 Research Article
 Open Access
3D Elbow Kinematics with Monoplanar Fluoroscopy: In Silico Evaluation
 L. Tersi^{1}Email author,
 S. Fantozzi^{1} and
 R. Stagni^{1}
https://doi.org/10.1155/2010/142989
© L. Tersi et al. 2010
 Received: 1 May 2009
 Accepted: 14 September 2009
 Published: 21 December 2009
Abstract
An insilico assessment of the performance of 3D videofluoroscopy for the analysis of the kinematics of long bones is proposed. A reliable knowledge of invivo joints kinematics in physiological conditions is fundamental in the clinical field. 3D videofluoroscopy theoretically permits a mm/deg level of accuracy in joint motion analysis, but the optimization algorithm for the pose estimation is highly dependent on the geometry of the bone segment analyzed. An automated technique based on distance maps and tangency condition was applied to the elbow bones. The convergence domain was explored to quantify and optimize measurement accuracy in terms of bias and precision. By conditioning the optimization algorithm using simple image features, the estimation error had small dispersion (interquartile range within 0.5 and 0.025 mm/deg for outofplane and inplane pose parameters, resp.), but with occasional bias and outliers. 3D videofluoroscopy produced promising results for the elbow joint, but further invitro validation studies should be carried out.
Keywords
 Bone Model
 Unscented Kalman Filter
 Bone Segment
 Intact Knee
 Convergence Domain
1. Introduction
3D videofluoroscopy is a technique for the evaluation of joint kinematics based on the alignment of 3D models of bones or prostheses and series of 2D radiographic images representing the relevant monoplanar or biplanar projections [1]. The joint kinematics is reconstructed calculating, independently for each videoframe, the 6 degrees of freedom (DOF) absolute pose (3 translations and 3 rotations) of each body segment, and then calculating the 6 DOFs of their relative pose.
Invivo knee tasks, such as squat, stair climbing, chair raising and sitting or step updown, were widely analysed with 3D videofluoroscopy in replaced [10–13] and intact knee [14]. 3D videofluoroscopy was also applied to quantify the invivo kinematics of ankle [15] and hip [16, 17] joints.
To estimate the 6 DOF of a bone segment in a frame acquired by videofluoroscopy, a 3D model of the bone is virtually moved until it is best aligned to the relevant 2D image. This automatic procedure is typically carried out by means of an iterative optimization algorithms. Different metrics have been used to quantify a cost or a fitness function for the optimization such as: (i) the euclidean distance between the contour of the virtual projection of the model and the contour extracted from the fluoroscopic image [8, 18], (ii) the root mean square distance between the projection lines and the model surface [2], (iii) similarity measures between the fluoroscopic image and digitally reconstructed radiographies [19–23].
Promising accuracy levels have been reported for the intact knee joint: 0.23 mm for translation, 1.2 deg for rotation with biplanar fluoroscopy [22]; and 0.42 mm for inplane and 5.6 mm for outofplane translations, 1.3 deg for rotation for monoplanar fluoroscopy [24]. However, these accuracies cannot a priori be considered valid for the other joints. Even if biplanar fluoroscopy is more robust, the present work focused on monoplanar fluoroscopy because it can investigate bigger volume with smaller Xray dose for the patient.
The performance of 3Dfluoroscopy is affected by the geometry of the bone segments analysed, and its accuracy could vary considerably because local minima, caused by symmetries of the models surfaces, or by occlusions, could severely interfere with a correct estimation of the pose. Therefore, the technique is highly dependent on the initial guess of the pose for the optimization which is typically specified manually. However, the extent to which the intervention of the operator can affect the final reliability of the pose estimation has not been clarified yet. Therefore, 3D videofluoroscopy might currently still be operatordependent [18] and the application of 3D videofluoroscopy is still too cumbersome to be suitable for routine clinical practice. The high potential of the method for routine clinical applications cannot be exploited without a concrete automation of the procedure, involving an automatic estimation of the initial guess of the pose. Therefore, the convergence domain of the optimization must be characterized in detail, exploring how different algorithms behave around reference poses.
