Department of Mechanical & Industrial Engineering, University of Massachusetts, Amherst, USA.
J Theor Biol. 2012 May 7;300:242-53. doi: 10.1016/j.jtbi.2012.01.031. Epub 2012 Jan 27.
We employed a probabilistic finite element analysis (FEA) method to determine how variability in material property values affects stress and strain values in a finite model of a Macaca fascicularis cranium. The material behavior of cortical bone varied in three ways: isotropic homogeneous, isotropic non-homogeneous, and orthotropic non-homogeneous. The material behavior of the trabecular bone and teeth was always treated as isotropic and homogeneous. All material property values for the cranium were randomized with a Gaussian distribution with either coefficients of variation (CVs) of 0.2 or with CVs calculated from empirical data. Latin hypercube sampling was used to determine the values of the material properties used in the finite element models. In total, four hundred and twenty six separate deterministic FE simulations were executed. We tested four hypotheses in this study: (1) uncertainty in material property values will have an insignificant effect on high stresses and a significant effect on high strains for homogeneous isotropic models; (2) the effect of variability in material property values on the stress state will increase as non-homogeneity and anisotropy increase; (3) variation in the in vivo shear strain values reported by Strait et al. (2005) and Ross et al. (2011) is not only due to variations in muscle forces and cranial morphology, but also due to variation in material property values; (4) the assumption of a uniform coefficient of variation for the material property values will result in the same trend in how moderate-to-high stresses and moderate-to-high strains vary with respect to the degree of non-homogeneity and anisotropy as the trend found when the coefficients of variation for material property values are calculated from empirical data. Our results supported the first three hypotheses and falsified the fourth. When material properties were varied with a constant CV, as non-homogeneity and anisotropy increased the level of variability in the moderate-to-high strains decreased while the level of variability in the moderate-to-high stresses increased. However, this is not the pattern observed when CVs calculated from empirical data were applied to the material properties where the lowest level of variability in both stresses and strains occurred when the cranium was modeled with a low level of non-homogeneity and anisotropy. Therefore, when constant material property variability is assumed, inaccurate trends in the level of variability present in modest-to-high magnitude stresses and strains are produced. When the cranium is modeled with the highest level of accuracy (high non-homogeneity and anisotropy) and when randomness in the material properties is calculated from empirical data, there is a large level of variability in the significant strains (CV=0.369) and a low level of variability in the modest-to-high magnitude stresses (CV=0.150). This result may have important implications with regard to the mechanical signals driving bone remodeling and adaptation through natural selection.
我们采用概率有限元分析（FEA）方法来确定皮质骨材料属性值的变化如何影响恒河猴颅骨有限模型中的应力和应变值。皮质骨的材料行为以三种方式变化：各向同性均匀、各向同性非均匀和正交各向异性非均匀。松质骨和牙齿的材料行为始终被视为各向同性和均匀的。颅骨的所有材料属性值均采用具有 0.2 或根据经验数据计算的变异系数（CV）的高斯分布进行随机化。拉丁超立方抽样用于确定有限元模型中使用的材料属性值。总共执行了 426 个确定性 FE 模拟。我们在这项研究中检验了四个假设：（1）材料属性值的不确定性对各向同性均匀模型的高应力影响不大，但对高应变影响显著；（2）材料属性值变化对应力状态的影响随着非均匀性和各向异性的增加而增加；（3）Strait 等人报告的体内剪切应变值的变化不仅是由于肌肉力和颅骨形态的变化，而且还由于材料属性值的变化；（4）假设材料属性值的均匀变异系数将导致中等至高应力和中等至高应变随非均匀性和各向异性程度变化的趋势与从经验数据计算材料属性值变异系数时发现的趋势相同。我们的结果支持前三个假设，并否定了第四个假设。当材料属性随常数 CV 变化时，随着非均匀性和各向异性的增加，中等至高应变的变化水平降低，而中等至高应力的变化水平增加。然而，当将从经验数据计算的 CV 应用于材料属性时，观察到的模式并非如此，在颅骨以低水平的非均匀性和各向异性建模时，应力和应变的变化水平最低。因此，当假设材料属性的变化恒定时，中等至高量级的应力和应变的变化水平会产生不准确的趋势。当颅骨以最高精度建模（高非均匀性和各向异性）并且从经验数据中计算材料属性的随机性时，显著应变的变化水平很大（CV=0.369），中等至高量级的应力的变化水平很低（CV=0.150）。这一结果可能对通过自然选择驱动骨重塑和适应的机械信号具有重要意义。
