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Leveraging the use of existing C-arms for Roentgen stereophotogrammetric analysis Chung, Vivian W.J. 2015

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     LEVERAGING THE USE OF EXISTING C-ARMS FOR  ROENTGEN STEREOPHOTOGRAMMETRIC ANALYSIS   by  Vivian W.J. Chung  B.A.Sc., Integrated Engineering  Western University, 2009   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Biomedical Engineering)  The University of British Columbia (Vancouver)  December 2015   © Vivian W.J. Chung, 2015  ii   Abstract “You can’t improve it unless you can measure it” is a common sentiment in engineering.  For total knee replacement patients, failed implants requiring revision surgery is a significant risk.  Our long-term goal, therefore, is to develop and evaluate a protocol that will allow us to accurately measure the full 3D position of an implant in the early post-surgical period in order to detect signs of relative motion occurring between implant and bone.   By doing this, we will be able to gain insights into the failure mechanisms behind total knee replacement implants. The 'gold standard' method for measuring relative motion is known as Roentgen Stereophotogrammetric Analysis (RSA) – a technique which extracts 3D information about the implant and bone positions from two roughly orthogonal radiographs.  This information can be used to quantify the migration of an implant over time to submillimeter accuracy, a metric that has been shown to reliably predict implant longevity in patients (Pijls 2012).  Unfortunately, commercial RSA systems are expensive, which has limited their use in clinical settings.  Our goal in this project was to develop an RSA protocol based on C-arm fluoroscopy machines, many of which already exist in most hospitals. We successfully developed such a protocol and evaluated its accuracies and precisions through a series of phantom-based verifications.   Results were highly promising:  accuracies ranged between -39 to 11 μm for translations and -0.025 to 0.029° for rotations, while system precisions ranged between 16 to 27 μm and 0.041 to 0.059°.   This performance was comparable to RSA systems in the literature, where traditional and more expensive radiographic equipment was typically employed.  In addition, inter-rater reliability tests also showed a high degree of correlation (ICC > 0.999) between two raters who were trained to use the protocol.  We conclude that we have developed an RSA protocol appropriate for measuring relative motion of knee replacement implants in phantoms and cadaveric specimens by leveraging the use of existing C-arm technology.  This research places us in position to further develop the protocol for use in extensive iii  prospective clinical assessments – research that can potentially drive future improvements in surgical technique and implant design. iv   Preface The work presented in this thesis was performed by the author, Vivian Chung, under the supervision of Dr. Antony Hodgson.  Dr. Bassam Masri provided guidance on the clinical application of the work documented.  Dr. Shahram Amiri, Dr. Carolyn Anglin and Dr. Robyn Newell provided guidance on technical directions.   Preliminary results of the work reported here were presented at the 15th annual meeting of the International Society for Computer Assisted Orthopaedic Surgery (CAOS) in Vancouver, Canada, in June 2015. Chapter 2 and Chapter 4 of the study were performed by the author with assistance by summer student, Jackie Yik.  The author designed all devices associated with the development and evaluation of the protocol.   Precision machining of device components were fabricated by machine shop staff at the BC Cancer Agency Joint Engineering Centre.  Coordinate measurements of the calibration cage were done through services rendered by the Manufacturing Automation Laboratory (MAL) of the Mechanical Engineering Department at the University of British Columbia.   The protocol developed in this thesis is in adherence (where applicable) to section 13 of ISO 16087:2013(E), which facilitates standardized outputs from RSA studies on orthopaedic implants.      v   Table of Contents Abstract ................................................................................................................................................. ii Preface .................................................................................................................................................. iv Table of Contents ................................................................................................................................. v List of Tables ....................................................................................................................................... ix List of Figures ....................................................................................................................................... x Acknowledgements ............................................................................................................................ xv Dedication .......................................................................................................................................... xvi Chapter 1   Introduction and Background ........................................................................................ 1 1.1 Total Knee Arthroplasty ........................................................................................................... 1 1.2 Predicting Implant Failures ...................................................................................................... 2 1.3 Assessing Implants with RSA .................................................................................................. 3 1.4 The Need for Low-Cost RSA ................................................................................................... 5 1.5 Cost Implications ..................................................................................................................... 6 1.5.1 Hardware .............................................................................................................................. 6 1.5.2 Software ................................................................................................................................ 7 1.6 RSA Process ............................................................................................................................. 9 1.6.1 Image Acquisition................................................................................................................. 9 1.6.2 Distortion Correction .......................................................................................................... 10 1.6.3 System Calibration ............................................................................................................. 12 1.6.4 Point Reconstruction........................................................................................................... 16 1.6.5 Migration Calculations ....................................................................................................... 16 1.7 Summary of Research Goals .................................................................................................. 17 1.8 Thesis Organization ............................................................................................................... 18 vi  Chapter 2   Investigating Spatial Constraints of a Dual C-Arm Setup ......................................... 19 2.1 Purpose ................................................................................................................................... 20 2.2 Materials and Methods ........................................................................................................... 21 2.3 Results .................................................................................................................................... 22 2.3.1 Orthogonality of C-Arms .................................................................................................... 22 2.3.2 Maintaining Calibration ...................................................................................................... 23 2.3.3 Imaging Field ...................................................................................................................... 24 2.4 Discussion .............................................................................................................................. 26 2.5 Limitations ............................................................................................................................. 27 2.6 Conclusions and Recommendations ...................................................................................... 27 Chapter 3   System Development ...................................................................................................... 29 3.1 System Overview ................................................................................................................... 29 3.2 Proposed RSA Protocol ......................................................................................................... 30 3.2.1 Image Acquisition............................................................................................................... 31 3.2.2 Distortion Correction .......................................................................................................... 31 3.2.3 System Calibration ............................................................................................................. 31 3.2.4 RSA Software ..................................................................................................................... 32 3.3 Image Acquisition .................................................................................................................. 33 3.3.1 Radiographic Equipment .................................................................................................... 34 3.3.2 Image Pre-Processing ......................................................................................................... 37 3.4 Distortion Correction ............................................................................................................. 38 3.4.1 Distortion Correction Device (Hardware) .......................................................................... 39 3.4.2 XROMM Distortion Correction Module (Software) .......................................................... 40 3.4.3 Distortion Correction Verification ...................................................................................... 41 3.5 System Calibration ................................................................................................................. 43 3.5.1 Calibration Cage Device (Hardware) ................................................................................. 