kinematics of natural knees with single-plane fluoroscopy
Tsung-Yuan Tsai, Tung-Wu Lu,a兲 and Chung-Ming Chen
Institute of Biomedical Engineering, National Taiwan University, No. 1, Sec. 1, Jen-Ai Road, Taipei 10051, Taiwan, Republic of China
Mei-Ying Kuo
Institute of Biomedical Engineering, National Taiwan University, No. 1, Sec. 1, Jen-Ai Road, Taipei 10051, Taiwan, Republic of China and Department of Physical Therapy, China Medical University,
Taichung 40402, Taiwan, Republic of China
Horng-Chaung Hsu
Department of Orthopaedic Surgery, China Medical University Hospital, Taichung 40447, Taiwan, Republic of China
共Received 14 July 2009; revised 8 December 2009; accepted for publication 7 January 2010; published 25 February 2010兲
Purpose: Accurate measurement of the three-dimensional共3D兲 rigid body and surface kinematics of the natural human knee is essential for many clinical applications. Existing techniques are limited either in their accuracy or lack more realistic experimental evaluation of the measurement errors. The purposes of the study were to develop a volumetric model-based 2D to 3D registration method, called the weighted edge-matching score 共WEMS兲 method, for measuring natural knee kinematics with single-plane fluoroscopy to determine experimentally the measurement errors and to compare its performance with that of pattern intensity共PI兲 and gradient difference 共GD兲 meth-ods.
Methods: The WEMS method gives higher priority to matching of longer edges of the digitally reconstructed radiograph and fluoroscopic images. The measurement errors of the methods were evaluated based on a human cadaveric knee at 11 flexion positions.
Results: The accuracy of the WEMS method was determined experimentally to be less than 0.77 mm for the in-plane translations, 3.06 mm for out-of-plane translation, and 1.13° for all rotations, which is better than that of the PI and GD methods.
Conclusions: A new volumetric model-based 2D to 3D registration method has been developed for measuring 3D in vivo kinematics of natural knee joints with single-plane fluoroscopy. With the equipment used in the current study, the accuracy of the WEMS method is considered acceptable for the measurement of the 3D kinematics of the natural knee in clinical applications. © 2010 American Association of Physicists in Medicine. 关DOI:10.1118/1.3301596兴
Key words: knee kinematics, 2D to 3D registration, x-ray fluoroscopy, genetic algorithm
I. INTRODUCTION
Knowledge of the three-dimensional 共3D兲 kinematics of the knee joint during activities is essential for understanding the normal function of the joint1and the diagnosis of pathology, such as ligament injury and osteoarthritis,2 and the evalua-tion of subsequent treatment, including surgery,3 rehabilitation,4 and the design of joint replacements.5 Mea-surement of knee kinematics is also an important component of biomechanical studies of the musculoskeletal system.6 Therefore, an accurate method for joint kinematics measure-ment is needed.
Several techniques are available for the measurement of 3D knee kinematics, but few allow noninvasive measure-ment with submillimeter accuracy. Imaging methods such as magnetic resonance imaging共MRI兲 and computed tomogra-phy共CT兲 may be used to provide 3D geometry and poses of knee, but they are limited to static and non-weight-bearing conditions. Skin marker-based methods, including stereopho-togrammetry and electromagnetic tracking systems, have
been widely used in clinical settings for the measurement of the 3D kinematics of the human body during functional activities.7–10 However, skin movement artifacts are difficult to prevent without the use of invasive bone pins.11The use of intracortical bone pins may not only expose the subjects to risks of infection but may also alter their kinematics. Al-though mathematical methods have successfully reduced the effects of skin movement artifacts in estimating bone positions,12 the measurement errors are still too large for an accurate description of the surface kinematics of the articular surfaces, i.e., rolling and sliding motions between surfaces.
Radiographic techniques provide the most direct way to measure joint motions without disturbances caused by soft-tissue artifacts. Surface model-based methods using dynamic fluoroscopy systems have been proposed for the accurate es-timation of the 3D poses共positions and orientations兲 of knee prosthesis components.13–17 All these approaches work by registering the known 3D CAD surface models of the pros-thesis component to the dynamic fluoroscopic images. The
surface model of a prosthesis component is projected onto the fluoroscopic image plane, and the pose of the component is then determined as the 3D pose of the component model that gives the best correspondence between the fluoroscopic image and the projected contours and/or areas of the model component. These methods have been shown to have high accuracy because the metallic components have precisely known geometric features and produce sharp edges in fluo-roscopic images. According to Banks and Hodge,16 the pre-cision values of total knee replacement共TKR兲 measurement were 0.21 mm, 3.9 mm, and 1.3° for the magnitude of in-plane, out-of-in-plane, and all rotations error, respectively. Im-provements and clinical applications of this type of method have been reported over the past decade.5,14,18,19The method has also been assumed to be applicable to natural knee joints13 even though bones differ fundamentally from TKR components in their form and internal structure. Fluoro-scopic images of bones look completely different compared to those of TKR components and have edges that are less well-defined. Bone edge attenuation has been suggested to be the primary factor limiting the theoretical accuracy of this type of method in measuring bone poses with single-plane fluoroscopy.20Under real life conditions, the accuracy would likely be worse. However, no well-documented experimental evaluation of these surface model-based methods for measur-ing natural knee kinematics exists.
Given the limitations in the accuracy of the existing sur-face model-based methods with single-plane fluoroscopy for the measurement of the kinematics of natural knees, methods using biplanar fluoroscopy21,22 have been proposed. Al-though more accurate than single-plane fluoroscopy, biplanar fluoroscopy methods inevitably increase radiation doses, un-acceptably constrain the motion of the patient,14 and some-times require surgical implantation of metal beads,22 restrict-ing their use to certain limited populations. These limitations also apply to approaches using multiviews of radiographs for bone pose reconstruction.23
More recently, slice-to-volume 共SV兲 registration methods24 have been developed for accurate intraoperative positioning of the body parts and the surgical instruments using either CT fluoroscopy or cine MRI. Theoretically, these methods can be further developed for the measurement of knee kinematics. However, CT fluoroscopy imaging is too slow for functional joint kinematics and the rotating C-arm also limits the imaging of the knee joint during weight-bearing functional activities. Fast cine MRI has been used to provide fast data of a single slice of the body parts for SV registration in image-assisted biopsies, interventional therapy of prostate cancer,25brain functional MRI,26postmortem pa-thology studies,27 and anatomical modeling.28Piecewise SV registration of MRI slices with segmented parts of a joint could be used for in vivo analysis of joint motion. However, since bones of a joint may move out of the imaging slice in most functional activities, the application of MR slice-volume registration will be limited in the study of functional weight-bearing activities such as level walking. Another limitation of an MR-based method is that integrated
mea-surements with devices including metal components and wire, for example, force plates, is not possible.
