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Single-Plane Radiographic Measurement of Mobile-Bearing Knee Motion Using an Unknown Distribution of Markers


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SINGLE-PLANE RADIOGRAPHIC MEASUR MENT OF MOBILE-BEARING KNEE MOTION USING AN UNKNOWN DISTRIBUTION OF MARKERS By SYDNEY M. MACHADO A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Sydney M. Machado

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iii ACKNOWLEDGMENTS I would like to thank my family for their continual love and support. I also would like to thank my advisor, Dr. Scott Banks, fo r his constant cheerleading and invaluable guidance during this process. In addition, I would like to thank my committee members, Dr. B.J. Fregly and Dr. Tony Schmitz, for their input and advice.

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iv TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iii LIST OF TABLES.............................................................................................................vi LIST OF FIGURES.........................................................................................................viii ABSTRACT....................................................................................................................... ..x CHAPTER 1 OVERVIEW.................................................................................................................1 Introduction................................................................................................................... 1 Background...................................................................................................................1 2 METHODS AND MATERIALS.................................................................................6 Images and Subjects.....................................................................................................6 Assumptions about Polyethyl ene Insert and Markers..................................................6 Recording 2-D Marker Projection Coordinates............................................................7 Shape Matching............................................................................................................8 Matching the Polyethylene Insert Markers...................................................................9 Optimization...............................................................................................................11 Computational Study..................................................................................................14 3 RESULTS...................................................................................................................18 Recording 2-D Marker Projection Coordinates..........................................................18 Image Matching..........................................................................................................18 Matching the Polyethylene Insert Markers.................................................................18 Computational Study..................................................................................................19 In Vivo Study..............................................................................................................24 4 DISCUSSION.............................................................................................................29 5 CONCLUSION...........................................................................................................35

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v APPENDIX A COMPUTATIONAL STUDY—COMPLETE RESULTS........................................36 Two Images................................................................................................................36 Three Images..............................................................................................................41 Four Images................................................................................................................47 B IN VIVO STUDY—ADDITIONAL RESULTS........................................................51 LIST OF REFERENCES...................................................................................................56 BIOGRAPHICAL SKETCH.............................................................................................58

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vi LIST OF TABLES Table page 3-1 Absolute error in : 2 images; constraint weighting, w = 10; no error in initial guess.........................................................................................................................2 1 3-2 Relative error in : 2 images, constraint weighti ng, w = 10; no error in initial guess.........................................................................................................................2 2 3-3 Absolute error in : 4 images, constraint weighti ng, w = 10, error (3) in initial guess.........................................................................................................................2 3 3-4 Relative error in : 4 images; constraint weighti ng, w = 10, error (3) in initial guess.........................................................................................................................2 3 3-5 Average rotations for all s ubjects for four activities................................................25 3-6 Average rotations for all subjects after correction for rotational bias......................25 A-1 Absolute error in : 2 images; constraint weighti ng, w = 1; no error in initial guess.........................................................................................................................3 6 A-2 Absolute error in : 2 images; constraint weight, w = 50; no error in initial guess.........................................................................................................................3 7 A-3 Relative error in : 2 images; constraint weighti ng, w = 1; no error in initial guess.........................................................................................................................3 8 A-4 Absolute error in : 2 images; constraint weight, w = 50; no error in initial guess.........................................................................................................................3 8 A-5 Absolute error in : 3 images; constraint weighti ng, w = 10; no error in initial guess.........................................................................................................................4 1 A-6 Absolute error in : 3 images; constraint weighting, w =10; error in initial guess..42 A-7 Relative error in : 3 images; constraint weighti ng, w = 1; no error in initial guess.........................................................................................................................4 3 A-8 Relative error in : 3 images; constraint weighting, w = 50; no error in initial guess.........................................................................................................................4 3

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vii A-9 Absolute error in : 4 images; constraint weighti ng, w = 10; no error in initial guess.........................................................................................................................4 7 A-10 Absolute error in : 4 images; constraint weighting, w = 10; error (3) in initial guess.........................................................................................................................4 7 A-11 Absolute error in : 4 images; constraint weig hting, w = 10; error (4) + rotational bias in initial guess...................................................................................48

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viii LIST OF FIGURES Figure page 2-1 The surgeon who performed the arthropl asty was given directions to distribute the markers roughly as shown....................................................................................6 2-2 Manual process of recording marker c oordinates from the radiographic images......7 2-3 Screen shot of shape-matching GUI with tibial component and marker model loaded......................................................................................................................... 8 2-4 Two different marker configurations would appear the same in a radiographic image. This is a problem for small im age sets, since increasing the number of available images minimizes this effect.....................................................................16 2-5 Procedure to determine marker locat ions and axial rotation of insert......................17 3-1 Absolute error between solution a nd known marker coordinates for 2, 3, 4 images as a function of the total insert rotation with respect to the viewing plane. Highlights show range of rotation that corresponds to the in vivo images..............20 3-2 Progression of marker c oordinate solutions in the x-z plane as the image rotation range increases from 4 to 52..................................................................................26 3-3 Top ( x z plane) and side ( x y plane) views of the marker locations after shapematching and optimization. Gray dots are the initial guess fr om shape-matching and black dots are the optimized solutions...............................................................27 3-4 Rotation trends for all subjects.................................................................................28 4-1 Knee kinematics for one subject with easily correctable rotational bias.................33 A-1 Image angle vs. error in marker coordi nates: 2 images; constraint weighting, w = 1; no error in initial guess.........................................................................................39 A-2 Image angle vs. error in marker coordi nates: 2 images constraint weighting, w = 50; no error in initial guess.....................................................................................40 A-3 Image angle vs. error in marker coordi nates: 3 images; constraint weighting, w = 1; no error in initial guess.........................................................................................44

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ix A-4 Image angle vs. error in marker coordi nates: 3 images; constraint weighting, w = 50; no error in initial guess.....................................................................................45 A-5 Image angle vs. error in marker coordi nates: 3 images; constraint weighting, w = 10; error applied to initial guess...............................................................................46 A-6 Image angle vs. error in marker coordi nates: 4 images; constraint weighting, w = 10; error and rotati onal bias applied to initial guess................................................49 A-7 Variation in principal distance vs. error in marker coordinates: 4 images; constraint weighting, w = 10; er ror applied to initial guess.....................................50

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x Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science SINGLE-PLANE RADIOGRAPHIC MEAS UREMENT OF MOBILE-BEARING KNEE MOTION USING AN UNKNOWN DISTRIBUTION OF MARKERS By Sydney M. Machado August 2006 Chair: Scott A. Banks Major Department: Mechanic al and Aerospace Engineering The two most common methods used to obtain precise in vivo measurements of knee kinematics for patients with total knee replacements are single-plane fluoroscopic shape-matching and roentgen stereophotogram metry. To determine the location of an unknown distribution of markers in the polyethyl ene insert of a rota ting-platform mobilebearing total knee replacement from single-pl ane radiographic images, these two methods must be combined. A guess about the marker distribution in each tibi al insert was made based on the known geometry of the insert a nd knowledge of the ge neral shape of the marker distribution. This guess was used to s eed a numerical optimi zation procedure that simultaneously sought to determine the 3D loca tions of the tantalum markers within the tibial insert and the rotation of the tibial inse rt with respect to the tibial baseplate. The quality of the numerical solution was a ssessed by mathematically projecting the estimated marker coordinates into the imag e plane and comparing those values to the values observed in the radiographic images A computational study was conducted to

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xi access the error propagation in this method before applying it to an in vivo study. It is estimated that with a four image set containi ng rotation with respect to the viewing plane of at least 10-20, this method can give a translation accuracy of 0.07in ( x -direction), 0008in ( y -direction), 0.2in ( z -direction), and a rotational accuracy of about 1.0. However, it was found that the method is subj ect to a rotational bias which must be corrected for either by use of a reference im age or by correcting the systematic error in data post-processing. The latter appro ach was taken in this investigation.

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1 CHAPTER 1 OVERVIEW Introduction The goal of this study is to determine the internal/external rotation of the polyethylene (PE) insert in the knees of 20 subjects who we re given rotating platform (RP) mobile-bearing total kn ee replacements (TKR). An unknown distribution of 4 tantalum markers were placed in the PE insert intra-operatively. The subjects were seen 3 times post-operatively and 4 single-plane radiographic images were taken during stance, kneel, deep kneel, and lunge activities. Mobile-bearing total knee replacements (MBTKR) have the potential to provide more natural knee kinematics and reduce wear related failure, and some in vitro wear studies (McEwen et al., 2005; McNulty 2002) su pport this idea. Howe ver, the clinical results for fixed and mobile-bearing knee impl ants appear to be equivalent (Callaghan 2001; Pagnano et al., 2004). In addition, only a small number of mobile bearing designs currently are available on the US market, all manufactured by the same company (Jones and Huo, 2006). For these reasons, it is hi ghly relevant to st udy these devices to determine their function and potential advantages over traditional TKR. Background Early fixed bearing TKR designs were high ly constrained, limiting range of motion and dramatically changing knee kinematic s (Ridgeway and Moskal, 2004). This constraint produced considerable shear at the component fixation interfaces leading to failure through loosening. In an effort to restore more natural knee motion and prevent

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2 loosening, designs with round-on-fl at or flat-on-flat articulatin g surfaces were introduced. These designs did have the intended effect, but soon a new predominate mode of failure emerged. Reducing the contact surface area su bjects the polyethylene to high levels of multi-directional contact stress and sliding, cau sing increased wear rates. The concept behind the mobile-bearing TKR was to reduc e wear by increasing conformity of the articulating surfaces, while sti ll allowing free range of moti on to restore more natural knee kinematics (Jones and Huo, 2006). Several in vitro studies (McEwen et al. 2005; McNulty, 2002) have shown that the polyethylene in RP TKR does exhibit wear rates superior to that in a fixed bearing counterpart. However, both of these studies tested only one RP design, the Sigma RP System (DePuy), and no study has yet found supe rior clinical results for RP TKR (Jones and Huo, 2006). In fact, one clinical st udy (Ridgeway and Moskal, 2004) reports 25 cases of clinical instability and pain fo llowing RP and meniscal bearing TKR. In light of the current inconclusive evidence regarding mobile-bearing TKR it is important that investigation in to this style of implant cont inue. To better understand the kinematics of a RP TKR there must be some way to determine the motion of the PE insert which appears transparent in radiographi c images. One way is to perform in vitro cadaver testing as has been done by Stukenbor g-Closeman et al. (2002), and D’Lima et al. (2001). However, in vivo studies are real ly needed to get a true idea of RP TKR performance during volitional, dynamic wei ght-bearing activities. In one study by Dennis et al.(2005), fluoroscopic shape matching techniques were used on a series of images taken in vivo of subjects with mobile-bear ing TKR. The distribution of markers was known and a CAD model of the PE insert was available. The shape matching technique

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3 is described in detail in Mahf ouz et al. (2003) and relies on image intensity and contour matching to hypothesize a pose that closely matche s that of an input fluoroscopic image. Once the femoral and tibial components were matched to the image, the authors simply aligned the CAD model of the in sert markers with the markers visible in the fluoroscopic image. In this study three different designs of RP TKR and one design that allowed for anterior-posterior translation as well as rotation were ev aluated. While the greatest amount of axial rotation reported was in the femu r with respect to the tibia, there was also significant rotation of the insert with respec t to the tibial compone nt. Rotation of the insert with respect to the femoral component also occurred, but was the smallest in magnitude. The results varied with implant design, but the overall trend was for the rotation of the PE insert to closely follow th at of the femoral component with respect to the tibia. These findings are in agreement with the in vitro findings of the D’Lima et al. (2001) study. Another in vitro study was done by St ukenborg-Closeman et al. (2001) on a rotating and AP translating TKR, the Interax IS A. In that study only 1.8 of axial rotation and small amounts of translati on were reported. These resu lts were confirmed by an in vivo study by Fantozzi et al (2004) on the same mobile-b earing design, which also found only small amounts of both translation and rotati on. Fantozzi et al. used a combination of image matching and roentgen stereophotogra mmetric analysis (RSA) to locate the tantalum markers and determine the motion of the insert. They found that the femoral component did not always drag the insert with it as hypothesized and that in some case the two components rotated independently.

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4 In a study by Delport et al. (2006) of th e Performance series of knee implants (Biomet), a posterior stabilized RP mobile-b earing design was also evaluated using shape matching. The tibiofemoral rotation reported wa s consistent with the study by Dennis et al. (2005). In addition, subjects with this design exhibited gr eater asymmetrical posterior femoral translation in flexion than previously reported in the literature. Delport et al. (2006) claimed the measured motions of the RP-TKR were closer to those found in the healthy natural knee. Other studies of mobile-bearing TKR ha ve relied on a known distribution of tantalum markers in the PE insert and use roentgen stereopho togrammetric (RSA) techniques to determine the marker orientat ions. Traditional RS A requires specialized and expensive equipment that is not widely av ailable and is somewhat labor intensive to use (Valstar et al., 2000). Bi-plane RSA equipment setups can also limit subject movement, and can cause the markers to become occluded during flexion-extension motion (Garling, et al. 2005). Because of this attempts have been made to make RSA more accessible by adapting it to a standard single-plane fluoroscopic imaging set up, developing a digital automated method, and de veloping a model-based form (Garling et al., 2005; Valstar et al., 2001, Ka ptein et al. 2003). One st udy by Yuan et al. (2002) incorporates RSA with single-plane fluorosc opy, but requires each 3-D marker to be assigned to the correct projection in each imag e so that the mathematical formulation can be solved directly. This turns out to be highly sensitive to number of control points, focus position, and object distance, which may pr ove difficult in a laboratory setting. Another study, Garling et al (2005), uses an approach called Model-Based Roentgen Fluoroscopic Analysis. In this study accurate models of marker geometry are created

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5 and to represent the insert and tibial component A global optimizer then uses the marker models to reconstruct th e location of the insert with respect to the tibial component based on a series of fluoroscopic images. At this point, only results of a phantom study have been reported for this method. Without a known marker model, however, a hybrid approach would have to be taken. This approach would start by using sh ape matching to determin e the orientation of the visible parts of the TKR. Once the pose of the tibial and femoral components are determined, they can be used to limit the degr ees of freedom available to the insert. Then trial and error shape matching could be used to determine a possible distribution of markers. Although the marker distribution is unknown, the locations can be described as points and are still related to the 2-D projec tions by the radiographi c projection geometry as described by Selvik (1989). Projecting the estimated marker coor dinates would give some error between the estimated and the know n, and this could be used as an overall nearness criterion to determine goodness of fit (Gordon and Herman, 1974). The criterion could then be optimized to determ ine the best possible lo cation and orientation of the markers within the constraints of the known dimensions of the PE insert.

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6 CHAPTER 2 METHODS AND MATERIALS Images and Subjects In the study 19 subjects were seen 3 times post-operatively at periods of 3, 6, and 12 months. One subject was seen at 2, 3, a nd 12 months. Each was asked to perform 4 activities: stand at full ex tension, 90 kneel, full flexi on kneel, and lunge. One singleplane radiographic image was taken for each ac tivity. This yielded between 4-12 images per subject. Assumptions about Polyethylene Insert and Markers Several simplifying assumptions can be made about the distribution of markers in the insert. The geometry of the insert is known from available CAD models. The markers were placed intra-operatively from the si de, thus it is likely that the markers are within a few millimeters of the insert periphe ry. The surgeon was instructed to place the markers roughly as shown in Figure 2-1, with the two medial markers being situated somewhat closer together, while the la teral markers are farther apart. Figure 2-1. The surgeon who performed the arthroplasty was given directions to distribute the markers roughly as shown.

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7 Finally, it is assumed in each image that fi ve degrees of freedom of the insert are constrained. The origin of the insert reference sy stem is co-incident with that of the tibial component, which is centered on the post a bout which the bearing rotates, and the inferior surface of the insert is co-planar with the tibial tra y. The only degree of freedom available is in axial rotation of the insert about the tibial post. Recording 2-D Marker Projection Coordinates The markers visible in each image were then selected and their 2-D projection coordinates recorded in a ma nual process (Figure 2-2). In order to quantify the error in this pro cess a series of 10 images were generated. Each image contained 20 markers which were distributed in uniform ly random locations. Noise and gray scale gradient was added to th e images, while the markers were subject to Gaussian blur ( = 0.5). The markers were then ma nually selected and the coordinates recorded. Figure 2-2. Manual process of recording ma rker coordinates from the radiographic images.

