasingly gained attention as an important complement to these methods. From the perspective of applying finite element models to study the dynamic characteristics of the spine, Kong and Goel proposed that the resonance frequency decreases as the number of motion segments increases5. Research by GUO and TEO indicates that an increase in the mass of the upper body leads to a reduction in the resonance frequency of the T12-S1 segment6. Keller et al. showed that the stress and strain induced by dynamic loads on spinal structures are 2 to 3 times greater than those caused by equivalent static loads7.

Griffin’s Handbook of Human Vibration8remains the most authoritative source summarizing the biodynamics of the seated human body exposed to whole-body vibration, including apparent mass, transmissibility, and resonance characteristics. In addition, his widely cited journal articles—for example, Griffin 9]on discomfort models for sinusoidal vibration, Mansfield & Griffin10 on nonlinear apparent mass and transmissibility, and Nawayseh & Griffin11,12 on power absorbed during vibration in different postures—have clarified the biomechanical pathways of vibration transmission through the lumbar spine.

Complementarily, Rakheja and collaborators have developed energy absorption and biodynamic response models for seated occupants under multi-axis vibration13, providing further insight into the mechanical loading of the lumbar spine under real-world occupational exposures.

Together, these studies highlight the significance of WBV research and contextualize the novelty of our work, which investigates lumbar spine responses at both the macro- and micro-scales using finite element modeling. We have acknowledged these foundational studies in our response, though we kept the manuscript focused on our finite element approach.

With advancements in technology, multi-scale analysis of bone has become widely applied in the biomedical field. However, research on the microstructure of bone mostly focuses on ex vivo experiments with long bones under relatively simple loading conditions14, while studies on irregular bones like the spine are comparatively scarce15. To date, no research has addressed the multi-scale dynamics of bone, particularly regarding the macro- and micro-scale time-domain dynamic responses of the entire lumbar spine system to external vibrational loads. This study added HU-based segmentation (cortical: 500–3,000 HU; trabecular: 100–500 HU) using Mimics, referencing Sadowska et al.16, and employed a submodel method based on Saint-Venant’s principle to conduct a dynamic analysis of lumbar cortical bone using a multi-scale finite element analysis method under different frequency loadings.

Materials and methods

Macro-modeling

The original data for the model were collected from the Imaging Department of Shanxi Provincial Traditional Chinese Medicine Hospital (Shanxi Provincial Institute of Traditional Chinese Medicine), Taiyuan, Shanxi, 030000, China. A healthy male volunteer, aged 25 years with no history of spinal disease, was recruited for this study. The study was approved by the Ethics Review Board of Shanxi Provincial Traditional Chinese Medicine Hospital (Approval Number: SZYLY2024KY-1001). All methods were carried out in accordance with relevant guidelines and regulations, and the study was conducted in compliance with the Declaration of Helsinki. Written informed consent was obtained from the participant prior to data collection and publication.

Spiral scanning of the entire lumbar spine was performed using computed tomography (CT). CT Scanner: Siemens dual-source CT; Tube Voltage (kV): 120 kV; Tube Current (mA): 240 mA; Scan Slice Thickness: 0.600 mm. A total of 347 images were obtained. The scan results were saved in Digital Imaging and Communications in Medicine (DICOM) format. Our methodology included CT imaging (reconstruction kernel: B30f; in-plane resolution: 0.6 mm), adaptive HU-based segmentation to minimize partial volume effects17.

The DICOM files of the volunteer were processed using Mimics medical image processing software to obtain the geometric models of each vertebral segment. These models were then saved as STL format files. The STL files were subsequently imported into Geomagic Studio 12.0 software for further processing. The vertebral geometric models, after being processed in Geomagic, were imported into HyperMesh software in IGES format. In HyperMesh, the vertebrae, intervertebral discs, and ligaments were meshed, ensuring that all component nodes were shared. Finally, finite element analysis calculations were performed using Abaqus 2021 (Simulia, USA) software.