From the above review, it appears that a single-plane fluo-roscopy method with high accuracy in measuring the in vivo 3D bone kinematics will contribute significantly to relevant clinical studies and applications. However, the accuracy of such a method must be established before application. The accuracy of the measurement of the 3D natural knee kine-matics using single-plane fluoroscopy has been reported,29,30 but few reports have clearly documented the technical as-pects of the methods and the experimental procedures in de-termining the accuracy, as these studies were mainly report-ing the results of the application of the methods.29,30 All these studies used methods similar to those for total knee replacements, but with the surface models of the bones of the normal knee generated from CT images. From these studies, the measurements for all axes of rotation and translation were accurate, but translation along the projection axis had much less accuracy mainly because the projection image of the surface models is much less sensitive to this out-of-plane translation than the in-plane translations. This is a major limitation in the study of the surface kinematics of the knee, such as contact point movements, because the motion of a point on the bone is determined not only by rotations and translations along the axes in the imaging plane but also by those along the projection axis. Komistek et al.29 had to dis-regard the measured translation of the contact point along the projection axis. Since surface models provide information only on the external structure of the bones, a possible means of improving the out-of-plane accuracy of single-plane fluo-roscopy methods may be to include more detailed represen-tations of the internal architecture of the bone models.29,30
A method of allowing information from CT images to be used in the 2D to 3D registration process is to register di-rectly the CT volumetric data to fluoroscopic images. Regis-tration of intraoperative fluoroscopic images with preopera-tive 3D CT images has been used in image-guided surgery.31 Lemieux et al.32proposed a registration method based on the comparison of biplanar radiographs and digitally recon-structed radiographs 共DRRs兲 obtained from the CT data. Subsequent developments have led to DRR-based registra-tion methods for aligning single-plane fluoroscopic images and CT scans for image-guided surgery.33–35Penney et al.34 compared the performances of six similarity measures for 2D to 3D registration of CT images and single fluoroscopic im-ages. They showed that pattern intensity 共PI兲 and gradient difference 共GD兲 measures were superior to the other mea-sures. In that study, a cadaveric lumbar spine was used for validation and the registration was performed for the whole spine as a rigid body. This is different from requirements for the study of joint kinematics in which each bone should be considered separately. In a later study, Penney et al.35 im-proved their method of registering one vertebra at a time. However, their method gave an out-of-plane translation error of 7.2 mm, which is still too great a difference for an accu-rate description of the knee joint kinematics. Direct evalua-tion of the accuracy of PI and GD on knee kinematics is needed. Registration using PI and GD relies on gray scales
and strong gradients associated with prominent bony fea-tures. Any noise that can contaminate the grayscales and gra-dients of the images of the bones, and thus the PI and GD, will affect the registration accuracy. For example, nonuni-form fluoroscopy intensity共e.g., owing to the x-ray tube an-ode heel effect兲 and the surrounding soft tissues are two such sources of noise during measurements. Thus, a similarity measure that takes into account the noise from these sources may produce a more accurate registration outcome.
The purposes of the study were to develop a new volu-metric model-based 2D to 3D registration method with a new similarity measure for measuring natural knee kinematics with single-plane fluoroscopy that takes into account of the noise from the abovementioned sources, to determine experi-mentally the measurement errors, and to compare the perfor-mance between PI, GD, and our new measure.
II. THE NEW VOLUMETRIC MODEL-BASED REGISTRATION METHOD
II.A. Overview
The general procedure for the proposed volumetric model-based CT-to-fluoroscopy image registration method is shown in Fig. 1. The CT data of a joint are used to create volumetric computer models for each of the bones. Fluoro-scopic images of the same joint under static conditions or during dynamic motion are also obtained. The 3D pose of the bone in each fluoroscopic image frame is then obtained by using an optimization procedure to search for the pose of the bone model whose DRR best matches the fluoroscopic image according to a similarity measure called the weighted edge-matching score 共WEMS兲. The registered poses of the bones for all image frames can then be used to obtain the 3D kine-matics of the joint. A more detailed description of the method is given below.
II.B. Bone model creation
The bones of the joint under study are scanned by CT. The CT data for each bone segment are then isolated from the original data set by applying a 3D region-growing method. The segmented volume is then checked manually slice-by-slice to ensure that the bony details are completely included in the volume with minimal remaining external soft tissues. In other words, no part of the bone generating gray-value edges and boundaries in the DRR is missing. Since the in-fluence of the remaining external soft tissues on the DRR is negligible, it is considered that the current segmentation pro-cess will not affect the results of the subsequent registration with fluoroscopic images.33 Bones with negligible relative motions, such as the tibia and fibula, may be considered as a single component during the registration procedure to im-prove the accuracy of the measurement.
II.C. Projection model of the fluoroscopy system The formation of a fluoroscopic image can be modeled as an ideal perspective projection of a point source x-ray onto a planar phosphor screen upon which the image is formed, as shown in Fig.2. Model parameters, including those for fluo-roscopic image distortion and the position of the x-ray source, are obtained through a calibration procedure using a 30⫻30 cm2 transparent calibration box. Two parallel sides of the box are a reseau plate and a star plate that are both marked with lead markers. During calibration, the box is placed with the reseau plate on the image intensifier of the fluoroscopy and its image is obtained. The reseau plate im-age is used to correct imim-age distortions, such as pincushion distortions and nonuniform scaling, via a modified polyno-mial method,36while that of the star plate is used to estimate the position of the point source x-ray by minimizing the projection errors of the control points on the plate. Given the parameters, a particular fluoroscopy system is modeled and can then be used for the generation of DRRs.
The DRRs are produced by casting rays through the CT volume and projecting them onto the image plane, as shown in Fig.2. Each of these rays goes through a number of voxels of the CT volume. The Hounsfield numbers of these voxels FIG. 1. The general procedure of the new registration method.