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8 Shape Matching The physical orientation of the knee wa s determined using a shape matching technique evolved from Banks and Hodge (1996) This technique extracts the contours of the TKR silhouette using a Canny edge detector, and then uses non-linear least squares to minimize the sum of Cartesian distances between the implant contour in the radiographic image with a contour of the pr ojected implant CAD model (Figure 2-3). The error in this process is know n to be less than 0.6mm in the sagittal plane and 1.08 for all rotations. Figure 2-3. Screen shot of sh ape-matching GUI with tibial component and marker model loaded. The two registered components easily can be checked to make sure they are not in a physically (impinging) or anatomically im possible position, and normally adjacent image frames can be checked to insure motion contin uity. However, in the current investigation there are no adjacent frames with which to ch eck for joint motion continuity. For this reason it was determined that a further com puter simulation of the method was necessary

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9 in order to fully quantify th e error. A computer simula tion was conducted by generating 10 views of the knee replacement used in th e study. The position and orientation were randomly generated from a uniform distribution over the ranges: -2.5 cm < x translation < 2.5 cm -2.5 cm < y translation < 2.5 cm 7.5 cm < z translation < 12.5 cm -16.0 deg < x rotation < 16.0 deg -16.0 deg < y rotation < 16.0 deg -90.0 deg < z rotation < 90.0 deg. This set of 10 images was given to 4 subj ects who were experienced with the image matching process and they were asked to provide an accurate shape registration. Student’s t-tests were used to determine wh ether the error found he re was significantly different from the error found previously. Matching the Polyethylene Insert Markers Based on the assumptions about the polye thylene insert and markers, the 3-D coordinates of the markers are hypothesized, and a CAD model is created. This model is then imported into the image matching softwa re. Since the marker model only has one degree of freedom, it is rotated axially un til the correct markers are aligned and/or occluded in all images of the current subject. If the mark ers do not align, the CAD model is modified and re-imported into the software This iterative process provides an initial guess for the marker distributions and insert rotation angles which then is refined through numerical optimization. The estimates of 3-D marker locations a nd insert axial rotation for each image can be checked against the know n 2-D projection coordinates simply by transforming them

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10 into the global coordinate system and inse rting them into the pr ojection equation. The transformation can be written as in Equation 2-1. g lobalglobaltibialcomponent inserttibialcomponentinsertTTT 2-1 This is where component tibial insertT is the transformation from the tibial coordinate system to the PE insert coordinate system, in which the markers are located (since the insert only has one degree of freedom with respect to the tibial component this will depend on the estimated axial rotation, ); global component tibialT is the transformation from the global coordinate system to the tibial component coordinate syst em (these are the rota tions and translations that are estimated by the shape matching); and global insertT is the transformation from the global coordinate system to the insert coor dinate system known only approximately. The transformation, global insertT, and the marker coordinates, x can be written as: ()()()cos()Tx (+ )-( )(+ )-( )Ty n ()-( )(nnnn nnn nnglobal insertT cccssscsssccsscccsscsssc csccssssscscccssccsscsscsc cscccsscs )+() Tz 0001 nn sccc 2-2 and global g lobalinsertinsert x Tx 2-3 where g lobalggg x xyz 2-4 and insertinsertinsertinsertxxyz 2-5 Cosine and sine are abbreviated as c and s respectively; , Tx, Ty, and Tz describe the known 3-D orientati on of the tibial component; and n is the estimated axial

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11 rotation for image n. The vector globalx is the estimated marker coordinates in the global reference frame and insertx is a vector of the estimated ma rker coordinates in the insert reference frame. Once globalx is calculated the 2-D projec tions of the markers can be determined using the known image setup geometry (Figure 2-4). The estimated 2-D projections, (uknown, vknown), can be written as: ()g est g x uC Cz 2-6 and ()g est g y vC Cz 2-7 where C is the principal distance. These marker projections, (uest, vest) typically do not match the true marker projections, (uknown, vknown) measured in the radiographic images. These differences, (uknown uest, vknown-vest) provide an error measure which can be minimized using numerical optimization techniques. Figure2-4. Imaging Geometry. Optimization For each subject’s set of images there are 3*m+n unknowns, where m is the total number of markers visible in e ach set of images and n is the number of images for which

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12 the insert rotation, needs to be measured. Each visi ble marker produces the 2 projection equations shown in section II(E) for each image in which it is visible. For example, if 8 images were available with 3 markers visible in each image, there would be 9 + 8 = 17 unknowns, and 2*3*8 = 48 possible projection equati ons if all images were considered simultaneously. This over-determined syst em is non-linear due to the unknown axial rotation, and the perspective proj ection transformation. One way to approach this problem is to use a non-linear least squares algorithm to minimize the error between the known 2-D projection coordinate s and the estimated 2-D proj ection coordinates. The Matlab optimization toolbox provides the lsqnonlin routine based on the interiorreflective Newton method, and requires a us er-defined cost function to compute and minimize a vector of error equa tions (Equation 2-8 and 2-9). 1 2() () () ()n f x f x Fx f x 2-8 2 2 211 ()() 22mini x iFxfx 2-9 In this case F(x) is the error vector and f1 through fn are the difference (uknown uest, vknown-vest) and can be expanded out to: 31323334 111213() ()() () () (globalglobalglobalglobal knowninsertinsertinsertinsertinsertinsertinsert globalglobalglobal insertinsertinsertinsertinsertg nknownknowngg gx fuCuCzCx Cz uCxyz Cxyz TTTT TTT14)global insertinsert T 2-10 and

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13 31323334 212221() ()() () () (globalglobalglobalglobal knowninsertinsertinsertinsertinsertinsertinsert globalglobalglobal insertinsertinsertinsertinsertg nknownknowngg gy fvCvCzCy Cz vCxyz Cxy TTTT TTT324)global insertinsertz T 2-11 In addition, assumptions about locations of the markers provide constraints for optimization. This is done by supplying to th e user-defined cost f unction a contour of the insert in the x-z (transverse) plane. This contour, C, is an l2 matrix where l is the number of discrete points along the contour and the column s are the x and z coordinates at each point. The distance, dcontour, between the contour and each marker’s (x, z) coordinates at the current iteration, k is stored in a matrix of the same size as the contour: []1,2,3...contouriinsertinsertk idCxzil 2-12 The minimum of the distance magnitude, d is then used to formulate a weighted dimensionless distance measure, D(x) The weight, w was determined during the computational study described in Chapter 2: Computational Study When D(x) is concatenated onto the error vector F(x) it implements a soft constraint penalizing marker location solutions that exceed some threshold distance, dcrit, from the insert periphery: (,1)(,2)22mincontourcontour iiddd 2-13 2()*critd Dxw d 2-14 () () () Fx Fx Dx 2-15

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14 The number of images evaluated at each ite ration can be specified by the user and the algorithm can randomly select that number of images from the available images. The optimization can then be run a selected number of times, such that a sufficient number of all possible combinations of images have b een selected or the so lution no longer changes by a threshold amount. The advantage of this, rather than evaluating all the images at once, is that if one or two images have greater amount of error in them (or are not sufficiently different to pr ovide novel information) a nd are causing the optimizer difficulty, the influence of those images can be reduced. Analogous to a Kalman filter, this is done by taking the sum of each solution divided by its residual, and then dividing by the sum of the residuals: j jr 1,2..., 1out jx x jn r 2-16 If the optimization is run n times, this will cause solutions with high residuals to have less effect on the total solution outx Computational Study A computer simulation study was performed to quantify the sensitivity of the estimation method to the number of images used and to the absolute amount of insert rotation with respect to the viewing frame of reference (an image set in which there is little insert rotation would not provide enough in formation for reliable estimates.). A set of synthetic marker data was created and was projected into 2-D coordinates for 46 images ranging from 0-90 of axial rotation using projection geometry similar to the in vivo images. The tibial component was assume d to be located at its origin, with zero

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15 rotation. A uniform distribution of noise wa s then added to the 2-D coordinates. The magnitude of this error was determ ined as described in Chapter 2: Recording 2-D Marker Projection Coordinates The insert and tibial component orientations were subjected to uniformly distributed error found by the proce ss described in section Chapter 2: Image Matching. For the first evaluation the known marker locations and axial rotati on served as the initial guess to the optimizer. Two, th ree, and four images were evaluated for a total axial rotation range of 0-90, 0-52, and 0-36 respectively. Each number of images was evaluated at weights of w = 1, 50, and 10 on the soft constraint (section II(F)) as well. The sets of 3 and 4 imag es were equally spaced throughout the total rotation range. Next, a random uniform error distribution of 3 was added to the axial rotation, and the marker coordinates were subject ed to random uniform distribution of error of the magnitude determined in Chapter 2: Matching the Polyethylene Insert Markers before being used as the initial guess. For this case only 3 and 4 images were evaluated. The rotational error in the initia l guess was then increased to 4 and the images evaluated again. In addition, the sens itivity to variation in the principal distance, C, was also evaluated. Finally, 3 and 4 image sets were evaluated with an additional rotational bias in the marker coordinates fo r both 3 and 4 in Since the markers are fixed to a rigid body that rotates only in-plane, this was done to simulate the situation where the relative position of the markers to each other can be determined while the absolute location within the insert remains ambiguous (Figure 2-4).

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16 Figure 2-5. Two different marker configurations would appear the same in a radiographic image. This is a problem for small im age sets, since increasing the number of available images minimizes this effect.

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17 Figure 2-5. Procedure to determine marker locations and axial rotation of insert.

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18 CHAPTER 3 RESULTS Recording 2-D Marker Projection Coordinates For the manual marker selection process the error in the u direction was found to be 1.0 0.3 pixels (range,-0.0 to 1.8 pixels), a nd 0.3 0.4 pixels (range, -0.6 to 1.6 pixels) in the v direction. For images of the size consider ed in this study, this is equivalent to 0.014in 0.005in (range, -0.001in to 0.025in) in the u direction and 0.004in 0.005in (range, -0.008in to 0.022in) in the v direction. Image Matching To quantify the error associated with th e shape matching process, four subjects experienced with the software were given a set of 10 synthetic images to match. Student’s t-tests were used to determine if the error found was significantly different from the error found in Banks and Hodge (1996). For x and y translation, and all rotations no significant difference was found (p = 0.05). However, a bias in the z translation resulted in an average erro r of -0.804in 0.041in (range, -1.763in to 0.171in). This was found to be highly signifi cantly different from the original reported error (p = .001). Matching the Polyethylene Insert Markers After the matching process was completed and an initial estimate determined for the marker model, the 3-D coordinates were mathematically projected onto the image plane and compared to the recorded 2-D coordinates from the actual images. The error between (uknown uest, vknown-vest) was found to be -0.00361in 0.082in (range, -0.876in-

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19 0.49in) in the u direction and -0.001in 0.067in (range, -0.570in0.574in) in the v direction. However, the large range valu es were found to correspond with only one 4 images sub-set of one subject’s images, out of a total of 151 images. With these 4 images removed the error in the u direction was 0.002in 0.000i n (range, -0.132in – 0.122in) and 0.00 0.022in (-0.141in – 0.179in). Computational Study A computational study was performed to asse ss the sensitivity of the method to the number of images used for estimation, the rotation range of the insert with respect to the viewing plane, constraint wei ghting, and random error in i nput and initial guesses. For two images, the average absolute error in the x and y direction increased with increasing rotation range (Figure 3-1). The z direction absolute error decreased rapidly as total rotation increased (Figure 3-1). For three and four images, average absolute error in the x direction increased with increasing rotation, but the error in the y and z directions showed decreasing errors with increasing rotation (F igure 3-1). In all cases, the change in x and y errors with increasing rotation range was much smaller than the change in z errors. For two images the insert ax ial rotation showed a consiste nt bias of about 5. By correcting for this bias, the relative error in the angle between the images was reduced from 7.06 1.96 to 1.70 1.96. In addi tion, the absolute error in the rotation for two images was 5.99 4.37 (Table 3-1 and 3-2) This value steadily increased as the rotation range increased. The rotational er ror decreased as the number of images increased, improving to -1.23 1.39 for four images. The relative rotational error between images for four images was 0. 15 1.15 (Tables 3-3 and 3-4).

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20 Figure 3-1. Absolute error between solution and known marker coordinates for 2, 3, 4 images as a function of the total insert rotation with respect to the viewing plane. Highlights show range of rotation that corresponds to the in vivo images.

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21 Table 3-1. Absolute error in : 2 images; constrai nt weighting, w = 10; no error in initial guess. Actual Found Error Actual Found Error 0 -2.1909 -2.1909 0 2.4519 2.4519 0 3.1747 3.1747 48 59.5943 11.5943 0 -0.9013 -0.9013 0 2.7858 2.7858 4 7.5629 3.5629 52 63.2585 11.2585 0 -0.4344 -0.4344 0 3.4182 3.4182 8 11.4426 3.4426 56 67.1996 11.1996 0 -0.1407 -0.1407 0 2.5802 2.5802 12 16.1124 4.1124 60 67.417 7.417 0 -0.0713 -0.0713 0 3.6068 3.6068 16 22.9429 6.9429 64 75.2629 11.2629 0 0.2621 0.2621 0 5.0742 5.0742 20 28.0633 8.0633 68 81.1398 13.1398 0 1.5036 1.5036 0 4.5062 4.5062 24 33.9424 9.9424 72 82.9359 10.9359 0 1.056 1.056 0 3.811 3.811 28 37.8924 9.8924 76 84.8924 8.8924 0 1.37 1.37 0 4.6463 4.6463 32 42.8302 10.8302 80 90.5222 10.5222 0 1.7608 1.7608 0 4.0709 4.0709 36 48.1659 12.1659 84 93.9927 9.9927 0 2.4276 2.4276 0 4.0592 4.0592 40 52.8619 12.8619 88 97.5466 9.5466 0 1.5947 1.5947 44 52.9173 8.9173 Average Error: 5.59 4.37

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22 Table 3-2 Relative error in : 2 images, constraint weighting, w = 10; no error in initial guess. Actual Found Corrected j i 2 1 2 1 0 5.3656 0 4 8.4642 3.0986 8 11.877 6.5114 12 16.2531 10.8875 16 23.0142 17.6486 20 27.8012 22.4356 24 32.4388 27.0732 28 36.8364 31.4708 32 41.4602 36.0946 36 46.4051 41.0395 40 50.4343 45.0687 44 51.3226 45.957 48 57.1424 51.7768 52 60.4727 55.1071 56 63.7814 58.4158 60 64.8368 59.4712 64 71.6561 66.2905 68 76.0656 70.7 72 78.4297 73.0641 76 81.0814 75.7158 80 85.8759 80.5103 84 89.9218 84.5562 88 93.4874 88.1218

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23 Table 3-3. Absolute error in : 4 images, constraint weighting, w = 10, error (3) in initial guess. Actual Found Error 0 -0.6436 -0.6436 2 2.8539 0.8539 4 3.8478 -0.1522 6 6.0921 0.0921 0 -1.0304 -1.0304 4 4.8672 0.8672 8 7.6452 -0.3548 12 12.2736 0.2736 0 -1.8479 -1.8479 6 5.6254 -0.3746 12 11.2466 -0.7534 18 18.0415 0.0415 0 -2.132 -2.132 8 7.1621 -0.8379 16 13.7297 -2.2703 24 21.6134 -2.3866 0 -2.5209 -2.5209 10 8.5415 -1.4585 20 17.4568 -2.5432 30 27.1436 -2.8564 0 -2.3112 -2.3112 12 11.0638 -0.9362 24 21.4104 -2.5896 36 32.9547 -3.0453 Average Error: -1.20 1.21 Table 3-4. Relative error in : 4 images; constraint weighting, w = 10, error (3) in initial guess. Actual Total Found Total Actual Found Found Found k i 4 1 j i 2 1 3 2 4 3 6 6.7357 2 3.4975 0.9939 2.2443 12 13.304 4 5.8976 2.778 4.6284 18 19.8894 6 7.4733 5.6212 6.7949 24 23.7454 8 9.2941 6.5676 7.8837 30 29.6645 10 11.0624 8.9153 9.6868 36 35.2659 12 13.375 10.3466 11.5443 Average Error ( j i): 0.14 1.15 Average Error ( k i): 0.43 1.04 The marker coordinates in the solution were shown to be fairly insensitive to constraint weighting. For three images, as the weight was increased from 1 to 10 the

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24 average absolute error in the x direction decreased from 0.167in 0.065in to 0.067in 0.046in. In the z direction the absolute average error decreased from 0.145in 0.126in to 0.102in 0.077in. The error in the y direction remained unchanged. This trend held for two and four images as well. The axial rotation of the insert in the solution showed more sensitivity to changes in the weighting. Fo r two images the relative error in the angle between the images decreased from 10.34 3.67 to 7.06 1.96. This method was found to be relatively inse nsitive to variatio n in the principal distance in x and y direction. In the z direction accuracy drop ped by about 0.1in for deviations of 5in from the known principal distance. The configuration most relevant to this study was four images, with both error and rotational bias in the initial guess, and 5-20 degrees of rotation as in the in vivo images. For this configuration the error in the x y and z directions was 0.041in 0.022in, 0.008in 0.002in, and 0.112in 0.049in respectively. The error in the insert rotation, was 0.67 3.27. Additional re sults are reported in Appendix A. In Vivo Study For the in vivo study the tantalum markers were found to be distributed in the polyethylene insert as shown in Figure 3-3. For the insert rotations, 9 of the 20 subject’s results contained varying magnitudes of rota tional bias as discussed in Chapter 2: Computational Study and demonstrated in Figure 2-4. These rotational biases were systematic and were corrected for during data post-processing. Before rotational bias was corrected for the average rotations for all subjects during the four activities were as shown in Table 3-5. After correcting rotati onal bias, the magnitude of femur-to-insert rotation was reduced from -12.33 15.92, to -6.12 3.29 (Table 3-6 and Figure 3-5A). These values are consistent with the design of the implant. At 12 months, the average

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25 external rotation of the tibial component with respect to the insert for all activities and subjects was -1.88 7.02, the external ro tation of the tibial com ponent with respect to the femoral component was -2.48 7.02, and the external rotation of the insert with respect to the femoral component was -0.41 2.46 (Fi gure 3-5B). Additional results are reported in Appendix B. Table 3-5. Average rotations for all subjects for four activities. No Corrections: TIBIA TO INSERT TIBIA TO FEMUR INSERT TO FEMUR Average () Std. Dev. () Average () Std. Dev. () Average () Std. Dev. () Max. Flex -3.020 7.30 Max. Flex-3.64 5.93 Max. Flex -0.62 5.20 Standing 2.05 5.88 Standing 1. 91 5.01 Standing -0.14 4.25 Squat -4.27 7.55 Squat -5.41 6.00 Squat -1.14 4.36 Kneeling -2.73 7.67 Kneeling -3.14 5.56 Kneeling -0.41 5.14 Max 15.92 Min -12.33 Table 3-6. Average rotations for all subj ects after correction for rotational bias. Rotational Bias Corrected: TIB/INS TIB/FEM INS/FEM Average () Std. Dev. () Average () Std. Dev. () Average () Std. Dev. () Max. Flex -3.57 7.09 Max. Flex-3.64 5.93 Max. Flex -0.74 2.40 Standing 1.49 5.07 Standing 1.91 5.01 Standing 0.45 2.03 Squat -4.83 6.50 Squat -5.41 6.00 Squat -0.79 2.35 Kneeling -3.30 6.18 Kneeling -3.14 5.56 Kneeling 0.00 2.83 Max 3.29 Min -6.12

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26 3 Images, w=1 -2 -1.5 -1 -0.5 0 0.5 1 1.5 -1-0.500.511.5X Z A 3 Images, w=10 -1.5 -1 -0.5 0 0.5 1 1.5 -1-0.500.511.5X Z B Figure 3-2. Progression of marker coordinate solutions in the x-z plane as the image rotation range increases from 4 to 52 with (A) constraint weight w = 1, and (B) w = 10. The solutions progress in a di rection towards the black dots that indicated the known marker positions.

