## 2 Materials and Methods
### Sensitivity Analysis
The obtained results, in terms of strains versus bone radius, are shown in Figure 3, where the red and green horizontal lines are drawn at the yield and ultimate level, allowing evaluating the position along the radius where the strain overcomes the thresholds. In particular, when the prescribed displacements \(u_{0}\) was less than \(u_{Y}^{T}\), the whole bone strain was contained below the elastic limit and hence the tissue can be assumed to be safe. When \(u_{0}\) was between \(u_{Y}^{T}\) and \(u_{Y}^{C}\), the von Mises strain in the trabecular tissue exceeded the ultimate value indicating the tissue is overstressed at the same time, the cortical tissue was completely preserved. Figure 3: Distribution of radial (**upper left**), circumferential (**middle left**), longitudinal (**lower left**) and Von Mises strain (**right**). The black curves refer to the displacement that produces the strain between yield and ultimate values in the cortical annulus, the orange curves describe the strain produced by displacement that causes the strain between the ultimate and the yield values for respectively trabecular and cortical tissues, the light blue indicates the strain in the range between the yield and the ultimate values within the cortical bone, boldface blue line denoting the effect of the displacement due to the insertion of the stem. The strain components in the cylindrical reference are depicted on the left of Figure 3. It can be noted that the main contribution to von Mises strain, \(\varepsilon_{vM}^{k}\), was given by the in-plane strains, \(\varepsilon_{g8}^{k}\) and \(\varepsilon_{rr}^{k}\). Moreover, the axial strain, \(\varepsilon_{zz}^{k}\), was one order smaller than the other components, and the radial strain, \(\varepsilon_{rr}^{k}\), result negative since the compression due to the insertion of the prosthesis. Comparison of the Fracture Risk Factor Results Obtained by Means of the Analytical and FEM Approaches  
As a first evaluation of the feasibility of the proposed procedure, the data obtained from the CT scan of two patients before the surgery were used [18]. The two case studies refer to patients with different post-surgical outcomes. The first suffered a fracture event during the implant and the second presented successful result. The CT data were used to obtain the mean value of the constitutive properties related to bone density. The shape of the bone was approximated to a hollow cylinder where the cross-section was assumed to have the same area of the actual one. The proposed analytical procedure was then applied to the two case studies and the FRF values were calculated. The results were compared to those obtained by means of FEM approaches from [PERSON] et al. [18]. For both patients, three femoral sections, proximal, medial, and distal, were selected in the middle of each Gruen zone, and the analytical analyses were performed by setting the average geometrical and constitutive properties calculated from CT data for each section, listed in Table 3.  
The results for each Gruen Zone, related to the fractured patient, are illustrated in Figure 4.  
On the left, the FEM results were shown as reported by [PERSON] et al. [18]. In the Figure middle, the von Mises strain was plotted along the radius and, on the right, the FRF was expressed in terms of traffic lights code, calculated by means of the analytical approach. Subsequently, the results were compared to those from the literature [18]. In fact, in this cited study, a patient that underwent an intra-operative periprosthetic fracture and another with a successful outcome were recruited from a cohort of 36 patients undergoing primary THA. In the cited paper, the finite element analyses are performed in order to investigate the distributions of stress and strain inside the femur and to calculate the FRF. The results reported in [PERSON] et al. [18] were obtained by detailed geometric reconstruction of the femurs and the prosthesis.