44 3.5.2  XROMM Calibration Module (Software) ......................................................................... 49 3.6 Point Reconstruction .............................................................................................................. 51 3.7 Relative Motion Calculations ................................................................................................. 52 3.7.1 Conventions, Definitions and Guidelines ........................................................................... 52 3.7.2 Software Development ....................................................................................................... 60 3.7.3 Reporting RSA Quality Metrics ......................................................................................... 69 3.8 Cost of System ............................................................................................................................ 71 Chapter 4   System Verification ........................................................................................................ 72 4.1 Introduction ............................................................................................................................ 72 4.2 Materials ................................................................................................................................. 73 vii  4.3 Methods .................................................................................................................................. 77 4.3.1 Relative Motion Calculations ............................................................................................. 77 4.3.2 Precision ............................................................................................................................. 78 4.3.3 Accuracy ............................................................................................................................. 79 4.3.4 Inter-Rater Reliability ......................................................................................................... 81 4.4 Results .................................................................................................................................... 82 4.4.1 Precision ............................................................................................................................. 82 4.4.2 Accuracy ............................................................................................................................. 85 4.4.3 Inter-Rater Reliability ......................................................................................................... 88 4.5 Discussion .............................................................................................................................. 89 4.5.1 System Performance ........................................................................................................... 89 4.5.2 Comparison with Other Studies .......................................................................................... 90 4.5.3 Clinical Relevance .............................................................................................................. 92 4.5.4 Limitations .......................................................................................................................... 92 4.5.5 Recommendations .............................................................................................................. 93 Chapter 5   Summary and Conclusions ........................................................................................... 95 5.1 Summary of Findings ............................................................................................................. 95 5.2 Novelty ................................................................................................................................... 97 5.3 Limitations ............................................................................................................................. 97 5.4 Future Directions .................................................................................................................... 98 5.4.1 Model-Based and Feature-Based RSA ............................................................................... 99 5.4.2 Additional Verifications ................................................................................................... 101 5.4.3 System Improvement Opportunities ................................................................................. 102 5.4.4 Measuring Polyethylene Wear .......................................................................................... 105 5.5 Conclusions .......................................................................................................................... 107 Bibliography ..................................................................................................................................... 109 Appendix A   Summary of Protocol ............................................................................................... 115 A.1 Image Acquisition ................................................................................................................ 115 A.2 Point Reconstruction with XROMM ................................................................................... 121 A.3 Reporting Relative Motions ................................................................................................. 123 Appendix B   MATLAB Programs ................................................................................................. 126 B.1 Batch Image Pre-Processing ................................................................................................ 126 B.2 Relative Motion and Rigid Body Error ................................................................................ 127 B.3 Coordinate System Conversion ............................................................................................ 130 B.4 Rigid Body Transforms ........................................................................................................ 131 viii  B.5 Condition Number ................................................................................................................ 132 B.6 Euler to Quaternion Conversion........................................................................................... 133 B.7 Relative Motion Calculator Batched Version .......................................................................... 134 Appendix C   RSA Algorithms ........................................................................................................ 137 C.1 Calibration ............................................................................................................................ 137 C.2 Point Reconstruction ............................................................................................................ 142 Appendix D   ICC Calculations ...................................................................................................... 144  ix   List of Tables Table 1: Cost estimates for major RSA hardware components .............................................................. 7 Table 2: Cost estimates for RSA software and image analysis services................................................. 8 Table 3: Multiple software modules were required in developing our RSA protocol.  This table summarizes the modules involved, most of which were available as open-source software, with the rest developed through in-house efforts. ....................................... 33 Table 4: Specifications of the single C-arm used in our RSA system to acquire radiographs. ............................................................................................................................ 37 Table 5: The residual values after distortion correction, expressed as the distance between midpoint of line drawn and the centroid of the nearest perforation to the midpoint.  Average residual value was 60.0 μm. .............................................................. 42 Table 6: Definitions of each of the six components of motion in the implant coordinate system.  Rotations are reported in Euler sequence of XYZ. ................................................... 57 Table 7: Summary of images acquired for each corresponding experiment used to verify accuracies, precisions, and inter-rater reliability of the RSA system ..................................... 80 Table 8: Precision results from the 66 paired-combinations of zero motion RSA exams, with precision defined as one standard deviation of any detected relative motions – which is ideally zero since no displacements were induced. ................................. 83 Table 9: Accuracy results, reported as the mean difference between RSA-detected relative motions and actual migrations induced using a micromanipulator (the ‘gold standard’). ..................................................................................................................... 85 Table 10: The ICC results between two raters tested using a two-way random effects model, single measures and absolute agreement.  Significance cut-off was set at p<0.007 (after multiple comparison correction) for each ICC.  Results suggest that the two raters produced highly correlated relative motion measures. ............................. 88 Table 11: Summary of accuracy and precision results obtained in this study in comparison with studies in the literature ................................................................................................... 91 Table 12: A comparison of protocol requirements for the three RSA approaches that may be used to measure TKA implant relative motions.  While marker-based and model-based RSA have been well studied, feature-based RSA remains a relatively unexplored technique for TKA relative motion measurements. ........................... 101  x   List of Figures Figure 1: An RSA reconstruction of a tibial implant and an artificial bone segment.  The lines shown are the projection lines from the X-ray source onto each radiograph.  The 3D location of each marker is found by identifying the point of intersection between the marker’s two projection lines (one onto each radiograph).  Using multiple markers, we can then represent the implant and bone as separate rigid bodies, thus enabling us to track their relative motion between two RSA exams. ................... 4 Figure 2: CAD model of two C-arms in a biplanar configuration ........................................................... 9 Figure 3: Examples of a (a) biplanar cage and (b) uniplanar cage (Valstar 2005). Reprinted from Acta Orthopaedic with permission from Taylor & Francis. ......................... 13 Figure 4: A standard biplanar calibration cage consists of four panels constructed of radio-translucent material.  