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27 Figure 3-3. Top ( x z plane) and side ( x y plane) views of the marker locations after shape-matching and optimization. Gray dots are the initial guess from shapematching and black dots are the optimized solutions.

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28 Average Rotation by Activity for All Subjects-6 -5 -4 -3 -2 -1 0 1 2 3 Max. FlexStandingSquatKneelingExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM A Average Rotation for All Subjects-5 -4 -3 -2 -1 0 1 2 3 months6 months12 monthsExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM B Figure 3-4. Rotation trends for all subjects over A) all times for each activity and B) all activities for each time point. External rotation is taken as positive.

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29 CHAPTER 4 DISCUSSION Roentgen Stereophotogrammetric Analysis (RSA) provides the standard by which other radiographic kinematic measurement tech niques can be judged. Small radio-opaque tantalum markers are placed directly into the bones and observe d using a pair of simultaneous radiographic exposures. These implanted fiducials are unaffected by motion of overlying muscle, fat and skin, and can provide joint motion measurements with uncertainties as low as 0.005mm for transla tions and 0.15 for ro tations (Selvik,1989). However, RSA equipment is expensive and not widely accessible (G arling et al., 2005), and marker placement requires a surgical proced ure. In addition, precise calibration must be done in the working volume before any measurements can be taken, and all subsequent measurements must be taken with in the calibrated volume. These constraints limit the space in which the subject can move, making it difficult to perform measurements of dynamic activities. RSA principals also have been a pplied to single-plane radiography or fluoroscopy, where greater experimental flexib ility and equipment availability are gained at the expense of larger measurement un certainties (especially for translations perpendicular to the image plane, Yuan et al ., 2002). Garling et al (2005) demonstrated a method for measuring the motions of a mob ile bearing TKR polyethylene insert having tantalum markers inserted in a precisely m easured three dimensional trapezoidal shape. Marker placement was designed to provide maximal non-collinearity for good visibility and well conditioned measurements.

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30 Single-plane fluoroscopy also has been used with 3D-to-2D shape registration techniques to study the motions of implan ts and bones (Banks and Hodge,1996; Mahfouz et al., 2003; Fregly et al., 2005) Fluoroscopes are available in most hospitals and provide the ability to acquire images at rates of 330 frames per second, which is adequate to study many relevant joint motions. The quality of the measurement is dependent upon the accuracy of the 3D model of the bone or impl ant being registered (Kaptein et al., 2003; Moro-oka et al., 2006), with implant models typically provided by the manufacturer or laser scans, and bone models typically being generated by CT or MR scans. This study provided a novel challeng e requiring adoption of a combined measurement approach. The task to meas ure the motions of the mobile-bearing TKR UHMWPe insert having an unknown distribut ion of four tantalum markers was accomplished using 3D-to-2D shape registrati on techniques for the femoral and tibial components and a novel technique motivated by RSA to determine the insert motions. The measurement approach was based on the co nstraint that the UHM WPe insert had one rotational degree of freedom with respect to the tibial baseplate, and that multiple views were available for each knee. Thus, 5 out of 6 degrees of freedom for the insert could be determined from the pose of the tibial baseplate, and the remaining motions determined by solution of an over-determine d set of projection equations from the series of images. This approach simultaneously estimates the unknown locations of th e tantalum markers and the unknown rotation of the UHMWPe insert. Application of this conceptual solution was complicated by several factors: 1) All four tantalum markers are not visible in each radiographic image (only one or two markers are visible in many images), 2) The total rotation range of the insert with respect to the viewing plane is between 5 and 20 for

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31 each knee, thus the observations provide a sp arse set of observations from which to determine estimates of all unknown parameters. A guess about the marker distribution in each tibial insert was made based on the known geometry of the insert and knowledge of the general sh ape of the marker distribution. This guess was used to seed a numerical optimization procedure that simultaneously sought to determine the 3D loca tions of the tantalum markers within the tibial insert and the rotation of the tibial inse rt with respect to the tibial baseplate. The quality of the numerical solution was assessed by mathematically projecting the estimated marker coordinates into the imag e plane and comparing those values to the values observed in the radiographic images. Measurement uncertainties were determined for each step in the measurement process, including measurement of tantalum ma rker locations in the radiographs (1.0 0.3 pixels in the u direction, and 0.3 0.4 pixels in the v direction), and measurement of the tibial baseplate pose. Tibial pose measurement e rrors for all rotations and (x, y) translations were not signif icantly different previously reported values (Banks and Hodge, 1996). However, z direction (out-of-plane) errors were found to be significantly different from previous reports due to a bias in the negative z direction. This measurement bias likely resulted from e dge attenuation in the synthetic images. The projections of estimated marker coor dinates were found to match very well, with average errors of (0.002i n 0.000in, 0.00 0.022in) in the (u, v) images. In one troublesome sub-set of images, the best match was extremely poor (errors of about 0.5in), but this accounted for only 4 out of 151 images. The hunch that this sub-set of images

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32 was an outlier was confirmed when the optimizer returned a non-solution for this sub-set only. In addition, the computational study indica ted that the optimization was primarily sensitive to the number of im ages, as the error in the z direction improved from approximately 0.30in with two images to about 0.115in with four images. The results also indicate that more rotation ma y not be better as error in the x direction is seen to markedly increase for rotations greater than 30 as it becomes the out-of-plane axis. A major difficulty with this measurement approach is that there is no unambiguous way to determine the absolute zero rotation of the tibial insert (Chapter 2, Figure 2-4). One possible way is to pick an image where the relative rotation between the tibial and femoral components is small (<2 ) and define that as zero in sert rotation. The markers could then be aligned in that image, with ro tations in other images defined relative to the reference image. A potential problem is that some subjects have no image with a small tibial/femoral component rotation. When this occurs, the only way to zero the rotation is to pick the image with the smallest compone nt rotation and assume that the insert has followed the femoral component as reported in the literature (De nnis et al, 2005). In reality, if the insert has not tracked with the femoral compone nt, this definition will result in a rotational bias. This correction approach was perfor med during data post-processing in this study. An example of an image set th at needed correction is shown in Figure 4-1, where the circles indicate images showing a sm all rotation between the tibial and femoral components, yet a surprisingly large rotati on between both components and the insert. This represents a physiologically and biom echanically improbable condition, which can

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33 be corrected by redefinition of the zero ro tation alignment of th e measured marker distribution. A B Figure 4-1. Knee kinematics for one subject w ith easily correctable rotational bias. A) Rotational offset that occurred using this method when a zero reference was not established. B) This can be corr ected by assuming neutral insert rotation when the tibial and femoral components are aligned (close to zero relative rotation). “TIB/INS” indicates axial ro tation between the tibial baseplate and the tibial insert, “TIB/FEM” indicates axial rotation between the tibial baseplate and the femoral component, and “INS/FEM” indicates axial rotation between the tibial insert and the femoral component. The corrected insert rotations appear to correlate well with the intended design of the implant, the in vitro study by Most et al. (2003), and other in vivo studies done on posterior stabilized rotating platform mob ile-bearing TKR designs (Dennis et al., 2005; Delport et al, 2006). The study by Most et al. (2003) on the same LPS TKR (Zimmer, Inc.) used in this investigation revealed a 55% restoration of the in ternal rotation of the tibial component (6.1 8.2 ) with muscle loads. This is similar to the magnitudes found

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34 here for squat and kneeling activities. De nnis et al. (2005) and Delport et al. (2006) found that the insert tends to follow the fe moral component, but De nnis et al. (2005) did report small amounts of rotation of the insert past the femoral component during small ranges of activities. Further investigation into this method woul d require that an attempt be made to ensure that each subject have a clear reference alignment image. A non-weight bearing image might serve this purpose, as Most et al. (2003) reported small internal rotation of the tibial component for unloaded testing (-0.12 6.2 at 30 flexion). Resolution of the rotation offset definition ambiguity would provide higher measurement certainties. As currently performed, the measurement process appears to determine the tantalum marker locations and insert rotations with sufficient accuracy to determine some useful information about mobile bearing knee replacement kinematics.

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35 CHAPTER 5 CONCLUSION A technique was developed to determine th e rotation of a polyet hylene insert in a rotating platform mobile-bear ing TKR from an unknown distribution of markers in single-plane radiographic images. Many levels of assumption were made in order to arrive at a solution. The major difficulty with the proposed method was that there was no unambiguous way to determine 0 rotation of the insert. Thus, some of the results turned out to have a systematic rotational offset. As a best guess, 0 was assumed to occur in an image with small (<2) tibial/femoral component rotation, if such an image existed, and the rotations were corrected in post processing. Despite this, useful information can still be extracted from the results. Once corrected, the results show component rotation that is consistent with the literature. Any future investigation into this method should include a richer image set, including a reference im age, and implement a more robust global optimizer.

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36 APPENDIX A COMPUTATIONAL STUDY—COMPLETE RESULTS Two Images Rotational Error: Table A-1. Absolute error in : 2 images; constraint weighting, w = 1; no error in initial guess Actual Found Error Actual Found Error 0 -1.6197 -1.6197 0 -2.8315 -2.8315 0 3.1075 3.1075 48 58.9269 10.9269 0 -1.1632 -1.1632 0 -1.0355 -1.0355 4 7.4789 3.4789 52 65.3768 13.3768 0 2.6249 2.6249 0 1.3973 1.3973 8 16.52 8.52 56 71.571 15.571 0 -0.5346 -0.5346 0 4.4683 4.4683 12 15.3502 3.3502 60 77.8549 17.8549 0 -1.1416 -1.1416 0 3.5788 3.5788 16 20.353 4.353 64 84.0316 20.0316 0 -2.006 -2.006 0 3.9181 3.9181 20 26.1008 6.1008 68 85.8808 17.8808 0 -2.3083 -2.3083 0 5.8344 5.8344 24 30.684 6.684 72 91.0406 19.0406 0 -2.4398 -2.4398 0 5.702 5.702 28 34.9818 6.9818 76 93.8884 17.8884 0 -2.6255 -2.6255 0 6.5901 6.5901 32 39.579 7.579 80 99.0792 19.0792 0 -2.8944 -2.8944 0 3.9863 3.9863 36 44.4742 8.4742 84 95.7458 11.7458 0 -2.7167 -2.7167 0 3.7608 3.7608 40 48.8154 8.8154 88 100.8134 12.8134 0 -2.5177 -2.5177 44 53.8935 9.8935 Average Error: 5.89 6.89

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37 Table A-2. Absolute error in : 2 images; constraint weight, w = 50; no error in initial guess Actual Found Error Actual Found Error 0 -1.8028 -1.8028 0 2.1091 2.1091 0 3.5153 3.5153 48 60.1034 12.1034 0 -0.97 -0.97 0 2.4913 2.4913 4 7.3013 3.3013 52 63.6336 11.6336 0 -0.476 -0.476 0 2.3236 2.3236 8 11.3543 3.3543 56 64.1328 8.1328 0 -0.5944 -0.5944 0 2.4988 2.4988 12 15.5643 3.5643 60 67.4903 7.4903 0 -1.007 -1.007 0 4.5735 4.5735 16 20.7849 4.7849 64 78.3378 14.3378 0 0.0431 0.0431 0 3.6 3.6 20 27.8864 7.8864 68 78.5927 10.5927 0 1.0142 1.0142 0 6.3712 6.3712 24 33.5256 9.5256 72 86.9858 14.9858 0 0.8303 0.8303 0 4.2361 4.2361 28 37.6981 9.6981 76 86.1621 10.1621 0 1.3096 1.3096 0 5.6359 5.6359 32 42.7857 10.7857 80 93.082 13.082 0 1.7342 1.7342 0 5.1215 5.1215 36 48.1488 12.1488 84 95.0952 11.0952 0 2.1299 2.1299 0 4.0395 4.0395 40 52.6214 12.6214 88 97.5943 9.5943 0 2.0622 2.0622 44 57.3912 13.3912 Average Error: 5.76 4.80

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38 Table A-3. Relative error in : 2 images; constraint weighting, w = 1; no error in initial guess Actual Found Corrected Actual Found Corrected j i 2 1 2 1 j i 2 1 2 1 0 4.7272 0 48 61.7584 57.0312 4 8.6421 3.9149 52 66.4123 61.6851 8 13.8951 9.1679 56 70.1737 65.4465 12 15.8848 11.1576 60 73.3866 68.6594 16 21.4946 16.7674 64 80.4528 75.7256 20 28.1068 23.3796 68 81.9627 77.2355 24 32.9923 28.2651 72 85.2062 80.479 28 37.4216 32.6944 76 88.1864 83.4592 32 42.2045 37.4773 80 92.4891 87.7619 36 47.3686 42.6414 84 91.7595 87.0323 40 51.5321 46.8049 88 97.0526 92.3254 44 56.4112 51.684 Average Error: 10.34 3.67 Average Corrected Error: 5.62 3.67 Table A-4. Absolute error in : 2 images; constraint weight, w = 50; no error in initial guess Actual Found Corrected Actual Found Corrected j i 2 1 2 1 j i 2 1 2 1 0 5.3181 0 48 57.9943 52.6762 4 8.2713 2.9532 52 61.1423 55.8242 8 11.8303 6.5122 56 61.8092 56.4911 12 16.1587 10.8406 60 64.9915 59.6734 16 21.7919 16.4738 64 73.7643 68.4462 20 27.8433 22.5252 68 74.9927 69.6746 24 32.5114 27.1933 72 80.6146 75.2965 28 36.8678 31.5497 76 81.926 76.6079 32 41.4761 36.158 80 87.4461 82.128 36 46.4146 41.0965 84 89.9737 84.6556 40 50.4915 45.1734 88 93.5548 88.2367 44 55.329 50.0109 Average Error: 7.41 2.29 Average Corrected Error: 2.10 2.29

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39 Translational error: 2 Images, W=1, IG Exact0 0. 02 0. 04 0. 06 0. 08 0. 1 0. 12 0. 14 0102030405060708090100Image Rotation Range (deg) A 2 Images, W=1, IG Exact0 0. 002 0. 004 0. 006 0. 008 0. 01 0. 012 0102030405060708090100Image Rotation Range (deg) B 2 Images, W=1, IG Exact0 0. 2 0. 4 0. 6 0. 8 1 1. 2 0102030405060708090100Image Rotation Range (deg) C Figure A-1. Image angle vs. error in marker c oordinates: 2 images; constraint weighting, w = 1; no error in initial guess. A) x -direction, B) y -direction, C) z -direction.

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40 2 Images, W= 50, IG Exact 0 0.02 0.04 0.06 0.08 0.1 0.12 0102030405060708090100 Image Rotati on Range (deg) A 2 Images,W= 50, IG Exact 0 0.002 0.004 0.006 0.008 0.01 0.012 0102030405060708090100Image Rotation Range (deg) B 2 Images, W= 50, IG Exact 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0102030405060708090100 Image Rotation Range (deg)Abs Error, Z (in)C Figure A-2. Image angle vs. error in marker c oordinates: 2 images; constraint weighting, w = 50; no error in initial guess. A) x -direction, B) y -direction, C) z -direction.