Currently, no fluoroscopic methods have been applied to the invivo kinematics of the elbow joint. This joint has, however, been characterized exvivo [25], with Roentgen Stereophotogrammetric Analysis (RSA) [26, 27], and electromagnetic tracking systems [28], as well as invivo with noninvasive technique such as infrared stereophotogrammetry [29], or nuclear magnetic resonance (MRI) [30]. Infrared stereophotogrammetry, however, suffers accuracy limits due to soft tissue artifacts (20–48% of loss of rotational motion of the upper arm [31, 32]), while MRI fails to detect the effect of the active contribution of muscles to the motion. Even though fluoroscopy has not yet been applied to the elbow, this joint is of particular interest for its validation because it is characterized by (i) a high degree of bone superimposition, (ii) being composed by thin long bones (in contrast with the typical morphology of knee prosthesis), and (iii) marked longitudinal cylindrical symmetries (especially for the radius). These aspects make the fluoroscopic analysis of the intact elbow considerably difficult, thus particularly suitable for a validation study.
To significantly improve the quality and the robustness of 3D videofluoroscopy, it is necessary to understand the effect of the various sources of errors on the final accuracy. For this purpose, an analytic approach is necessary in order to find appropriate solutions to each single defect of the method.
In the present study, we analysed, by means of computer simulation, the convergence property of a modified version of the pose estimation algorithm proposed by Lavalleé and Szelinsky [33] and based on adaptive distance maps (ADM), in order to better understand the influence of local minima and to optimize the pose estimation in terms of accuracy and precision. This algorithm was chosen because of its light computational weight and because it permits to achieve good accuracies even with incomplete contours [2], which can arise from occlusions or image blurring due to the bone motion.
In this evaluation study we considered (i) the geometric characteristic of the bone models, (ii) the resolution of the fluoroscopic projections, and (iii) the resolution of ADM as the only sources of errors. Confounding effects caused by the geometric distortions, by errors in the calibration of the fluoroscopic models, or by the incompleteness of the bone contours were disregarded in the present study.
The aim of the present study was to investigate the suitability of 3D videofluoroscopy for the analysis of elbow bones kinematics, through a detailed exploration of the convergence domain of the minimization algorithm, in order to quantify and optimize measurement accuracy in terms of bias and precision.
2. Material and Methods
2.1. Pose Estimation Algorithm
An established technique was implemented to estimate the 3D pose of an object of known 3D geometry given its monoplanar fluoroscopic projection [2]. The algorithm was originally proposed by Lavallée and Szeliski [33] for biplanar projection, and is based on 3D adaptive distance maps. In brief, (i) the fluoroscope is virtually modelled with a perspective projection model; (ii) the 3D pose estimation is obtained with an iterative procedure that finds the best alignment between a bone surface model and its 2D fluoroscopic projection (typically a 1024 1024 DICOM image). In the present study, the bone surface was modelled with triangles meshes, however different representation can be used (i.e., cloud of points, NURBS). The resolution of the meshes was not relevant, because the images for the alignment were generated insilico projecting exactly the same models. There was, thus, a perfect correspondence between the 3D model and the projection independently from the meshes properties. The quality of the alignment is represented by a cost function defined as:
RMSD is the root mean square distance between the surface of the model positioned in the pose = ( , , , ) and projection lines . The projection lines represent the Xrays that generated the edge points of the bone segment projection extracted by a Canny edge detector [34] in the fluoroscopic image and is expressed in parametric form
where = is a point representing the Xray source position (focus), = is the th of the points of the contour, both expressed in the fluoroscopic system of reference, and is their distance:
To quantify RMSD, is sampled and, for each sampling point , the distance from is computed. The distance of the projection line from the surface is then defined as the minimum distance among those of the line sampling points:
The best alignment condition is finally identified finding the values of the pose p that minimize the RMSD with the Levenberg–Marquardt (LM) method [35]:
The VXL [36] implementation of LM was used in the present work.
For a faster quantification of the distance between the line and the model surface, and to define the sampling step of , adaptive distance maps (ADM) of the models surface were precomputed and stored. Briefly, the ADM is an octreebased representation of an object [37]. In this representation, the volume outside and inside the surface of the object is nonuniformly discretized. The map assigns to each point of the discretization the corresponding signed distance from the surface of the model: positive if outside, negative if inside the object. The distance is computed as the minimum distance between the discretization point and the surface of model of the bone. The structure of the ADM is an octree which is built with an iterative procedure which subdivides a cube (also called octant) iteratively in other 8 halfside octants only if it contains at least one point of the mesh. The octree is then refined to avoid discontinuities between the levels of subdivision of two adjacent octants. The vertices of the octants are the volume discretization points. The distance of a generic point from the surface is then computed with a trilinear interpolation of the distances of the 8 vertices of the smallest octant containing the point. The octant side dimension gets smaller closer to the surface, thus the interpolation error becomes negligible. In the present work, the resolution of the octree (smallest octant side) will be referred as DMR (Distance Map Resolution). For a further improvement of the algorithm speed, also the sampling step of the projection lines is adaptive. The sampling step varies accordingly to the local resolution of the ADM and gets smaller closer to the surface. If is the side of the smallest octant containing the sampled point then the next point to evaluate the distance will be
Finally, is resampled around the closest point to the surface with a uniform step length ten times smaller than DMR.
