INSERM, UMR 1033, F-69008, Lyon, France.
J Mech Behav Biomed Mater. 2013 Feb;18:200-12. doi: 10.1016/j.jmbbm.2012.08.012. Epub 2012 Nov 21.
Numerical simulation using finite element models (FEM) has become more and more suitable to estimate the mechanical properties of trabecular bone. The size and kind of elements involved in the models, however, may influence the results. The purpose of this study is to analyze the influence of hexahedral elements formulation on the evaluation of mechanical stress applied to trabeculae bone during a compression test simulation. Trabecular bone cores were extracted from 18 L2 vertebrae (12 women and 6 men, mean age: 76 ± 11, BV/TV=7.5 ± 1.9%). Samples were micro-CT scanned at 20 μm isotropic voxel size. Micro-CT images have been sub-sampled (20, 40 and 80 μm) to create 5.6 mm cubic FEM. For each sample, a compression test FEM has been created, using either 8-nodes linear hexahedral elements with full or reduced integration or 20-nodes quadratic hexahedral elements fully integrated, resulting in nine models per samples. Bone mechanical properties have been assumed isotropic, homogenous and to follow a linear elastic behavior law (Young modulus: 8 GPa, Poisson ratio: 0.3). Despite micro-architecture modifications (loss of connectivity, trabeculae thickening) due to voxel size increase, apparent mechanical properties calculated with low resolution models are significantly correlated with high resolution results, no matter the element formulation. However, stress distributions are more sensitive to both resolution and element formulation modifications. With linear elements, increasing voxel size leads to an alteration of stress concentration areas due to stiffening errors. On the opposite, the use of reduced integration induces severe smoothing and underestimation of stress fields resulting in stress raisers loss. Notwithstanding their high computational cost, quadratic elements are most appropriate for stress prediction in low resolution trabecular bone FEM. These observations are dependent on trabecular bone micro-architecture, and are more significant for low density sample displaying low trabecular thickness. In conclusion, we found that element formulation is almost important as element size when evaluating trabecular bone mechanical behavior at trabeculae scale. Therefore, element type should be chosen carefully when evaluating trabecular bone behavior using FEM.
使用有限元模型（FEM）进行数值模拟已经越来越适合于估计松质骨的力学性能。然而，模型中涉及的单元的大小和类型可能会影响结果。本研究的目的是分析六面体单元格式对压缩试验模拟中松质骨机械应力评估的影响。从 18 个 L2 椎体中提取松质骨核心（12 名女性和 6 名男性，平均年龄：76±11，BV/TV=7.5±1.9%）。样本以 20μm 各向同性体素大小进行微 CT 扫描。微 CT 图像经过亚采样（20、40 和 80μm）以创建 5.6mm 立方有限元模型。对于每个样本，使用全积分或减缩积分的 8 节点线性六面体单元或完全积分的 20 节点二次六面体单元创建一个压缩试验有限元模型，导致每个样本有 9 个模型。假设骨力学性能具有各向同性、均匀性并遵循线性弹性行为定律（杨氏模量：8GPa，泊松比：0.3）。尽管由于体素大小增加导致微结构发生了修改（连通性丧失，骨小梁变厚），但使用低分辨率模型计算的表观力学性能与高分辨率结果显著相关，无论单元格式如何。然而，应力分布对分辨率和单元格式修改更为敏感。使用线性单元，体素尺寸的增加会导致由于硬化误差而改变应力集中区域。相反，使用减缩积分会导致严重的平滑和低估应力场，从而导致应力升高器的损失。尽管二次单元具有较高的计算成本，但在低分辨率松质骨有限元中最适合预测应力。这些观察结果取决于松质骨微结构，对于显示低骨小梁厚度的低密度样本更为显著。总之，我们发现，在评估小梁尺度的松质骨力学行为时，单元格式几乎与单元大小一样重要。因此，在使用有限元法评估松质骨行为时，应仔细选择单元类型。
