## 4 Conclusions  
In this work, an analytical-based procedure is presented with the aim of predicting the intra-operative fracture risk in the case of THA in a direct, straightforward way. The methodology makes use of a simplified model to predict the mechanical interaction between the prosthesis and the femur bone, with the advantage of resulting in immediate, concise and intelligible maps of fracture risk which we felt, if implemented in a friendly code or as an app, could constitute a practical tool to help the medical decision-making process, in particular with respect to the choice of adopting cemented or cementless implant. The maps were expressed in term of traffic lights color code, obtained by transferring into a direct measure of fracture risk, based on the comparison between in situ von Mises strain and corresponding bone-dependent strain thresholds. The effectiveness of the proposed procedure was confirmed by comparing the obtained results in terms of strain with those from more precise FE models that consider elasto-plastic material model and large displacements. The strain calculation was done with reference to two actual patient cases whose CT data were used to define geometrical and constitutive parameters. The obtained strain maps through the analytical formulation were interpreted following the proposed criteria where red, yellow, and green zones were used as a flag for fracture risk evaluation. In particular, the eventuality the red zone reached the cortical bone is assumed as a high fracture risk index. The strain level distribution was checked by comparison with numerical calculation on a very accurate geometrical finite element model in the literature ([PERSON] et al.2018) that assumes a non-linear constitutive model of the bone tissue whose data are obtained by CT tests. It has to be stressed that the possibility of building up an accurate FE model and performing non-linear calculations has a very high cost in terms of computing time. To build up a 3D model of a femur with a usual PC from CT images and to perform calculation of non-linear model costs an order of magnitude time of days. The analytical approximate solution takes a few seconds. The proposed methodology is able to provide direct information with a limited cost in terms of computational time and interpretation of the results. Due to the simplicity of the implementation, the methodology could be applied to different sections of the femur and extended to a multi-phase model--by dividing the section into more than two annuli--with no significant effort. It has to be noted that both gender and age differences were taken into account in their analyses and in the approach, since the mechanical quality id directly derived from patient-specific CT data: this implies that at least from the mechanical point of view, any mechanically relevant difference in the bone stiffness and strength due to gender or age is automatically considered because determined on the basis of the peculiar bone mineral properties of the selected patient. This is de facto the force of the proposed method. Nevertheless, the initially bone mineral density information and the associated mechanical properties could vary differently in time and with age in male and female: this \"evolutive\" aspect, which would imply to introduce growth and remodeling in the proposed modeling strategy, is obviously beyond the scope of the present paper. Moreover, even if the statistical relevance of the data at this time available is not yet established, the authors focused on the mechanical consistency of the procedure and on its potential as a friendly as well as accurate complementary tool for helping the surgeon in his/her medical choices, postponing the validation of the method to a subsequent study. In conclusion, the proposed analytical procedure gives confident strain maps of the femur sections and are quite comparable to more sophisticated numerical results in the sense to give global information on the possibility of perioperative femur fracture risk.