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

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Abstract

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).

Introduction

Subjects with a normal hip joint have 120° flexion in common activities of daily living (ADL) (Johnson and Smidt, 1970). The reduced range of motion (ROM) resulting from total hip replacement (THR) leads to frequent prosthetic impingement, which may restrict ADL and cause subluxation and dislocation. It may also increase polyethylene debris and contribute to prosthetic loosening. Therefore, to know the ROM of THR is very important in both short- and long-term clinical situations and in the design of prostheses. Its ROM can be obtained by very expensive Computer Graphics: a 3D CAD model from each manufacturer of each of their total hip designs and sizes (3D-CG) or hip joint models, but it takes a lot of time and labor (Jaramaz et al., 1998; Kummer et al., 1999; Robinson et al., 1997; Scifert et al., 1998; Seki et al., 1998; D’Lima et al., 2000). Our objective is to create a mathematical formula that is able to calculate the ROM of THR in a very easy and accurate way, to reveal the effect on ROM of the oscillation angle and the interaction of ROM with cup abduction, anterior opening and neck anteversion.

Section snippets

Materials and methods

The ROM of THR is governed by the following five factors. (1) Prosthetic ROM (oscillation angle), θ. (2) Cup abduction, α. (3) Cup anterior opening, β. (4) The angle of the neck position from the horizontal plane, a. (5) The anteversion of neck around the vertical axis (long body axis) from coronal plane, b.

The limit of the neck motion due to the impingement in the cup with a flat surface is described as a cone (the prosthetic ROM cone). The vertical angle of this cone is named as prosthetic

Results

  • 1.

    A θ greater than 120° seems to be necessary to fulfill an acceptable ROM. When a was fixed to 52° and b was fixed to 15°, FL and ER were calculated for different oscillation angles (θ=100°–135°) in three cup positions (α=35°,β=10°), (α=45°,β=20°), (α=55°,β=30°). A 120° oscillation angle gives only 99° FL and 80° ER in (α=35°,β=10°), 139° FL and only 28° ER in (α=55°,β=30°), and 119° FL and 51° ER even in (α=45°,β=20°).

  • 2.

    We created the 3D graphs showing the interaction of α and β to FL, ER and EXT

Discussion

These formulas are a very accurate, fast and easy way to get the ROM for a THR with a flat surface cup and a symmetric neck. The oscillation angle is taken from the company data or measured from a blueprint of the cup, head and neck shape. The values of α and β of the cup position and a and b of the neck position can be measured or calculated from an accurate anterior–posterior X-ray view, and an axial CT view of the THR (see Appendix C at Website).

Lewinnek proposed a cup position with

Supplementary data

Acknowledgements

We are most grateful to Ms. K. Ohkuni for the data of ROM of THR by the 3D-CG.

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