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41 Three Images Rotational error: Table A-5. Absolute error in : 3 images; constraint weighti ng, w = 10; no error in initial guess. Actual Found Error Actual Found Error 0 -0.633 -0.633 0 7.2912 7.2912 2 4.5263 2.5263 16 25.32559.3255 4 4.4858 0.4858 32 39.05977.0597 0 -0.6709 -0.6709 0 4.0689 4.0689 4 7.3625 3.3625 18 24.198 6.198 8 8.9238 0.9238 36 40.335 4.335 0 -0.2665 -0.2665 0 6.3237 6.3237 6 8.8093 2.8093 20 28.38688.3868 12 13.2609 1.2609 40 45.77045.7704 0 0.763 0.763 0 5.2056 5.2056 8 11.8751 3.8751 22 29.16677.1667 16 18.1219 2.1219 44 48.75654.7565 0 1.0716 1.0716 0 5.2356 5.2356 10 13.7674 3.7674 24 31.01947.0194 20 22.5468 2.5468 48 52.68684.6868 0 2.2766 2.2766 0 6.1742 6.1742 12 16.8327 4.8327 26 33.64987.6498 24 27.0697 3.0697 52 57.43185.4318 0 5.7357 5.7357 14 21.9787 7.9787 28 33.9036 5.9036 Average Error: 4.25 2.66

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42 Table A-6. Absolute error in : 3 images; constraint weighti ng, w =10; error in initial guess. Actual Found Error Actual Found Error 0 -1.5769 -1.5769 0 6.2072 6.2072 2 3.4663 1.4663 16 24.29498.2949 4 3.4855 -0.5145 32 38.08016.0801 0 -1.3302 -1.3302 0 5.1881 5.1881 4 6.4947 2.4947 18 25.21537.2153 8 8.107 0.107 36 41.26335.2633 0 0.606 0.606 0 6.3239 6.3239 6 9.6256 3.6256 20 28.38688.3868 12 14.0306 2.0306 40 45.77045.7704 0 -0.1891 -0.1891 0 5.2051 5.2051 8 10.9973 2.9973 22 29.16587.1658 16 17.3053 1.3053 44 48.75534.7553 0 3.3628 3.3628 0 5.2338 5.2338 10 15.8487 5.8487 24 31.01967.0196 20 24.4705 4.4705 48 52.689 4.689 0 1.0848 1.0848 0 5.2302 5.2302 12 15.7409 3.7409 26 32.78566.7856 24 26.0517 2.0517 52 56.65324.6532 0 3.4432 3.4432 14 19.8582 5.8582 28 31.9193 3.9193 Average Error: 3.96 2.65

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43 Table A-7. Relative error in : 3 images; constraint weighting, w = 1; no error in initial guess. Actual Total Found Total Actual Found Found k i 3 1 j i 2 1 3 2 4 5.2837 2 6.6476 -1.3639 8 9.2543 4 7.8223 1.432 12 12.4132 6 8.3696 4.0436 16 16.0225 8 10.2859 5.7366 20 19.47 10 11.5541 7.9159 24 22.8282 12 13.4423 9.3859 28 26.3193 14 15.2137 11.1056 32 29.774 16 16.9423 12.8317 36 33.4835 18 18.6503 14.8332 40 36.5714 20 20.5289 16.0425 44 40.2441 22 22.2384 18.0057 48 43.8764 24 23.9671 19.9093 52 47.4544 26 25.5844 21.87 Average Error ( j i): -0.81 2.61 Average Error ( k i): -1.62 2.02 Table A-8. Relative error in : 3 images; constraint weighting, w = 50; no error in initial guess. Actual Total Found Total Actual Found Found k i 3 1 j i 2 1 3 2 4 5.2691 2 5.4026 -0.1335 8 9.5991 4 7.8306 1.7685 12 13.4879 6 9.0499 4.438 16 17.2886 8 11.062 6.2266 20 21.1853 10 12.53 8.6553 24 24.4717 12 14.3732 10.0985 28 28.111 14 16.2115 11.8995 32 31.708 16 17.9979 13.7101 36 35.7361 18 19.8475 15.8886 40 38.9733 20 21.8143 17.159 44 43.3022 22 23.8314 19.4708 48 47.1667 24 25.6398 21.5269 52 51.0954 26 27.3935 23.7019 Average Error ( j i): 0.13 2.37 Average Error ( k i): 0.26 1.00

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44 Translational error: 3 Images, W=1, IG Exact0 0. 05 0. 1 0. 15 0. 2 0. 25 0102030405060Image Rotation Range (deg) A 3 Images, W=1, IG Exact0 0. 0 02 0. 0 04 0. 0 06 0. 0 08 0 01 0 012 0 014 0102030405060Image Rotation Range (deg) B 3 Images, W=1, IG Exact0 0.1 0.2 0.3 0.4 0.5 0.6 0102030405060Image Rotation Range (deg)Abs Error, Z (in)C Figure A-3. Image angle vs. error in marker c oordinates: 3 images; constraint weighting, w = 1; no error in initial guess. A) x -direction, B) y -direction, C) z -direction.

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45 3 Images, W= 50, IG Exact0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0102030405060Image Rotation Range (deg)Abs Error, X (in)A 3 Images, W= 50, IG Exact0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0102030405060Image Rotation Range (deg)Abs Error, Y (in)B 3 Images, W= 50, IG Exact0 0.05 0.1 0.15 0.2 0.25 0.3 0102030405060Image Rotation Range (deg)Abs Error, Z (in)C Figure A-4. Image angle vs. error in marker c oordinates: 3 images; constraint weighting, w = 50; no error in initial guess. A) x -direction, B) y -direction, C) z direction.

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46 3 Images, W=10, Error in IG0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0102030405060Image Rotation Range (deg)Abs Error, X (in)A 3 Images, W=10, Error in IG0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0102030405060Image Rotation Range (deg)Abs Error, Y (in)B 3 Images, W=10, Error in IG0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0102030405060Image Rotation Range (deg) C Figure A-5. Image angle vs. error in marker c oordinates: 3 images; constraint weighting, w = 10; error applied to initial guess. A) x -direction, B) y -direction, C) z direction.

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47 Four Images Rotational error: Table A-9. Absolute error in : 4 images; constraint weighti ng, w = 10; no error in initial guess. Actual Found Error Actual Found Error 0 -0.0484 -0.0484 0 -2.49 -2.49 2 3.5978 1.5978 8 6.879 -1.121 4 4.5798 0.5798 16 13.4985 -2.5015 6 6.7568 0.7568 24 21.4155 -2.5845 0 -0.9709 -0.9709 0 -2.5738 -2.5738 4 4.8758 0.8758 10 8.4846 -1.5154 8 7.7243 -0.2757 20 17.4 -2.6 12 12.4662 0.4662 30 27.0878 -2.9122 0 -2.3132 -2.3132 0 -2.3048 -2.3048 6 5.0493 -0.9507 12 11.0676 -0.9324 12 10.5654 -1.4346 24 21.4118 -2.5882 18 17.2911 -0.7089 36 32.9533 -3.0467 Average Error: -1.23 1.39 Table A-10. Absolute error in : 4 images; constraint weight ing, w = 10; error (3) in initial guess. Actual Total Found Total Actual Found Found Found k i 4 1 j i 2 1 3 2 4 3 6 6.7357 2 3.4975 0.9939 2.2443 12 13.304 4 5.8976 2.778 4.6284 18 19.8894 6 7.4733 5.6212 6.7949 24 23.7454 8 9.2941 6.5676 7.8837 30 29.6645 10 11.0624 8.9153 9.6868 36 35.2659 12 13.375 10.3466 11.5443 Average Error ( j i): 0.14 1.15 Average Error ( k i): 0.43 1.04

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48 Table A-11. Absolute error in : 4 images; constraint weight ing, w = 10; error (4) + rotational bias in initial guess. Relative Error in 4 Images, W=10, Error (4) in IG Actual Total Found Total Actual Found Found Found k i 4 1 j i 2 1 3 2 4 3 6 6.7794 2 1.6179 0.7639 4.3976 12 13.0111 4 4.5533 2.6873 5.7705 18 19.8871 6 6.7062 5.583 7.5979 24 23.7482 8 7.8849 6.5687 9.2946 30 29.7466 10 9.6958 8.9404 11.1104 36 35.2571 12 11.5415 10.3439 13.3717 Average Error ( j i): 0.13 1.25 Average Error ( k i): 0.40 0.99

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49 4 Images, W=10, Error + Rotational Bias in IG0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0510152025303540Image Rotation Range (deg)Abs Error, X (in)A 4 Images, W=10, Error + Rotational Bias in IG0 0.002 0.004 0.006 0.008 0.01 0.012 0510152025303540Image Rotation Range (deg)Abs Error, Y (in)B 4 Images, W=10, Error + Rotational Bias in IG0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0510152025303540Image Rotation Range (deg)Abs Error, Z (in)C Figure A-6. Image angle vs. error in marker c oordinates: 4 images; constraint weighting, w = 10; error and rotational bias applied to initial guess. A) x -direction, B) y direction, C) z -direction.

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50 4 Images, 12deg Image Rotation Range0 0.01 0.02 0.03 0.04 0.05 0.06 -25-20-15-10-50510152025Variation in Principal Distance (in)Abs Error, X (in)A 4 Images, 12deg Image Rotation Range0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 -25-20-15-10-50510152025Variation in Principal Distance (in)Abs Error, Y (in)B 4 Images, 12deg Image Rotation Range0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 -25-20-15-10-50510152025Variation in Principal Distance (in)Abs Error, Z (in)C Figure A-7. Variation in prin cipal distance vs. error in marker coordinates: 4 images; constraint weighting, w = 10; er ror applied to initial guess. A) x -direction, B) y -direction, C) z -direction.

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51 APPENDIX B IN VIVO STUDY—ADDITIONAL RESULTS Figure B-1. Marker di stributions in the x-z plane for all knees. Gray dots indicate initial guess made during shape matching and black dots indicate optimizer solution.

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52 Figure B-2. Marker di stributions in the x-y plane for all knees. Gray dots indicate initial guess made during shape matching and black dots indicate optimizer solution.

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53 TSR3 Months-10 -8 -6 -4 -2 0 2 Max. FlexStandingSquatKneelingExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM A TSR6 Months -14 -12 -10 -8 -6 -4 -2 0 Max. FlexStandingSquatKneelingExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM B TSR12 Months -12 -10 -8 -6 -4 -2 0 2 4 Max. FlexStandingSquatKneelingExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM C Figure B-3. TKR component rotation for subject TS at A) 3 months, B) 6 months, C) 12 months.

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54 SSL3 Months -10 -8 -6 -4 -2 0 2 4 6 Max. FlexStandingSquatExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM A SSL6 Months -15 -10 -5 0 5 10 Max. FlexStandingSquatKneelingExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM B SSL12 Months -12 -10 -8 -6 -4 -2 0 2 4 6 Max. FlexStandingSquatKneelingExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM C Figure B-4. TKR component rotation for subject SL at A) 3 months, B) 6 months, C) 12 months.

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55 KML3 Months -8 -6 -4 -2 0 2 4 6 8 10 Max. FlexStandingSquatKneelingExternal Rotation (deg .) TIB/INS TIB/FEM INS/FEM cA KML6 Months 0 1 2 3 4 5 6 7 8 9 Max. FlexStandingSquatKneelingExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM B KML12 Months -20 -15 -10 -5 0 5 10 Max. FlexStandingSquatKneelingExternal Rotation (deg.) TIB/INS TIB/FEM INS/FEM C Figure B-5. TKR component rotation for subject KM at A) 3 months, B) 6 months, C) 12 months.

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56 LIST OF REFERENCES Banks, S.A., Hodge, W.A., 1996. Accurate measurement of three dimensional knee replacement kinematics using single-pla ne fluoroscopy. IEEE Transactions on Biomedical Engineering 43, 638-649. Callaghan, J.J., 2001. Mobile bearing knee repl acement: clinical resu ltsa survey of the literature. Clinical Orthopaedics and Related Research 392, 221-225 DLima, D.D., Trice, M., Urquhart, A.G ., Colwell Jr., C.W., 2001. Tibiofemoral conformity and kinematics of rotatin g-bearing knee prostheses. Clinical Orthopaedics and Related Research 386, 235-242. Delport, H.P., Banks, S.A., De Schepper, Bellemans, M.D., 2006. A kinematic comparison of fixedand mobile-bearing knee replacements. Journal of Bone and Joint Surgery [British], Accepted 25 April. Dennis, D.A., Komistek, R.D., Mahfouz, R.M ., Outten, J.T., Sharma, A., 2005. Mobilebearing total knee arthropl asty: do the polyethylene bearings rotate? Clinical Orthopaedics 440, 88-95. Fantozzi, S., Leardini, A., Banks, S.A., Ma rcacci, M., Gianni, S., Catani, F, 2004. Dynamic in-vivo tibio-femoral and bear ing motions in mobile bearing knee arthroplasty. Knee Surgery, Sports Traumatology, Arthroscopy 12, 144-151. Garling, E.H., Bart, L.K., Geleijins, K., Nelissen, R.G.H.H., Valstar, E.R., 2005. Marker configuration model-based roentgen fluoroscopic analysis. Journal of Biomechanics 38, 893-901. Gordon, R., Herman, G.T., 1974. Three-dimens ional reconstruction from projections: a review of algorithms. Inte rnational Review of Cy tologyA Survey of Cell Biology 38,111-151. Jones, R.E., Huo, M.H., 2006. Rotating platform knees: an emerging c linical standard in the affirmative. Journal of Arthroplasty 21, 33-36. Kaptein, B.L., Valstar, E.R., Stoel, B.C., Rozing, P.M., Reiber, J.H.,2003. A new modelbased RSA method validated using CAD models and models from reversed engineering. Journal of Biomechanics 36, 873.

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57 Mahfouz, M.R., Hoff, W.A., Komisteck, R.D ., Dennis, D.A., 2003. A robust method for registration of three-dimensional knee implant models to two-dimensional fluoroscopy. IEEE Transactions on Medical Imaging 22, 1561-1574. McEwen, H.M.J., Barnett, P.I., Bell, C.J., 2005. The influence of design and materials in kinematics on the in vitro wear of total knee replacements. Journal of Biomechanics 38, 357-365 McNulty, D., 2002. The effect of cross-linking UHMWPE on in vitro wear rates of fixed and mobile-bearing knees. Abstracts of the Symposium on ultra-high molecular weight polyethylene for joint replacements. Miami Beach, Florida. Moro-oka T., Hamai S., Miura, H., Higaki H., Fregly B.J., Iwamoto Y., Banks S.A., 2006. Can MR derived bone models be used for accurate motion measurement with single-plane 3D shape registrati on? Journal of Orthopaedic Research, Submitted March. Most, E., Li, G., Schule, S., Sultan, P., Park, S., Zayontz, S., Rubash, H., 2003. The Kinematics of Fixedand Mobile-Beari ng Total Knee Arthroplasty. Clinical Orthopaedics & Related Research 416,197-207 Pagnano, M.W., Trousdale, R.T., Stuart, M.J ., 2004. Rotating platform knees did not improve patellar tracking. A prospectiv e randomized study of 240 primary total knee arthroplasties. Clinical Orthopae dics and Related Research 428, 221-227 Ridgeway, S., Moskal, J.T., 2004. Early in stability with mobile-bearing total knee arthroplasty: a series of 25 cases. Journal of Arthroplasty 19, 686-693. Selvik, G., 1989. Roentgen stereophotogram metry: a method for the study of the kinematics of the skeletal system. Acta Orthopaedica Scandinavica Supplementum 232, 1-51. Stukenborg-Colsman, C., Ostermeier, S., We nger, K.H., Wirth, C.J.,2002. Relative motion of a mobile bearing inlay after total knee arthroplastydynamic in virto study. Clinical Biomechanics 17, 49-55. Valstar, E.R., Vrooman, H.A., Toksvig-Larsen S. Ryd, L., Nelissen, R.G., 2000. Digital automated RSA compared to manually ope rated RSA. Journal of Biomechanics 33, 1593-1599. Yuan, X., Ryd, L., Tanner, K.E., Lidgren, L., 2002. Roentgen single-plane photogrammetric analysis (RSPA)—a new approach to the study of musculoskeletal movement. Journal of B one and Joint Surgery [British] 84B, 908914

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58 BIOGRAPHICAL SKETCH Sydney Machado is originally from central California and attended University of California, Santa Barbara, where she earned he r B.S. in mechanical engineering. She then moved to Florida to pursue her master’s degree in mechanical engineering at the University of Florida. During her time at th e University of Florida, Sydney has worked for Dr. Scott Banks in the Orthopaedic Biomechanics Lab in the Mechanical and Aerospace Engineering Department. Sydney has also volunteered for several outreach engineering programs, such as Eye on Engi neering, Project Athe na, and MESA, in both California and Florida.


Permanent Link: http://ufdc.ufl.edu/UFE0015920/00001

Material Information

Title: Single-Plane Radiographic Measurement of Mobile-Bearing Knee Motion Using an Unknown Distribution of Markers
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015920:00001

Permanent Link: http://ufdc.ufl.edu/UFE0015920/00001

Material Information

Title: Single-Plane Radiographic Measurement of Mobile-Bearing Knee Motion Using an Unknown Distribution of Markers
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015920:00001


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SINGLE-PLANE RADIOGRAPHIC MEASUREMENT OF MOBILE-BEARING KNEE
MOTION USING AN UNKNOWN DISTRIBUTION OF MARKERS















By

SYDNEY M. MACHADO


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA


2006





























Copyright 2006

by

Sydney M. Machado















ACKNOWLEDGMENTS

I would like to thank my family for their continual love and support. I also would

like to thank my advisor, Dr. Scott Banks, for his constant cheerleading and invaluable

guidance during this process. In addition, I would like to thank my committee members,

Dr. B.J. Fregly and Dr. Tony Schmitz, for their input and advice.
