FIG. 2. Perspective projection model of the fluoroscopy system. X and Y axes define the image plane. Rays of the point source x-ray passing through the CT volume are projected onto the image plane to form a DRR.
are then integrated along the ray and projected onto an im-aging plane to obtain a DRR image resembling a radiograph.34DRRs are synthetic x-ray images with detailed internal information of the bony structure.
II.D. Similarity measures
During the search for the 3D pose of a bone model whose DRR image best matches the fluoroscopic image, a similarity measure is necessary to describe the level of similarity be-tween the DRR共IDRR兲 and the fluoroscopic image 共Ifl兲 共Fig. 3兲. The PI and GD previously proposed in literature33,35 and the new WEMS are described as follows.
II.D.1. PI
The pattern intensity measure33 attempts to subtract the DRR共IDRR兲 from the fluoroscopic image 共Ifl兲 so that in the optimal registration position the structure from the bone will vanish. The measure is described mathematically as a func-tion of a scaling factor s as follows:
PIr,共s兲 =
兺
i,j d2
兺
ⱕr22
2+共I
diff共i, j兲 − Idiff共v,w兲兲2
, 共1兲
where
d2=共i − v兲2+共j − w兲2, 共2兲
Idiff= Ifl− sIDRR. 共3兲
The PI method operates on a single difference image 共Idiff兲, which is created by subtracting the DRR from the fluoro-scopic image using a suitable scaling factor, s关Eq.共3兲兴. Pixel values are only compared if the distance 共d兲 between the pixels is less than or equal to a radius r 关Eq.共2兲兴. The con-stant is used to ensure that the measure remains near its maximum value when small deviations in intensity共such as those caused by noise兲 occur. As suggested by Penney et al.,34 = 10 and r = 3 are used in the current study. For each new calculation of the similarity measure PI共s兲, the op-timum scaling factor s is determined by maximizing PI共s兲 using a one-dimensional gradient descent-type search strat-egy.
II.D.2. GD
For the calculation of the GD according to Penney et al.,34 Ifl and IDRRare transformed by applying horizontal and ver-tical Sobel templates to obtain four gradient images, namely,
IDRR/i, IDRR/j, Ifl/i, and Ifl/j. These gradient im-ages represent the partial derivatives of the DRR and those of the fluoroscopic image intensities along the two orthogonal axes of the images. Vertical and horizontal gradient differ-ence images, Idiff Vand Idiff H, are then calculated by subtract-ing the DRR gradient image intensities from the fluoroscopic gradient image intensities as follows:
Idiff V共i, j兲 = Ifl i − s IDRR i , Idiff H共i, j兲 =Ifl j − s IDRR j , 共4兲
where s is a scaling factor and is so chosen that at the opti-mal registration position, the structure of the bone will van-ish. The gradient difference measure is defined as
GD共s兲 =
兺
i,jAv
Av+共Idiff V共i, j兲兲2
+
兺
i,j
Ah
Ah+共Idiff H共i, j兲兲2 , 共5兲 where Avand Ahare constants that are set to the variance of the pixel intensities in the respective共vertical and horizontal兲 gradient fluoroscopic image. By using both vertical and hori-zontal gradient images, the similarity measure is able to compare the directions of the gradients as well as their mag-nitudes.
II.D.3. WEMS
For the calculation of the WEMS, the Canny method37is first used to obtain edge images Efland EDRRfor Ifland IDRR, respectively 共Fig. 4兲. The Canny method uses a high and a low threshold to detect strong and weak edges and includes the weak edges in the output only if they are connected to strong edges. This makes it less likely to be fooled by noise and more likely to detect true weak edges, which is different from other edge-detection methods. After edge detection, Efl is then dilated to become a dilated edge image A1by apply-ing a 3⫻3 cross-shaped structuring element M to each pixel of the edges once, where
M =
冤
0 1 0 1 1 1 0 1 0冥
, 共6兲 A1= A0丣M = dilate共A0, M兲, 共7兲 A0= Efl. 共8兲Applying M once again to the dilated image A1gives a new dilated image A2. Repeating the operation d times will pro-duce d dilated images, i.e., A1, A2, . . . , Ad. Summation of these dilated images and Eflwill produce a final dilated edge image Bfl with a dilation band of 共2d+1兲 pixels in width 关Fig.4共c兲兴 as follows:
(a)
(b)
FIG. 3. 共a兲 A fluoroscopic image 共Ifl兲 of a knee and 共b兲 a DRR of its femur
Bfl= 1 d + 1
兺
i=0d
Ai, An+1= An丣M . 共9兲
In order to maintain the values of the pixels in the original edge to unity, the dilated image Bfl is divided by共d+1兲. In
Bfl, the value of each pixel in the band, i.e., pixels within a distance 共d兲 to the original edge, is set to be inversely pro-portional to its shortest distance from the original edge. In the current study, d = 3 was chosen empirically. The use of a dilated edge image enables the DRR edges within the band to be considered in the WEMS value, which improves the convergence.
Similar to manual registration, long edges are given greater weightings in the similarity score. The weightings are determined based on EDRR. The longer the edge to which 1 pixel belongs, the greater the weighting of the pixel. In order to determine the weightings, the 8-connected length of each edge in terms of the number of pixels is calculated first. That is, two pixels adjacent to each other vertically, horizontally or diagonally are considered to be on the same edge. The 8-connected length of each edge to the pth power is then stored in each of the corresponding pixels in LDRRthat has the same size as EDRR. In the current study, p was selected as 0.5 based on a systematic search using computer simulations. This weighting method is designed to give longer edges higher weightings, while maintaining the weightings within a reasonable range. Each pixel of the dilated image Bflis then multiplied by the value of the corresponding pixel of LDRRto produce image Ereg 关Fig.4共d兲兴 as follows:
Ereg共x,y兲 = LDRR共x,y兲Bfl共x,y兲. 共10兲
That is, the resulting Eregis formed by the overlapping edges of LDRRand Bfl, and the value of a pixel in Ereg provides a measure of the distance of the corresponding pixel in EDRR from the nearest pixel in the overlapped Efl. The smaller the value, the greater the distance. WEMS is then defined as the total area of Eregas a percentage of the total area of LDRRas follows:
WEMSd,p=
兺共x,y兲Ereg共x,y兲 兺共x,y兲LDRR共x,y兲
. 共11兲
The WEMS value ranges from 0 to 1. The greater the WEMS value, the closer EDRRis matched with Eflbecause there is a greater percentage of matched weighted edges in LDRR. A WEMS value of 0 indicates that no edges are matched, whereas the maximum WEMS value of 1 indicates that all edges are matched.
