Probabilistic finite element analysis of a craniofacial finite element model.
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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.
