A survey of formal methods for determining the centre of rotation of ball joints

https://doi.org/10.1016/j.jbiomech.2005.10.002Get rights and content

Abstract

The determination of an accurate centre of rotation (CoR) from segment marker positions is of interest across a wide range of applications, but particularly for clinical gait analysis and for estimating the hip joint centre during surgical intervention of the knee, for limb alignment purposes. For the first time in this survey of formal methods, we classify, analyse and compare different methods (geometric, algebraic, bias compensated algebraic, and Pratt sphere fit methods, as well as the centre transformation technique, the Holzreiter approach, the helical pivot technique, the Schwartz transformation techniques, the minimal amplitude point method and the Stoddart approach) for the determination of spherical joint centres from marker position data. In addition, we propose a new method, the symmetrical CoR estimation or SCoRE, in which the coordinates of the joint centre must only remain constant relative to each segment, thus not requiring the assumption that one segment should remain at rest.

For each method, 1000 CoR estimations were analysed with the application of isotropic, independent and identically distributed Gaussian noise (standard deviation 0.1 cm) to each of the marker positions, to all markers on the segment simultaneously and the two in combination. For the test conditions used here, most techniques were capable of determining the CoR to within 0.3 cm, as long as the spherical range of motion (RoM) of the joint was 45° or more. Under the most stringent conditions tested, however, the SCoRE was capable of best determining the CoR, to within approximately 1.2 mm with a RoM of 20°. The correct selection and application of these methodologies should help improve the accuracy of surgical navigation and clinical kinematic measurement.

Introduction

The determination of joint kinematics during clinical motion analysis often includes assumptions regarding the point about which two segments move relative to one another. The determination of this so-called centre of rotation (CoR) can often be difficult to measure in vivo (Cappozzo et al., 2005; Croce et al., 2005), but knowledge of its exact location is important in clinical gait analysis settings, where the calculation of hip joint moments may form the basis of therapy. In addition, the ability to establish the hip joint centre for determining lower limb alignment axes during surgical intervention (Kinzl et al., 2004) of the knee is becoming increasingly important with the increasing popularity of navigation systems, with accuracy a premium.

Although specific bone landmarks and joint positions can be measured using techniques such as digital roentgen stereophotogrammetric (Vrooman et al., 1998) and video fluoroscopy analysis (Dennis et al., 1998), reflective marker positions determined using infra-red optical systems allow non-invasive measurement of kinematics in real time. During gait analysis or surgery, such markers may be fastened to the skin or more directly attached to the bone segments. The CoR at the hip may then be calculated from the three-dimensional (3D) coordinates of the markers, measured during manipulation of the femur relative to the pelvis. During measurement of these marker positions, the relative motion between the markers and bone (from, e.g. local shifting and deformation of the skin) and the accuracy of the measurement system have been shown to be main causes of error, or artefact (Leardini et al., 2005; Stagni et al., 2005; Taylor et al., 2005). Any computational method for determining the CoR must therefore be able to provide accurate results from noisy marker positions.

The most widely used methods for estimating the positions of joint centres are based on simplified models for the specific joint in question. Most of these approaches then apply empirical relations between externally palpable bone landmarks and the joint centres themselves (Bell et al., 1990; Kadaba et al., 1990; Davis et al., 1991; Frigo and Rabuffetti, 1998; Vaughan et al., 1999). The results of such approaches, termed predictive or regression methods, for human hip joints have been shown to be accurate to approximately 2 cm when compared with X-ray measurements (Bell et al., 1990; Neptune and Hull, 1995).

Recently, so-called formal methods have been proposed that do not refer to empirical correlations. The underlying mathematical approaches can be divided into two categories. The first includes variants of sphere fitting methods, where the centre and the radii of spheres are optimised to fit the trajectories of marker positions (Cappozzo, 1984; Bell et al., 1990; Shea et al., 1997; Silaghi et al., 1998; Leardini et al., 1999; Piazza et al., 2001; Gamage and Lasenby, 2002; Halvorsen, 2003). The second type of approach considers the distance between markers on each joint segment fixed (apart from skin artefacts and measurement errors), to enable the definition of a local coordinate system (Stoddart et al., 1999; Marin et al., 2003; Piazza et al., 2004; Schwartz and Rozumalski, 2005; Siston and Delp, in press). Appropriate transformation of these local systems for all time frames into a common reference system enables approximation of the joint centre at a fixed position. These techniques are considered “coordinate transformation methods”.

