## 2 Materials and Methods
### A Straight Definition of Fracture Risk Factor  
A simple and direct estimation of the parameter that defines the fracture risk during the implant is described hereafter. As a measure of the material behavior, in the literature both stress and strain levels within the bone were considered [31; 32; 33; 34]. In this work, as a measure of Fracture Risk Factor (FRF), the von Mises equivalent strain, \(\varepsilon_{vM}(r)\), was calculated as a function of the radius, r, and compared with the admissible values from the literature.  
In particular, the equivalent strain was compared to the yield strain, \(\varepsilon_{Y}\), and the ultimate strain \(\varepsilon_{ult}\) whose values are deduced from literature. In detail, it was assumed that the yield strain was 0.0069 [23] and the ultimate strain 0.0145 [24] for both tissue. The assumed criterion for evaluating the risk factor is therefore set as follows:  
Preliminary classification of the strain level used for defining the admissibility strain domain:  
Level 1. \(\varepsilon_{vM}(r)\leq\varepsilon_{Y}\), the bone tissue is safe;  
Level 2. \(\varepsilon_{Y}<\varepsilon_{vM}(r)<\varepsilon_{ult}\) the bone tissue has to be considered yielded but not crashed;  
Level 3. \(\varepsilon_{ult}<\varepsilon_{vM}(r)\) the bone tissue is crashed.  
The admissibility domain constitutes the set of admissible solutions for local strain and should be integrated into global structural behavior considering the collapse mechanism corresponding to the typical PPFs [35]. In the actual proposed procedure, the PPF is considered to occur when the over yield strain extends from the inner constrained boundary through the tissue and reaches the cortical bone. Finally, as the Fracture Risk, (FR) factor, the level of the cortical bone strain was considered, namely: when the Level 3 of the strain involves completely the tissue up to the cortical zone the Fracture Risk is considered at a high level, when the cortical bone strain belongs to Level 2 the Fracture Risk has to be considered moderate, when the cortical tissue strain belongs to Level 1 the bone can be assumed with a low fracture risk.  
The above-mentioned procedure was applied to the slices located at the middle of each Gruen Zones [36], and the obtained FR classification was expressed through a simple and intelligible \"traffic lights\" color code in order to provide immediate and concise information to surgeons. In other words, the obtained results were plotted along the radius, and colored in green indicating the strain belonged to Level 1, in orange Level 2 and in red Level 3. Moreover, colored plots of the sections were shown in order to provide direct and concise information to the medical operators. We refer to this depiction scheme as \"Fracture Risk Factors\" (Table 1).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Total Von [PERSON] Mechanical Strain** & **Fracture Risk Factors** & **Traffic Lights Color Code** \\ \hline \(\varepsilon_{VM}<\varepsilon_{ult}\) & HIGH RISK OF FRACTURE \\ \hline \(\varepsilon_{Y}<\varepsilon_{VM}<\varepsilon_{ult}\) & LOW RISK OF FRACTURE \\ \hline \(\varepsilon_{VM}<\varepsilon_{Y}\) & SAFE \\ \hline \hline \end{tabular}
\end{table}
Table 1: Traffic Lights Color Code.