A kinematic model for calculating cup alignment error during total hip arthroplasty

Reduced range of motion, prosthetic impingement, and joint dislocation can all result from misalignment of the acetabular component (i.e. cup alignment) in patients undergoing total hip arthroplasty. Most methods for acetabular component alignment are designed to provide 45-50 degrees abduction and 15-25 degrees of operative anteversion (also known as flexion) with respect to the anterior pelvic plane coordinate system. Yet in most cases, this coordinate system is not assigned properly, due to differences in patient anatomy and improper positioning in the operating room. This misalignment can result in an error in the cup alignment, which can cause the above-mentioned consequences. This work presents a complete mathematical formulation for the analysis of the inaccuracies related to the anterior pelvic plane axes (APPA) definition and their effect on final cup orientation. We do this by introducing a method taken from Kinematics of Mechanisms, and by representing the errors in the APPA as three concurrent axes of rotation, followed by the version and abduction rotations which are defined relative to the previous rotations. We also present a sensitivity analysis of the results by introducing differential changes between sequential coordinate frames, which simulates the errors in the APPA and their effect on cup orientation. Finally, we demonstrate a computational method which provides corrected version and abduction angles to achieve the desired cup orientation, given that the actual measurement errors are known.