Development of a Mathematical Model to Determine the Stability of Osteosynthesis in Pertrochanteric Fractures

Background and Objectives: Determining the mechanical stability of osteosynthesis in pertrochanteric fractures remains a critical challenge in orthopedic biomechanics. The aim of this study was to develop a mathematical model for quantifying the stability of osteosynthesis and to establish criteria for its evaluation under physiological loading conditions. Materials and Methods: A mathematical model describing the biomechanical behavior of a proximal femur with a pertrochanteric fracture stabilized using a cephalomedullary nail (CMN) was developed. The model integrates force equilibrium, stress–strain relationships, and loading conditions representative of early functional rehabilitation. The theoretical framework was implemented in MATLAB/Simulink R2025b and complemented by finite element analysis to determine stress distribution, deformation patterns, and stability-related parameters of the bone–implant system. Results: The developed mathematical model enabled a quantitative assessment of osteosynthesis stability through the evaluation of key mechanical indicators, including displacement, stress distribution, and safety factor within the fixation system. Critical stress zones in the implant and surrounding bone were identified, allowing analysis of load transfer mechanisms. Finite element simulations showed that improved fixation mechanics reduced peak implant stresses, limited displacement at the fracture site, and increased the safety factor of the fixation construct, resulting in a more uniform load distribution in the surrounding bone and enhanced overall stability of the osteosynthesis system. Conclusions: The proposed mathematical model provides a systematic approach for determining the stability of osteosynthesis in pertrochanteric fractures. It offers a theoretical basis for optimizing implant design and fixation strategies, with potential applications in preclinical evaluation and surgical planning.