Each panel is embedded with a series of radio-opaque markers, and all markers on the cage have known positions relative to one another.  The panel closest to the X-ray source is called the 'control plane', while the one closest to the image intensifier is the 'fiducial plane’. ..................................... 14 Figure 5: Full calibration cage example from XROMM (Brainerd 2010).  Reprinted from Journal of Experimental Zoology with permission from John Wiley & Sons, Inc. .......................................................................................................................................... 15 Figure 6: Fluorescent strings helped us visualize the orientation of X-ray beams ................................ 21 Figure 7: Two spatial configurations of the C-arms suitable for imaging knees: with both gantries positioned above platform (left) or alternatively, with one gantry positioned below platform and the other above (right).  We refer to these two as the AA (above-above) and AB (above-below) configurations. .............................................. 22 Figure 8: Recommended gantry positioning (AB configuration), with a platform for subject to stand upon in weight bearing conditions ................................................................ 23 Figure 9: To position the subject’s knee at the intersection of the two X-ray beams, we found it helpful to mark out the imaging space on the platform using tape ........................... 25 Figure 10: The entry width to the imaging space was approximately 15" (indicated by double arrows) ........................................................................................................................ 25 Figure 11: Workflow overview of the RSA protocol.  Each RSA exam involved acquiring four radiographs (step #1) followed by RSA software analysis, which included: correcting distortions on radiographs (#2), calibrating the two X-ray views (#3) xi  and reconstructing the 3D position of implant and bone markers (#4).  Lastly, relative motion was calculated between each pair of RSA exams (#5). ................................. 30 Figure 12: The isocentric design of the Siemens Arcadis C-arm allows the calibration cage to remain roughly equidistant to the image intensifier after the C-arm is rotated 90-degrees orbitally from one position (left) to the next (right). ............................................ 34 Figure 13: The image acquisition workflow of a dual C-arm (above) involves (a) obtaining distortion correction images of both X-rays, (b) obtain calibration images, and (c) obtain images of the subject.  In contrast, the use of a single C-arm (below) requires a sequence which ensures the cage and subject are not moved between shots from the two X-ray views, and that a unique distortion image is obtained for each X-ray view. ............................................................................................................... 35 Figure 14: Two types of test patterns were used to measure the system’s spatial resolution.  A radiographic ruler (top) provides pixels/mm measurements, while the two square patterns (bottom) provide line-pairs/mm measurements.  By taking radiographs of these test patterns, we determined the spatial resolution of our C-arm to be 1.8 lp/mm and 5 pixels/mm. ................................................................................... 36 Figure 15: The distortion correction device is made of a perforated sheet metal that is mounted onto a piece of carbon fiber.  Pictured here is the device before it is mounted onto the image intensifier using two spring-loaded latches. .................................... 39 Figure 16: Before (left) and after (right) applying the distortion correction.   When one views the same images in a reduced size (shown at the bottom of each image), the pincushion distortion effects become more apparent to the naked eye. ........................... 40 Figure 17: To quantify the level of distortion which remains after distortion correction, lines are drawn between the centroids of two perforations chosen at random.  If the image is free of distortion, then the midpoint of this line should intersect perfectly with a centroid of a third perforation.  Since we expect some level of distortion to remain, the distance between the midpoint and this third centroid can be defined as the residual. ................................................................................................ 42 Figure 18: Through empirical experimentation, we determined the dimensions of the conical imaging volume of our C-arm.  Show here is the CAD model of a biplanar C-arm configuration along with the corresponding imaging volume.  This approach allowed us to determine appropriate marker placements for each panel of the calibration cage (also shown). ............................................................................ 45 Figure 19: Several cages were prototyped for the purpose of testing marker placement.  With an early stage two-panel cage prototype (top left) and its resulting radiograph (top right), we found that support structures (shown by arrows) can potentially occlude markers.  We also discovered that having an excessive number of markers extending beyond the C-arm's field of view made it difficult to match each marker with its corresponding 3D coordinates.  The bottom two figures show a later prototype which addressed these two issues by reducing the number of markers and placing sufficient distances between markers and support structure. .................................................................................................................... 47 Figure 20: Shown here is the CAD model of a cage panel, with pre-drilled tunnels at each location where markers are placed.  The tunnels are narrower than the 2 mm markers, thus allowing portions of each marker surface to remain exposed.  This exposure facilitates CMM verification at later stages. ........................................................... 49 xii  Figure 21: A screenshot of the XROMM Calibration Module.  After importing the 3D coordinates of all 50 calibration markers, the user identifies each marker's location on the radiograph.  Residual values (in pixels) are then calculated and plotted to provide information on the quality of calibration.  Typically, an average residual of 0.1 to 0.3 pixels is expected (y-axis of plot on right). ............................. 50 Figure 22: Screenshot of the XROMM Digitization Module.  Here, an implant marker was identified by the user on the ML radiograph (on left).  XROMM then guides the user in identifying the same marker’s location on the AP radiograph (right) by providing a line upon which it expects the marker to lie. ....................................................... 51 Figure 23: Photo of the CMM-verified calibration cage (left), showing the locations of the origin and the orthogonal axes of the cage coordinate system.  Also shown is a visualization of cage marker locations with respect to the cage coordinate system (right).  Markers 1 to 25 are used to calibrate the first X-ray view, markers 26 to 50 for the second view. .................................................................................... 53 Figure 24: A total of four markers were embedded within the polyethylene portions of the tibial component (left).  Three markers were implanted within the liner atop the tibial tray, and a fourth was inserted into the tibial plug.  Using these four markers, the implant coordinate system (right) is defined.  First, the y-axis is defined as the vector drawn from the rigid body centroid (‘Pc’) location to its projection (‘N’) on the plane defined by the first three markers. Then, the z-axis is defined as the vector from ‘N’ to the first marker (‘P1’).  Lastly, the x-axis is calculated as the cross product from y-axis to z-axis. ............................................................ 54 Figure 25: Reporting MTPM using physical marker locations is susceptible to the effect of marker loosening.  For example, marker #3 on the implant pictured here has loosened between the two RSA exams.  Thus, its point motion would have been erroneously reported as the MTPM. ....................................................................................... 58 Figure 26: The fictive marker approach to reporting MTPM works to reduce the effect markers loosening has on MTPM by first finding the rigid body transforms between the two RSA exams.  Note that in the example here, marker #3 was loosened at followup, but its effect was minimized during MTPM calculation. .................... 59 Figure 27: The overall architecture of our custom Relative Motion Calculator program.  It uses the coordinate files generated by XROMM’s Digitization Module as the input and then calculated relative motion of the implant in terms of the six components of motion and as the Maximum Total Point Motion (MTPM).  The whole process is comprised of the four major calculation steps shown. ................................ 61 Figure 28: Screenshot of the Relative Motion Calculator showing T1 (left) and T2 (right) marker clusters ....................................................................................................................... 62 Figure 29: Our relative motion calculations use a Singular Value Decomposition approach.  Here, the implant is represented by a triangle, and the bone is shown as a square.   We use the bone as the reference segment in order to solve the relative motion of the implant segment between the baseline and followup RSA exams.  The dashed lines indicate the position of implant and bone at T2, after transforming each T2 marker using Rref and dref such that the bone marker clusters at T2 are matched to those from T1 using a least squares approach. ........................ 65 xiii  Figure 30: Screenshot of the Relative Motion Calculator, reporting relative motion results of an implant that was micromanipulated 3mm proximally.  The six components of motion along with the MTPM are provided. ...................................................................... 68 Figure 31: The phantom box (right) fitted with the translational micromanipulator and bone ‘A’.  Bone ‘B’ (middle) had some bone removed, primarily on the posterior aspects, to accommodate extreme rotations (>5°) of the implant during ML and AP rotational accuracy tests.  A rotational micromanipulator (left) is also used to induce rotational motions. .................................................................................................. 74 Figure 32: The AP and ML radiographs of the phantom, showing locations of the markers embedded in the implant (double circle) and artificial bone (single circle).  Note that although 16 beads were implanted in the bone, on average, 8.7 bone markers were used between each RSA-pair during our verifications.  