Study of the vibrational behaviour of a healing tibia using finite element modelling

Two finite element models of a fractured tibia with healing callus were developed. In the first model, the callus was modelled at the middle of the diaphysis, while in the second one the callus was located at two-thirds of the length, distal from the knee. From these two models the static torsional stiffness as well as the resonant frequencies and mode shapes of the first four vibration modes were calculated for a series of increasing values of Youngʹs modulus of the callus. Two situations were considered. In the first situation, the geometry of the callus was kept constant, while in the second, the dimensions of the callus were reduced while its Youngʹs modulus was increased. The resonant frequencies were found to increase with increasing stiffness of the callus. The single bending modes were found to be more sensitive when the callus was at the middle of the diaphysis, whereas the double bending modes were more sensitive when the callus was situated distally. Mode shapes were similar to those for the intact tibia when the stiffness of the callus was 5% of the stiffness of the intact bone or higher. A basically linear relation was found between the torsional stiffness and the resonant frequencies. A theoretical relation between resonant frequencies and torsional stiffness was evaluated and found to be valid if the Youngʹs modulus of the callus was cqual to or greater than 5% of the Youngʹs modulus of the intact bone. The present results support the quantitative interpretation of vibration analysis measurements for the assessment of tibial fracture healing.