II.E. Optimization
For the search of a model pose such that its DRR best matches, the measured fluoroscopic image, an optimization procedure is used. The three translations and three rotations of the bone model are taken as the design variables, and the optimization problem is to find the optimum design that minimizes −PI共s兲, −GD共s兲 or −WEMSd,p关Eqs.共1兲,共5兲, and 共11兲, respectively兴. Since the feasible region of the problem is nonlinear, the genetic algorithm共GA兲 is chosen to find the global minimum.38 The GA is a zero-order optimization al-gorithm and has been shown to be robust in searching for the global optimum.39The initial guess of the design variables is obtained through a manual positioning procedure using a graphical user interface 共GUI兲 program described in Sec. II F. Given the initial guess, an initial population of a size of 50 for the GA is then randomly created within a range of 30 and 100 mm for in-plane and out-of-plane translation, and 20° for all rotations. The maximum number of generations is 100. The current optimization parameters are determined based on computer simulations, considering both conver-gence rate and optimum results. A further increase in the number of generations and population size may slightly im-prove the optimum design, but the time required for conver-gence will increase excessively.
II.F. GUI program
The GUI program was developed using MATLAB 共The Mathworks, Inc., USA兲, which allows the user to construct a virtual fluoroscopy system using system parameters obtained from the experimental system calibration. The program also allows the user to visualize the corrected fluoroscopic images and the CT-derived computer bone models. In the virtual world, the view point is set to the x-ray focus position, which allows the user to visualize the projected bone model on top of the fluoroscopic image. With this arrangement, and through convenient mouse control of the bone pose, the user can quickly find an initial pose of the model for the subse-quent registration using optimization. This process is re-(c)
(a) (b)
(d)
FIG. 4. Image processing steps for WEMS.共a兲 Edge image of the DRR in
Fig.3共b兲: EDRR.共b兲 Edge image of the fluoroscopic image in Fig.3共a兲: Efl.
共c兲 Bflobtained by dilation of Eflwith a small band.共d兲 Edge image 共Ereg兲,
which is formed by the overlapping edges of Bfland LDRR. The inlet figures
in共a兲, 共c兲, and 共d兲 show the detailed images of the marked areas. See main text for detailed definitions of the symbols.
peated for all the bones of the joint. After registration of the bones for a series of image frames, the 3D motion of the bones can be viewed from any view point.
III. EVALUATION OF ACCURACY
The accuracy of the WEMS method for the kinematics of the knee was evaluated and compared to that obtained with the PI and GD methods, both theoretically using computer simulation and experimentally with an in vitro approach. The former determines the theoretical accuracy of the method, while the latter gives the accuracy in realistic situations. All the analyses were performed on a notebook computer with an Intel Core 2 Duo T7300 CPU and 3 Gbytes of RAM.
III.A. Computer simulation
Computer simulation allows the evaluation of registration methods without the presence of noise that can contaminate the images of the bones and thus affect the registration ac-curacy. This noise includes the fluoroscopic image distortion, nonuniform fluoroscopy intensity, and effects of the sur-rounding soft tissues.
The computer simulation was performed using CT bone models of a cadaveric knee, namely, the femur, tibia, and fibula, within a projection model of the fluoroscopy system used in the in vitro experiment to be described in Sec. III B. The focal length from the x-ray source to the center of the image plane was 1260 mm. The image plane was a circular plate with a diameter of 300 mm. The CT model of the knee was first placed at a known 共true兲 pose in the middle of the imaging space, and the DRR of the CT model was generated to give a simulated fluoroscopic image with a size of 1020 ⫻932 pixels. Thirty different poses of the knee in the imag-ing space were then created by addimag-ing random deviations to each degree of freedom of the original pose of the knee while the tibia remained stationary. The ranges of the random de-viations were 10 mm for all translations, 145° for flexion, and 25° for abduction/adduction and internal/external rotations.40 Thirty simulated fluoroscopic images corre-sponding to the 30 poses were also generated. The PI, GD, and WEMS methods were then used to register each of the bone models to each of the simulated fluoroscopic images using the previously described optimization procedure with the same initial guess obtained through a quick manual po-sitioning. The registration errors were then calculated for each of the methods.
III.B. Experimental evaluation
III.B.1. Experimental procedure
An in vitro experiment was performed to evaluate the measurement errors of the PI, GD, and WEMS methods us-ing a fresh-frozen intact normal left knee, a fluoroscopy sys-tem 共Angiography, Advantx LCA, GE, France兲, a motion capture system共Vicon, Oxford Metrics, UK兲, and a CT sys-tem 共CT, PQ-5000, Picker International, USA兲. The knee specimen had a length of 25 cm proximal to the joint line and 25 cm distal. The proximal femur and the distal tibia
were fixed with bone cement on a plastic frame made of acrylonitrile butadiene styrene plastic, which is highly rigid 共Fig.5兲. The plastic frame was built to allow flexion of the knee at 11 different angles up to 100° at 10° intervals and to ensure that at each flexion position, the bones were fixed relative to each other and to the frame. In order to track the poses of the bones and the frame relative to the laboratory coordinate system, two clusters, each with four infrared ret-roreflective markers, were attached to both the proximal fe-mur and the distal tibia, and eight additional markers were placed on the frame共Fig.5兲.
At each of the 11 flexion positions defined by the fixation frame, the knee specimen was imaged by CT and fluoros-copy separately. The CT data had 512⫻512⫻600 voxels, each with a size of 0.78125⫻0.78125⫻1 mm3. With the CT data, the spatial relationships of the femur, tibia, and the infrared retroreflective markers were accurately determined. During fluoroscopic imaging, the positions of the retroreflec-tive markers on the specimen/fixation frame construct were also measured using the Vicon system. For each flexion po-sition, 20 trials of simultaneous fluoroscopic imaging and Vicon measurements were performed. The CT data at the extended position were used to create the bone models as described in Sec. II B and to define the local coordinate sys-tems for the femur and tibia, which were chosen to be par-allel to the CT coordinate system but with origins at the centroid of each bone. The pose of the bone in space was described in terms of homogeneous transformation matrices that are functions of the three translations and the three ro-tations. Homogeneous transformation matrices for the femur and tibia were defined with respect to the CT coordinate system, namely, CTTf and CTTt, respectively. It is noted that other flexion positions can also be used for bone model cre-ation, as a model of a bone is independent of its position relative to other bones. With the bone models, poses of the femur relative to the tibia under the CT coordinate system, f
Tt, at each flexion angle were then determined by minimiz-ing the root mean squared distance between the points of FIG. 5. Setup of the experiment with the specimen fixed on the plastic frame at about 20° of flexion.
