A mathematical formula to calculate the theoretical range of motion for total hip replacement

The reduced range of motion (ROM) resulting from total hip replacement (THR) leads to frequent prosthetic impingement, which may restrict activities of daily living and cause subluxation and dislocation. Therefore, to know the ROM of THR is very important in clinical situations and in the design of prostheses. THR involves a pure ball and socket joint. We created a mathematical formula to calculate the theoretical ROM of THR limited by the prosthetic impingement. The ROM of THR is governed by the following five factors, (1) The prosthetic ROM (oscillation angle: obtained from company data), (2) cup abduction (3) cup anterior opening, (4) the angle of the femoral neck component from the horizontal plane, and (5) the femoral neck anteversion. The last 4 factors are able to be obtained from anterior–posterior, axial X-rays and CT of the patient’s THR. The objective was to create mathematical formulas that could accurately and quickly calculate the ROM of THR. By entering the five values into a computer programmed with the formulas, one could obtain the ROM for the THR. This reveals the effect on ROM of the oscillation angle and the interaction of ROM with cup abduction, anterior opening and neck anteversion. Furthermore this readily would enable a clinical evaluation of the possibility of postoperative dislocation and help in postoperative rehabilitation. The calculated numerical values of ROM by these mathematical formulas were successfully compared with the ROMs obtained from 3-dimensional computer graphics (3D-CG).