Fig. 1
Fig. 1
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Macroscopic finite element model of the lumbar spine.

The complete nonlinear finite element model, as shown in Fig. 1, includes the cortical bone, cancellous bone, posterior structures, intervertebral discs, endplates, and ligaments. The cortical bone and cancellous bone are considered to be orthotropic materials18. The thickness of the cortical bone and endplates is set to 1 mm. These are defined as C3D8R elements19,20. The endplate mesh is connected to the vertebral body and intervertebral disc meshes using a shared node method21. The intervertebral disc consists of the nucleus pulposus, fibrous matrix, and annulus fibrosus, with the nucleus pulposus occupying approximately 44% of the volume of the intervertebral disc22,23. The annulus fibrosus is composed of 6 layers of T3D2 elements, with stiffness decreasing from the outer to the inner layers. The ligaments include seven models: capsular ligament (CL), intertransverse ligament (ITL), supraspinous ligament (SL), interspinous ligament (ISL), ligamentum flavum (LF), anterior longitudinal ligament (ALL), and posterior longitudinal ligament (PLL). The ligament models are truss models that can only withstand tension24,25. The detailed material parameters for each component were determined based on the literature18,19,20,21,22,23,24,25, as shown in Table 1.

Table 1 Material properties of the spinal components.
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Micro-modeling

As shown in Fig. 2, this study established a multi-scale finite element model covering from the macroscopic scale to the microscopic scale, focusing on the dynamic characteristics of the lumbar cortical bone. At the microscopic scale, the bone unit is primarily composed of Haversian canals, which are elongated channels responsible for transporting blood and nutrients, supporting the vital activities of the bone tissue; lamellar bone matrix, consisting of layers of bone tissue surrounding the Haversian canals, providing the bone with its strength and toughness through their specific arrangement; and cement lines, which are boundary lines connecting different bone units and helping to distribute stress, thus preventing fractures. The interactions and structural characteristics of these components collectively determine the biomechanical properties of the cortical bone, providing a foundation for understanding the mechanical behavior of bone under various physiological and pathological conditions.

Fig. 2
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Microscopic model of lumbar cortical osteon.

According to the description in reference26, the number of concentric lamellae surrounding the Haversian canal ranges from 4 to 20 layers, and the lamellar density of cortical bone in cross-sections ranges between 11 and 20 per unit area. Based on statistical sample analysis27, this study defines a single bone unit as having six concentric lamellae, with a density of 16 per unit area. Specific dimensional parameters include: a bone unit diameter of 200 \({\text{um}}\), a Haversian canal diameter of 60 \({\text{um}}\), a lamella thickness of 10 \({\text{um}}\), a matrix thickness of 1 \({\text{um}}\), and a cement line thickness of 5 \({\text{um}}\). The material properties of the various structures within the bone unit are shown in Table 226,27.

Table 2 Unit types and material parameters for each part of the osteon.
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Boundary conditions of the model

In this study, a finite element model of the human lumbar spine (L1-S1) was established for further analysis, as shown in Fig. 3. A 40 kg mass point was added to the top surface of the upper vertebra to simulate the mass of the upper body28. Additionally, a 400 N follower load (200 N on each side) was applied through temperature-coupled trusses on both sides of each vertebra. The equivalent damping ratio used for transient analysis was set to 0.08, based on modal analysis. Briefly, all degrees of freedom on the lower surface of the sacral vertebra were fixed, and a sinusoidal axial force of 40 N was applied to the top surface of the vertebra, with frequencies of 5, 7, and 9 Hz29,30.

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Boundary conditions of the model.