TABLE OF CONTENTS



A C K N O W L E D G M E N T S ......... .................................................................................... iii

LIST OF TABLES ........ ...................... ..... .... ............ ............ ........ vi

L IST O F F IG U R E S .......................................................................... ..... viii

A B ST R A C T ................. .......................................................................................... x

CHAPTER

1 O V E R V IE W ......................................................................... 1

In tro d u ctio n ............................................ ............................... 1
B background ................................. ................................................ .1

2 M ETHODS AND M ATERIALS ........................................ ........................... 6

Im ages and Subjects ................. .......................... .......... ....... .. ........ .. 6
Assumptions about Polyethylene Insert and Markers ...............................................6
Recording 2-D Marker Projection Coordinates..........................................................7
Shape M watching .................................. ......... ................... .......... 8
Matching the Polyethylene Insert Markers................................... ...............9
O p tim iz atio n ................................ ........... .......... ............................ 1 1
C om putational Study .......................................... ................... .... .. ... 14

3 R E SU L T S ....................... ............................................................................18

Recording 2-D Marker Projection Coordinates ..................................................... 18
Im age M watching ............................................................................ 18
Matching the Polyethylene Insert Markers................... ....... ..... ............... 18
C om putational Study .......................................... ................... .... .. ... 19
In Vivo Study ..................................... ................................ .......... 24

4 DISCUSSION ....................................................... ........... .............. 29

5 C O N C L U SIO N .......... ......................................................................... ........ .... .. 35










APPENDIX

A COMPUTATIONAL STUDY-COMPLETE RESULTS .......................................36

T w o Im a g e s ........................................................................................................... 3 6
T three Im ages .................................................................. 4 1
F ou r Im ag e s ................................................................4 7

B IN VIVO STUDY-ADDITIONAL RESULTS...............................................51

L IST O F R E F E R E N C E S .......................................................................... ....................56

B IO G R A PH IC A L SK E T C H ...................................................................... ..................58











































v
















LIST OF TABLES

Table p

3-1 Absolute error in 0: 2 images; constraint weighting, w = 10; no error in initial
gu ess. ................................................................................ 2 1

3-2 Relative error in 0: 2 images, constraint weighting, w = 10; no error in initial
gu ess. ............................................................................... 2 2

3-3 Absolute error in 0: 4 images, constraint weighting, w = 10, error (+3) in initial
gu ess. ............................................................................... 2 3

3-4 Relative error in 0: 4 images; constraint weighting, w = 10, error (+3) in initial
gu ess. ............................................................................... 2 3

3-5 Average rotations for all subjects for four activities. ......... ............. ................25

3-6 Average rotations for all subjects after correction for rotational bias ....... ........ 25

A-i Absolute error in 0: 2 images; constraint weighting, w = 1; no error in initial
g u e ss ....................................................................................... . 3 6

A-2 Absolute error in 0: 2 images; constraint weight, w = /50; no error in initial
g u e ss ...................................... .................................................... 3 7

A-3 Relative error in 0: 2 images; constraint weighting, w = 1; no error in initial
g u e ss ...................................... .................................................... 3 8

A-4 Absolute error in 0: 2 images; constraint weight, w = /50; no error in initial
g u e ss ...................................... .................................................... 3 8

A-5 Absolute error in 0: 3 images; constraint weighting, w = 10; no error in initial
gu ess. ............................................................................... 4 1

A-6 Absolute error in 0: 3 images; constraint weighting, w =10; error in initial guess..42

A-7 Relative error in 0: 3 images; constraint weighting, w = 1; no error in initial
gu ess. ................................................................................ 4 3

A-8 Relative error in 0: 3 images; constraint weighting, w = /50; no error in initial
gu ess. ................................................................................ 4 3









A-9 Absolute error in 0: 4 images; constraint weighting, w = 10; no error in initial
gu ess. ................................................................................ 4 7

A-10 Absolute error in 0: 4 images; constraint weighting, w = 10; error (+3) in initial
gu ess. ................................................................................ 4 7

A-11 Absolute error in 0: 4 images; constraint weighting, w = 10; error (4) +
rotational bias in initial guess ......... ............................................... ...................48















LIST OF FIGURES


Figure page

2-1 The surgeon who performed the arthroplasty was given directions to distribute
the m arkers roughly as shown .................................. ............... ............... 6

2-2 Manual process of recording marker coordinates from the radiographic images. .....7

2-3 Screen shot of shape-matching GUI with tibial component and marker model
lo a d e d ............................................................. ................ 8

2-4 Two different marker configurations would appear the same in a radiographic
image. This is a problem for small image sets, since increasing the number of
available images minimizes this effect ......... ...... .. .......... ............... 16

2-5 Procedure to determine marker locations and axial rotation of insert ................ 17

3-1 Absolute error between solution and known marker coordinates for 2, 3, 4
images as a function of the total insert rotation with respect to the viewing plane.
Highlights show range of rotation that corresponds to the in vivo images ............20

3-2 Progression of marker coordinate solutions in the x-z plane as the image rotation
range increases from 4 to 52 ............................................ ...................26

3-3 Top (x-z plane) and side (x-y plane) views of the marker locations after shape-
matching and optimization. Gray dots are the initial guess from shape-matching
and black dots are the optimized solutions............... ............................................ 27

3-4 R otation trends for all subjects ...................................................... .............. 28

4-1 Knee kinematics for one subject with easily correctable rotational bias. ................33

A-i Image angle vs. error in marker coordinates: 2 images; constraint weighting, w =
1; no error in initial guess ................... .............. .. ...... .....................39

A-2 Image angle vs. error in marker coordinates: 2 images constraint weighting, w =
50; no error in initial guess.......................................................... ............... 40

A-3 Image angle vs. error in marker coordinates: 3 images; constraint weighting, w =
1; no error in initial guess ................................................ .. .. .... .. ........ .... 44









A-4 Image angle vs. error in marker coordinates: 3 images; constraint weighting, w =
50; no error in initial guess.......................................................... ............... 45

A-5 Image angle vs. error in marker coordinates: 3 images; constraint weighting, w =
10; error applied to initial guess.......................................... ......................... 46

A-6 Image angle vs. error in marker coordinates: 4 images; constraint weighting, w =
10; error and rotational bias applied to initial guess.. ........................................... 49

A-7 Variation in principal distance vs. error in marker coordinates: 4 images;
constraint weighting, w= 10; error applied to initial guess.................................50















Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

SINGLE-PLANE RADIOGRAPHIC MEASUREMENT OF MOBILE-BEARING
KNEE MOTION USING AN UNKNOWN DISTRIBUTION OF MARKERS

By

Sydney M. Machado

August 2006

Chair: Scott A. Banks
Major Department: Mechanical and Aerospace Engineering

The two most common methods used to obtain precise in vivo measurements of

knee kinematics for patients with total knee replacements are single-plane fluoroscopic

shape-matching and roentgen stereophotogrammetry. To determine the location of an

unknown distribution of markers in the polyethylene insert of a rotating-platform mobile-

bearing total knee replacement from single-plane radiographic images, these two methods

must be combined. A guess about the marker distribution in each tibial insert was made

based on the known geometry of the insert and knowledge of the general shape of the

marker distribution. This guess was used to seed a numerical optimization procedure that

simultaneously sought to determine the 3D locations of the tantalum markers within the

tibial insert and the rotation of the tibial insert with respect to the tibial baseplate. The

quality of the numerical solution was assessed by mathematically projecting the

estimated marker coordinates into the image plane and comparing those values to the

values observed in the radiographic images. A computational study was conducted to









access the error propagation in this method before applying it to an in vivo study. It is

estimated that with a four image set containing rotation with respect to the viewing plane

of at least 10-20o, this method can give a translation accuracy of 0.07in (x-direction),

0008in (y-direction), 0.2in (z-direction), and a rotational accuracy of about 1.0.

However, it was found that the method is subject to a rotational bias which must be

corrected for either by use of a reference image or by correcting the systematic error in

data post-processing. The latter approach was taken in this investigation.














CHAPTER 1
OVERVIEW

Introduction

The goal of this study is to determine the internal/external rotation of the

polyethylene (PE) insert in the knees of 20 subjects who were given rotating platform

(RP) mobile-bearing total knee replacements (TKR). An unknown distribution of 4

tantalum markers were placed in the PE insert intra-operatively. The subjects were seen

3 times post-operatively and 4 single-plane radiographic images were taken during

stance, kneel, deep kneel, and lunge activities.

Mobile-bearing total knee replacements (MB-TKR) have the potential to provide

more natural knee kinematics and reduce wear related failure, and some in vitro wear

studies (McEwen et al., 2005; McNulty 2002) support this idea. However, the clinical

results for fixed and mobile-bearing knee implants appear to be equivalent (Callaghan

2001; Pagnano et al., 2004). In addition, only a small number of mobile bearing designs

currently are available on the US market, all manufactured by the same company (Jones

and Huo, 2006). For these reasons, it is highly relevant to study these devices to

determine their function and potential advantages over traditional TKR.

Background

Early fixed bearing TKR designs were highly constrained, limiting range of motion

and dramatically changing knee kinematics (Ridgeway and Moskal, 2004). This

constraint produced considerable shear at the component fixation interfaces leading to

failure through loosening. In an effort to restore more natural knee motion and prevent









loosening, designs with round-on-flat or flat-on-flat articulating surfaces were introduced.

These designs did have the intended effect, but soon a new predominate mode of failure

emerged. Reducing the contact surface area subjects the polyethylene to high levels of

multi-directional contact stress and sliding, causing increased wear rates. The concept

behind the mobile-bearing TKR was to reduce wear by increasing conformity of the

articulating surfaces, while still allowing free range of motion to restore more natural

knee kinematics (Jones and Huo, 2006).

Several in vitro studies (McEwen et al. 2005; McNulty, 2002) have shown that

the polyethylene in RP TKR does exhibit wear rates superior to that in a fixed bearing

counterpart. However, both of these studies tested only one RP design, the Sigma RP

System (DePuy), and no study has yet found superior clinical results for RP TKR (Jones

and Huo, 2006). In fact, one clinical study (Ridgeway and Moskal, 2004) reports 25

cases of clinical instability and pain following RP and meniscal bearing TKR.

In light of the current inconclusive evidence regarding mobile-bearing TKR it is

important that investigation into this style of implant continue. To better understand the

kinematics of a RP TKR there must be some way to determine the motion of the PE

insert which appears transparent in radiographic images. One way is to perform in vitro

cadaver testing as has been done by Stukenborg-Closeman et al. (2002), and D'Lima et

al. (2001). However, in vivo studies are really needed to get a true idea ofRP TKR

performance during volitional, dynamic weight-bearing activities. In one study by Dennis

et al.(2005), fluoroscopic shape matching techniques were used on a series of images

taken in vivo of subjects with mobile-bearing TKR. The distribution of markers was

known and a CAD model of the PE insert was available. The shape matching technique









is described in detail in Mahfouz et al. (2003) and relies on image intensity and contour

matching to hypothesize a pose that closely matches that of an input fluoroscopic image.

Once the femoral and tibial components were matched to the image, the authors simply

aligned the CAD model of the insert markers with the markers visible in the fluoroscopic

image. In this study three different designs ofRP TKR and one design that allowed for

anterior-posterior translation as well as rotation were evaluated. While the greatest

amount of axial rotation reported was in the femur with respect to the tibia, there was also

significant rotation of the insert with respect to the tibial component. Rotation of the

insert with respect to the femoral component also occurred, but was the smallest in

magnitude. The results varied with implant design, but the overall trend was for the

rotation of the PE insert to closely follow that of the femoral component with respect to

the tibia. These findings are in agreement with the in vitro findings of the D'Lima et al.

(2001) study.

Another in vitro study was done by Stukenborg-Closeman et al. (2001) on a

rotating and AP translating TKR, the Interax ISA. In that study only 1.80 of axial rotation

and small amounts of translation were reported. These results were confirmed by an in

vivo study by Fantozzi et al. (2004) on the same mobile-bearing design, which also found

only small amounts of both translation and rotation. Fantozzi et al. used a combination of

image matching and roentgen stereophotogrammetric analysis (RSA) to locate the

tantalum markers and determine the motion of the insert. They found that the femoral

component did not always drag the insert with it as hypothesized and that in some case

the two components rotated independently.









In a study by Delport et al. (2006) of the Performance series of knee implants

(Biomet), a posterior stabilized RP mobile-bearing design was also evaluated using shape

matching. The tibiofemoral rotation reported was consistent with the study by Dennis et

al. (2005). In addition, subjects with this design exhibited greater asymmetrical posterior

femoral translation in flexion than previously reported in the literature. Delport et al.

(2006) claimed the measured motions of the RP-TKR were closer to those found in the

healthy natural knee.

Other studies of mobile-bearing TKR have relied on a known distribution of

tantalum markers in the PE insert and use roentgen stereophotogrammetric (RSA)

techniques to determine the marker orientations. Traditional RSA requires specialized

and expensive equipment that is not widely available and is somewhat labor intensive to

use (Valstar et al., 2000). Bi-plane RSA equipment setups can also limit subject

movement, and can cause the markers to become occluded during flexion-extension

motion (Garling, et al. 2005). Because of this, attempts have been made to make RSA

more accessible by adapting it to a standard single-plane fluoroscopic imaging set up,

developing a digital automated method, and developing a model-based form (Garling et

al., 2005; Valstar et al., 2001, Kaptein et al. 2003). One study by Yuan et al. (2002)

incorporates RSA with single-plane fluoroscopy, but requires each 3-D marker to be

assigned to the correct projection in each image so that the mathematical formulation can

be solved directly. This turns out to be highly sensitive to number of control points,

focus position, and object distance, which may prove difficult in a laboratory setting.

Another study, Garling et al. (2005), uses an approach called Model-Based Roentgen

Fluoroscopic Analysis. In this study accurate models of marker geometry are created









and to represent the insert and tibial component. A global optimizer then uses the marker

models to reconstruct the location of the insert with respect to the tibial component based

on a series of fluoroscopic images. At this point, only results of a phantom study have

been reported for this method.

Without a known marker model, however, a hybrid approach would have to be

taken. This approach would start by using shape matching to determine the orientation of

the visible parts of the TKR. Once the pose of the tibial and femoral components are

determined, they can be used to limit the degrees of freedom available to the insert. Then

trial and error shape matching could be used to determine a possible distribution of

markers. Although the marker distribution is unknown, the locations can be described as

points and are still related to the 2-D projections by the radiographic projection geometry

as described by Selvik (1989). Projecting the estimated marker coordinates would give

some error between the estimated and the known, and this could be used as an overall

nearness criterion to determine goodness of fit (Gordon and Herman, 1974). The

criterion could then be optimized to determine the best possible location and orientation

of the markers within the constraints of the known dimensions of the PE insert.















CHAPTER 2
METHODS AND MATERIALS

Images and Subjects

In the study 19 subjects were seen 3 times post-operatively at periods of 3, 6, and

12 months. One subject was seen at 2, 3, and 12 months. Each was asked to perform 4

activities: stand at full extension, 900 kneel, full flexion kneel, and lunge. One single-

plane radiographic image was taken for each activity. This yielded between 4-12 images

per subject.

Assumptions about Polyethylene Insert and Markers

Several simplifying assumptions can be made about the distribution of markers in

the insert. The geometry of the insert is known from available CAD models. The

markers were placed intra-operatively from the side, thus it is likely that the markers are

within a few millimeters of the insert periphery. The surgeon was instructed to place the

markers roughly as shown in Figure 2-1, with the two medial markers being situated

somewhat closer together, while the lateral markers are farther apart.

Medial



*





Lateral


Figure 2-1. The surgeon who performed the arthroplasty was given directions to
distribute the markers roughly as shown.








Finally, it is assumed in each image that five degrees of freedom of the insert are

constrained. The origin of the insert reference system is co-incident with that of the tibial

component, which is centered on the post about which the bearing rotates, and the

inferior surface of the insert is co-planar with the tibial tray. The only degree of freedom

available is in axial rotation of the insert about the tibial post.

Recording 2-D Marker Projection Coordinates
The markers visible in each image were then selected and their 2-D projection

coordinates recorded in a manual process (Figure 2-2).

In order to quantify the error in this process a series of 10 images were generated.

Each image contained 20 markers which were distributed in uniformly random locations.

Noise and gray scale gradient was added to the images, while the markers were subject to

Gaussian blur (a = 0.5). The markers were then manually selected and the coordinates

recorded.









il





Figure 2-2. Manual process of recording marker coordinates from the radiographic
images.









Shape Matching

The physical orientation of the knee was determined using a shape matching

technique evolved from Banks and Hodge (1996). This technique extracts the contours

of the TKR silhouette using a Canny edge detector, and then uses non-linear least squares

to minimize the sum of Cartesian distances between the implant contour in the

radiographic image with a contour of the projected implant CAD model (Figure 2-3). The

error in this process is known to be less than 0.6mm in the sagittal plane and 1.080 for all

rotations.





















Figure 2-3. Screen shot of shape-matching GUI with tibial component and marker model
loaded.

The two registered components easily can be checked to make sure they are not in a

physically (impinging) or anatomically impossible position, and normally adjacent image

frames can be checked to insure motion continuity. However, in the current investigation

there are no adjacent frames with which to check for joint motion continuity. For this

reason it was determined that a further computer simulation of the method was necessary









in order to fully quantify the error. A computer simulation was conducted by generating

10 views of the knee replacement used in the study. The position and orientation were

randomly generated from a uniform distribution over the ranges:

-2.5 cm < x translation < 2.5 cm

-2.5 cm < y translation < 2.5 cm

7.5 cm < z translation < 12.5 cm

-16.0 deg < x rotation < 16.0 deg

-16.0 deg < y rotation < 16.0 deg

-90.0 deg < z rotation < 90.0 deg.

