Elastic Properties of Human Osteon and Osteonal Lamella Computed by a Bidirectional Micromechanical Model and Validated by Nanoindentation

Abstract

Knowledge of the anisotropic elastic properties of osteon and osteonal lamellae provides a better understanding of various pathophysiological conditions, such as aging, osteoporosis, osteoarthritis, and other degenerative diseases. For this reason, it is important to investigate and understand the elasticity of cortical bone. We created a bidirectional micromechanical model based on inverse homogenization for predicting the elastic properties of osteon and osteonal lamellae of cortical bone. The shape, the dimensions, and the curvature of osteon and osteonal lamellae are described by appropriately chosen curvilinear coordinate systems, so that the model operates close to the real morphology of these bone components. The model was used to calculate nine orthotropic elastic constants of osteonal lamellae. The input values have the elastic properties of a single osteon. We also expressed the dependence of the elastic properties of the lamellae on the angle of orientation. To validate the model, we performed nanoindentation tests on several osteonal lamellae. We compared the experimental results with the calculated results, and there was good agreement between them. The inverted model was used to calculate the elastic properties of a single osteon, where the input values are the elastic constants of osteonal lamellae. These calculations reveal that the model can be used in both directions of homogenization, i.e., direct homogenization and also inverse homogenization. The model described here can provide either the unknown elastic properties of a single lamella from the known elastic properties at the level of a single osteon, or the unknown elastic properties of a single osteon from the known elastic properties at the level of a single lamella.

Effects of geometry and loading on the elastic behavior of a single osteon via analytical structural mechanics approaches

“…The homogenized procedures were applied to several sets of elastic constants of collagen fiber available in the literature to estimate the equivalent elastic properties of the secondary osteon. For both DMO and OMO models the elastic constants proposed by Korsa et al 56 at the lamellae scale show better agreement with reported experimental data on the osteon at the micro-level.…”

“…As shown in Figure 4 , the results of case 2 do not meet the second criterion, while tha t of case 5 fall within the experimental ranges. It can be concluded that for both of the theoretical models of an osteon, the mechanical properties of the mineralized collagen fibril sheet proposed by Korsa et al 56 are closely matched with the experiments. Then, this set of mechanical properties is taken for the parametric studies.…”

“…The great discrepancies among predicted elastic moduli and anisotropy ratios results in Figure 4 may be attributed to either different compositional assumptions or homogenization techniques applied in previous works. [52][53][54][55][56] First, the predicted axial moduli for the five sets of elastic constants summarized in Table 1 are compared with those reported by Ascenzi and Bonucci 60 who conducted extensive experimental studies on mechanical properties of the osteon. They found that the axial modulus ranges from 4.81 GPa to 17.46 GPa, thus the results for cases 1, 3, and 4 considerably overestimate the actual modulus.…”

“…At the sub-micro level, that is, single lamellae level, some of the important predictions relevant to the effective elastic constants can be found in references [52][53][54][55][56] which are listed in Table 1.…”

Different diameters of titanium dioxide nanotubes modulate Saos-2 osteoblast-like cell adhesion and osteogenic differentiation and nanomechanical properties of the surface

“…Thus, the cells are in contact with biocompatible material of same stiffness as an osteointegrating bone. [59][60][61] The stiffness increased toward to bulk cpTi material. We can assume that by removing the mechanically incompatible transition we extend the life of the implant by limiting the micro-motions occurring on the ramp change of stiffness at the interface between bone and bulk titanium.…”