For the boundary conditions of the micro-scale bone unit model, a specific boundary displacement method based on Saint-Venant’s principle was employed. In submodel analysis, the displacement results from the global model, particularly those at the submodel boundaries, were used as boundary conditions for the submodel. The boundary conditions of the submodel were directly derived from the global model results, ensuring consistency and continuity between the two models. In this method, the driving variable is typically displacement, where the displacement results at the submodel boundaries from the global model are used to drive the submodel analysis. This allows the submodel to perform a more detailed analysis on a refined mesh, obtaining precise mechanical response information for the local region of the model, including important parameters such as stress and strain. The macroscopic model of the human lumbar spine L1-S1 finite element model is considered the global model, while the bone units are regarded as submodels within this structure. The simulation results of the human lumbar spine L1-S1 model are used as the loading and boundary conditions for the cortical bone micro-scale bone unit model for further calculation.

Result

Model validation results

In the field of biomechanics research and engineering applications, ensuring the accuracy and reliability of finite element models is crucial. The validity of this study’s model was verified by comparing its predicted results with experimental data to assess whether the model can accurately predict the biomechanical behavior of the lumbar spine and other spinal structures. Specifically, the accuracy of the model was tested by comparing the same parameter values under identical loading conditions. The L4 and L5 segments of the model were selected for validation. Constraints were applied to the lower surface of the L5 vertebra, fixing its six degrees of freedom, and axial loads of 500, 1,000, 1,500, and 2,000 N were uniformly applied to the upper surface of the L4 vertebra to observe its axial displacement31.

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Comparison of the axial displacement of the L4 and L5 segments under axial load in the model with the results in Reference32,33,34,35.

The computed results were then compared with experimental data to validate the model. Under uniform axial loads of 500, 1,000, 1,500, and 2,000 N, the axial displacements of the vertebra were 0.40, 0.61, 0.81, and 1.1 mm, respectively. As shown in Fig. 4, the axial load-displacement curve was compared with experimental results from the literature32,33,34,35. The close agreement between in vitro experiments and finite element analysis under the same conditions confirmed the model’s reliability and validity.

Macro osteon calculation results

Modal analysis results

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The first three modal shapes of the model. (a) 1st, (b) 2nd, (c) 3rd.

According to existing research36, vibration response frequencies that cause serious harm to the human body are generally below 30 Hz. As shown in Fig. 5, the first three modal shapes of the entire lumbosacral spine (L1-S1) are lateral movement, forward-backward movement, and axial stretching, with corresponding natural frequencies of 1.09 Hz, 1.25 Hz, and 10.37 Hz, respectively.

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Dynamic Von-Mises stress response in macro osteon; (a) L1 segment osteon, (b) L2 segment osteon, (c) L3 segment osteon, (d) L4 segment osteon, (e) L5 segment osteon.

Under external vibrational loads, the macro-scale bone units in each segment of the lumbar spine exhibit cyclic stress responses. As shown in Fig. 6, when the load frequency increases from 5 Hz to 7 Hz and then to 9 Hz, the maximum Von-Mises stress in the L1 vertebral bone unit rises from 0.344 MPa to 0.511 MPa and finally to 0.890 MPa. The closer the excitation frequency approaches the first-order axial natural frequency of 10.37 Hz, the greater the stress response amplitude.The first axial resonance frequency (10.27 Hz) aligns with the range of whole-body vibration exposure in occupational settings (5–20 Hz). Near this frequency, stress amplification in L1 macro-units increased by 158% (5 Hz → 9 Hz), suggesting heightened injury risk during resonance.

For dynamic responses at different frequencies, the L1 vertebral bone unit shows the largest response amplitude, while the L3 vertebral bone unit exhibits the smallest. Under 5 Hz sinusoidal excitation, the Von-Mises stress amplitudes in the L1 and L3 vertebrae are 0.344 MPa and 0.187 MPa, respectively; under 9 Hz sinusoidal excitation, the stress amplitudes increase to 0.890 MPa and 0.490 MPa for the L1 and L3 vertebrae, respectively.

