## 2 Materials and Methods
### Sensitivity Analysis
Moreover, the material properties were described using non-linear behavior considering elastic-plastic constitutive properties of the bone as a function of the local density. The results from the FEM analysis performed by [PERSON] et al. [18] were used as a benchmark of the present analytical simplified calculations. The results related to the fractured patient are shown in Figure 4. The FEM output from [PERSON] et al. [18] is reported on the left. The strain maps are reported on the 3D model of the femur and showed that the proximal Gruen Zone 1-7 was interested by strain that reaches the ultimate limit as it was highlighted by the wide red zone. It has to be highlighted that the presence of the red zone within the great part of the cross-section especially with the high strained part of the bone that has reached the external geometrical boundary, \begin{table}
\begin{tabular}{c c c c c c c} \hline  & \multicolumn{5}{c}{**Fractured Patient**} \\ \hline  & \multicolumn{3}{c}{**Cortical Bone**} & \multicolumn{3}{c}{**Trabecular Bone**} & \multicolumn{3}{c}{**Prosthesis**} \\ \cline{2-7}  & \multicolumn{1}{c}{**Young Modulus**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Young Modulus**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Prescribed**} \\  & \multicolumn{1}{c}{**(MPa)**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Displacement (mm)**} \\ \hline Proximal & 6784.2852 & 17.6796 & 4799.6436 & 15.4339 & 8.4708 & 0.4666 \\ \hline Medical & 8166.8600 & 13.7304 & 7438.5087 & 12.0371 & 5.4743 & 0.3016 \\ \hline Distal & 8288.6042 & 13.8508 & 7516.2335 & 11.5322 & 3.6661 & 0.2020 \\ \hline  & \multicolumn{3}{c}{**Un-Fractured Patient**} & \multicolumn{3}{c}{**Un-Fractured Patient**} \\ \hline  & \multicolumn{3}{c}{**Cortical bone**} & \multicolumn{3}{c}{**Trabecular bone**} & \multicolumn{3}{c}{**Prosthesis**} \\ \cline{2-7}
**Slice** & \multicolumn{1}{c}{**Young Modulus**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Young Modulus**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Prescribed**} \\  & \multicolumn{1}{c}{**(MPa)**} & \multicolumn{1}{c}{**(MPa)**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Radius (mm)**} & \multicolumn{1}{c}{**Displacement (mm)**} \\ \hline Proximal & 7611.5127 & 21.2760 & 4715.3445 & 18.0206 & 8.4126 & 0.4634 \\ \hline Medical & 8829.9168 & 15.5881 & 7927.6850 & 12.8490 & 5.5006 & 0.3030 \\ \hline Distal & 9221.9785 & 14.1060 & 8680.2633 & 11.7860 & 3.6128 & 0.1990 \\ \hline \end{tabular}
\end{table} 
Table 3: Geometric and constitutive properties for fractured and un-fractured patients, obtained from CT data. can be assumed as an index of a high risk of fracture, a fact confirmed by the observation that the patient actually presented intraoperative fracture. Besides the FEM results, the actual calculation results are represented as a strain diagram versus the radius of the section of the bone. It can be seen that even with the simplified approach, a high risk of fracture was evaluated for the proximal section of the femur. In fact, the von Mises strain resulted greater than the ultimate strain value for the whole bone tissue. On the right, a schematic image of the cross-section was shown where a color code highlighted the FRF: therein, red saturated the section, this meaning that the bone district was subjected to high risk of fracture. In addition, it can be seen that in the Gruen zone 2-6, while the trabecular bone resulted at a high level of strain, in the cortical bone it was low hence it can be assumed that the risk of fracture in the zones is low as well. In the Gruen zone 3-5, the cortical bone resulted safe, while the trabecular bone presented a high risk of fracture around the prosthesis and a low risk of fracture extending from the half part of the trabecular bone until the interface between trabecular and cortical bone.