each model surface to the closest points on the corresponding bone model at the extended position using the iterative clos-est point algorithm41to obtainCTT
⬘
f andCTTt⬘
at each flexion position. These were taken as the “true” poses of the knee joint at that flexion angle. The root mean squared errors of the distances between corresponding points of the bone mod-els reconstructed from a different CT data set at each flexion angle were all less than 0.025 mm. The true bone poses relative to the laboratory coordinate system were then deter-mined using the positions of the markers of the clusters cap-tured by the Vicon system. The variance of the repeated mea-surements of the positions of a single static marker by the Vicon system was determined to be less than 0.12 mm. The fluoroscopy system and motion capture system were inte-grated through simultaneous measurement of the accurately positioned lead markers and five infrared retroreflective markers on the calibration box to enable the description of the bone poses relative to the same coordinate system, ob-tained either by the Vicon system or by the 3D registration procedure.III.B.2. Convergence test
The robustness of the WEMS method was evaluated by a convergence test using a fluoroscopic image of the extended cadaveric knee. For each femur and tibia, 30 initial guesses of the pose of the bone were created by adding random de-viations to each degree of freedom of the true pose of the bone. The translation of the bone was 8 mm with a random direction in the image plane, i.e., the X-Y plane, and either 40 or⫺40 mm along the out-of-plane direction, i.e., the Z axis. The deviation in rotation was 8° along a random axis of rotation. For each initial guess, the errors of the registered poses of the bones and the knee were obtained for subse-quent statistical analysis.
III.B.3. Registration with PI, GD, and WEMS
For each of the 220 experimental trials, the bone models were registered to the fluoroscopic image one by one using each of the PI, GD, and WEMS methods with the same ini-tial guess obtained through a quick manual positioning. The registration errors were then calculated for each of the meth-ods.
III.C. Statistical analysis
For the computer simulation, the means and standard de-viations of the errors in the poses of the bones and knee across the 30 simulated trials were calculated for each method, i.e., PI, GD, and WEMS, to determine the theoreti-cal accuracy of the methods. For the experimental tests, the errors in the poses of the bones and knee were ensemble averaged over the 220 trials of the 11 flexion positions for PI, GD, and WEMS, respectively. The mean values indicate the bias of the method, while the standard deviations indicate the precision. For convergence test, the means and standard de-viations of the errors of the 30 initial guesses were calculated to determine the robustness of the WEMS method. For the computer simulation and experimental tests, the means and standard deviations of the time required for registrations were also calculated.
IV. RESULTS
IV.A. Computer simulation
Computer simulation results showed that the errors in the poses of the knee and its bones were comparable for all methods, with those of the WEMS method being slightly lower 共TableI兲. For the WEMS method, the magnitudes of the means 共standard deviations兲 for the femoral pose errors were less than 0.004共0.10兲 mm for the in-plane translation components 共X and Y兲, 0.26 共1.32兲 mm for the out-of-plane translation共Z兲, and 0.15° 共0.23°兲 for all the angular compo-nents. The corresponding values for the GD method were 0.08 共0.14兲 mm, 0.73 共1.86兲 mm, and 0.28° 共0.48°兲, respec-tively. For the PI method, they were 0.07 共0.16兲 mm, 0.68 共1.90兲 mm, and 0.26° 共0.46°兲, respectively. Tibial pose errors were slightly greater than those of the femur for all three methods共TableI兲.
For knee poses, the WEMS method gave the magnitudes of the means 共standard deviations兲 of less than 0.12 共0.24兲 mm for the in-plane translation components, 0.20共1.95兲 mm for the out-of-plane translation, and 0.11°共0.42°兲 for all the angular components. The corresponding values for the GD method were 0.15 共0.20兲 mm, 0.26 共2.88兲 mm, and 0.11° 共0.68°兲, respectively 共TableI兲. For the PI method, they were 0.15 共0.26兲 mm, 0.86 共3.12兲 mm, and 0.31° 共0.86°兲, respec-TABLEI. Means共standard deviations兲 of the errors of the bone and knee poses estimated using the PI, GD, and WEMS methods, and corresponding run time over 30 computer simulated trials. Directions are with respect to the fluoroscopic coordinate system, where X and Y are parallel to the image plane and Z is perpendicular to the image plane.
WEMS GD PI
Femur Tibia Knee Femur Tibia Knee Femur Tibia Knee
X共mm兲 0.00共0.10兲 0.09共0.15兲 0.12共0.24兲 ⫺0.08 共0.10兲 0.08共0.11兲 ⫺0.15 共0.20兲 ⫺0.07 共0.12兲 ⫺0.12 共0.16兲 0.15共0.25兲 Y共mm兲 0.00共0.06兲 0.04共0.21兲 ⫺0.02 共0.16兲 ⫺0.05 共0.14兲 0.05共0.15兲 0.06共0.14兲 0.06共0.16兲 ⫺0.04 共0.22兲 0.03共0.26兲 Z共mm兲 ⫺0.26 共1.32兲 0.21共1.73兲 0.20共1.95兲 ⫺0.73 共1.86兲 ⫺0.61 共1.93兲 0.26共2.88兲 ⫺0.68 共1.90兲 0.44共2.11兲 0.86共3.12兲 x共deg兲 0.03共0.22兲 0.02共0.29兲 0.10共0.40兲 ⫺0.04 共0.48兲 0.04共0.51兲 ⫺0.11 共0.50兲 0.02共0.46兲 0.05共0.61兲 0.12共0.86兲 y共deg兲 ⫺0.15 共0.23兲 ⫺0.13 共0.30兲 ⫺0.03 共0.42兲 ⫺0.28 共0.40兲 ⫺0.29 共0.43兲 ⫺0.02 共0.68兲 ⫺0.26 共0.32兲 ⫺0.22 共0.53兲 ⫺0.31 共0.70兲 z共deg兲 ⫺0.03 共0.15兲 0.01共0.32兲 0.11共0.41兲 0.07共0.14兲 0.05共0.14兲 0.04共0.38兲 0.05共0.21兲 0.07共0.35兲 0.16共0.49兲 Run time共s兲 277共50兲 266共65兲 338共64兲 354共74兲 289共44兲 329共67兲
tively共TableI兲. When comparing all three methods, the run time of the GD method took the longest, while the WEMS method was the fastest, both for the femur and for the tibia registration共TableI兲.