Microscopic osteon calculation results

The stress at the micro-scale in the cortical bone of the lumbar spine is significantly higher than that at the macro-scale. As shown in Fig. 7, the cyclic stress response of the cement line in the micro-scale bone units shows a consistent response trend across different spatial locations. Under 5 Hz, 7 Hz, and 9 Hz loads, the maximum Von-Mises stress in the cement line of the L1 vertebral micro-scale bone unit is 5.34 MPa, 8.23 MPa, and 13.57 MPa, respectively, while in the L4 vertebral micro-scale bone unit, the maximum Von-Mises stress is 0.47 MPa, 0.70 MPa, and 1.19 MPa, respectively.

Fig. 7
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Dynamic Von-Mises stress response in micro osteon cement line; (a) L1 segment osteon, (b) L2 segment osteon, (c) L3 segment osteon, (d) L4 segment osteon, (e) L5 segment osteon.

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Dynamic Von-Mises stress response in micro osteon bone plate medium; (a) L1 segment osteon, (b) L2 segment osteon, (c) L3 segment osteon, (d) L4 segment osteon, (e) L5 segment osteon.

As shown in Fig. 8, the cyclic stress response of the bone plate matrix in the micro-scale bone units increases as the frequency rises, with higher frequencies resulting in greater response amplitudes. Under 5 Hz, 7 Hz, and 9 Hz loads, the maximum Von-Mises stress in the bone plate matrix of the L1 vertebral micro-scale bone unit is 0.48 MPa, 0.90 MPa, and 1.47 MPa, respectively, while in the L4 vertebral micro-scale bone unit, the maximum Von-Mises stress is 0.11 MPa, 0.17 MPa, and 0.29 MPa, respectively.

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Dynamic Von-Mises stress response in micro osteon bone plate; (a) L1 segment osteon, (b) L2 segment osteon, (c) L3 segment osteon, (d) L4 segment osteon, (e) L5 segment osteon.

As shown in Fig. 9, the cyclic stress response of the bone plate in micro-scale bone units increases as the frequency rises, with higher frequencies resulting in greater response amplitudes. Under 5 Hz, 7 Hz, and 9 Hz loads, the bone plate stress in the micro-scale bone units of the L1 vertebra is the highest, while that of the L4 vertebra is the lowest, with a gradual decrease from L1 to L4. The maximum Von-Mises stress in the bone plate of the L1 vertebral micro-scale bone unit is 3.88 MPa, 7.25 MPa, and 11.92 MPa, respectively, while in the L4 vertebral micro-scale bone unit, the maximum stress is 0.74 MPa, 1.23 MPa, and 2.19 MPa. As shown in Fig. 10, the maximum stress at the micro- and macro-scales differs significantly, with the stress in the micro-scale bone units being approximately 2.38 to 17.33 times greater than that in the macro-scale units. This stress ratio varies depending on operating conditions and the spatial position of the bone unit, with little correlation to the excitation force frequency.

Fig. 10
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Von Mises stress ratio of cortical bone micro/macro osteon in the L1-L5 segment of the lumbar spine.

Discussion

This study analyzes the dynamic characteristics of bone units using the specific boundary displacement method based on Saint-Venant’s principle. It calculates the macro- and micro-scale dynamic characteristics of lumbar spine bone units under different frequencies of sinusoidal excitation loads. The results show that both macro- and micro-scale bone units exhibit cyclic stress response characteristics. Under the same frequency loading, the stress response amplitude of macro-scale bone units varies depending on their spatial location, decreasing from L1 to L5 and then increasing, with the minimum values observed at the L3 and L4 vertebrae and the maximum at the L1 vertebra. The closer the frequency approaches the first-order axial natural frequency, the greater the stress response amplitude of the macro-scale bone units, consistent with the findings of Fan et al.37.

At the micro-scale, the stress in the micro-scale bone unit’s cement line, bone plate matrix, and bone plates also shows cyclic response behavior. Among them, the cement line and the outermost bone plate exhibit the highest stress, due to the differences in Young’s modulus between micro- and macro-scale bone units. The various components of the micro-scale bone unit have different Young’s moduli, with the cement line and outermost bone plate having higher values, leading to larger stress amplitudes. This aligns with the theory that stiffer components in nonlinear systems bear more stress, consistent with Wang’s38findings.The observed micro-scale stress concentrations (2.38–17.33×macro-scale) corroborate findings by Gaziano et al.39, who reported 3–15×stress gradients in trabecular bone under dynamic loading. The dominance of cement line stress aligns with Nobakhti et al.40, who identified cement lines as strain amplifiers in cortical bone.