A global reference frame was defined with the and axis parallel and axis perpendicular to the image plane, with the origin in the centre of the image plane. The Euler convention was used for rotations. The field of view was represented with a diameter of 400 mm. The Xray source was virtually placed in = (0, 0, 1000) mm, representing a typical distance of a standard fluoroscope, and pixel spacing was fixed at 0.34 mm. The effects on the final accuracy of the errors associated to the identification of the principal point (  coordinates of the Xray source) and principal distance (  coordinate of the Xray source) were disregarded in the present study as already quantified elsewhere [38]. The Xrays were represented by a beam of straight lines and the effect of the geometrical image distortions, caused by the electronics of the image formation chain of real fluoroscopes [39], was not considered in the present study because, dealing with real images, the geometrical distortion can be efficiently corrected using a global spatial warping techniques [40, 41].
In the implementation of the LM minimization algorithm, 3 parameters must be specified: (i) the convergence tolerance on the RMSD ( ), (ii) the convergence tolerance on the 6 DOF of the pose ( ), and (iii) the step length for forward (FD) Jacobian computation (eps).
2.2. Algorithm Convergence Properties
High resolution models of humerus, ulna, and radius were downloaded from the official site of the European project VAKHUM (contract #IST199910954 [42]) and used in the performed simulations. For each model two ADM were calculated and stored with DMR equal to 0.5 and 1 mm.
where is the value of the generic pose parameter correspondent to the minimum of RMSD.
A detailed convergence domain analysis was carried out for different sets of simulation parameters. The minimization was started from 1000 randomlychosen initialcondition poses among 117649 permutations obtained varying the translations ( , , ) and rotations ( , , ) between the values , , , 0, 3, 7, 10 mm/deg, respectively, around the reference pose. The simulations parameters were initially varied using different values for DMR (0.5–1 mm), and eps ( mm/deg) while and were both kept fixed at mm/deg.
2.3. Algorithm Conditioning
Different authors stressed that the minimization of RMSD is affected by local minima and by the large differences in the sensitivity to the various DOF [13, 24]. If the minimization starts from initial conditions inside a local minima basin, the pose estimation will be incorrect. Two different solutions were implemented and compared in order to better deal with the problem of local minima.
The first, proposed by Fregly et al. [24], involved the sequential (seq) minimization of the DOF in order of sensitivity. For this purpose, three groups of DOF were formed: (i) inplane pose parameters ( , , ); (ii) outofplane rotations ( , ); (iii) outofplane translation (Tz). After this three sequential minimization, the RMSD was finally further minimized with respect to the 6 DOF simultaneously.
The second solution (feat) involved the use of features extracted from the image to get closer to the real pose before starting the minimization. Two features were calculated on the bonecontour points: (i) the direction of maximum variance of the distribution, and (ii) the farthest point from the field of view border among the projections of the bonecontour point along the maximum variance axis. The first feature was used to evaluate a first guess for the bone model orientation around the projection axis ( ) , while the inplane translation components ( , ) were estimated using the second feature. The three DOF were modified iteratively until convergence, and then the minimization was started as previously described.
The analysis was repeated for each bone model, using seq, feat and seqfeat together, with 2 values of eps ( mm/deg) and DMR = 0.5 mm.
2.4. Data Analysis
For each set of parameters, the final deviations between estimated and reference poses, and the relevant residual RMSD were quantified. Bias and precision of the final estimates of the pose were quantified calculating for each DOF the median (m) and the interquartile range (iqr). For bias results, a Student's test ( ) was performed to determine if the values were statistically different from zero, indicating the presence of a systematic error. Moreover, to measure how outlierprone the distributions were, the kurtosis ( ) was also calculated.
To investigate the effects of the different minimization parameters (DMR, eps, seq, feat, bone models and initial deviations) on the final estimates and RMSD, an nway analysis of variance (ANOVA) was performed considering a significance level = 0.05 (with Bonferroni adjustment for multiple comparisons).
3. Results
3.1. Sensitivity Analysis
The cusp shape of RMSD with respect to each DOF, except , was verified, while with respect to , RMSD showed a rounder trend. In particular the trend of the cost function for the humerus is shown in Figure 3. Similar trends were obtained also for the ulna and the radius models.