IV.B. Experimental evaluation
Experimental results showed that the errors of the global poses of the bones using the WEMS method were generally smaller than those from the PI and GD methods. The WEMS method had means 共standard deviations兲 of the absolute tibial pose errors of less than 0.34 共0.86兲 mm, 0.38 共4.61兲 mm, and 0.81° 共1.21°兲 for the in-plane translation compo-nents, out-of-plane translation component, and all angular components, respectively. The corresponding values for the GD method were 0.77共1.32兲 mm, 2.83 共6.03兲 mm, and 0.50° 共0.93°兲, and those for the PI were 0.85 共1.63兲 mm, 6.28 共7.39兲 mm, and 0.60° 共1.33°兲 共Table II兲. For the femur, the WEMS method had means 共standard deviations兲 of the ab-solute tibial pose errors of less than 0.68 共0.72兲 mm, 0.94 共3.23兲 mm, and 0.91° 共0.83°兲 for the in-plane translation components, out-of-plane translation component, and all an-gular components, respectively. The corresponding values for the GD method were less than 0.55 共0.87兲 mm, 1.64 共3.70兲 mm, and 0.46° 共0.92°兲, and those of the PI were 1.01 共1.38兲 mm, 3.37 共5.17兲 mm and 0.65° 共0.99°兲 共Table II兲. Again, the run time of the GD method was the longest, while that of the WEMS was the shortest, both for the femur and for the tibia 共Table II兲. For a better understanding of the differences between the two methods in the measurement errors, especially the out-of-plane translation, cost function values for both the femur and tibia as a function of the out-of-plane translation for these three methods were generated for all test positions and a typical case is given in Fig. 9.
With the measurement errors of the femur and tibia, the errors of the poses of knee kinematics using the WEMS method were also less than those from the GD method, es-pecially those of the out-of-plane component. The means 共standard deviations兲 of the knee pose errors using the WEMS method were less than 0.24共0.77兲 mm, 0.41 共3.06兲 mm, and 0.59° 共1.13°兲 for the in-plane translation compo-nents, out-of-plane translation component, and all the angu-lar components, respectively, while the corresponding values
for the GD method were 0.60 共0.86兲 mm, 3.12 共6.02兲 mm, and 0.83°共1.60°兲, and those for the PI were 0.83 共0.98兲 mm, 11.39共8.55兲, and 0.61° 共2.05°兲 共TableII兲.
IV.C. Convergence test
In the analysis of the experimental data, the GA optimi-zation procedure successfully converged to an optimum for each of the thirty different initial guesses. Over the genera-tions, when the mean fitness value approaches the best fit-ness value, a local minimum is reached 共Fig.6, generation 18兲. The GA was able to escape the local minimum as indi-cated by the sudden separation of the mean fitness value from the best value in Fig.6. With this ability, the GA could further reduce the fitness value and finally converge to the final optimum solution. For example, at generation 3 of one of the 30 trials, the current best pose of the femur is still far from the true position, as shown by the mismatch between the femoral model and the underlying fluoroscopic image in Fig. 7共a兲. As the number of the generation increases, the current best fitness pose of the femur approaches the final TABLEII. The means共standard deviations兲 of the errors of the bone and knee poses estimated using the PI, GD, and WEMS methods over 220 experimental trials. Directions are with respect to the fluoroscopic coordinate system, where X and Y are parallel to the image plane and Z is perpendicular to the image plane.
WEMS GD PI
Femur Tibia Knee Femur Tibia Knee Femur Tibia Knee
X共mm兲 ⫺0.42 共0.56兲 ⫺0.34 共0.86兲 0.24共0.77兲 ⫺0.34 共0.72兲 ⫺0.77 共1.32兲 ⫺0.60 共0.71兲 ⫺1.01 共1.38兲 ⫺0.85 共1.63兲 0.83共0.76兲 Y共mm兲 0.68共0.72兲 ⫺0.24 共0.61兲 0.04共0.75兲 0.55共0.87兲 ⫺0.46 共0.85兲 ⫺0.31 共0.86兲 0.66共1.16兲 ⫺0.44 共0.94兲 ⫺0.59 共0.98兲 Z共mm兲 ⫺0.94 共3.23兲 ⫺0.38 共4.61兲 0.41共3.06兲 1.64共3.70兲 ⫺2.83 共6.03兲 ⫺3.12 共6.02兲 3.37共5.17兲 6.28共7.39兲 11.39共8.55兲 x共deg兲 ⫺0.52 共0.77兲 0.38共0.57兲 0.51共0.73兲 0.44共0.70兲 ⫺0.42 共0.93兲 ⫺0.61 共1.26兲 ⫺0.02 共0.99兲 ⫺0.50 共1.08兲 0.58共1.84兲 y共deg兲 0.91共0.83兲 0.13共1.21兲 0.59共1.10兲 ⫺0.35 共0.92兲 0.10共0.71兲 ⫺0.83 共1.60兲 ⫺0.43 共0.96兲 0.60共1.33兲 ⫺0.61 共2.05兲 z共deg兲 0.55共0.75兲 ⫺0.81 共0.49兲 ⫺0.40 共1.13兲 ⫺0.46 共0.72兲 ⫺0.50 共0.68兲 0.04共0.76兲 0.65共0.51兲 ⫺0.31 共0.81兲 0.60共0.36兲 Run time共s兲 280共42兲 259共74兲 331共92兲 363共83兲 295共52兲 325共68兲 0 10 20 30 40 50 60 70 -0.85 -0.8 -0.75 -0.7 -0.65 -0.6 -0.55 -0.5 Generation Fi tn ess V al ue Best:-0.81141 Mean:-0.81141 Mean Fitness Best Fitness
FIG. 6. Fitness values共i.e., negative WEMS values兲 over generations during the optimization process of the registration of the femur in a typical experi-mental trial using GA. The best and mean fitness values across all the individuals for each generation are indicated by solid and open circles, re-spectively. The fitness values decreased with increased generations and fi-nally converged to the final solution.
optimum solution关Figs.7共b兲and7共c兲兴, which almost exactly covers the area of the femur in the fluoroscopic image 关Fig. 7共d兲兴.