This set of 10 images was given to 4 subjects who were experienced with the image

matching process and they were asked to provide an accurate shape registration.

Student's t-tests were used to determine whether the error found here was significantly

different from the error found previously.

Matching the Polyethylene Insert Markers

Based on the assumptions about the polyethylene insert and markers, the 3-D

coordinates of the markers are hypothesized, and a CAD model is created. This model is

then imported into the image matching software. Since the marker model only has one

degree of freedom, it is rotated axially until the correct markers are aligned and/or

occluded in all images of the current subject. If the markers do not align, the CAD model

is modified and re-imported into the software. This iterative process provides an initial

guess for the marker distributions and insert rotation angles which then is refined through

numerical optimization.

The estimates of 3-D marker locations and insert axial rotation for each image can

be checked against the known 2-D projection coordinates simply by transforming them










into the global coordinate system and inserting them into the projection equation. The

transformation can be written as in Equation 2-1.

Global Tglobal tmbial component 2-1
insert tbial component insert


This is where Thbal component is the transformation from the tibial coordinate system


to the PE insert coordinate system, in which the markers are located (since the insert only

has one degree of freedom with respect to the tibial component this will depend on the

estimated axial rotation, ); al comonent is the transformation from the global coordinate


system to the tibial component coordinate system (these are the rotations and translations

that are estimated by the shape matching); and Tglobal is the transformation from the


global coordinate system to the insert coordinate system known only approximately. The

transformation, Tglobal and the marker coordinates, x, can be written as:

Global
insert
c6, (cac sass y) ,-0(casy+sascy) -c/sa s6(cac-casfsy) cosln(casy+sascy ) Tx 2-2
cn(sacy+ca sp sy)-s0n (sa sy-ca sp cy) cac/ sOn(sacy+ca sp sy)-cOn(sa sy-ca sp cy) Ty
(-cP sy)ce0(c,- cy)s8n sP (-p s )sn+(cp cy) co, TI
0 0 0 1

and

globl global ns 2-3
Xglobal L_ ^insert ] Xinsert -3

where


Global =[g Yg zg] 2-4


and


insert =[Xrnsert Yinsert Znsert] 2-5

Cosine and sine are abbreviated as c and s, respectively; a, 0, y, Tx, Ty, and Tz

describe the known 3-D orientation of the tibial component; and On is the estimated axial









rotation for image n. The vector global is the estimated marker coordinates in the global

reference frame and x..t is a vector of the estimated marker coordinates in the insert

reference frame. Once xglob, is calculated the 2-D projections of the markers can be

determined using the known image setup geometry (Figure 2-4).

The estimated 2-D projections, knownw, Vknown), can be written as:

x
~st = C g 2-6
(C zg)

and
Vet = C Y 2-7
(C- z )

where C is the principal distance.

These marker projections, (uest, Ves), typically do not match the true marker

projections, knownw, Vknown, measured in the radiographic images. These differences,

(Uknown Uest, Vknown-Vest) provide an error measure which can be minimized using

numerical optimization techniques.



MMaraerl

laZrkr N----


C

Figure2-4. Imaging Geometry.

Optimization

For each subject's set of images there are 3*m+n unknowns, where m is the total

number of markers visible in each set of images and n is the number of images for which









the insert rotation, 0, needs to be measured. Each visible marker produces the 2 projection

equations shown in section II(E) for each image in which it is visible. For example, if 8

images were available with 3 markers visible in each image, there would be 9 + 8 = 17

unknowns, and 2*3*8 = 48 possible projection equations if all images were considered

simultaneously. This over-determined system is non-linear due to the unknown axial

rotation, 0, and the perspective projection transformation. One way to approach this

problem is to use a non-linear least squares algorithm to minimize the error between the

known 2-D projection coordinates and the estimated 2-D projection coordinates. The

Matlab optimization toolbox provides the Isqnonlin routine based on the interior-

reflective Newton method, and requires a user-defined cost function to compute and

minimize a vector of error equations (Equation 2-8 and 2-9).




f,(x)
F(x) = .() 2-8




minIF(x)1 Zf(x)2 2-9
S2 2 ,


In this case F(x) is the error vector andfi throughfn are the difference knownn est,

vknown-Vest), and can be expanded out to:

(x,)
fn = known C (- = Uknown (C- Z )- C(X )


Snow ( ert 1 31 Inert Inmer t ert ltnsetT 3 mner t m nert T ) 4 1

C( T insert,, + g OTY inert 2Y ntT13Z-,nsert + tT14)









(y,)
+ = known C (y = known (C )- C(y,)
(C- z )


= vo (C gob,,31, + m to, J + ot, ,toTZ + glob, T34) 2-11


SC ( er 21 iert et 22Y imert t23Z -rt ert24

In addition, assumptions about locations of the markers provide constraints for

optimization. This is done by supplying to the user-defined cost function a contour of the

insert in the x-z (transverse) plane. This contour, C, is an 1x2 matrix where 1 is the

number of discrete points along the contour and the columns are the x and z coordinates

at each point. The distance, d"Jntour, between the contour and each marker's (x, z)

coordinates at the current iteration, k, is stored in a matrix of the same size as the

contour:

d.o.'- = C, [x.n,, z It]k i 1,2, 3...1 2-12


The minimum of the distance magnitude, d, is then used to formulate a weighted

dimensionless distance measure, D(x). The weight, w, was determined during the

computational study described in Chapter 2: Computational Study. When D(x) is

concatenated onto the error vector F(x), it implements a soft constraint penalizing marker

location solutions that exceed some threshold distance, dcnt, from the insert periphery:


d= ,min (d'"o-) +(d') o"' 2-13
(h_\ ,1) \ (1,2)

2
D(x)= w* 2-14


FF(x)2
F'(x)= = 2-15
LDO:)J









The number of images evaluated at each iteration can be specified by the user and

the algorithm can randomly select that number of images from the available images. The

optimization can then be run a selected number of times, such that a sufficient number of

all possible combinations of images have been selected or the solution no longer changes

by a threshold amount. The advantage of this, rather than evaluating all the images at

once, is that if one or two images have greater amount of error in them (or are not

sufficiently different to provide novel information) and are causing the optimizer

difficulty, the influence of those images can be reduced. Analogous to a Kalman filter,

this is done by taking the sum of each solution divided by its residual, and then dividing

by the sum of the residuals:




out= j = 1,2...,n 2-16



If the optimization is run n times, this will cause solutions with high residuals to

have less effect on the total solution xout *

Computational Study

A computer simulation study was performed to quantify the sensitivity of the

estimation method to the number of images used and to the absolute amount of insert

rotation with respect to the viewing frame of reference (an image set in which there is

little insert rotation would not provide enough information for reliable estimates.). A set

of synthetic marker data was created and was projected into 2-D coordinates for 46

images ranging from 0-90 of axial rotation using projection geometry similar to the in

vivo images. The tibial component was assumed to be located at its origin, with zero









rotation. A uniform distribution of noise was then added to the 2-D coordinates. The

magnitude of this error was determined as described in Chapter 2: Recording 2-D Marker

Projection Coordinates. The insert and tibial component orientations were subjected to

uniformly distributed error found by the process described in section Chapter 2: Image

Matching. For the first evaluation the known marker locations and axial rotation served

as the initial guess to the optimizer. Two, three, and four images were evaluated for a

total axial rotation range of 00-90, 0o-52o, and 00-36 respectively. Each number of

images was evaluated at weights ofw = 1, /50, and 10 on the soft constraint (section

II(F)) as well. The sets of 3 and 4 images were equally spaced throughout the total

rotation range. Next, a random uniform error distribution of 3 was added to the axial

rotation, 0, and the marker coordinates were subjected to random uniform distribution of

error of the magnitude determined in Chapter 2: Matching the Polyethylene Insert

Markers before being used as the initial guess. For this case only 3 and 4 images were

evaluated. The rotational error in the initial guess was then increased to +40 and the

images evaluated again. In addition, the sensitivity to variation in the principal distance,

C, was also evaluated.

Finally, 3 and 4 image sets were evaluated with an additional rotational bias in the

marker coordinates for both +3 and +4 in 0. Since the markers are fixed to a rigid body

that rotates only in-plane, this was done to simulate the situation where the relative

position of the markers to each other can be determined while the absolute location

within the insert remains ambiguous (Figure 2-4).







16


Medial
*










Lateral



Figure 2-5. Two different marker configurations would appear the same in a radiographic
image. This is a problem for small image sets, since increasing the number of
available images minimizes this effect.




















































Figure 2-5. Procedure to determine marker locations and axial rotation of insert.














CHAPTER 3
RESULTS

Recording 2-D Marker Projection Coordinates

For the manual marker selection process the error in the u direction was found to be

1.0 + 0.3 pixels (range,-0.0 to 1.8 pixels), and 0.3 0.4 pixels (range, -0.6 to 1.6 pixels)

in the v direction. For images of the size considered in this study, this is equivalent to

0.014in + 0.005in (range, -0.001in to 0.025in) in the u direction and 0.004in + 0.005in

(range, -0.008in to 0.022in) in the v direction.

Image Matching

To quantify the error associated with the shape matching process, four subjects

experienced with the software were given a set of 10 synthetic images to match.

Student's t-tests were used to determine if the error found was significantly different

from the error found in Banks and Hodge (1996). For x and y translation, and all

rotations no significant difference was found (p = 0.05). However, a bias in the z

translation resulted in an average error of-0.804in + 0.041in (range, -1.763in to -

0.171in). This was found to be highly significantly different from the original reported

error (p = .001).

Matching the Polyethylene Insert Markers

After the matching process was completed and an initial estimate determined for

the marker model, the 3-D coordinates were mathematically projected onto the image

plane and compared to the recorded 2-D coordinates from the actual images. The error

between knownn Uest, Vknown-Vest) was found to be -0.00361in + 0.082in (range, -0.876in-









0.49in) in the u direction and -0.001in 0.067in (range, -0.570in- 0.574in) in the v

direction. However, the large range values were found to correspond with only one 4

images sub-set of one subject's images, out of a total of 151 images. With these 4 images

removed the error in the u direction was 0.002in 0.000in (range, -0.132in 0.122in)

and 0.00 + 0.022in (-0.141in 0.179in).

Computational Study

A computational study was performed to assess the sensitivity of the method to the

number of images used for estimation, the rotation range of the insert with respect to the

viewing plane, constraint weighting, and random error in input and initial guesses. For

two images, the average absolute error in the x and y direction increased with increasing

rotation range (Figure 3-1). The z direction absolute error decreased rapidly as total

rotation increased (Figure 3-1). For three and four images, average absolute error in the x

direction increased with increasing rotation, but the error in they and z directions showed

decreasing errors with increasing rotation (Figure 3-1). In all cases, the change in x and y

errors with increasing rotation range was much smaller than the change in z errors.

For two images the insert axial rotation showed a consistent bias of about 5. By

correcting for this bias, the relative error in the angle between the images was reduced

from 7.060 1.960 to 1.700 1.960. In addition, the absolute error in the rotation for two

images was 5.990 4.370 (Table 3-1 and 3-2). This value steadily increased as the

rotation range increased. The rotational error decreased as the number of images

increased, improving to -1.230 1.390 for four images. The relative rotational error

between images for four images was 0.150 1.150 (Tables 3-3 and 3-4).

















2 IMAGES


3 IMAGES


"_____________________


4 IMAGES


Figure 3-1. Absolute error between solution and known marker coordinates for 2, 3, 4

images as a function of the total insert rotation with respect to the viewing

plane. Highlights show range of rotation that corresponds to the in vivo

images.


''


; "
;





~ ~.ur~nl ul


I "


''''


-rsr~ ~rr.l


'
i
iJ


L-kfP r-Fiiri.liw)


r*-~- ~-.rl


-ma nn r


'"







.










Table 3-1. Absolute error in 0: 2 images; constraint weighting, w = 10; no error in initial
guess.
Actual 0 Found 0 Error Actual 0 Found 0 Error
0 -2.1909 -2.1909 0 2.4519 2.4519
0 3.1747 3.1747 48 59.5943 11.5943
0 -0.9013 -0.9013 0 2.7858 2.7858
4 7.5629 3.5629 52 63.2585 11.2585
0 -0.4344 -0.4344 0 3.4182 3.4182
8 11.4426 3.4426 56 67.1996 11.1996
0 -0.1407 -0.1407 0 2.5802 2.5802
12 16.1124 4.1124 60 67.417 7.417
0 -0.0713 -0.0713 0 3.6068 3.6068
16 22.9429 6.9429 64 75.2629 11.2629
0 0.2621 0.2621 0 5.0742 5.0742
20 28.0633 8.0633 68 81.1398 13.1398
0 1.5036 1.5036 0 4.5062 4.5062
24 33.9424 9.9424 72 82.9359 10.9359
0 1.056 1.056 0 3.811 3.811
28 37.8924 9.8924 76 84.8924 8.8924
0 1.37 1.37 0 4.6463 4.6463
32 42.8302 10.8302 80 90.5222 10.5222
0 1.7608 1.7608 0 4.0709 4.0709
36 48.1659 12.1659 84 93.9927 9.9927
0 2.4276 2.4276 0 4.0592 4.0592
40 52.8619 12.8619 88 97.5466 9.5466
0 1.5947 1.5947
44 52.9173 8.9173

Average Error: 5.590 4.370











Table 3-2 Relative error in 0: 2 images, constraint weighting, w = 10; no error in initial guess.
Actual Found Corrected
AOj-AOi A02-A01 A02-A01
0 5.3656 0
4 8.4642 3.0986
8 11.877 6.5114
12 16.2531 10.8875
16 23.0142 17.6486
20 27.8012 22.4356
24 32.4388 27.0732
28 36.8364 31.4708
32 41.4602 36.0946
36 46.4051 41.0395
40 50.4343 45.0687
44 51.3226 45.957
48 57.1424 51.7768
52 60.4727 55.1071
56 63.7814 58.4158
60 64.8368 59.4712
64 71.6561 66.2905
68 76.0656 70.7
72 78.4297 73.0641
76 81.0814 75.7158
80 85.8759 80.5103
84 89.9218 84.5562
88 93.4874 88.1218










Table 3-3. Absolute error in 0: 4 images, constraint weighting, w = 10, error (3) in initial
guess.
Actual 0 Found 0 Error
0 -0.6436 -0.6436
2 2.8539 0.8539
4 3.8478 -0.1522
6 6.0921 0.0921
0 -1.0304 -1.0304
4 4.8672 0.8672
8 7.6452 -0.3548
12 12.2736 0.2736
0 -1.8479 -1.8479
6 5.6254 -0.3746
12 11.2466 -0.7534
18 18.0415 0.0415
0 -2.132 -2.132
8 7.1621 -0.8379
16 13.7297 -2.2703
24 21.6134 -2.3866
0 -2.5209 -2.5209
10 8.5415 -1.4585
20 17.4568 -2.5432
30 27.1436 -2.8564
0 -2.3112 -2.3112
12 11.0638 -0.9362
24 21.4104 -2.5896
36 32.9547 -3.0453


Average Error: -1.200 1.210


Table 3-4. Relative error in 0: 4 images; constraint weighting, w
guess.
Actual Total Found Total Actual Found Found Found
AOk AOi A04 AO1 Oij- A0i A2 A1 A03 A02 A04 A03
6 6.7357 2 3.4975 0.9939 2.2443
12 13.304 4 5.8976 2.778 4.6284
18 19.8894 6 7.4733 5.6212 6.7949
24 23.7454 8 9.2941 6.5676 7.8837
30 29.6645 10 11.0624 8.9153 9.6868
36 35.2659 12 13.375 10.3466 11.5443
Average Error (AOj AOi): 0.14 1.150
Average Error (AOk Ai): 0.43 1.040


10, error (3) in initial


The marker coordinates in the solution were shown to be fairly insensitive to

constraint weighting. For three images, as the weight was increased from 1 to 10 the









average absolute error in the x direction decreased from 0.167in + 0.065in to 0.067in +

0.046in. In the z direction the absolute average error decreased from 0.145in 0.126in to

0.102in + 0.077in. The error in they direction remained unchanged. This trend held for

two and four images as well. The axial rotation of the insert in the solution showed more

sensitivity to changes in the weighting. For two images the relative error in the angle

between the images decreased from 10.34 3.670 to 7.060 1.96.

This method was found to be relatively insensitive to variation in the principal

distance in x andy direction. In the z direction accuracy dropped by about 0. lin for

deviations of +5in from the known principal distance.

The configuration most relevant to this study was four images, with both error and

rotational bias in the initial guess, and 50-20o degrees of rotation as in the in vivo images.

For this configuration the error in the x, y, and z directions was 0.041in 0.022in, 0.008in

0.002in, and 0.112in 0.049in respectively. The error in the insert rotation, 0, was

0.670 3.270. Additional results are reported in Appendix A.

In Vivo Study

For the in vivo study the tantalum markers were found to be distributed in the

polyethylene insert as shown in Figure 3-3. For the insert rotations, 9 of the 20 subject's

results contained varying magnitudes of rotational bias as discussed in Chapter 2:

Computational Study and demonstrated in Figure 2-4. These rotational biases were

systematic and were corrected for during data post-processing. Before rotational bias

was corrected for the average rotations for all subjects during the four activities were as

shown in Table 3-5. After correcting rotational bias, the magnitude of femur-to-insert

rotation was reduced from -12.330 15.920, to -6.120 3.290 (Table 3-6 and Figure 3-5A).