The average number of markers suggested in the literature for each bone structure is 6-9 (Valstar 2005). ................................................................................................................. 76 Figure 33: Experiment setup for testing system accuracy when the implant was translated axially in the PD and ML directions.  The same setup was also used during zero motion exams to assess precisions.  The phantom box was placed in the middle of the calibration cage, with the axial directions nominally aligned between the cage and implant coordinate systems. .................................................................................... 76 Figure 34: Box plots of precision results.  Each box encloses the interquartile range (IQR), with the upper and lower edges marking the 25th percentile (q1) and the 75th percentile (q3).  The central line of each box represents the median.  Each set of whiskers corresponds to the limits of what we have defined as outliers; with upper limit set at q3 + 1.5*(IQR) and lower limit as q1 – 1.5*(IQR).  Outliers are shown as crosses.  Each bar indicates the mean error, while error bars indicate one standard deviation of the error. All reported relative motions are in the implant coordinate system.  Note that MTPM values are unsigned (unlike the six components of motion), thus its plotted bias can only be positive. ............................ 84 Figure 35: Bland-Altman plot of the accuracy test results for implant relative motion expressed in terms of MTPM.  Results show slight bias with the zero line positioned beyond the 95% CI boundaries of the mean. ........................................................ 86 Figure 36: Bland-Altman plot of the accuracy test results for implant relative motion along the six degrees of freedom.  Results show low degrees of error overall, but slight bias on PD translation, PD rotation and AP translation. ............................................... 87 Figure 37:  Flowchart of future developments ............................................................................................ 99 Figure 38: To illustrate the effect material thickness has on the quality of distortion correction, a cross-sectional view of the perforated sheet metal is shown here as rectangles and the projection center of the X-ray beams is represented by the circle.  When beams (dashed lines) pass through each perforation, a thick sheet material will produce radiographs where the centroid representation of perforation ‘B’ poorly reflects its true location.  Since the distortion correction algorithm relies on distances between perforation centroids, this effect can reduce the quality of correction. ........................................................................................... 104 Figure 39: Historically, the primary mechanisms of failure for TKAs are typically polyethylene wear, loosening, instability and infection.  In this chart, we summarize percentages of revisions that were attributed to these failure xiv  mechanisms as reported by six large-scaled studies.  The patient data included in these studies varied between 200 to 300,000 revision surgeries performed between 1997 and 2012. ....................................................................................................... 106 xv   Acknowledgements I would like to thank my supervisor, Dr. Antony Hodgson for his unwavering support and mentorship.  The opportunity you have given me is one I will always cherish.  The knowledge and experience I have gained throughout this research was more than I could ever have imagined.   Further thanks are extended to Dr. Bassam Masri and Dr. Shahram Amiri for providing insights on overarching goals.  Many thanks are owed to Dr. Robyn Newell and Dr. Carolyn Anglin for guiding my through countless technical challenges; they are brilliant mentors that I have always been able to count on for honest and unbiased guidance. I am also indebted to Dr. Angela Kedgley for her contributions.  To my surprise, she took me under her wings after a single Skype call from the other side of the world.  The generosity she has shown me, I will always pay forward.  I would like to thank Dr. Thomas Oxland, Dr. David Wilson, Jackie Yik (our incredible summer student) and the Smart-C team:  Andrew Meyer, Tiffany Ngo, Michele Touchette, Masashi Karasawa, Mohammad Amini, Hooman Esfandiari, Luke Haliburton and Renee Bernard for their crucial feedback and contributions.   Many mentors and peers have also made this research project possible, to whom I extend my heartfelt gratitude:  Dr. Maureen Ashe, Dr. Meghan Winters, Dr. Christine Voss, Dr. Peter Cripton, Christina Thiele, Mark Semple, Angela Melnyk, Mike Brenneman, Paul Drexler and the Centre for Hip Health and Mobility (CHHM) operations team, the Engineers-in-Scrubs fellows, St. John’s College fellows, and my supportive lab mates Reza Nickmanesh, Amanda Frazer, Agnetha De Sa, Erin Bussin, Rachel Wong, Nolan Lee, and Kan Cheung.      xvi   Dedication To my family and friends,    whose presence can be felt in my every pursuit.     You are my home and the reason I strive.    To my UBC Thunder family,    who redefined my understanding of limits.     More than teammates, in you, I found new strengths.   1   Chapter 1   Introduction and Background 1.1 Total Knee Arthroplasty Knee replacement demand is ballooning in North America.  In the US, primary Total Knee Arthroplasty (TKA) is projected to grow to 3.5 million cases by 2030 – an astounding 6-to-7-fold growth over 25 years (Kurtz 2007), which will make it challenging to treat all those who need this surgery.  Even today, one in four knee replacements in Canada fails to meet the benchmark for acceptable wait times, and similar trends are emerging in Australia and UK (Ackerman 2011; Canadian Institute for Health Information (CIHI) 2013).  The Canadian Institute for Health Information reported in 2013 that joint replacements may be “rising at a rate that is outpacing the ability of health systems to keep up” (Canadian Institute for Health Information (CIHI) 2013).  While increasing the supply of surgeries is one way to address this problem, it will be even more effective to find ways to decrease demand. One key element driving demand is the rate of revision surgeries.  Currently, revisions account for over 8% of all knee replacement procedures (Kurtz 2007).  Logically, we can reduce demand arising from revision procedures by improving the implant and procedure such that the original implants last longer and therefore need to be replaced less frequently. While only about 15% of implants will have been replaced by 20 years after the original procedure, up to 60% of revised implants will have failed in the first 5 years (Fehring 2001). 2  Infection and instability are the major causes of early revision (Fehring 2001), with wear and aseptic loosening playing a more significant role in later revision.  Both implant design (Gøthesen 2013) and implant alignment (Fang 2009) are known to affect survival.   1.2 Predicting Implant Failures  Given that the average knee implant survival rate currently stands at 10-20 years, it is very difficult to use revision rates to guide innovations.  What is needed is a mechanism that can provide feedback on new surgical techniques and technologies within a short timeframe (2-5 years) to steer progress.  In other words, we need a method that predicts long term revision rates in TKA.  To do so, we must examine the primary reasons for revisions, which has been identified to be aseptic loosening, polyethylene wear, infection and instability (Sharkey 2002; Dalury 2013; Schroer 2013).  These failures modes were the reason behind more than 75% of TKA revisions reported by hospital centers (Dalury 2013; Fehring 2001; Sharkey 2002) and are categorized as either late-stage (more than 2-5 years post-surgery) or early-stage failure modes.  We now recognize that infection and instability tend to occur within the first few years, while aseptic loosening and polyethylene wear tend to be the cause behind revisions occurring ≥2 years after surgery (Dalury 2013; Sharkey 2014).  The ability to predict these late-stage failures are of great interest to researchers and clinicians, since they are difficult to identify through short-term clinical trials.  Specifically, we would like to predict implant loosening – the failure mode behind nearly a quarter of all revisions reported (Dalury 2013; Sharkey 2002).   The term ‘relative motion’ refers to detectable implant motions as a result of either implant micromotion or migration.  Migration describes a shift in implant position over a period of time, while micromotion refers to implant movement as a result of loading the knee joint under various conditions.  Both migration and micromotion may lead to implant failure due to aseptic loosening:  the failure of the bone-to-implant fixation interface in the absence of infection.   Ryd et al. (1995) first found that implant loosening can be predicted through RSA-identified migration.  Significant evidence now exists that implant migration on the scale of 1-3 mm in the early postoperative period is a powerful predictor of premature implant loosening (Pijls 2012; Ryd 1995; Kärrholm 1994).  Specifically, Ryd (1995) suggested that more than 0.2 mm of migration in 3  the tibial component after two postoperative years can be used to predict patients at risk of early revision.  This was later echoed in a systematic review by Pijls et al. (2012), who reported that migrations greater than 0.5 mm within the first postoperative year identified at-risk patients (i.e. revision rates higher than 5% at 10 years) while migrations of 1.6 mm or more were unacceptable.  In fact, every 1 mm increase in migration correlates with an 8% increase in revision rates (Pijls 2012). There is therefore, a need to develop a protocol that will enable measurement of implant relative motions (specifically migrations) as small as 500 microns.  While such a protocol may be adapted to measure micromotion by imposing various loading conditions to the knee, our interest remains with migration measurements over time, which has been shown to be a predictor of early implant failure due to aseptic loosening (Pijls 2012).  Through this protocol, clinicians and researchers can more effectively drive changes in techniques and processes, instead of depending on less-sensitive and less specific long-term outcome measures (e.g., survival rate).     1.3 Assessing Implants with RSA First developed over 25 years ago (Selvik 1989), Roentgen Stereophotogrammetric Analysis (RSA) is a technique that uses stereo X-ray images, in conjunction with small (>1mm diameter) biocompatible, radio-opaque tantalum markers that are injected into the bone during the TKA procedure, to determine the relative three-dimensional locations of bones and implants.  By tracking the location of each marker on the two radiographs (Figure 1), RSA finds the precise 3D location of each marker.  