Given the nonlinear nature of the registration problem, as shown by the changes in the fitness values of the WEMS measure of the femoral poses in a neighborhood of 5 mm and 5° around the converged optimum pose 共Fig. 8兲, the final optimum poses obtained were not strongly dependent on ini-tial guesses. For the randomly generated 30 iniini-tial positions, the magnitudes of the means 共standard deviations兲 of the femoral pose errors were less than 0.31 共0.13兲 mm, 0.20 共1.31兲 mm, and 0.55° 共0.33°兲 along the in-plane direction, the out-of-plane direction, and all the angles, respectively. For the tibial pose errors, the magnitudes of the means 共stan-dard deviations兲 were less than 0.42 共0.31兲 mm, 1.14 共2.06兲 mm, and 0.26°共0.72°兲 along the in-plane direction, the out-of-plane direction, and all the angles, respectively. For the knee pose measurements, the magnitudes of the means 共stan-dard deviations兲 of the errors were less than 0.47 共0.50兲 mm, 1.72 共2.97兲 mm, and 0.70° 共0.83°兲 along the in-plane direc-tion, the out-of-plane direcdirec-tion, and all the angles, respec-tively共TableIII兲.
V. DISCUSSION
A new volumetric model-based 2D to 3D registration method has been developed for measuring 3D in vivo kine-matics of natural knee joints with single-plane fluoroscopy, taking into account of the influences from tissues other than the targeted bones, and noise from the fluoroscopy system itself. The method was shown experimentally to have mea-surement errors less than 0.24共0.77兲 mm for in-plane
trans-lations, 0.41 共3.06兲 mm for out-of-plane translations, and 0.59°共1.13°兲 for all rotations, which were smaller than those of the PI and GD methods, two of the best registration meth-ods suggested in literature.34 The error for the out-of-plane translation by the WEMS is smaller than previously pub-lished errors of other bone registration methods.30,42 Under the same conditions, no obvious failed registrations for all three methods were encountered in our analyses. The error levels of the WEMS method are comparable to the reported errors for TKR registration methods.14–16Similar to the TKR registration methods, the WEMS method should be consid-ered acceptable for most clinical applications.
Apart from the definitions of the similarity measures, the accuracy of the bone pose obtained using the current 2D to 3D registration methods is also affected by factors such as resolution of the CT data, pixel size of the fluoroscopic im-age and the projection geometry of the fluoroscopy system. Because of the different effects from these factors, the accu-racy of the registration methods can be higher than the res-olution of the CT data of the bone. In the current study, the
TABLEIII. Means共standard deviations兲 of the errors of the WEMS method in the convergence test. Directions are with respect to the fluoroscopic co-ordinate system, where X and Y are parallel to the image plane and Z is perpendicular to the image plane.
Femur Tibia Knee
X共mm兲 ⫺0.17 共0.13兲 ⫺0.42 共0.31兲 ⫺0.47 共0.50兲 Y共mm兲 ⫺0.31 共0.08兲 ⫺0.23 共0.25兲 ⫺0.12 共0.28兲 Z共mm兲 0.20共1.31兲 ⫺1.14 共2.06兲 ⫺1.72 共2.97兲 x共deg兲 0.09共0.25兲 ⫺0.17 共0.47兲 0.21共0.53兲 y共deg兲 ⫺0.55 共0.33兲 ⫺0.16 共0.72兲 ⫺0.70 共0.83兲 z共deg兲 ⫺0.14 共0.22兲 0.26共0.31兲 ⫺0.38 共0.39兲 -5 0 5 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 Fi tn es s V a lue X (mm) -5 0 5 -0.9 -0.8 -0.7 -0.6 -0.5 Y (mm) -5 0 5 -0.82 -0.8 -0.78 Z (mm) -5 0 5 -0.82 -0.8 -0.78 θX(°) Fi tn ess Val ue -5 0 5 -0.82 -0.8 -0.78 θY(°) -5 0 5 -0.82 -0.8 -0.78 θZ(°) FIG. 8. Changes in the fitness values共i.e., negative WEMS values兲 for each component of the femoral poses in the neighborhood of 5 mm or 5° around the converged optimum pose while keeping the other components un-changed. The fitness values at the optimum pose are very close to the mini-mum fitness values. Directions are with respect to the fluoroscopic coordi-nate system, where X and Y are parallel to the image plane and Z is perpendicular to the image plane.
(a)
(d)
(c)
(b)
FIG. 7. Best fitted poses of the femur in the same trial as used in Fig.6at generation 3共a兲, 5 共b兲, 26 共c兲 and at the final converged optimum pose 共d兲. The reduction in the differences between the bone and the fluoroscopic image over generations can be seen visually within the marked areas.
performance of the three different similarity measures was compared, while the other factors were controlled. As in other previous studies,42 the errors of the registered poses were expressed statistically in terms of the associated means and standard deviations. The WEMS method had mean val-ues smaller than the CT resolution. Since mean valval-ues indi-cate the bias of the error distribution, they can be much smaller than the CT resolution.42 The WEMS method also had standard deviations less than the CT resolution, which may be attributed to two reasons. First, the volumetric CT data of a bone were used to represent the bone as an object, with several hundred thousand voxels. The origin of the bone was then defined as the centroid of the bone volume. Thus, the accuracy of the centroid can be higher than the CT res-olution. The second reason is the projection geometry of the fluoroscopy system, which gives the fluoroscopic image high sensitivity to the bone pose change in space. With the current resolution of the fluoroscopy system, this projection geom-etry enables a spatial resolution of the bone pose higher than the CT resolution. It is also noted that if the errors of all tests are close to the CT resolution, the standard deviation can be smaller than the CT resolution.
