## 2 Materials and Methods
### Mathematical Modeling
The insights from this modeling approach are crucial for optimizing the design and application of the derotation plate, ensuring that it provides the required biomechanical support while minimizing the risk of complications, such as stress shielding, loosening, or fatigue failure. In the finite element model used in this study, the interactions between the bone, prosthesis, and plate/screws were carefully defined to ensure realistic biomechanical behavior. A frictional contact with a coefficient of 0.3 was applied at the bone-prosthesis interface, allowing limited movement to simulate micromotion during the early postoperative period before full osseointegration. The bone-plate interface was modeled with a surface-to-surface contact using a friction coefficient of 0.42, ensuring realistic load transmission while securing the osteotomized segment with screws to prevent displacement. The screw-bone \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Elasticity Modulus (MPa)** & **Poisson Ratio** \\ \hline Cortical bone & 15,000 & 0.3 \\ Spongy bone & 1000 & 0.3 \\ Plate, prosthesis, screws (titanium) & 110,000 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical–mechanical parameters of the modeled systems. interface was defined as frictional contact with a coefficient of 0.4, with locking screws modeled as interlocking constraints to eliminate micromotion. At the screw-plate interface, a fully bonded contact was applied to simulate the rigid engagement of locking screws within the plate holes, preventing displacement or loosening under loading conditions. These interface conditions allowed for accurate modeling of the stress distribution, load transfer, and mechanical stability within the finite element analysis framework, aligning with validated biomechanical models commonly used in orthopedic research. This mathematical modeling lays the groundwork for further experimental validation and clinical applications, emphasizing the importance of integrating computational methods in the development of advanced orthopedic implants. Computer 3D models are shown in Figure 1.  
The FEA model of the femur with the installed prosthesis and derotation plate represents a critical step in evaluating the biomechanical performance of the system. Using advanced software tools, such as SolidWorks 2023, Autodesk Inventor Professional 2023, and KOMPAS-3D v19.0 (APM FEM), the 3D geometric structure was discretized into a finite element mesh, enabling a detailed stress-strain analysis. The model consists of 52,704 linear tetrahedral elements, providing an accurate representation of the complex geometry and material properties of the system. A total of 86,463 nodes were defined in the mesh, ensuring sufficient resolution for precise calculations. The incorporation of high-resolution meshing facilitates the simulation of load transfer, stress distribution, and displacement behavior under applied physiological forces. This comprehensive modeling approach is particularly important for understanding the interaction among the prosthesis, the femur, and the derotation plate. Such insights are crucial for optimizing the implant design, minimizing mechanical complications, and improving clinical outcomes. The finite element mesh for the prosthesis-femur-plate system is illustrated in Figure 2.  
Figure 1: Computer model of the studied “endoprosthesis–femur–derotation plate” system: (**a**) system view with anatomical context (femur, screws, plate, spokes), (**b**) system view with an emphasis on the structural fixation elements (derotation plate, screws, spokes). The finite element mesh serves as the foundation for subsequent simulations, enabling the evaluation of the biomechanical stability and performance of the implant system under realistic loading conditions. By capturing the intricate details of the system's geometry and material properties, this modeling framework ensures that the results are both accurate and clinically relevant.