With a sufficient number of markers representing the bone and implant, the information can be used to reliably assess relative motion between a bone segment and an implant to well under a millimetre.   4   Figure 1: An RSA reconstruction of a tibial implant and an artificial bone segment.  The lines shown are the projection lines from the X-ray source onto each radiograph.  The 3D location of each marker is found by identifying the point of intersection between the marker’s two projection lines (one onto each radiograph).  Using multiple markers, we can then represent the implant and bone as separate rigid bodies, thus enabling us to track their relative motion between two RSA exams.    The value of RSA for improving implant design was demonstrated by Nelissen et al. (2011), who reported that implant designs that had gone through an RSA assessment cycle and been revised based on the resulting data had a 22% to 35% reduction in revision volumes compared with designs that had not been evaluated using RSA.   One specific example demonstrating the value of an RSA-based design cycle was the discontinuation of Zimmer’s ProxiLock hip stem after a clinical study in the Netherlands exposed its excessive migration of up to 4.7 mm and 12.2° in patients over a short followup period of 2 years using RSA.  This independent study led the manufacturer to remove the product from market (Luites 2006).   Similarly, another clinical RSA study in 1998 on Stryker-Howmedica’s uncoated, uncemented Interax Total Knee flagged the design as high risk due to RSA-identified excessive migrations (Nelissen 1998).  The efficiency of RSA in providing critical insight was demonstrated in a 2012 systematic review of 50 studies, in which Pijls et al. found that RSA-based studies on total knee prostheses using only 20-60 patients and a short followup of 1 year was able to arrive at the same conclusions as those drawn using data from national joint registries that included over 1000 patients with 5-10 years 5  followup (Pijls 2012).  With mounting evidence of the predictive power of RSA, the use of short-term clinical RSA trials prior to the introduction of new implants have been advocated by researchers, clinicians and regulatory bodies (Nelissen 2011).   While RSA is a proven technique for accurately measuring implant migration and predicting early failure in patients, its use both clinically and in research has been quite limited.  This is expected, since few hospitals are equipped with RSA capability at a clinical level.  Systems that have been commissioned are either developed in-house or purchased from a small number of companies that market commercial RSA equipment, software or analysis services.   To our knowledge, there are fewer than ten RSA systems available for clinical use across North America.  These include the Hospital for Special Surgery (New York, NY), Midwest Orthopaedics at Rush (Chicago, IL) and Hôpital Maisonneuve-Rosemont (Montréal, Québec).  Aside from hospitals, a few research institutes including the Orthopaedic Innovation Centre (Winnipeg, Manitoba) and Robarts Research Institute (London, Ontario) also offer RSA services.   1.4 The Need for Low-Cost RSA Much of the limited implementation of clinical RSA systems may be attributed to the high costs associated with purchasing a commercial RSA suite, which can be as much as $750K US dollars in radiographic equipment, renovations and software licensing fees.  We therefore see a need to develop an RSA protocol that leverages low-cost resources.  In the broader picture, Nelissen et al. (2011) have argued that if the cost of a TKA is assumed to be $37,000 USD, the annual cost savings to the US healthcare system from eliminating revisions due to non-RSA-tested TKA is estimated to be over $400 million.  As such, even a modest reduction in revision rates would outweigh concerns over the cost of running RSA trials.   However, this does not necessarily make a financial case to the implant manufacturer, especially if the cost of RSA testing falls upon the implant manufacturer while the majority of cost savings as a result of reduced revision rates are received by the healthcare system.  We speculate that for RSA testing on new implants to become a standard, it will either need to be made mandatory by regulatory bodies or sufficient incentives will need to be provided to implant manufacturers.  Without 6  regulations or incentives to do so, companies that use RSA-testing may incur higher initial development costs that could increase the price of their implants and so put them at a competitive disadvantage.  In any case, it remains in the best interest of patients to minimize the cost of RSA testing, such that it becomes more affordable to researchers and clinicians – either independent or those funded by implant manufacturers.   1.5 Cost Implications In this section, we will provide cost estimates for RSA systems in terms of hardware and software components.  We will contrast different ways of constructing an RSA system and the cost and benefits associated with each component option.       1.5.1 Hardware The quickest but mostly costly way of obtaining RSA capability is to purchase a commercial RSA suite, which typically includes two X-ray machines (positioned by overhead gantries) and the necessary equipment to calibrate these machines.  A system like this is estimated to cost roughly $750,000, which includes the cost of renovation (~$70,000) required to shield a room against radiation and installation of the X-ray machines together with the imaging bed and calibration device1.  Alternatively, a research center can construct an RSA system by acquiring two traditional X-ray machines or two C-arms.  C-arms are portable X-ray units that use much lower dosages of radiation than traditional machines, and are capable of capturing either still radiographs or video fluoroscopy.   To compare the cost of these three options, let us assume the cost of renovation is equal for each of the options in Table 1.  For non-commercial RSA systems, the X-ray machines would also require calibration; we therefore have included the cost of a commercial RSA calibration device for these systems.   7    COMMERCIAL SYSTEM1 IN-HOUSE SYSTEM USING CONVENTIONAL  X-RAY MACHINES  IN-HOUSE SYSTEM  USING C-ARMS Dual X-Ray Machines $680,000 $175,000/unit2 x 2  $100,000/unit3  x 2  Calibration Device Included $15,0004 $15,0004 Total $680,000 $365,000 $215,000 Table 1: Cost estimates for major RSA hardware components   From these estimates, it is clear that at least the initial cost of commercial RSA systems can be much more costly (+200%) than in-house developed systems (though there may be additional advantages to purchasing a commercial system that have not been considered here, such as technical support, training, a well-established workflow, and a system that has already been verified for accuracies – advantages that may have cost saving implications).  For in-house systems, using C-arms could potentially cost 40% less than using conventional X-ray machines.  In addition, most hospitals will likely have a number of C-arm machines that could potentially be used on an occasional basis to support RSA procedures; in these situations, no additional costs would be incurred by the institution.  Institutions typically have in-house x-ray machines, but these are normally installed in place and are not mobile, so it would be difficult to reconfigure these to be used for RSA investigations.  The most relevant comparison, therefore, would be between the $15,000 cost of purchasing a calibration device for use with existing C-arms, versus the comparatively much higher costs ($365-680k) of buying a new system, whether based on conventional X-ray machines or a dedicated RSA system. 1.5.2 Software On the software side, we have three primary options:                                                    1 R. Harmon (personal communication, February 24 2014) 2 Block Imaging. (2015). 3 Price Points for Digital X-ray Equipment Options. Retrieved from  http://info.blockimaging.com/bid/72486/3-Price-Points-for-Digital-X-ray-Equipment-Options  3 Block Imaging. (2015). C-Arm Cost Price Guide 2015. Retrieved from http://info.blockimaging.com/c-arm-cost-price-guide 4 P. Grundström (personal communication, February 25 2014) 8  (1) Outsource analysis to an off-site provider that charges on a per-exam basis (2) Purchase commercial RSA software and analyze exams in-house (3) Develop in-house RSA software and analyze exams in-house   For each of these options, we provide the corresponding cost estimates in Table 2 for a total of one hundred patients.  Similar to the available hardware options, savings are substantial (up to 80% per patient) if a research centre has access to trained staff that can analyze RSA exams either using software developed in-house or after purchasing commercial software.  Using open-source software can result in essentially zero up-front costs, which makes it much less expensive to begin an RSA program.    COMMERCIAL SERVICE1 IN-HOUSE ANALYSIS USING COMMERCIAL SOFTWARE  IN-HOUSE ANALYSIS USING IN-HOUSE SOFTWARE SERVICE MODEL Off-site analysis In-house analysis In-house analysis SOFTWARE LICENSE FEE none $26,000/license2 none ANALYSIS COST PER PATIENT (2 EXAMS EACH) $600 $1203 $1203 TOTAL COST PER 100 PATIENTS $60,000 $17,2004 $12,000 Table 2: Cost estimates for RSA software and image analysis services   When considering these options, one should keep in mind that using a commercial service comes with added benefits over the in-house analysis models, such as well-established analysis and quality control protocols and no training costs.  On the other hand, in-house analysis could potentially provide quicker turn-around time.                                                      1 R. Harmon (personal communication, February 24 2014) 2 B. Kaptein (personal communication, July 10 2014) 3 Based on an estimate of 2 hours per RSA exam at $30/hour 4 Assuming that the license is renewed/updated every five years 9  1.6 RSA Process Aside from cost, there are also technical considerations behind selecting the elements which make up an RSA system.  In this section, we provide an overview of the components required within an RSA process, which involves four main subprocesses:  acquiring the images, correcting distortions (if required), calibrating the imaging field, and computing the relative positions of markers and implant. 1.6.1 Image Acquisition An RSA system comprises a biplanar radiographic setup that enables simultaneous image acquisition of a subject from two perspectives.  Most existing RSA systems use traditional X-ray units for this purpose, which involves a combination of ceiling mounted and/or mobile X-ray units.  An alternative method of image acquisition is to employ C-arms (Figure 2), which are portable X-ray units that are generally lower in cost and use lower radiation dose than traditional X-ray machines.  