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