The stress in micro-scale bone units is approximately 2.38 to 17.33 times greater than that in macro-scale bone units. Under different frequency excitations, the micro-to-macro stress ratio remains essentially the same across different spatial locations, indicating that varying frequency excitation mainly affects the stress amplitude of bone units without altering the micro-to-macro stress ratio. The material properties of macro-scale bone units are consistent with cortical bone, while micro-scale bone units consist of bone plates, bone plate matrix, and cement lines, with the cement line bearing the highest stress. Under a 7 Hz excitation load, the stress ratio between micro-scale bone plate matrix and corresponding macro-scale cortical bone reaches a maximum of 17.33. Under the same boundary conditions, uneven distribution of material properties is more likely to cause stress concentration. According to Wolff’s law, bone grows where needed and resorbs where not needed. High-stress areas in bone units promote the conversion of bone cells into osteoblasts, increasing the density and hardness of bone in these regions. Stress in the bone plate matrix is significantly lower than that in the bone plates, and this stress discontinuity plays a positive role in protecting bone tissue.

First, the observation that stress amplitudes increase substantially near the resonance frequency of the lumbar spine (10.27 Hz) offers biomechanical insights into the etiology of vibration-induced spinal disorders. For example, occupational exposure to whole-body vibration in this frequency range (e.g., among drivers, pilots, or heavy equipment operators) may accelerate localized stress accumulation, thereby increasing the risk of intervertebral disc herniation, microdamage, and eventual degeneration. Second, the multiscale stress amplification identified in our study provides a theoretical basis for the design of fatigue-resistant spinal implants or dynamic stabilization devices. The amplification factor (up to 17.33×) indicates that microscale structures such as cement lines may serve as mechanical weak points or damage initiation sites under dynamic loading conditions—knowledge that is crucial for personalized orthopedic interventions and implant material optimization.

Furthermore, the relatively consistent micro-to-macro stress ratio across frequencies and vertebral levels may serve as a predictive indicator in patient-specific modeling, supporting early diagnosis or intervention planning. These findings thus not only enhance our understanding of lumbar biomechanics under vibrational environments but also have the potential to inform both clinical risk assessment and biomechanical device design.

There are certain limitations to this study: the model did not account for factors such as ribs, skin, muscles, and the spinal cord, which may have influenced the simulation results to some extent. Nevertheless, during model validation, we compared and validated the computational simulation results with existing experimental data. Future work should integrate muscle forces and soft tissues, validate models with in vivo strain measurements, and explore frequency-dependent damage thresholds across diverse populations.

Conclusion

This study constructed and validated macro- and micro-scale models of the lumbar spine to explore the dynamic characteristics of bone units under different frequencies of sinusoidal excitation. The results show that lumbar bone units exhibit cyclic stress response characteristics under various frequency loads, and as the excitation frequency approaches the first-order axial natural frequency of the lumbar spine, the stress response amplitude of bone units increases significantly. Moreover, the stress amplitude ratio between macro- and micro-scale bone units remains relatively stable under different frequency excitations, indicating that frequency primarily affects stress amplitude rather than the ratio. In the microstructure, the cement line and the outermost bone plate bear the most stress, which is related to the differences in material properties of the micro-scale bone units and the Young’s modulus of the macro-scale bone units. Previous studies have not examined the dynamic characteristics of the human lumbar spine from both macro- and micro-scale perspectives. This study fills that gap, providing a theoretical basis for further understanding the dynamic behavior of the lumbar spine under vibrational loads and laying the groundwork for protecting the human lumbar spine from vibrational injury.