2.5.2. Bone Properties

The biomechanical behavior of bone is extremely complex due to its anisotropic and viscoelastic nature. However, it exhibits elastic behavior under usual mechanical conditions. The femur being a long bone, the analysis was performed considering the transversely isotropic properties of cortical bone and, according to the literature [53], it has been assumed that the trabecular bone presents a large-scale isotropy. Bone properties were estimated using the apparent density (*ρapp*), which was obtained employing the "Segment Statistics" tool of 3D Slicer® [18] and considering its relationship with the Hounsfield units (HU). Rho et al. [54] determined a linear relationship between HU and apparent density for the proximal femur:

$$
\rho\_{app} = 131/1000 + 1.067 \text{Hll} / 1000 \text{[g/cm}^3] \tag{16}
$$

The Young's modulus of cortical bone in the longitudinal direction (*Ez*,*cortical*) and the stiffness of trabecular one (*Etrabecular*) were estimated using the equation described by Keyak et al. [55] and rectified by Schileo et al. [56]:

$$E\_{z,overall} = E\_{trabeam} = 14.900 \left( 0.6 \rho\_{app} \right)^{1.86} \text{[MPa]} \tag{17}$$

In addition, the Young's modulus (*Ex*, *Ey*) and shear modulus (*Gyz*, *Gzx*) in the transverse direction, for cortical bone, were calculated using Pithioux's laws [57]:

$$E\_x = E\_y = 0.6E\_z \tag{18}$$

$$\mathbf{G}\_{yz} = \mathbf{G}\_{z\mathbf{x}} = \mathbf{0}.\mathbf{25E}\_{z} \tag{19}$$

Poisson's coefficients in the longitudinal (*νyz*, *νzx*) and transverse (*νxy*) directions of cortical bone were obtained from the literature, being 0.25 and 0.4, respectively [58]; the value of 0.3 for the Poisson's coefficient (*ν*) of trabecular bone was taken from experimental data [59]. The shear modulus in the longitudinal direction (*Gxy*) of cortical bone [60,61] and the shear modulus (*G*) of trabecular bone were obtained from the following equations:

$$G\_{xy} = \frac{E\_x}{2\left(1 + \upsilon\_{xy}\right)}\tag{20}$$

$$G = \frac{E}{2(1+\nu)}\tag{21}$$

Table 1 summarizes the physical and mechanical properties of both bones for GC1 and GC2.


**Table 1.** Physical and mechanical properties of the cortical and trabecular bone of each geometric case.