A number of studies have compared these different predictive and formal methods (Bell et al., 1990; Seidel et al., 1995; Leardini et al., 1999; Camomilla et al., in press), but reached unclear conclusions, since testing has been performed under different conditions. Many approaches may only be used under the assumption that one body segment is at rest relative to the other. Whilst the appropriate transformation of one set of markers into the local coordinate system of the second is nearly always possible, the measurements of the segment at rest can lead to systematic errors, and the use of techniques that consider this disadvantage is therefore necessary. The goal of this study was to perform a systematic survey of formal techniques for estimating the CoR. In addition, we propose a new method of joint centre determination that is capable of the dynamic assessment of two body segments moving simultaneously.

Section snippets

The virtual hip joint

For appropriate, direct comparisons between different approaches to determine the CoR, including their statistical analyses, numerical simulations in which the exact joint position is known are most suitable. This allows the fast simulation of various geometric situations, marker numbers and placement, and error conditions. We have consequently used a virtual joint with positions that could easily represent markers used during gait analysis or lower limb surgery: using marker sets approximately

Results

For all approaches, the RMS errors increased approximately exponentially with decreasing RoM (Fig. 3, top) when one segment was held stationary and noise was applied independently to each marker on the moving segment to simulate skin elasticity conditions and measurement errors. Since movement was only applied to a single segment, the SCoRE approach reduces to the same formulation as the CTT and the two methods therefore delivered identical results. Under these conditions, the algebraic

Discussion

The ability to accurately determine the CoR is of importance across a number of disciplines, but particularly in the field of orthopaedics, where knowledge of the hip joint centres can lead to improved clinical assessment, kinematic measurement, surgical navigation or positioning and orientation of replacement components. Whilst biomechanical literature has paid considerable attention to the issue of determining the CoR, this is, to the authors’ knowledge, the first time that a complete,

Acknowledgements

This study was supported by a grant of the German Research Foundation number KFO 102/1. Figures were created with the help of the AMIRA software (Stalling et al., 2005).

References (49)

  • K. Halvorsen et al.

    A new method for estimating the axis of rotation and the center of rotation

    Journal of Biomechanics

    (1999)
  • S. Holzreiter

    Calculation of the instantaneous centre of rotation for a rigid body

    Journal of Biomechanics

    (1991)
  • A. Leardini et al.

    Validation of a functional method for the estimation of hip joint centre location

    Journal of Biomechanics

    (1999)
  • A. Leardini et al.

    Human movement analysis using stereophotogrammetry Part 3. Soft tissue artifact assessment and compensation

    Gait & Posture

    (2005)
  • R.R. Neptune et al.

    Accuracy assessment of methods for determining hip movement in seated cycling

    Journal of Biomechanics

    (1995)
  • S.J. Piazza et al.

    Accuracy of the functional method of hip joint center location: effects of limited motion and varied implementation

    Journal of Biomechanics

    (2001)
  • S.J. Piazza et al.

    Assessment of the functional method of hip joint center location subject to reduced range of hip motion

    Journal of Biomechanics

    (2004)
  • M.H. Schwartz et al.

    A new method for estimating joint parameters from motion data

    Journal of Biomechanics

    (2005)
  • G.K. Seidel et al.

    Hip joint center location from palpable bony landmarks—a cadaver study

    Journal of Biomechanics

    (1995)
  • K.M. Shea et al.

    Validation of a method for location of the hip joint centre

    Gait & Posture

    (1997)
  • I. Söderkvist et al.

    Determining the movements of the skeleton using well-configured markers

    Journal of Biomechanics

    (1993)
  • R. Stagni et al.

    Quantification of soft tissue artefact in motion analysis by combining 3D fluoroscopy and stereophotogrammetry: a study on two subjects

    Clinical Biomechanics (Bristol, Avon)

    (2005)
  • D. Stalling et al.

    Amira: a highly interactive system for visual data analysis

  • W.R. Taylor et al.

    On the influence of soft tissue coverage in the determination of bone kinematics using skin markers

    Journal of Orthopaedic Research

    (2005)
  • Cited by (305)

    View all citing articles on Scopus
    View full text