To decide which radiographic setup is most appropriate when developing an RSA system, there are several considerations:  Radiation dose  Cost & accessibility   Spatial constraints associated when the size of the X-ray units  Ability to capture biplanar radiographs simultaneously  Figure 2: CAD model of two C-arms in a biplanar configuration 10   While the amount of radiation dose varies from system to system, traditional RSA systems have used exposure settings as high as 100 kV and 320 mA (Ishida 2012).  On the low end, 50-70 kV and 4-10 mA were used (Cai 2008; Solomon 2010).  RSA systems using C-arms typically use lower levels of exposure:  between 49-80 kV and 0.3- 5 mA (Brainerd 2010; Kedgley 2009).    In terms of costs, building an RSA system using traditional X-ray units will likely be more expensive than C-arms.   Since intraoperative X-ray imaging is now a staple tool in orthopaedic surgery, having C-arm access is quite common in most hospitals across developed countries.  Thus in our protocol development, we aim to take advantage of our access to C-arm technology.   The downside of using C-arms, however, is the spatial constraint it imposes.  Because the space between the image intensifier and X-ray source is less than a metre wide, it can be challenging to position two C-arms in a biplanar configuration.  The spatial constraint of the RSA system is further complicated by its use on TKA patients, where the patient will need sufficient space in order to stand within the imaging field.  Spatial constraint is less of an issue with traditional X-ray units which are typically ceiling-mounted, thus allowing the imaging field size to be easily adjusted.    Lastly, it is important that the two biplanar radiographs are captured almost simultaneous to one another.  Otherwise, there is a high likelihood that patient motion will occur between shots, resulting in unacceptable errors and voiding the analysis.  Mechanical systems may be built to simultaneously trigger the exposures between the two X-ray units.  Alternatively, one may also use video fluoroscopy and devise a moving object that can be seen on both videos.  The two videos can then synchronized afterwards by matching up the position of the object on both videos at any given point in time (Kedgley 2009).  1.6.2 Distortion Correction  For both traditional and C-arm X-ray equipment, there are generally two types of X-ray detectors that are used to produce radiographic images from the emitted X-ray photons:  flat panel (FP) and image intensifier (II).  When an image intensifier is used, there will be a likelihood of significant image distortion that needs to be addressed before the images can be used in analysis.  In this section, we will detail what causes this distortion.  Within an image intensifier, there are three main layers where the X-ray energy is converted: the input phosphor layer, photocathode layer, and output phosphor layer (Wang 2000).  When the 11  emitted X-ray photons first arrive at the II, they will strike the input phosphor layer, converting the energy into visible light photons.  These light photons then in turn strike the photocathode layer, causing the release of electrons into the evacuated tube of the II.   The electrons follow a trajectory towards the end of the tube and strike the output phosphor layer, converting the electrons back into light photons.  It is these light photons that are finally focused by optical lenses and converted into a digital image. During the electrons’ travel inside the tube, their trajectories are guided by electrodes towards an anode target that is placed behind the output phosphor layer.  However, due to the electromagnetic interference (EMI) that exists in the environment, the electrons travelling closest to the perimeter of the tube have the highest tendency to stray from their trajectory, causing image distortions that are known as the pin-cushion effect.  Sources of EMI may include the earth’s gravitational poles and any presence of electronic equipment in the room. In a flat plate design, X-ray photons released from the X-ray tube (the source) will strike a scintillation layer first, converting the X-ray photons into light photons.  These photons then cause a photo diode layer to release of electrons.  The electrons, without traversing through an evacuated tube, will immediately strike an amorphous silicon layer, turning the electrons directly into electronic image data.  As a result, the paths of the electrons are not significantly affected by EMI.  When purchasing radiographic equipment, FP-based devices are almost always more costly than an II-based setup.   As such, one will find that II-based C-arms are more readily available at most hospital centers in Canada.  Unfortunately, the radiographs produced by an II typically contain distortions that require correction before any image analysis may proceed.   Distortion correction is most often achieved by imaging a device containing a uniform radiopaque pattern (referred to as the ‘distortion correction device’).  By observing how the pattern presents itself in the radiograph, we can derive the transforms needed to reduce distortions in subsequent images obtained under the same image acquisition conditions.    12  1.6.3 System Calibration General Approach to Calibration In essence, RSA systems extract 3D information from pairs of 2D coordinates seen on the two biplanar radiographs – a process commonly referred to as ‘point reconstruction’.  But before this can be done, one must first establish the positional relationships between each of the 2D radiographs and the 3D space being imaged through a calibration protocol. To calibrate an RSA system, we first image an object, referred to as the ‘calibration cage’, which contains a series of markers at known 3D positions.  We then derive the 2D-to-3D transforms by observing the locations of these cage markers on each radiograph and fitting these to a model, which subsequently enables us to calculate the 3D locations of points based on their locations in a pair of 2D radiographs.    Calibration Cage Design There are three main types of cage designs in the literature:  uniplanar, biplanar and full cage.  This section will detail the pros and cons of these three classes of designs. Uniplanar Cage A uniplanar cage design (Figure 3) is commonly used when the X-ray sources are not physically fixed to the image intensifiers (II) by gantries, allowing each of the II to be oriented in non-parallel configurations to its corresponding X-ray source.  This design is also applicable when X-ray cassettes are used in place of IIs.   With the uniplanar setup, the cage is typically placed in front of the IIs (or cassettes) and the subject is placed between the cage and the source.  By having the subject and cage radiographed simultaneously, this method reduces X-ray exposure to the radiologist and reduces image processing time.    13   Figure 3: Examples of a (a) biplanar cage and (b) uniplanar cage (Valstar 2005). Reprinted from Acta Orthopaedic with permission from Taylor & Francis.  Some commercially available RSA suites such as the Halifax SR Suite 1.0 (Halifax Biomedical, Mabou, Canada) employ this method, with the calibration cage mounted directly under a platform on which the patient would lie upon.       Biplanar Cage Biplanar cages are also available commercially through companies such as RSA Biomedical (Umeå, Sweden) and Tilly Medical (Lund, Sweden).  These cages are designed to surround the subject for simultaneous exposure of both cage and subject – a method referred to as “self-calibration” (Choo 2003).  However, these cages can also be used in a “pre-calibration” manner, where the cage and subject are each independently imaged, in order to: (1) Increase volume of space available for subject to maneuver (2) Reduce marker occlusion    Traditionally, a biplanar calibration cage is a radio-translucent box with stainless steel marker embedded on all four of its side panels (Figure 4).   Each X-ray beam needs to pass through at least two panels to provide sufficient calibration information.  The panel that is closest to the image 14  intensifier is traditionally referred to as the ‘fiducial plane’, while the one that is closer to the X-ray source is called the ‘control plane’.    X-RAYPROJECTIONCENTERRADIOGRAPHCALIBRATION CAGEFIDUCIAL PLANECONTROL PLANE Figure 4: A standard biplanar calibration cage consists of four panels constructed of radio-translucent material.  Each panel is embedded with a series of radio-opaque markers, and all markers on the cage have known positions relative to one another.  The panel closest to the X-ray source is called the 'control plane', while the one closest to the image intensifier is the 'fiduc ial plane’.   It has been demonstrated that biplanar cages produce more accurate point reconstruction than a uniplanar cage (Choo 2003).  Choo theorized that this is because a biplanar cage calibrates the volume in which the subject is (or will) be placed, thus point reconstruction is accomplished through an interpolation approach.   With a uniplanar cage, the subject is placed beyond the calibration volume, so the point reconstruction is done through extrapolation.  It has been well documented in the literature that interpolative calibration methods tend to result in lower reconstruction errors than extrapolation (Yuan 2000; Hinrichs 1995; Choo 2003).  In an RSA guideline published by Valstar et al., both uniplanar and biplanar cages were advocated (Valstar 2005).   Full Cage  If the pre-calibration approach is taken, a full cage design would also be appropriate (Figure 5).  Instead of having the two distinct fiducial and control planes, a full cage design uses 3 or more planes of markers to calibrate each camera, taking advantage of the entire volume of the cage for marker placement.  Due to a lack of space within the full cage design to place a subject, these cages may only be used to pre-calibrate and not self-calibrate.  Brown University has developed an open-source 15  RSA software, XROMM, which was designed to work with full calibration cages, but the software can easily adapt to any cage design (Brainerd 2010).   Figure 5: Full calibration cage example from XROMM (Brainerd 2010).  Reprinted from Journal of Experimental Zoology with permission from John Wiley & Sons, Inc.   Calibration Algorithm  The earliest RSA calibration approach involves a 2-step optimization process often referred to as the Fiducial-Control Planes (FCP) method (Choo 2003; Selvik 1989).  First, an 8-parameter transform is derived for each X-ray beam by relating 2D coordinates on the radiograph to 2D coordinates on the fiducial plane markers.  At least four fiducial markers are needed for this purpose.  Note that all fiducial markers must be coplanar with one another, since the 8-parameter transform relies on this assumption.   16  In the next step, the location of the projection center is found using at least two control markers.  