## 2 Materials and Methods
### Sensitivity Analysis  
As a first application, a sensitivity analysis was performed by applying increasing constraints on the inner boundary of the structural section and evaluating the strain and its spreading within the tissue. The reference section was the central section of the Gruen Zone 1. The Zone is situated in the proximal region of the femur where the fractures' phenomena are typically located. The material properties and the geometry of the sample specimen were obtained from a young patient [37]; the case studied represents a typical reference in terms of material densities and constitutive properties for a successful surgical scenario. The cortical and the trabecular radii were \(r_{T}=21\) mm and \(r_{C}=18\) mm, moreover the Young moduli were set equal to \(20,000\) MPa and \(4000\) MPa for the cortical and trabecular tissue. The moduli correspond to apparent densities of \(2\ \frac{\mathrm{g}}{\mathrm{cm}^{3}}\) and \(0.7\ \frac{\mathrm{g}}{\mathrm{cm}^{3}}\), respectively. The yield strain was \(\varepsilon_{Y}=0.0069\)[37] and the ultimate strain \(\varepsilon_{ult}=0.0145\)[24] for both tissue. The radius of the prosthesis was \(r_{0}=8.5\) mm, and the prescribed displacement, \(u_{0}\), producing a volumetric increment equal to \(10\%\), was \(u_{p}=0.4683\) mm. The data of the analysis were collected in Table 2.  
For the analysis of the sensitivity of the formulation, the components of the stress and strain (radial, circumferential, longitudinal and von Mises's) were calculated for trabecular and cortical phases. The analytical approach allows computing what assigned constraint, in terms of prescribed boundary displacement, one should apply for obtaining significant von Mises strain distribution. In particular, it was possible to define two different displacement constraints for each bone tissue zones, i.e.,  
\[ \begin{array}{ll}u_{Y}^{k}:&\varepsilon_{\mathrm{vM}}^{k}=\varepsilon_{Y}\\ u_{ult}^{k}:&\varepsilon_{\mathrm{vM}}^{k}=\varepsilon_{ult}\end{array} \ \ \  
inserted. Moreover, the strain decreases rapidly from the inner stem surface toward the bone cortex. The behavior of the strain allows us to consider that the inmost tissue, subjected to the effect of stem implant will undergo severe values of strain that decreases with an almost exponential way. The high strain level does not represent the indicator of collapse, i.e., a fracture, but a local crisis that is almost inevitable. It has to be stressed that the collapse occurs when the strain overcame the yield limit in an entire portion of the structure till cortex. It is thus important to consider that it is impossible that the strain in the neighborhood of the stem is under the yield value, but this will not affect the prosthesis safety during daily activities. It is worth highlighting that the displacement producing the limit value in the cortical phase, \(u_{Y}^{C}\), was more than two times greater than the prescribed displacement producing the ultimate strain in the trabecular tissue, \(u_{ult}^{T}\), as well as we found that \(u_{ult}^{C}\approx 5u_{ult}^{T}\). In other words, the prescribed displacement \(u_{0}^{PRO}\) due to the insertion of the stem component-assumed to be \(u_{0}^{PRO}=0.4683\) mm-produces strains which overcome the elastic limits of both cortical and spongy bone, while the ultimate strain is attained and overcame only in the trabecular region. This means that the cortical tissue did not undergo permanent strain and, even if high stress levels tend to approach the material threshold, no failure is therein expected, while trabecular tissue appears as at high fracture risk level. As mentioned above, in the sensitivity analysis, several values of displacements at the innermost surface, \(u_{0}\), were prescribed, running from \(u_{Y}^{T}\) to \(u_{ult}^{C}\).