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Mathematical formulae to calculate the theoretical range of motion of prosthetic hip implants with non-circular neck geometry

Component design and position are major factors in achieving adequate range of motion post hip replacement. It is therefore important to understand how implant design and positioning impact the motion available before impingement. The Yoshimine–Ginbayashi equations allow the theoretical range of motion of total hip replacements to be calculated accurately and quickly. However, a significant limitation of these equations is that they can only be applied to femoral neck geometry with a circular cross section. The objective of the present study was to reformulate the equations to allow fast and accurate calculation of the range of motion for prostheses with any femoral neck geometry. Using vector analysis, formulae were derived such that the range of motion could be calculated from the cross section of the neck at the point of impingement, cup radius at the point of impingement, cup inclination angle, cup anteversion angle, neck angle of the femoral component from the transverse plane and neck anteversion. The range of motion was calculated for circular, truncated circle and free-form femoral neck geometry under a range of component positions. These values were successfully compared with those measured from analysis of solid models in three-dimensional computer-aided design software. The method of analysis presented offers a powerful new technique enabling hip prostheses to be analysed in an accurate and efficient manner, therefore facilitating optimisation of the design geometry, assessment of existing pre-clinical designs and clinical pre- or postoperative evaluation.