Sensitivity of the RMSD to the variation of the 6 DOF.
Model  [mm ]  [mm ]  [mm ]  [mm ]  [mm ]  [mm ] 

humerus  5.5 −1  9.6 −2  1.5 −3  8.2 −3  1.6 −2  8.0 −1 
radius  5.7 −1  7.6 −2  0.5 −3  1.0 −3  0.3 −2  9.2 −1 
ulna  5.3 −1  1.6 −1  1.5 −3  5.0 −3  2.0 −2  8.4 −1 
3.2. Distance Map Resolution
Effects of the distance map resolution on the pose parameters estimates for the radius model.
DOF  DMR = 1 mm  DMR = 0.5 mm  

 iqr 

 iqr 
 
[mm]  0.03  0.01  20.2 
 0.01  50.7 
[mm]  0.26  0.75  8.4  0.21  0.48  9.7 
[mm]  0.79  7.15  16.6 
 4.41  47.3 
[deg]  0.05  8.5  76.8  0.11  5.5  74.1 
[deg]  0.22  0.12  140.2  0.19  0.1  293.7 
[deg]  0.02  0.02  307.8  0.01  0.02  866.2 
3.3. Sequential Alignment
Using eps= mm/deg, the optimization of , and resulted in median 0.04 mm/deg,iqr 0.10 mm/deg even if had a large number of outliers. had a median 0.1 mm/deg, withiqr 0.50 mm/deg: Figure 4 shows the outliers gathered together only for positive values of . Especially for the radius, the minimization process of and tended to prematurely stop during its descent towards the global minimum, resulting in large dispersions (median 0.05 mm/deg,iqr = 5.10 deg and iqr = 4.10 mm, respectively).
Increasing the step eps to mm/deg, for all the three models, the median values for and were smaller than 0.08 mm/deg, with a dispersion lower than 0.45 mm/deg.
3.4. Features
3.5. Features and Sequential Alignment
Final accuracy in terms of median ( ), interquartile range (iqr) and kurtosis ( ) of the pose estimations, for the three bone models with the seqfeat alignment.
DOF  Humerus  Ulna  Radius  

 iqr 

 iqr  k 
 iqr 
 
[mm]  0.003  0.004  14.0  0.005  0.008  3.0  0.021  0.006  4.8 
[mm]  0.087  0.005  73.7  0.018  0.025  2.2  0.051  0.025  3.1 
[mm]  0.766  0.153  5.0  0.132  0.155  3.3  0.692  0.260  3.1 
[deg]  0.045  0.090  4.9  0.071  0.474  2.0  0.017  0.317  3.1 
[deg]  0.011  0.032  5.6  0.044  0.086  3.2  0.195  0.073  4.2 
[deg]  0.001  0.011  26.8  0.007  0.019  2.9  0.007  0.003  4.9 
4. Discussion
In this work, a sensitivity analysis and a convergence domain analysis of the minimization algorithm for the pose estimation in 3D videofluoroscopy were addressed.
The sensitivity analysis showed that the cost function (RMSD) varies differently with each DOF: the inplane pose parameters have a sensitivity at least one order of magnitude larger than the outofplane pose parameters. Moreover, the performed simulation showed that the cost function could have different behaviours depending on the analysed segment (Figure 3): considering , we found a clear local minimum for the humerus but not for ulna and radius. For all three models, the RMSD with respect to showed an asymmetric trend. That is due to the fact that only a partial part of the bone is included in the imaging field (Figure 2), thus, given the long diaphysis of the bone, two different scenarios occur while moving the model along the  axis: if the models moves further into the imaging field, the projection lines coming from the bone contour points intersect or pass near to the model surface, continuing to give only a little contribution to the increment of the cost function. On the other hand, if the model moves further out of the imaging field, there is no model surface for part of the projection lines to intersect with, increasing their contribution to the cost function. The findings about the RMSD explained the behaviour of the unconditioned LM algorithm, which was found to be noticeably sensitive to the local minima of the RMSD. Given the longitudinal cylindrical symmetry, the estimations of the pose of long bones were affected by large dispersions not only for , as previously thoroughly reported for the knee prostheses, but also for (see Table 2). Moreover, for the DOF with relative small iqr such as the inplane pose parameters, the values of were high ( 8), that is the distributions were affected by large number of outliers. The ANOVA confirmed the hypothesis that the higher is the resolution of the distance map (DMR) the more the accuracy and the precision of the technique increase ( ). However, given the limits of the resolution of the fluoroscopic image and of the surface model mesh, a further increasing of DMR would be unnecessary. For the unconditioned minimization, varying the step for the FD Jacobian computation (eps) from to mm/deg induced a further instability.