These values are consistent with the design of the implant. At 12 months, the average










external rotation of the tibial component with respect to the insert for all activities and

subjects was -1.88 + 7.02, the external rotation of the tibial component with respect to

the femoral component was -2.48 7.02, and the external rotation of the insert with

respect to the femoral component was -0.41 + 2.460 (Figure 3-5B). Additional results are

reported in Appendix B.

Table 3-5. Average rotations for all subjects for four activities.
No Corrections:
TIBIA TO INSERT TIBIA TO FEMUR INSERT TO FEMUR
Average Std. Dev. Average Std. Dev. Average Std. Dev.
(o) (0) (0) (0) (0) (0)
Max. Flex -3.020 7.30 Max. Flex -3.64 5.93 Max. Flex -0.62 5.20
Standing 2.05 5.88 Standing 1.91 5.01 Standing -0.14 4.25
Squat -4.27 7.55 Squat -5.41 6.00 Squat -1.14 4.36
Kneeling -2.73 7.67 Kneeling -3.14 5.56 Kneeling -0.41 5.14
Max 15.920
Min -12.33o


Table 3-6. Average rotations for all subjects after correction for rotational bias.
Rotational Bias Corrected:
TIB/INS TIB/FEM INS/FEM
Average Std. Dev. Average Std. Dev. Average Std. Dev.
(o) (o) (o) (o) (o) (o)
Max. Flex -3.57 7.09 Max. Flex -3.64 5.93 Max. Flex -0.74 2.40
Standing 1.49 5.07 Standing 1.91 5.01 Standing 0.45 2.03
Squat -4.83 6.50 Squat -5.41 6.00 Squat -0.79 2.35
Kneeling -3.30 6.18 Kneeling -3.14 5.56 Kneeling 0.00 2.83


Min -6.120


Max


3.290













3 Images, w=1


3 Images, w=10


Figure 3-2. Progression of marker coordinate solutions in the x-z plane as the image
rotation range increases from 4 to 520 with (A) constraint weight w = 1, and
(B) w = 10. The solutions progress in a direction towards the black dots that
indicated the known marker positions.















Top View
i .i .*.: i .' -. i '



i~j
0 -*"'' "'- *
.

*?- .l -i'















Top view
/ *'" : I ''* ":'

: ~ ~ ~ ~ ~ ;.'* "'- '-
*.^ .
i(* ;*
o *



-lO -D-5 o.c- 3.5 1.0- .



______ Top view ___


KMR

I.c


TSR


Side View










*

* -.


Side View

Side View


Figure 3-3. Top (x-z plane) and side (x-y plane) views of the marker locations after

shape-matching and optimization. Gray dots are the initial guess from shape-
matching and black dots are the optimized solutions.


:" r 2


QJr


U -




t *


r.T. 1.0


.1


- 0.5


-0.5 O.I? P.1 1.0 :.5 -1.-0 -0.- O.I?











Average Rotation by Activity for All
Subjects


MTIB/INS
TIB/FEM
OINS/FEM


Average Rotation for All Subjects


MTIB/INS
TIB/FEM
OINS/FEM


B
Figure 3-4. Rotation trends for all subjects over A) all times for each activity and B) all
activities for each time point. External rotation is taken as positive.


2

1

0
0

w-2

E -3
-5
Lu -4

-5














CHAPTER 4
DISCUSSION

Roentgen Stereophotogrammetric Analysis (RSA) provides the standard by which

other radiographic kinematic measurement techniques can be judged. Small radio-opaque

tantalum markers are placed directly into the bones and observed using a pair of

simultaneous radiographic exposures. These implanted fiducials are unaffected by motion

of overlying muscle, fat and skin, and can provide joint motion measurements with

uncertainties as low as 0.005mm for translations and 0.15 for rotations (Selvik,1989).

However, RSA equipment is expensive and not widely accessible (Garling et al., 2005),

and marker placement requires a surgical procedure. In addition, precise calibration must

be done in the working volume before any measurements can be taken, and all

subsequent measurements must be taken within the calibrated volume. These constraints

limit the space in which the subject can move, making it difficult to perform

measurements of dynamic activities.

RSA principals also have been applied to single-plane radiography or

fluoroscopy, where greater experimental flexibility and equipment availability are gained

at the expense of larger measurement uncertainties (especially for translations

perpendicular to the image plane, Yuan et al., 2002). Garling et al. (2005) demonstrated

a method for measuring the motions of a mobile bearing TKR polyethylene insert having

tantalum markers inserted in a precisely measured three dimensional trapezoidal shape.

Marker placement was designed to provide maximal non-collinearity for good visibility

and well conditioned measurements.









Single-plane fluoroscopy also has been used with 3D-to-2D shape registration

techniques to study the motions of implants and bones (Banks and Hodge,1996; Mahfouz

et al., 2003; Fregly et al., 2005). Fluoroscopes are available in most hospitals and provide

the ability to acquire images at rates of 3-30 frames per second, which is adequate to

study many relevant joint motions. The quality of the measurement is dependent upon the

accuracy of the 3D model of the bone or implant being registered (Kaptein et al., 2003;

Moro-oka et al., 2006), with implant models typically provided by the manufacturer or

laser scans, and bone models typically being generated by CT or MR scans.

This study provided a novel challenge requiring adoption of a combined

measurement approach. The task to measure the motions of the mobile-bearing TKR

UHMWPe insert having an unknown distribution of four tantalum markers was

accomplished using 3D-to-2D shape registration techniques for the femoral and tibial

components and a novel technique motivated by RSA to determine the insert motions.

The measurement approach was based on the constraint that the UHMWPe insert had one

rotational degree of freedom with respect to the tibial baseplate, and that multiple views

were available for each knee. Thus, 5 out of 6 degrees of freedom for the insert could be

determined from the pose of the tibial baseplate, and the remaining motions determined

by solution of an over-determined set of projection equations from the series of images.

This approach simultaneously estimates the unknown locations of the tantalum markers

and the unknown rotation of the UHMWPe insert. Application of this conceptual solution

was complicated by several factors: 1) All four tantalum markers are not visible in each

radiographic image (only one or two markers are visible in many images), 2) The total

rotation range of the insert with respect to the viewing plane is between 5 and 200 for









each knee, thus the observations provide a sparse set of observations from which to

determine estimates of all unknown parameters.

A guess about the marker distribution in each tibial insert was made based on the

known geometry of the insert and knowledge of the general shape of the marker

distribution. This guess was used to seed a numerical optimization procedure that

simultaneously sought to determine the 3D locations of the tantalum markers within the

tibial insert and the rotation of the tibial insert with respect to the tibial baseplate. The

quality of the numerical solution was assessed by mathematically projecting the

estimated marker coordinates into the image plane and comparing those values to the

values observed in the radiographic images.

Measurement uncertainties were determined for each step in the measurement

process, including measurement of tantalum marker locations in the radiographs (1.0 +

0.3 pixels in the u direction, and 0.3 0.4 pixels in the v direction), and measurement of

the tibial baseplate pose. Tibial pose measurement errors for all rotations and (x, y)

translations were not significantly different previously reported values (Banks and

Hodge, 1996). However, z direction (out-of-plane) errors were found to be significantly

different from previous reports due to a bias in the negative z direction. This

measurement bias likely resulted from edge attenuation in the synthetic images.

The projections of estimated marker coordinates were found to match very well,

with average errors of (0.002in + 0.000in, 0.00 + 0.022in) in the (u, v) images. In one

troublesome sub-set of images, the best match was extremely poor (errors of about 0.5in),

but this accounted for only 4 out of 151 images. The hunch that this sub-set of images









was an outlier was confirmed when the optimizer returned a non-solution for this sub-set

only.

In addition, the computational study indicated that the optimization was primarily

sensitive to the number of images, as the error in the z direction improved from

approximately 0.30in with two images to about 0.115in with four images. The results

also indicate that more rotation may not be better as error in the x direction is seen to

markedly increase for rotations greater than 300 as it becomes the out-of-plane axis.

A major difficulty with this measurement approach is that there is no unambiguous

way to determine the absolute zero rotation of the tibial insert (Chapter 2, Figure 2-4).

One possible way is to pick an image where the relative rotation between the tibial and

femoral components is small (<2) and define that as zero insert rotation. The markers

could then be aligned in that image, with rotations in other images defined relative to the

reference image. A potential problem is that some subjects have no image with a small

tibial/femoral component rotation. When this occurs, the only way to zero the rotation is

to pick the image with the smallest component rotation and assume that the insert has

followed the femoral component as reported in the literature (Dennis et al, 2005). In

reality, if the insert has not tracked with the femoral component, this definition will result

in a rotational bias. This correction approach was performed during data post-processing

in this study. An example of an image set that needed correction is shown in Figure 4-1,

where the circles indicate images showing a small rotation between the tibial and femoral

components, yet a surprisingly large rotation between both components and the insert.

This represents a physiologically and biomechanically improbable condition, which can












be corrected by redefinition of the zero rotation alignment of the measured marker


distribution.


KIL- 12 Months (Corrected)

20
15
10
S5
5 U TIBANS
0 TIBFEM
I Fie- ZiaO.d,-. OaI I-neeIin. INSFEM

1 10
-15
-20
B
Figure 4-1. Knee kinematics for one subject with easily correctable rotational bias. A)
Rotational offset that occurred using this method when a zero reference was
not established. B) This can be corrected by assuming neutral insert rotation
when the tibial and femoral components are aligned (close to zero relative
rotation). "TIB/INS" indicates axial rotation between the tibial baseplate and
the tibial insert, "TIB/FEM" indicates axial rotation between the tibial
baseplate and the femoral component, and "INS/FEM" indicates axial rotation
between the tibial insert and the femoral component.


The corrected insert rotations appear to correlate well with the intended design of


the implant, the in vitro study by Most et al. (2003), and other in vivo studies done on


posterior stabilized rotating platform mobile-bearing TKR designs (Dennis et al., 2005;


Delport et al, 2006). The study by Most et al. (2003) on the same LPS TKR (Zimmer,


Inc.) used in this investigation revealed a 55% restoration of the internal rotation of the


tibial component (6.1 8.2) with muscle loads. This is similar to the magnitudes found


KIL- 12 Months

20
15




n nM FI. isanl.-i.9 Suai i K '"ell."* INSFEM
-5- -
-10
-15
-20









here for squat and kneeling activities. Dennis et al. (2005) and Delport et al. (2006)

found that the insert tends to follow the femoral component, but Dennis et al. (2005) did

report small amounts of rotation of the insert past the femoral component during small

ranges of activities.

Further investigation into this method would require that an attempt be made to

ensure that each subject have a clear reference alignment image. A non-weight bearing

image might serve this purpose, as Most et al. (2003) reported small internal rotation of

the tibial component for unloaded testing (-0.12o + 6.2 at 30 flexion). Resolution of the

rotation offset definition ambiguity would provide higher measurement certainties. As

currently performed, the measurement process appears to determine the tantalum marker

locations and insert rotations with sufficient accuracy to determine some useful

information about mobile bearing knee replacement kinematics.














CHAPTER 5
CONCLUSION

A technique was developed to determine the rotation of a polyethylene insert in a

rotating platform mobile-bearing TKR from an unknown distribution of markers in

single-plane radiographic images. Many levels of assumption were made in order to

arrive at a solution. The major difficulty with the proposed method was that there was no

unambiguous way to determine 0 rotation of the insert. Thus, some of the results turned

out to have a systematic rotational offset. As a best guess, 0 was assumed to occur in an

image with small (<2) tibial/femoral component rotation, if such an image existed, and

the rotations were corrected in post processing. Despite this, useful information can still

be extracted from the results. Once corrected, the results show component rotation that is

consistent with the literature. Any future investigation into this method should include a

richer image set, including a reference image, and implement a more robust global

optimizer.















APPENDIX A
COMPUTATIONAL STUDY-COMPLETE RESULTS

Two Images

Rotational Error:

Table A-1. Absolute error in 0: 2 images; constraint weighting, w = 1; no error in initial
guess
Actual 0 Found 0 Error Actual 0 Found 0 Error
0 -1.6197 -1.6197 0 -2.8315 -2.8315
0 3.1075 3.1075 48 58.9269 10.9269
0 -1.1632 -1.1632 0 -1.0355 -1.0355
4 7.4789 3.4789 52 65.3768 13.3768
0 2.6249 2.6249 0 1.3973 1.3973
8 16.52 8.52 56 71.571 15.571
0 -0.5346 -0.5346 0 4.4683 4.4683
12 15.3502 3.3502 60 77.8549 17.8549
0 -1.1416 -1.1416 0 3.5788 3.5788
16 20.353 4.353 64 84.0316 20.0316
0 -2.006 -2.006 0 3.9181 3.9181
20 26.1008 6.1008 68 85.8808 17.8808
0 -2.3083 -2.3083 0 5.8344 5.8344
24 30.684 6.684 72 91.0406 19.0406
0 -2.4398 -2.4398 0 5.702 5.702
28 34.9818 6.9818 76 93.8884 17.8884
0 -2.6255 -2.6255 0 6.5901 6.5901
32 39.579 7.579 80 99.0792 19.0792
0 -2.8944 -2.8944 0 3.9863 3.9863
36 44.4742 8.4742 84 95.7458 11.7458
0 -2.7167 -2.7167 0 3.7608 3.7608
40 48.8154 8.8154 88 100.8134 12.8134
0( -2.5177 -2.5177
44 53.8935 9.8935

Average Error: 5.890 6.890














Table A-2. Absolute error in 0: 2 images; constraint weight, w = 550; no error in initial
guess
Actual 0 Found 0 Error Actual 0 Found 0 Error
0 -1.8028 -1.8028 0 2.1091 2.1091
0 3.5153 3.5153 48 60.1034 12.1034
0 -0.97 -0.97 0 2.4913 2.4913
4 7.3013 3.3013 52 63.6336 11.6336
0 -0.476 -0.476 0 2.3236 2.3236
8 11.3543 3.3543 56 64.1328 8.1328
0 -0.5944 -0.5944 0 2.4988 2.4988
12 15.5643 3.5643 60 67.4903 7.4903
0 -1.007 -1.007 0 4.5735 4.5735
16 20.7849 4.7849 64 78.3378 14.3378
0 0.0431 0.0431 0 3.6 3.6
20 27.8864 7.8864 68 78.5927 10.5927
0 1.0142 1.0142 0 6.3712 6.3712
24 33.5256 9.5256 72 86.9858 14.9858
0 0.8303 0.8303 0 4.2361 4.2361
28 37.6981 9.6981 76 86.1621 10.1621
0 1.3096 1.3096 0 5.6359 5.6359
32 42.7857 10.7857 80 93.082 13.082
0 1.7342 1.7342 0 5.1215 5.1215
36 48.1488 12.1488 84 95.0952 11.0952
0 2.1299 2.1299 0 4.0395 4.0395
40 52.6214 12.6214 88 97.5943 9.5943
0 2.0622 2.0622
44 57.3912 13.3912

Average Error: 5.760 4.800














Table A-3. Relative error in 0: 2 images; constraint weighting, w = 1; no error in initial
guess
Actual Found Corrected Actual Found Corrected
A6j A6i A02 A1 A02- A1 AOj Ai A02 AO1 A2 AO1
0 4.7272 0 48 61.7584 57.0312
4 8.6421 3.9149 52 66.4123 61.6851
8 13.8951 9.1679 56 70.1737 65.4465
12 15.8848 11.1576 60 73.3866 68.6594
16 21.4946 16.7674 64 80.4528 75.7256
20 28.1068 23.3796 68 81.9627 77.2355
24 32.9923 28.2651 72 85.2062 80.479
28 37.4216 32.6944 76 88.1864 83.4592
32 42.2045 37.4773 80 92.4891 87.7619
36 47.3686 42.6414 84 91.7595 87.0323
40 51.5321 46.8049 88 97.0526 92.3254
44 56.4112 51.684
Average Error: 10.34 3.670
Average Corrected Error: 5.62 3.67

Table A-4. Absolute error in 0: 2 images; constraint weight, w = I50; no error in initial
guess
Actual Found Corrected Actual Found Corrected
AOj AOi A02 AO1 A02 AOI AOj AOi A02 AO1 A02 AO1
0 5.3181 0 48 57.9943 52.6762
4 8.2713 2.9532 52 61.1423 55.8242
8 11.8303 6.5122 56 61.8092 56.4911
12 16.1587 10.8406 60 64.9915 59.6734
16 21.7919 16.4738 64 73.7643 68.4462
20 27.8433 22.5252 68 74.9927 69.6746
24 32.5114 27.1933 72 80.6146 75.2965
28 36.8678 31.5497 76 81.926 76.6079
32 41.4761 36.158 80 87.4461 82.128
36 46.4146 41.0965 84 89.9737 84.6556
40 50.4915 45.1734 88 93.5548 88.2367
44 55.329 50.0109
Average Error: 7.410 2.290
Average Corrected Error: 2.100 2.290













Translational error:


2 Images, W=1, IG Exact


Image Rotation Range (deg)


2 Images, W=1, IG Exact


Image Rotation Range (deg)


2 Images, W=1, IG Exact


Image Rotation Range (deg) C

Figure A-1. Image angle vs. error in marker coordinates: 2 images; constraint weighting,

w = 1; no error in initial guess. A) x-direction, B)y-direction, C) z-direction.








































Image Rotation Range (deg)


Image Rotatn Rnge (deg)


Figure A-2. Image angle vs. error in marker coordinates: 2 images; constraint weighting,
w = 150; no error in initial guess. A) x-direction, B)y-direction, C) z-direction.


