## 2 Materials and Methods
### Mathematical Modeling  
Mathematical modeling plays a critical role in understanding the SSS and mechanical behavior of complex, multicomponent systems with heterogeneous structures. The FEM is widely regarded as the most reliable and effective numerical approach for these analyses. FEM divides a 3D geometric model into discrete finite elements of simple shapes, which collectively form a continuous finite element mesh. The iterative nature of FEM ensures that solutions--such as stresses, deformations, and deformation rates--calculated at each step serve as the basis for subsequent computations, providing a dynamic and precise understanding of the system's behavior under applied loads. For this study, the FEM analysis was implemented using software platforms, such as KOMPAS-3D (APM FEM), Autodesk Inventor PRO, and SolidWorks. These tools were employed to construct and evaluate a detailed model of the biotechnical system, comprising the prosthesis, femur, and derotation plate. The physical and mechanical properties of the system components were sourced from established literature to ensure accurate simulations. Table 1 presents the physical-mechanical parameters of the modeled system's components. The properties included the elasticity modulus and Poisson ratio, which are critical for accurately representing the behavior of cortical bone, spongy bone, and titanium components under loads. These parameters serve as the foundation for the finite element model, ensuring that the biomechanical properties of each material are accurately represented in the simulations. To replicate physiological conditions, the main functional load was modeled as a vertical force (F1 = 1000 N) applied to the center of the femoral head (or prosthetic head). Additionally, a horizontal force (F2 = 200 N) was included to analyze rotational displacements. These forces were selected to simulate typical biomechanical stresses experienced by the hip joint during weight-bearing activities. Using the finite element method, the stress-strain state and biomechanical behavior of the system--comprising the prosthesis, femur, and derotation plate--were evaluated under these loading conditions. The simulation provided insights into the load distribution, stress concentrations, and overall mechanical performance of the system. In this study, a safety factor analysis was performed to assess the mechanical reliability of the prosthesis-femur-derotation plate system under physiological loading conditions. The safety factor (SF) was determined using the von [PERSON] stress criterion, where the yield strength of each material was compared to the maximum stress observed in the finite element (FE) analysis. The SF was calculated using the following Equation (1):  
\[SF=\frac{\sigma_{yield}}{\sigma_{max}} \tag{1}\]  
The SF was calculated using the von [PERSON] stress criterion, where \(\sigma_{yield}\) represents the yield strength of the material and \(\sigma_{max}\) denotes the maximum von Mises stress obtained in the simulation. For this study, the following yield strength values were used: titanium (plate, screws, prosthesis)--900 MPa, cortical bone--130 MPa, spongy bone--5 MPa. The SF distribution was mapped across the implant system to identify potential failure zones. Regions with SF > 1 indicate safe stress levels, whereas SF < 1 suggests areas at risk of mechanical failure under loading conditions. This analysis is essential to confirm that the implant design meets the mechanical safety requirements for clinical applications. The table underscores the distinct mechanical behaviors of the materials involved in the system. The high elasticity modulus of titanium ensures stability and durability for the derotation plate, while the lower modulus values for cortical and spongy bone reflect their respective flexibility and load-absorbing properties. These distinctions are critical for ensuring the compatibility of the implant with the surrounding biological structures and minimizing stress shielding.