To solve the convergence problems of the LM algorithm, the effect of the seq alignment on the minimization process was evaluated. When the initial conditions are too far from the reference pose, if not conditioned, the pure algorithm tries to explore the value of the cost function varying all the 6 DOF simultaneously, risking to move the less sensitive DOF away from the global solution. With the seq alignments, instead, the DOF with larger convergence domain are aligned in a first step, while the more critical (outofplane) DOF are minimized only when closed to the reference pose. Although this technique leads to an improvement of the precision of the estimate (Figure 4), the algorithm is still sensitive to local minima, or to local lowsensitivity areas of the cost function that can occur also for the inplane pose parameters, such as for (Figure 3). This could be the cause for the still high number of outliers (mean of ) obtained with the seq alignment. Varying eps from to mm/deg permitted to improve the performance of the optimization avoiding the early convergence for the less sensitive DOF ( and , Figure 4). However the problems of convergence are not completely solved, probably because two different DOF could have a correlated effect on the RMSD, thus, a sequential minimization could interfere with a proper descending to the global minimum.
The use of features completely avoided the problems of the seq alignment, because, differently from seq, the feat alignment is completely independent from the RMSD and, thus, from its local minima. For our simulations, we used two simple features: (i) the direction of maximum variance of the distribution of bonecontour points to extrapolate , and (ii) a characteristic point such as the farthest from the field of view border for , . These features are particularly suitable for long bones, which cannot be completely included in the field of view; for short bones or prosthesis, however, the mean of the coordinates of the contour points can equally be used. These minimization settings, together with a FD Jacobian step eps = (bigger enough to filter small noisy fluctuation of the RMSD), permitted to significantly reduce , iqr, and .
Even if the combined effects of feat and seq did not introduced further improvements ( value 0.05), with a fine tuning of the minimization parameters, a high level of precision can be achieved (iqr 0.025 mm/deg for inplane pose parameters, iqr 0.5 mm/deg for outofplane pose parameters) but with a high variability between the models (i.e., considering , iqr equal to 0.09 deg for the humerus, and equal to 0.47 deg for the ulna), confirming the hypothesis that the performances of the method should be assessed for each bone model to be analysed. The results also showed that the final estimate is biased. This is due to the intrinsic limitations of the technique given by the resolution of the fluoroscopic projections and of the distance maps. Moreover, in spite of the fine tuning of the optimization parameters, the local minima showed in Figure 3 of the RMSD versus for the humerus, seldom caused the LM minimization algorithm to detect false poses. These false poses can generally be easily identified by an operator with a visual feedback of the alignment and, in such cases, the minimization can be repeated starting from a different initial condition.
The results of this study confirm that the accuracy and the precision that can be achieved, especially with the feat alignment, allow the technique to be suitable for the kinematic study of the elbow, but without excluding the complete independence by the operator.
5. Conclusions
In the present work, we proposed an analytical process to evaluate the performance of 3D videofluoroscopy for its application to the analysis of kinematics of long bones, by means of insilico simulations. The effects of the dominant sources of error such as bone symmetries, distance map resolution and image spacing, were investigated. Solutions were proposed to improve the accuracy and the precision of the method.
Given the high variability of the morphology of the bones or prostheses that could be analysed with 3D videofluoroscopy, different performance assessment studies should be carried out before undertaking any new application of the technique, especially for clinical purpose. We focused on the elbow because it allowed to investigate different characteristics of 3D videofluoroscopy applied to long bones. Compared to knee prosthesis, which have been deeply studied during the past years with 3D fluoroscopy [10–12], the marked cylindrical longitudinal symmetries of long bones constitute a dominant disturbing factor that could interfere with the final accuracy of the technique.
The robustness of the alignment algorithm applied to the elbow joint was completely characterized. Even if the convergences to local minima was not completely avoided, with a proper conditioning and a fine tuning of the minimization algorithm parameters, excellent results can be achieved in term of low bias and high precision. Moreover, the methods and the findings addressed in this work focusing on monoplanar projections, can easily be extended also to biplanar 3D videofluoroscopy.
Again, all these considerations are to confirm that for application to the kinematics of any particular joint, a detailed validation study should be carried out, especially for clinical studies. The accuracy and the precision achieved with the feat alignment, allow the technique to be suitable for the kinematic study of the elbow, and most likely even of other long bones, however further invitro validation must be carried out.
Authors’ Affiliations
References
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