This is done by projecting the control markers onto the fiducial plane, which involves observing their 2D coordinates on the radiograph and using the 8-parameter transform.  Then, for each control marker, a line is drawn from its projection on the fiducial plane to its physical location on the control plane.  The point at which all lines intersect is the projection center (or focal point), which is found using a least squares approach.    In 1971, Abdel-Aziz and Kara introduced the Direct Linear Transform (DLT) method as an alternative to FCP.  By directly relating the 2D image coordinates to 3D object coordinate using least squares approach, DLT requires only a single-step optimization as opposed to the two-step FCP method.   We will cover how this is done is the next section. More complex variants of the 11-DLT method also exist, involving up to 12 to 16 parameters.  These additional parameters reduce calibration errors by accounting for optical distortion de-centering distortions (Marzan 1975).  We provide the derivations to the 11-DLT described by Kwon (1998) in Appendix C.1. 1.6.4 Point Reconstruction Post calibration, we have two sets of 11-DLT parameters, one for each X-ray view.  With this information, we can then use the 2D coordinates of any object point that is observable on both radiographs, and solve for the point’s corresponding 3D coordinates in object space.  These equations have also been presented by Kwon (1998) and are provided in Appendix C.2. 1.6.5 Migration Calculations After 3D coordinates of markers on the implant and bone have been calculated, we can then obtain their relative position to one another by modelling each as a rigid body represented by its corresponding markers.   Mathematically, one can check the quality of these representations (Valstar 2005) by evaluating the condition numbers (CN) and rigid body errors (RBE) resulting from the RSA, both of which will be detailed later in section 3.7.3.    Once two RSA exams are done to quantify location of the implant relative to the bone at two different times, we can then calculate relative motion that has occurred between implant and bone 17  (Söderkvist 1993).  The calculations for relative motion will be later detailed in the software development section (3.7.2). The use of RSA in joint replacement implants have been well studied in the literature, as a result, guidelines set in ISO16087:2013 have set a standardized approach to reporting RSA findings on migration.  Specifically, data should be: (1) Reported with respect to the implant itself (i.e., the use of an implant coordinate system) (2) Reported in terms of anatomical axes (i.e., ML/AP/PD axis of the implant) (3) Reported in terms of translation and angular rotation in all six degrees of freedom, (4) Accompanied by definition of the point(s) used to measure migration,  (5) Accompanied by CN and RBE results, with cut-off limits stated.   In this thesis, we use these guidelines to report detected relative motion between a tibial TKA implant and artificial bone.    1.7 Summary of Research Goals Our primary research goal is to move towards creating an RSA facility at Vancouver General Hospital (VGH) for eventual assessment of total knee arthroplasties (both clinical and research applications).  The scope of this thesis is limited to developing the protocol based on the use of a single X-ray unit, and the extent of system verification will be limited to phantom-based experiments.   We aim to construct a fully functional low-cost RSA system that is capable of measuring relative motion of a tibial TKA component relative to artificial bones.  To do so, a marker-based approach was used, where metal beads were embedded into both the bone segment and the implant.  Future developments beyond the scope of this thesis will involve a model-based approach, where no markers will be embedded into the implant and implant positions will be tracked using computer models of the implants instead (Valstar 2001; Kaptein 2007; Hurschler 2009).     In this thesis, we present the design and verification of our system using a realistic phantom, quantifying both system accuracy and precision following standards set out in ISO 16087:2013, which is a guideline on the application of RSA to assess migration of orthopaedic implants.  18  1.8 Thesis Organization Chapter 1 provided an overview of relevant literature on total knee arthroplasty outcomes and the role of RSA in striving for improved surgical outcomes.  The need for affordable and accessible RSA systems was described in this section. We then provided both cost and technical considerations to commissioning an RSA system through a cost analysis followed by detailed background on how RSA systems work. Chapter 2 details a qualitative investigation into the feasibility of the X-ray setup that we have proposed for our RSA protocol.  We provide observations with regards to spatial constraint concerns associated with the setup in the context of a clinical environment.   Chapter 3 presents the elements within our proposed protocol, followed by a detailed description of the development process by elaborating on background information provided in chapter 1 and 2. Chapter 4 details the verifications carried out on the developed RSA protocol through a series of phantom-based studies. Chapter 5 concludes with our findings and discusses the novelty and contribution of our developed RSA protocol.  Limitations, improvement opportunities and future directions to further the development of the protocol to make it ready for clinical use are also discussed. 19   Chapter 2   Investigating Spatial Constraints of a Dual C-Arm Setup In our proposed RSA protocol, we have opted to use a single C-arm for image acquisition.   However, for this protocol to be adaptable to clinical applications in the future, it also must work with two C-arms.  While a single C-arm is sufficient for phantom and cadaveric work where the specimen will not move between shots, in-vivo subjects cannot remain sufficiently stationary.  Thus, a dual C-arm setup is a must in the clinical environment to simultaneously acquire the two images.   In this chapter, we began our protocol development by first establishing that a dual C-arm setup would be feasible for administering RSA exams, as this is a central assumption within our protocol.    Specifically, we set out to investigate the primary challenge involved in using a dual C-arm setup – the issue of spatial constraints.    The use of C-arm for RSA exams have been demonstrated in the literature.  In an in-vivo RSA study, Kedgley (2009) seated subjects within a dual C-arm setup to examine glenohumeral joint kinematics.  Balsdon (2012) instructed participants to walk across the platform of a dual C-arm RSA system, enabling fluoroscopic video acquisition on foot kinematics.  Similarly, Brainerd (2010) also described a dual C-arm setup for zoological kinematic research using open-source software.  In terms of RSA studies on implants using C-arms, Amiri (2012) measured TKA knee kinematics by reorienting a single C-arm to obtain biplanar views of artificial and cadaveric knees.   These studies were primarily focused on employing a C-arm RSA system for the purpose of research.  To our 20  knowledge, there has yet been any study that investigates spatial considerations associated with using a dual C-arm system to administer RSA exams in clinical practice, specifically on TKA patients.    2.1 Purpose  To understand the clinical feasibility of using dual C-Arms to administer RSA exams, it is important that the possible spatial configurations of the two C-Arms are briefly explored.   In clinical practice, a few obvious considerations arise: (1) Orthogonality of C-arms:  Can the two C-arms be positioned substantially orthogonal to one another while keeping the X-ray beams parallel to ground?  In theory, an orthogonal X-ray setup produces the highest system accuracy.  (2) Maintaining Calibration:  Can the C-Arms be calibrated, and then remain reasonably stationary throughout the duration of the exam such that calibration accuracies and precisions remain acceptable? (3) Imaging Field:  Will the subject have sufficient space to enter and stand within the imaging field in weight-bearing stance?  We evaluated possible C-arm configurations with the above criteria.  First, we investigated possible C-arm configurations that roughly align the orthogonal X-ray beams parallel to a standing subject’s transverse plane (in order to capture AP and ML views).  In marker-based RSA, markers embedded in the polyethylene liner can be easily occluded by the metal artefact of the tibial implant due to their close proximity, using X-ray beams that are parallel to the transverse plane reduces the likelihood of this occlusion.   With each C-arm configuration, we then discussed whether we would likely be able to keep the C-arms stationary during each RSA exam once calibration has been done.  If the C-arms move post calibration, we run the risk of compromising the accuracy of our calibration.    Lastly, we evaluated the imaging field by focusing on two measurements:  the entry width into the imaging field and the size of the imaging field.  The subject must have enough space to enter one foot at a time into the C-arm setup and then stand in weight-bearing stance within the field.    21  2.2 Materials and Methods To answer these questions, we obtained two C-Arms that were available to us (GE Stenoscop II, GE Healthcare, Buckinghamshire, UK and Siemens Siremobil 2000, Siemens AG, Erlangen, Germany) and explored potential spatial configurations suitable for the clinical studies we anticipate conducting (primarily on the knee).  To help us visualize the orientation of the X-ray beams without the use of radiation, a piece of fluorescent-coloured string was taped to each C-arm from the X-ray source to the image intensifier (Figure 6).  These strings assisted us in positioning the C-arms such that the center of the two X-ray beams approximately intersected one another.      Figure 6: Fluorescent strings helped us visualize the orientation of X-ray beams    We also used a wooden table which served as the platform for the patient to stand upon. It was necessary for us to elevate the patient with a platform since the lowest position we could position either C-arms while keeping the beam roughly parallel to ground still resulted in a 25” gap between the beam and ground – exceeding the ground-to-knee height for most patients in upright stance.   22  2.3 Results 2.3.1 Orthogonality of C-Arms We found that there are two practical means of achieving substantially orthogonal biplanar configurations with the two C-arms (Figure 7) by manipulating the position and orientation of each C-arm’s gantries (i.e. the “C” shaped portion of the C-arm): (1) Above-Above (AA) Configuration: Position both gantries above the platform (2) Above-Below (AB) Configuration: Position one gantry below the platform and one above   Figure 7: Two spatial configurations of the C-arms suitable for imaging knees: with both gantries positioned above platform (left) or alternatively, with one gantry positioned below platform and the other above (right).  