A data-driven framework for developing a unified density-modulus relationship for the human lumbar vertebral body

Despite the broad agreement that bone stiffness is heavily dependent on the underlying bone density, there is no consensus on a unified relationship that applies to both cancellous and cortical compartments. Bone from the two compartments is generally assessed separately, and few mechanical test data are available for samples from the transitional regions between them. In this study, we present a data-driven framework integrating experimental testing and numerical modeling of the human lumbar vertebra through an energy balance criterion, to develop a unified density-modulus relationship across the entire vertebral body, without the necessity of differentiation between trabecular and cortical regions. A dataset of 25 spinal segments harvested from fresh-frozen human spines consisting of L1 vertebrae with adjacent intervertebral disks and neighboring T12 and L2 endplates was examined through a systematic process. Each specimen was subjected to axial compression using a custom-designed radiolucent device, and the deformation at multiple points during the ramp was quantified using digital volume correlation applied to the time-lapse series of microcomputed tomography images acquired during loading. A finite element model of each specimen was constructed from quantitative computed tomography images, with the experimental displacement fields imposed to replicate the observed deformation. The optimal density-modulus relationship, both in exponential and polynomial forms, was then determined by using data-driven techniques to match the numerical strain energy with the experimental external work. The resulting relationships effectively recovered bone tissue modulus at the microscale. Subsequently, the unified relationships were applied to investigate the vertebral structure-property correlations at the macroscale: as expected, compressive stiffness exhibited a moderate correlation with bone mineral density, whereas bending stiffness was revealed to correlate strongly with bone mineral content. These findings support the accuracy of the developed density-modulus relationships for the vertebral body and indicate the potential of the proposed framework to extend to other properties of interest such as vertebral strength and toughness.