A concise gradient tensor model in quantitatively describing the anisotropic properties of lumbar trabecular bone.

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This study aimed to evaluate the feasibility of image gradient methods for quantifying directional heterogeneity in lumbar trabecular bone. A total of 326 male patients' lumbar CT images were collected and grouped by age (20-39 years, 40-59 years, and ≥ 60 years). The tangential gradient calculation was performed on the mid-sagittal plane of the L4 vertebra, and the ratio of the longitudinal to transverse gradient (ratio_σ) was computed to construct a gradient tensor model. Statistical analysis was conducted to assess the differences in ratio_σ among the groups. Individuals with the same bone mineral density were selected to construct finite element models to validate the value of different tensor models for fracture risk assessment. Our study results indicate that there is significant directional heterogeneity in the lumbar trabecular core across all age groups, which becomes more pronounced with advancing age, as evidenced by the gradual increase in ratio_σ values (1.44±0.29 for 20-39 years group, 1.86±0.19 for 40-59 years group, and 2.10±0.33 for ≥ 60 years group). Finite element analysis results at the same bone mineral density show that the high tensor group has a higher proportion of fracture risk elements compared to the low tensor group. The trabecular core shows marked directional heterogeneity with age-dependent variations. Using directional gradient decomposition, we developed gradient tensor models to quantify direction-specific remodeling states in bone microstructures. This tensor-based approach also enables fracture risk assessment at comparable bone mineral density levels.