Bone orientation and position estimation errors using Cosserat point elements and least squares methods: Application to gait

Abstract

The aim of this study was to analyze the accuracy of bone pose estimation based on sub-clusters of three skin-markers characterized by triangular Cosserat point elements (TCPEs) and to evaluate the capability of four instantaneous physical parameters, which can be measured non-invasively in vivo, to identify the most accurate TCPEs. Moreover, TCPE pose estimations were compared with the estimations of two least squares minimization methods applied to the cluster of all markers, using rigid body (RBLS) and homogeneous deformation (HDLS) assumptions. Analysis was performed on previously collected in vivo treadmill gait data composed of simultaneous measurements of the gold-standard bone pose by bi-plane fluoroscopy tracking the subjects' knee prosthesis and a stereophotogrammetric system tracking skin-markers affected by soft tissue artifact. Femur orientation and position errors estimated from skin-marker clusters were computed for 18 subjects using clusters of up to 35 markers. Results based on gold-standard data revealed that instantaneous subsets of TCPEs exist which estimate the femur pose with reasonable accuracy (median root mean square error during stance/swing: 1.4/2.8 deg for orientation, 1.5/4.2 mm for position). A non-invasive and instantaneous criteria to select accurate TCPEs for pose estimation (4.8/7.3 deg, 5.8/12.3 mm), was compared with RBLS (4.3/6.6 deg, 6.9/16.6 mm) and HDLS (4.6/7.6 deg, 6.7/12.5 mm). Accounting for homogeneous deformation, using HDLS or selected TCPEs, yielded more accurate position estimations than RBLS method, which, conversely, yielded more accurate orientation estimations. Further investigation is required to devise effective criteria for cluster selection that could represent a significant improvement in bone pose estimation accuracy.

Introduction

In the process of the estimation of a bone orientation and position (pose) from a marker cluster attached to the skin using optoelectronic stereophotogrammetry, the accuracy is greatly compromised by the relative motion between the skin markers and the underlying bone, which is defined in the literature as the soft tissue artifact (STA) (Leardini et al., 2005). STA is caused by a combination of skin stretching and sliding, muscle contractions, gravity and inertial effects (Leardini et al., 2005, Peters et al., 2010). Commonly used bone pose estimators (BPEs) define an appropriate marker cluster model and match the model with the measured marker trajectories at each time step by solving a least squares (LS) minimization problem constraining the cluster transformation to a translation and a rotation, with or without uniform scaling (Cappello et al., 1996, Challis, 1995, Söderkvist and Wedin, 1993, Spoor and Veldpaus, 1980). This technique has been implemented in numerous studies according to the framework of Procrustes analysis, and will be referred to in this paper as rigid body least squares (RBLS). Alternatively, the transformation can be approximated by a translation and a general non-singular tensor, which permits homogeneous deformation (including stretching and shearing). This approach was implemented in several studies under the framework of affine mapping (Ball and Pierrynowski, 1998, Dumas and Chèze, 2009, Solav et al., 2014), and will be referred to in this paper as homogeneous deformation least squares (HDLS). RBLS and HDLS generally obtain different results (Dumas and Chèze, 2009, Rubin and Solav, 2016, Solav et al., 2014).

The STA effects on a cluster can be decomposed into a rigid and a non-rigid component (Grimpampi et al., 2014). Various studies have shown that the rigid component is greater than the non-rigid component (Andersen et al., 2012, Barré et al., 2013, Benoit et al., 2015). Moreover, the rigid component has been demonstrated to be the only one impacting pose estimation accuracy when using RBLS (Bonci et al., 2015, Dumas et al., 2015, Grimpampi et al., 2014). This explains why commonly used BPEs based on RBLS, which compensate only for the non-rigid component (Cappozzo et al., 1997, Grimpampi et al., 2014, Heller et al., 2011, Söderkvist and Wedin, 1993, Veldpaus et al., 1988), are insufficient to fully compensate for the STA effects (Cereatti et al., 2006, Stagni and Fantozzi, 2009).

A recent work by some of the authors (Solav et al., 2016), presented a method based on the continuum mechanics theory of Cosserat, which evaluated the different poses estimated by all possible combinations of three-marker clusters characterized by triangular Cosserat point elements (TCPEs). At each time step, the strain in each TCPE, and the difference in the orientation and position estimations between pairs of TCPEs, were used to define three scalar TCPE parameters (E, $\varphi$, and T, respectively). These parameters are measured directly from the skin-marker trajectories and do not require gold-standard data. Although the TCPE parameters vanish when the entire cluster moves as a rigid body, and are therefore quantifying the non-rigidity of the entire cluster, the parameters $\varphi$ and T contain information regarding the relative rigid transformation between sub-clusters. Therefore, they should not be strictly included in the same category as other STA non-rigid component definitions. It was hypothesized that TCPEs having small values of the parameters $\{E\text{,}\varphi \text{,}T\}$ are more likely to accurately estimate the underlying bone pose. The correlations between each parameter and the pose estimation errors were evaluated using ex-vivo data measured on the lower limb of three cadavers having 12 skin-markers on the thigh and bone pins as gold-standard (Cereatti et al., 2009). The parameter $\varphi$ showed to be the most correlated with the orientation error and the parameter T with the position error. Therefore, it was concluded that $\{\varphi \text{,}T\}$ have the potential to identify TCPEs which more accurately estimate the bone orientation and position, respectively. However, the method was evaluated on a small ex-vivo dataset, and further research using comprehensive in vivo data was required to better understand whether this approach can be effectively used for developing an accurate BPE.

The goal of this study is twofold: first, to explore using in vivo gait data different criteria for non-invasive cluster selection based on the TCPE approach; and second, to compare the TCPE estimates with two different LS methods. The selection criteria are based upon four independent instantaneous parameters. Three are the above-mentioned parameters $\{E\text{,}\varphi \text{,}T\}$. Since previous studies (Bonci et al., 2014, Camomilla et al., 2015, Camomilla et al., 2013) suggested that STA amplitudes linearly increase with proximal and distal joint kinematics, we hypothesized that the distance between the TCPE location and the most active joint may be used as a criterion for the selection of TCPEs least affected by STA. Therefore, a fourth parameter, named active joint distance parameter (D) was added to the analysis.

The evaluation was performed using gait data collected on 19 subjects, with simultaneous measurements of a stereophotogrammetric system tracking skin-markers and a bi-plane fluoroscopic system tracking the subjects' knee prosthesis as gold-standard reference (Barré et al., 2013, Barré et al., 2015). Since during gait the largest STA occurs on the thigh (Akbarshahi et al., 2010, Andersen et al., 2010, Barré et al., 2015, Benoit et al., 2006, Reinschmidt et al., 1997), the present methodology was evaluated by focusing on femur pose estimation.