V*"*,


Image Rot atOn Range (de)


2 Images, w- 0, IG Exact












Three Images

Rotational error:

Table A-5. Absolute error in 0: 3 images; constraint weighting, w = 10; no error in initial
guess.
Actual 0 Found 0 Error Actual 0 Found 0 Error
0 -0.633 -0.633 0 7.2912 7.2912
2 4.5263 2.5263 16 25.3255 9.3255
4 4.4858 0.4858 32 39.0597 7.0597
0 -0.6709 -0.6709 0 4.0689 4.0689
4 7.3625 3.3625 18 24.198 6.198
8 8.9238 0.9238 36 40.335 4.335
0 -0.2665 -0.2665 0 6.3237 6.3237
6 8.8093 2.8093 20 28.3868 8.3868
12 13.2609 1.2609 40 45.7704 5.7704
0 0.763 0.763 0 5.2056 5.2056
8 11.8751 3.8751 22 29.1667 7.1667
16 18.1219 2.1219 44 48.7565 4.7565
0 1.0716 1.0716 0 5.2356 5.2356
10 13.7674 3.7674 24 31.0194 7.0194
20 22.5468 2.5468 48 52.6868 4.6868
0 2.2766 2.2766 0 6.1742 6.1742
12 16.8327 4.8327 26 33.6498 7.6498
24 27.0697 3.0697 52 57.4318 5.4318
0 5.7357 5.7357
14 21.9787 7.9787
28 33.9036 5.9036 _
Average Error: 4.25 2.660














Table A-6. Absolute error in 0: 3 images; constraint weighting, w =10; error in initial
guess.
Actual 0 Found 0 Error Actual 0 Found 0 Error
0 -1.5769 -1.5769 0 6.2072 6.2072
2 3.4663 1.4663 16 24.2949 8.2949
4 3.4855 -0.5145 32 38.0801 6.0801
0 -1.3302 -1.3302 0 5.1881 5.1881
4 6.4947 2.4947 18 25.2153 7.2153
8 8.107 0.107 36 41.2633 5.2633
0 0.606 0.606 0 6.3239 6.3239
6 9.6256 3.6256 20 28.3868 8.3868
12 14.0306 2.0306 40 45.7704 5.7704
0 -0.1891 -0.1891 0 5.2051 5.2051
8 10.9973 2.9973 22 29.1658 7.1658
16 17.3053 1.3053 44 48.7553 4.7553
0 3.3628 3.3628 0 5.2338 5.2338
10 15.8487 5.8487 24 31.0196 7.0196
20 24.4705 4.4705 48 52.689 4.689
0 1.0848 1.0848 0 5.2302 5.2302
12 15.7409 3.7409 26 32.7856 6.7856
24 26.0517 2.0517 52 56.6532 4.6532
0 3.4432 3.4432
14 19.8582 5.8582
28 31.9193 3.9193
Average Error: 3.960 2.650












Table A-7. Relative error in 0: 3 images; constraint weighting, w = 1; no error in initial
guess.
Actual Total Found Total Actual Found Found
A6k A6i A63 A61 AOj A6i A62 A61 A63 A62
4 5.2837 2 6.6476 -1.3639
8 9.2543 4 7.8223 1.432
12 12.4132 6 8.3696 4.0436
16 16.0225 8 10.2859 5.7366
20 19.47 10 11.5541 7.9159
24 22.8282 12 13.4423 9.3859
28 26.3193 14 15.2137 11.1056
32 29.774 16 16.9423 12.8317
36 33.4835 18 18.6503 14.8332
40 36.5714 20 20.5289 16.0425
44 40.2441 22 22.2384 18.0057
48 43.8764 24 23.9671 19.9093
52 47.4544 26 25.5844 21.87
Average Error (AOj AOi): -0.81o + 2.61
Average Error (AOk AOi): -1.62 + 2.02


Table A-8. Relative error in 0: 3 images; constraint weighting, w = /50; no error in initial


guess.
Actual Total Found Total


Actual


Found


Found


A6k AOi A63 AO1 AOj AOi A62 AO1 AO3 A2
4 5.2691 2 5.4026 -0.1335
8 9.5991 4 7.8306 1.7685
12 13.4879 6 9.0499 4.438
16 17.2886 8 11.062 6.2266
20 21.1853 10 12.53 8.6553
24 24.4717 12 14.3732 10.0985
28 28.111 14 16.2115 11.8995
32 31.708 16 17.9979 13.7101
36 35.7361 18 19.8475 15.8886
40 38.9733 20 21.8143 17.159
44 43.3022 22 23.8314 19.4708
48 47.1667 24 25.6398 21.5269
52 51.0954 26 27.3935 23.7019
Average Error (AOj AOi): 0.13 2.370
Average Error (AOk AOi): 0.260 1.000























Translational error:



3 Images, W=1, IG Exact


Image rotation Eange (dg)


3 images, W=1, IG Exact


I mage Rotaton Range (deg)


images, W=1, IG Exact


Image Rotation Range (deg) C

Figure A-3. Image angle vs. error in marker coordinates: 3 images; constraint weighting,

w = 1; no error in initial guess. A) x-direction, B)y-direction, C) z-direction.















3 Images, W= 50, IG Exact


Image Rotaton Range (deg)


3 Images, W= 50, IG Exact


Icaga Ratat,., Racga (dag)


3 Images, VW=50, IG Exact


Image Rotation Range (deg)


Figure A-4. Image angle vs. error in marker coordinates: 3 images; constraint weighting,

w = /50; no error in initial guess. A) x-direction, B) y-direction, C) z-

direction.














3 Images, W=10, Error in IG


SImages, W10, Error in IG

3 Images, W=10, Error in IG


Image Rotation Range (deg)

3 Images, W=10, Error in IG


n ^ .


0 10 20 30 40 50 60
Image Rotation Range (deg)

C

Figure A-5. Image angle vs. error in marker coordinates: 3 images; constraint weighting,

w = 10; error applied to initial guess. A) x-direction, B)y-direction, C) z-

direction.









Four Images


Rotational error:


Table A-9. Absolute error in 0: 4 images; constraint weighting, w =
guess.
Actual 0 Found 0 Error Actual 0 Found 0 Error
0 -0.0484 -0.0484 0 -2.49 -2.49
2 3.5978 1.5978 8 6.879 -1.121
4 4.5798 0.5798 16 13.4985 -2.5015
6 6.7568 0.7568 24 21.4155 -2.5845
0 -0.9709 -0.9709 0 -2.5738 -2.5738
4 4.8758 0.8758 10 8.4846 -1.5154
8 7.7243 -0.2757 20 17.4 -2.6
12 12.4662 0.4662 30 27.0878 -2.9122
0 -2.3132 -2.3132 0 -2.3048 -2.3048
6 5.0493 -0.9507 12 11.0676 -0.9324
12 10.5654 -1.4346 24 21.4118 -2.5882
18 17.2911 -0.7089 36 32.9533 -3.0467
Average Error: -1.23 1.390


Table A-10. Absolute error in 0: 4 images; constraint weighting, w
initial guess.
Actual Total Found Total Actual Found Found Found
A0k A0i A04 A01 Aj A0i A02 AO1 A03 A02 A04 A03
6 6.7357 2 3.4975 0.9939 2.2443
12 13.304 4 5.8976 2.778 4.6284
18 19.8894 6 7.4733 5.6212 6.7949
24 23.7454 8 9.2941 6.5676 7.8837
30 29.6645 10 11.0624 8.9153 9.6868
36 35.2659 12 13.375 10.3466 11.5443
Average Error (AOj AOi): 0.14 1.150
Average Error (A0k A0i): 0.43 1.04


10; no error in initial


10; error (+3) in






48


Table A-11. Absolute error in 0: 4 images; constraint weighting, w = 10; error (+4) +
rotational bias in initial guess.
Relative Error in 0, 4 Images, W=10, Error (+4) in IG
Actual Total Found Total Actual Found Found Found
Ak Ai A04 A1 A A6i A62 A61 A03 A02 A04 A03
6 6.7794 2 1.6179 0.7639 4.3976
12 13.0111 4 4.5533 2.6873 5.7705
18 19.8871 6 6.7062 5.583 7.5979
24 23.7482 8 7.8849 6.5687 9.2946
30 29.7466 10 9.6958 8.9404 11.1104
36 35.2571 12 11.5415 10.3439 13.3717
Average Error (AOj AOi): 0.13 1.250
Average Error (A6k A6i): 0.400 + 0.990














4 Images, W=10, Error + Rotational Bias in IG


4 Img, 10, ERo n Rtinal Ba IG


4 Images W10 EBoor + Rotational Bias In IG


Image Rotation Range (deg)


4 Images, W10, Eor + Rotational Bias in IG


Image Rotaon Range (deg)

C

Figure A-6. Image angle vs. error in marker coordinates: 4 images; constraint weighting,

w = 10; error and rotational bias applied to initial guess. A) x-direction, B)y-

direction, C) z-direction.


















4 Images, 12deg Image Rotation Range


4 Images, 12deg Image Rotation Range


van enon In Pncipal Dltance (n)



4 Images, 12deg Image Rotation Range


VanlOn in Pnncipal D nce (n) C

Figure A-7. Variation in principal distance vs. error in marker coordinates: 4 images;

constraint weighting, w = 10; error applied to initial guess. A) x-direction, B)

y-direction, C) z-direction.























APPENDIX B

IN VIVO STUDY-ADDITIONAL RESULTS


AHR












KNL

*


AKR












KMR


KFR_








MWR


NYR


SKL
* *


TSHR
* e


TSR


YHR


YISR


Figure B-1. Marker distributions in the x-z plane for all knees. Gray dots indicate initial

guess made during shape matching and black dots indicate optimizer solution.


RNR












T.L












YIR
t


*










AHR







S *


KML


6-(


I O I


TKL






l S I


* g


I -f


KMR


S I -.


TSHR




-o o' Jo


TSR







- I I,.


YISR






,1 *


Figure B-2. Marker distributions in the x-y plane for all knees. Gray dots indicate initial
guess made during shape matching and black dots indicate optimizer solution.


KFR


I"' I0


MWR


I I I


KIL










NYR


I* to


YHR


'YR






S S


I *o


0 <


YIR












TSR- 3 Months


*TIB/INS
TIB/FEM
INS/FEM


TSR- 6 Months


*TIB/INS
TIBIFEM
INSIFEM


TSR- 12 Months


STIB/INS
TIB/FEM
INS/FEM


Figure B-3. TKR component rotation for subject TS at A) 3 months, B) 6 months, C) 12
months.


3 .. I ... ...
0



I-II
-2

-4

-6

-8

10
Im






10--------------------------


-2
0)
-4

2 -6
S
S-8
FU
C
. -10
x
LU
-12

-14


j noing 1ql'I3l l I-ing












SSL- 3 Months


6

4
d)


.o 0

S-2

-41


-8

-10



SSL- 6 Months

10


5





S5


w -10


-15



SSL-12 Months


6

4

2

0






i -8

-10

-12


* TIB/INS
TIB/FEM
INS/FEM


*TIB/INS
TIB/FEM
INS/FEM


STIB/INS
TIB/FEM
INS/FEM


C

Figure B-4. TKR component rotation for subject SL at A) 3 months, B) 6 months, C) 12
months.










KML- 3 Months


I Fe.. Stanl in


I
S'LIal Kneelin,


TIB/INS
TIB/FEM
INS/FEM

I


Squat Kneeling


KML- 12 Months


*TIB/INS
TIB/FEM
INS/FEM


C
Figure B-5. TKR component rotation for subject KM at A) 3 months, B) 6 months, C) 12
months.


10
8-
S6-
c 4
4-
.2
5 2
z 0-
E -2
; -4
-U


*TIB/INS
TIB/FEM
INS/FEM


KML- 6 Months


9
8

0
a 7
I 6
5
I 4
3
2
1
0


Max. Flex


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LIST OF REFERENCES


Banks, S.A., Hodge, W.A., 1996. Accurate measurement of three dimensional knee
replacement kinematics using single-plane fluoroscopy. IEEE Transactions on
Biomedical Engineering 43, 638-649.

Callaghan, J.J., 2001. Mobile bearing knee replacement: clinical results-a survey of the
literature. Clinical Orthopaedics and Related Research 392, 221-225

D'Lima, D.D., Trice, M., Urquhart, A.G., Colwell Jr., C.W., 2001. Tibiofemoral
conformity and kinematics of rotating-bearing knee prostheses. Clinical
Orthopaedics and Related Research 386, 235-242.

Delport, H.P., Banks, S.A., De Schepper, Bellemans, M.D., 2006. A kinematic
comparison of fixed- and mobile-bearing knee replacements. Journal of Bone and
Joint Surgery [British], Accepted 25 April.

Dennis, D.A., Komistek, R.D., Mahfouz, R.M., Outten, J.T., Sharma, A., 2005. Mobile-
bearing total knee arthroplasty: do the polyethylene bearings rotate? Clinical
Orthopaedics 440, 88-95.

Fantozzi, S., Leardini, A., Banks, S.A., Marcacci, M., Gianni, S., Catani, F, 2004.
Dynamic in-vivo tibio-femoral and bearing motions in mobile bearing knee
arthroplasty. Knee Surgery, Sports Traumatology, Arthroscopy 12, 144-151.

Garling, E.H., Bart, L.K., Geleijins, K., Nelissen, R.G.H.H., Valstar, E.R., 2005. Marker
configuration model-based roentgen fluoroscopic analysis. Journal of
Biomechanics 38, 893-901.

Gordon, R., Herman, G.T., 1974. Three-dimensional reconstruction from projections: a
review of algorithms. International Review of Cytology-A Survey of Cell
Biology 38,111-151.

Jones, R.E., Huo, M.H., 2006. Rotating platform knees: an emerging clinical standard in
the affirmative. Journal of Arthroplasty 21, 33-36.

Kaptein, B.L., Valstar, E.R., Stoel, B.C., Rozing, P.M., Reiber, J.H.,2003. A new model-
based RSA method validated using CAD models and models from reversed
engineering. Journal of Biomechanics 36, 873-882.









Mahfouz, M.R., Hoff, W.A., Komisteck, R.D., Dennis, D.A., 2003. A robust method for
registration of three-dimensional knee implant models to two-dimensional
fluoroscopy. IEEE Transactions on Medical Imaging 22, 1561-1574.

McEwen, H.M.J., Barnett, P.I., Bell, C.J., 2005. The influence of design and materials in
kinematics on the in vitro wear of total knee replacements. Journal of
Biomechanics 38, 357-365

McNulty, D., 2002. The effect of cross-linking UHMWPE on in vitro wear rates of fixed
and mobile-bearing knees. Abstracts of the Symposium on ultra-high molecular
weight polyethylene for joint replacements. Miami Beach, Florida.

Moro-oka T., Hamai S., Miura, H., Higaki H., Fregly B.J., Iwamoto Y., Banks S.A.,
2006. Can MR derived bone models be used for accurate motion measurement
with single-plane 3D shape registration? Journal of Orthopaedic Research,
Submitted March.
Most, E., Li, G., Schule, S., Sultan, P., Park, S., Zayontz, S., Rubash, H., 2003. The
Kinematics of Fixed- and Mobile-Bearing Total Knee Arthroplasty. Clinical
Orthopaedics & Related Research 416,197-207

Pagnano, M.W., Trousdale, R.T., Stuart, M.J., 2004. Rotating platform knees did not
improve patellar tracking. A prospective randomized study of 240 primary total
knee arthroplasties. Clinical Orthopaedics and Related Research 428, 221-227

Ridgeway, S., Moskal, J.T., 2004. Early instability with mobile-bearing total knee
arthroplasty: a series of 25 cases. Journal of Arthroplasty 19, 686-693.

Selvik, G., 1989. Roentgen stereophotogrammetry: a method for the study of the
kinematics of the skeletal system. Acta Orthopaedica Scandinavica
Supplementum 232, 1-51.

Stukenborg-Colsman, C., Ostermeier, S., Wenger, K.H., Wirth, C.J.,2002. Relative
motion of a mobile bearing inlay after total knee arthroplasty- dynamic in virto
study. Clinical Biomechanics 17, 49-55.

Valstar, E.R., Vrooman, H.A., Toksvig-Larsen, S. Ryd, L., Nelissen, R.G., 2000. Digital
automated RSA compared to manually operated RSA. Journal of Biomechanics
33, 1593-1599.

Yuan, X., Ryd, L., Tanner, K.E., Lidgren, L., 2002. Roentgen single-plane
photogrammetric analysis (RSPA)-a new approach to the study of
musculoskeletal movement. Journal of Bone and Joint Surgery [British] 84B, 908-
914















BIOGRAPHICAL SKETCH

Sydney Machado is originally from central California and attended University of

California, Santa Barbara, where she earned her B.S. in mechanical engineering. She

then moved to Florida to pursue her master's degree in mechanical engineering at the

University of Florida. During her time at the University of Florida, Sydney has worked

for Dr. Scott Banks in the Orthopaedic Biomechanics Lab in the Mechanical and

Aerospace Engineering Department. Sydney has also volunteered for several outreach

engineering programs, such as Eye on Engineering, Project Athena, and MESA, in both

California and Florida.