# Biomechanical Study of Proximal Femur for Designing Stems for Total Hip Replacement
## 2 Materials and Methods
### Finite Element Model  
NX 10(r) (Siemens PLM Software Solutions, Plano, Texas, US) was used for performing finite element analyses (FEA) employing its Nastran solver. Each bone was meshed using 3D tetrahedral elements (CTETRA 4 and CTETRA 10 elements from NX) with an overall element size of 1.87 mm, which was selected after performing convergence analyses using both the p-method (progressive increase in the higher order of the shape functions used) and the h-method (progressive increase of the number of elements). Accordingly, tests with different element sizes (0.94 mm, 1.45 mm and 1.87 mm) and with different form functions (4-node and 10-node tetrahedral elements) were performed and the differences in results below 2% were verified. Finally, a compromise in terms of calculation speed and quality of results, with 1.87 mm elements and the 10-node tetrahedral elements, was chosen. Cortical bone was meshed with 75,160 elements and 132,598 nodes, while trabecular bone was meshed with 96,815 elements and 145,684 nodes. Both meshes were bonded using gluing surface-to-surface tools and employing a search distance of 0.2 mm. The FEA was developed considering orthotropic properties of bone, according to previous research about its mechanical features. Elastic properties and yield limits of cortical and trabecular bone were estimated using the apparent density (kg\(\cdot\)m\({}^{-3}\)), which was obtained employing the \"Segment Statistics\" tool of 3D Slicer(r) (see Figure 3) and considering its relationship with the Hounsfield units (HU). This relationship was based on the linear relation described by [PERSON] et al. (2018):  
\[\rho_{app}=0.523 HU+1000 \tag{1}\]  
Figure 3: Trabecular bone segment statistics. Figure 2: (**A**) The segmentation of right femur using 3D Slicer®. (**B**) The process to obtain the trabecular bone using Logical Operators tool. [PERSON]'s moduli (E\({}_{z}\)) and yield limits (\(\sigma_{z}\)) of cortical and trabecular bone in the longitudinal (compressive) direction, were estimated using the equations described by [PERSON] et al. (2016):  
\[E_{z}=14900\rho_{app}^{1.86} \tag{2}\]  
\[\sigma_{z}^{C}=102\rho_{app}^{1.8} \tag{3}\]  
Furthermore, [PERSON]'s modulus and shear modulus in the transverse direction, for both bone types, were estimated using [PERSON]'s laws ([PERSON], 2016). We considered shear modulus for cortical bone to be the same as for trabecular one:  
\[E_{x}=E_{y}=0.6E_{z} \tag{4}\]  
\[G_{yz}=G_{zx}=0.25E_{z} \tag{5}\]  
Shear modulus in the longitudinal direction was obtained from the experimental equation of [PERSON] et al. (2016) that relates it to the transverse shear modulus:  
\[G_{xy}=0.7G_{yz} \tag{6}\]  
Compressive yield limits in the transverse direction depend of longitudinal compressive limit, as also happens with the tensile and shear limits in ith and jth directions, which are related to the compressive ones by the empiric correlation described by [PERSON] ([PERSON], 2016):  
\[\sigma_{x}^{C}=\sigma_{y}^{C}=0.6\sigma_{z}^{C} \tag{7}\]  
\[\sigma_{i}^{C}=2\sigma_{i}^{T} \tag{8}\]  
Figure 3:0  
Table 1 summarizes the elastic properties of both bones. The Poisson's ratios in the longitudinal and transverse directions were obtained from the literature ([PERSON]  
analyzing fracture risk regions and changes in the biomechanics of proximal cortical femur. The general form for Tsai-Wu criteria for transversely isotropic material is:  
\[ \begin{split} f(\sigma)=[F_{1}\sigma_{zz}+F_{2}\Big{(}\sigma_{xx}+ \sigma_{yy}\Big{)}+F_{11}\sigma_{zz}^{2}+F_{22}\Big{(}\sigma_{xx}^{2}+\sigma_{ yy}^{2}\Big{)}+F_{66}\tau_{xy}^{2}+F_{44}\Big{(}\tau_{yz}^{2}+\tau_{xx}^{2}\Big{)}\\ +2F_{12}\Big{(}\sigma_{zz}\sigma_{xx}+\sigma_{zz}\sigma_{yy}\Big{)} +2F_{23}\sigma_{xx}\sigma_{yy}\Big{]}^{1/2}\end{split}  \tag{10}\]  
where \(F_{1}\); \(F_{2}\); \(F_{11}\); \(F_{22}\); \(F_{44}\); \(F_{66}\) are coefficients calculated from the yield limits of cortical bone (Table 3) and \(F_{12}\); \(F_{23}\) are determined experimentally. In this study, they were considered to be equal to zero.