In Vitro Characterisation of Resonant Behaviour of Separated Ni–Ti Endodontic File Fragments Using a Biomechanical Modelling Approach

INTRODUCTION: Instrument separation is a recognised complication of endodontic therapy, particularly with nickel-titanium (Ni-Ti) rotary instruments. Ultrasonic retrieval is widely used; however, its effectiveness depends on efficient transmission of vibrational energy to the separated fragment. Delivering energy closer to the fragment's resonant frequency may enhance energy transfer and fragment mobilisation. The resonant behaviour of fractured Ni-Ti instruments, however, remains poorly understood. This study analysed the resonant frequency of separated Ni-Ti rotary endodontic files using a biomechanical modelling approach. METHODS: Fractured Ni-Ti instruments were modelled as apically fixed cantilever beams. Flexural and longitudinal resonant frequencies were calculated using beam theory. Material properties were derived from the literature, while fragment length and diameter were varied across 8736 simulated configurations generated using a customised computational model. Multiple linear regression analysis assessed the influence of geometric and material parameters on effective resonant frequency. RESULTS: Predicted resonant frequencies ranged from 11 to 116 kHz, exceeding the operating frequencies of most commercially available ultrasonic devices. Geometric parameters, particularly fragment length and diameter, were significant predictors of resonant frequency. Increasing fragment length and decreasing effective diameter consistently reduced resonant frequency. CONCLUSIONS: Fragment geometry primarily determines the resonant frequency of fractured Ni-Ti instruments. Most fragments resonate at frequencies higher than those generated by current ultrasonic systems, suggesting a potential frequency mismatch that may influence retrieval efficiency.