A Viscoelastic Modeling for Failure Analysis of Human Vertebral Bone Undergoing Quasi-Static and Dynamic Compression

Vertebral fractures are among the most common skeletal injuries and present significant clinical and biomechanical challenges, particularly in older adults and individuals with low bone density. Accurate prediction of vertebral mechanical response and failure under varying loading conditions is essential for improving understanding of spinal injury mechanisms. This study develops a density-dependent viscoelastic analytical model to predict the stiffness and fracture force of human vertebral specimens subjected to different compression rates. The vertebral body is represented as a composite structure consisting of a cortical shell and a trabecular core. Cortical bone is modeled as a linear elastic material, whereas trabecular bone is described using a Kelvin-Voigt viscoelastic formulation. Density-dependent constitutive relationships are incorporated for the elastic modulus and viscous coefficient of trabecular bone. Unknown material parameters are identified through optimization using the Nelder-Mead algorithm, based on experimental compression data from cadaveric vertebral specimens tested under quasi-static and dynamic loading conditions. The calibrated model reproduced the overall trend of specimen-to-specimen mechanical variation observed experimentally. Predicted stiffness values were in reasonable agreement with measured data. Fracture force predictions showed moderate agreement for dynamically tested specimens (R² = 0.60), which improved to R² = 0.88 after exclusion of one statistically identified outlier. Compared with a purely linear elastic formulation, the proposed viscoelastic model demonstrated modest improvement in stiffness prediction and more substantial improvement in fracture force prediction. These findings indicate that incorporating density-dependent viscoelastic effects improves representation of vertebral mechanical behavior, particularly at higher loading rates. Owing to its simplicity and computational efficiency, the proposed model requires only limited imaging input and may be useful for future biomechanical investigations, rapid screening, and injury risk prediction.