Previous studies have used “clean” images of objects out-side the body15,19or synthetic data42when evaluating 2D to 3D registration methods. With clean images, both the PI, GD, and WEMS methods give very small errors 共Table I兲, suggesting that the above similarity measures are well de-fined for the optimization algorithm to reach an optimum solution that is close to the true solution. The resulting errors can be taken as the lower bound of errors for natural knee kinematics measurements under realistic conditions. How-ever, with real life fluoroscopic images that are subject to noise from the fluoroscopy system itself and from tissues other than the targeted bones, the performance of these meth-ods will depend on their ability to reduce the effects of this noise on the similarity measures, that is, whether the opti-mum design corresponding to the cost function remains un-changed in the presence of errors. The introduction of soft-tissue structures into the phantom image has been shown to cause a large effect on the performance of some similarity
measures previously applied to 2D to 3D image
registration.34 Therefore, the performance of these methods should be evaluated based on more realistic experimental data. While experimental evaluations of the accuracy of TKR registration methods have been reported,14–16 to the best of our knowledge, the current study is the first well-documented experimental assessment of registration methods for measur-ing natural knee kinematics with smeasur-ingle-plane fluoroscopy.
Establishing the ground truth is essential for the experi-mental evaluation of the performance of different registration methods. Ideally, the ground truth data should be at least as accurate as the method being tested and should be obtained using an independent method other than fluoroscopy. Previ-ous studies have used fiducial markers in x-ray images,21,22 but this is not an independent method. Other approaches, such as mechanical positioning systems, are difficult to use on bones located within the body. Infrared stereophotogram-metry has also been used to provide an independent means of
determining the ground truth data of total knee implants with an accuracy of 0.2893 mm.14 In the current study, infrared stereophotogrammetry was used to provide the global poses of the bones using markers attached to the bones through cortical pins. The accuracy of a single point determined by the Vicon system was less than 0.1290 mm. The poses of the knee joint, i.e., tibia relative to the femur, were determined by a CT scan. The use of the fixation frame enabled us to take advantage of the high accuracy of both the CT machine and the Vicon system.
The convergence study showed that the WEMS method, combined with the GA, has the ability to overcome various local minima encountered during the search process to obtain consistent results, regardless of where the search began, as long as the initial guesses are within the tested range共Table III兲. For most daily activities, the tested region in the current study should encompass the movement range of the knee within the time between two consecutive images of the cur-rent fluoroscopy system, i.e., 1/30 s. This suggests that the registered pose for one image can be used as the initial guess for the next image, facilitating the autoregistration of the bone poses for the rest of the image series. This feature helps minimize manual involvement in the registration of image series for in vivo knee kinematics during test or functional activities.
The difference between the model-generated DRR and the fluoroscopic image is potentially the most important factor that affects the performance of volumetric model-based reg-istration methods. Superimposed on the signals of the tar-geted bones in a fluoroscopic image is noise from sources such as overlying and underlying soft-tissue structures and bones. Other fluoroscopy-related sources of noise may in-clude nonuniformity of the intensifier response and geomet-ric distortions.35 Therefore, the accuracy of a registration method depends on its ability to extract the common features of both the DRR and fluoroscopic images and include them in its cost function. Failure to do this will lead to incorrect cost functions giving wrong optimum solutions. As shown in Fig.9, the greater errors with the GD method appear to be a result of errors in the cost function curve owing to noise in the fluoroscopic image and not necessarily the failure of the optimization method in reaching a global minimum. The ef-fect of noise in the PI and GD methods was more pro-nounced in the case of the tibia, with translation errors of 7.4 and 6.7 mm, respectively, in the out-of-plane direction共Fig. 9兲. The differences between the cost function curves of the femur and tibia suggest that the PI and GD methods are sensitive to the targeted bone as, with different gray values, shape, and structure, the contrast between the bone and the surrounding tissues in the fluoroscopic image may be differ-ent. The new registration method improved the accuracy, es-pecially in the out-of-plane translation component, by taking a different approach, namely, WEMS based on the matching of longer edges of the DRR and fluoroscopic edge images. Longer edges seem to be less sensitive to the abovemen-tioned noise and targeted bones so they are more likely to be common features of the DRR and fluoroscopic images. The inclusion of weightings to the edges enabled the optimization
process to give higher priority to matching long edges, simi-lar to a manual registration process. Numerically, the use of a dilated bandwidth creates slopes on each side of the edge so that the optimization problem is better defined and easier to solve in the bandwidth. With these carefully devised strate-gies, the WEMS method successfully improves the existing methods in reliably reaching an optimum that is also close to the true solution.
The accuracy of the WEMS method determined in the current study may be further improved if it is used with more advanced fluoroscopy and CT systems that provide
higher-quality fluoroscopic images and bone models. More ad-vanced computers or computational techniques, such as par-allel distributed computing, will be able to reduce the run time. The method can also be easily combined with biplanar fluoroscopy for higher accuracy. However, the use of a single-plane fluoroscopy system may be preferred as these systems are cheaper and more common in hospitals, and no modification of the hardware is needed to apply the current 2D to 3D registration method for clinical applications. Al-though there was not much difference in the performance of the WEMS method in registering the femur and tibia com-pared to the GD method, experimental evaluation of the ac-curacies is suggested before application to other bones. Com-pared to SV methods, the current approach adopts high speed single-plane fluoroscopy, which enables the measurement of fast motions within a relatively large space. Incorporating forceplates with the current method will allow for measure-ments of in vivo knee joint kinetics during functional activi-ties.
VI. CONCLUSIONS
A new volumetric model-based 2D to 3D registration method has been developed for measuring natural knee kine-matics with single-plane fluoroscopy. It mimics the manual registration process that gives higher priority to matching of longer edges in the DRR and dilated fluoroscopy images. The accuracy of the WEMS method was determined experi-mentally to be less than 0.77 mm for the in-plane transla-tions, 3.06 mm for out-of-plane translation, and 1.13° for all rotations, which is better than that of the PI and GD methods. The current study is the first well-documented experimental evaluation study of 2D to 3D registration methods for mea-suring natural knee kinematics with single-plane fluoroscopy. With the equipment used in the current study, the accuracy of the WEMS method is considered acceptable for the measure-ment of the 3D kinematics of the natural knee in clinical applications.
ACKNOWLEDGMENTS
The work presented in this paper was partially supported by the National Science Council of Taiwan共Grant No. NSC-94-2213-E-002-112兲. The authors are grateful for the assis-tance of Professor Ming-Jia Jou, Mrs. Jen-Fen Chen, and Dr. Chien-Chung Kuo during the experiment. Thanks go also to the person who donated his body to medical science. The authors 共T.-Y.T. and T.-W.L.兲 contributed equally to this work.
a兲Author to whom correspondence should be addressed. Electronic mail:
twlu@ntu.edu.tw; Telephone: 33653335; Fax: ⫹886-2-33653335.
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