We refer to these two as the AA (above-above) and AB (above-below) configurations.      The AA configuration allows the C-arms to be easily removed and stored away when not in use, but it was not possible to orient the X-ray beams perfectly parallel to the subject’s transverse plane due to the overlapping gantries.  We were only able to achieve orthogonality with both beams inclined approximately 10 degrees relative to platform.  As such, AA was not a suitable configuration for marker-based RSA study on TKA.   With the AB configuration, we were able to position the X-ray beams orthogonal to one another with each beam parallel to the subject’s transverse plane.  The downside of this setup is that ABOVE-ABOVE (AA) ABOVE-BELOW (AB) 23  one of the gantries must be maneuvered into position underneath the platform.  However, the second gantry was easily positioned, since the two gantries do not overlap (Figure 8).     Figure 8: Recommended gantry positioning (AB configuration), with a platform for subject to stand upon in weight bearing conditions  2.3.2 Maintaining Calibration  As discussed in section 1.6.3, there are two possible workflows for calibrating an RSA system:  pre-calibration and self-calibration.  In the pre-calibration approach, the calibration cage is imaged separately from the subject of interest.  Alternatively, the cage and subject can be imaged simultaneously with the self-calibration approach. From our experience administering RSA on phantoms, we recommend pre-calibration over self-calibration.  The reasoning is that while simultaneous capture with self-calibration would intuitively be more accurate, it is difficult to achieve in practice.  Particularly with small sized cages, 24  positioning the cage around a patient knee’s can be time consuming and risks distorting the cage through accidental contact.  However, if a pre-calibration approach is taken, the C-arms must be protected against being accidentally moved after pre-calibration.  Regardless of C-arm configuration, we would suggest constructing a radiolucent barrier between the C-arms and the patient - for example, a box could be bolted to the platform that surrounds the imaging space and provides a 'hall' through which the subject could enter the central imaging field.     If one would like to avoid the requirement of pre-calibration, then the cage must be sufficiently large to accommodate a patient’s knee easily.  The material and construction of the cage must also be stiff enough such that its size does not cause submillimetric warping of the cage and so that contact with the cage does not distort the calibrated grid.  2.3.3 Imaging Field Using our two C-arms, we were able to create an imaging space approximately 2’x2’ wide (Figure 9).  The width of this space is marked out on the platform using white tape, and an ‘X’ denotes roughly where the two X-ray beams intersect.   When administering an RSA exam, the radiology technician would instruct the patient to place their foot directly onto the ‘X’ prior to more refined alignment of the patient’s knee using the laser guidance system built into most C-arm units.   In terms of entry into the imaging space, the opening between the two C-arms for the subject to enter was approximately 15” wide (Figure 10).  25   Figure 9: To position the subject’s knee at the intersection of the two X-ray beams, we found it helpful to mark out the imaging space on the platform using tape    Figure 10: The entry width to the imaging space was approximately 15" (indicated by double arrows)     26  2.4 Discussion During our initial investigations, we quickly discovered that for our C-arm models, it was not possible to position either gantries such that the X-ray beams are positioned low enough to capture the knee of a patient who was standing at ground level.  Thus, we found it was necessary to use an elevated platform to raise the height of the patient’s knee relative to ground.   We used a standard table to serve this purpose. Using the platform, we have found two configurations (AA and AB) that can achieve our aforementioned criterion of orthogonality between the X-ray beams; however, only configuration AB allowed both beams to be positioned parallel to the transverse plane of the patient – a crucial criteria to reduce marker occlusion.  In Kedgley’s study (2009), the authors imaged the glenohumeral joint of a subject using a configuration similar to that of AA.  Comparable to our findings, their setup did not allow beams to be positioned parallel to ground, but the configuration remained a viable option to the authors since they were not imaging metal implants, and thus did not have the issue of metal implants occluding markers.   To maintain calibration accuracy, we note that pre-calibration and self-calibration are both viable options.  In the literature, we have found two in-vivo dual C-arm RSA studies – both of which used pre-calibration (Brainerd 2010; Kedgley 2009).  We speculate that with the spatial constraints of a dual C-arm setup, it may be difficult to position a subject with a calibration cage to be imaged simultaneously.  While companies such as Tilly Medical Products (Lund, Sweden) have manufactured biplanar cages purposed for RSA imaging on knees using a self-calibration workflow, to our knowledge, there has yet been a published study that uses the self-calibration approach with a dual C-arm RSA system.     If a pre-calibration approach is adopted, sufficient care must be exercised to avoid moving the C-arms once calibration is complete.  It should be noted that the effects of accidental C-arm movements on calibration accuracy in a pre-calibrated RSA system has yet been investigated in the literature. Kedgley (2009) recommended that if either C-arms were accidentally touched during examination, the C-arms must be re-calibrated.    In terms of spatial constraints, Brainerd (2010) described a similar setup to our AB configuration, which was used to study the kinematics of small animals (e.g., miniature pigs).  Since 27  these animals were much smaller in comparison to a human subject, spatial limitations was less of a concern.  For our purposes, configuration AB was most appropriate, but we still needed to be cognisant of spatial limitations imposed on our subjects.  Thus, we evaluated the entry width and the imaging field size of our AB configuration. To conclude whether there would be sufficient space within the imaging space for a person to maintain a weight-bearing stance, we compared against width of existing medical imaging devices.  The industry-standard aperture diameter is 17.7” for fluoroscopy, 27.6” for CT, and 23.6” for MRI (Uppot 2007).  Although these devices typically image the patient in supine position, they offer a crude comparison as to acceptable limits for a dual C-arm RSA system.   Thus, we targeted an imaging space of at least 18” wide.  Since our AB configuration had a clearance of approximately 24” x 24” within the imaging field, we conclude that a dual C-arm RSA system would likely provided sufficient imaging space for an average patient.  The entry width into this imaging field was narrower (15”), but close to the aperture for industry-standard fluoroscopy (17.7”).   2.5 Limitations We should note that the findings described in this chapter may not be universal to all C-arm models, as C-arms vary greatly in dimensions depending on manufacturer and model.  Additional factors that we did not explore which could affect the decision in choosing an appropriate C-arm configuration include:  average size and height of subjects, ability of subjects to stand upright without handrails / support structures, and whether authors will be using additional testing rigs to simulate alternative knee loading conditions.    2.6 Conclusions and Recommendations In this chapter, we carried out a qualitative study on the possible spatial configurations of C-arms for a dual C-arm RSA system.  Based on our findings, we concluded that the AB configuration is most appropriate when imaging TKA patients using a marker-based dual C-arm RSA system.  With this configuration, the orthogonality of the X-ray beams can be achieved while keeping both beams 28  roughly parallel to the subject’s transverse plane.  This allowed the knee to be imaged in AP and ML views, thus reducing the possibility of marker occlusion.   The downside of using the AB configuration was that it positions one of the gantries below a platform, which would be difficult to maneuver into place.  It is thus recommended that for hospital centre interested in building an RSA system in-house to have two C-arms dedicated to administering RSA, permanently positioned within an examination room with proper radiation shielding.    Furthermore, we also recommend that a pre-calibration approach is taken as opposed to self-calibration due to the limited space within the imaging field.   To maintain calibration accuracy with a pre-calibration workflow, it is crucial that the C-arms remain stationary until the patient has been imaged.  Thus, building barriers between the C-arms and the imaging space is also recommended.  With the AB configuration, we were able to achieve sufficient space for a subject to enter the imaging field and then maintain an upright stance within the field.  However, if barriers were to be built to protect the C-arm from being moved post calibration, we recommend that studies maintain an imaging space of at least 18” wide.    29   Chapter 3   System Development  3.1 System Overview In this chapter, we detail the development process for our RSA protocol.  We cover each element of our RSA system, including both the hardware and software components.  Figure 11 provides an overview of the protocol workflow.  In order to measure relative motions, the RSA protocol requires two independent RSA exams administered at different times.  Clinically, this time period could be minutes or years, depending on the study design.  While RSA exams are typically administered at different postoperative followup periods to measure implant migration over time, same-day RSA exams can measure implant loosening by imposing different weight-bearing conditions on the knee between exams. Each RSA exam has an image acquisition component and a software analysis component.  For each of the two X-ray views, image acquisition involves obtaining two radiographs:  one capturing the distortion correction device (as previously mentioned in section 1.6.2) and the other capturing our subject of interest together with the calibration cage (section 1.6.3).  These four radiographs are then imported into an RSA software to correct for image distortion, calibration each X-ray view, and lastly, reconstruct 3D position of implant and bone markers.   30  Once an RSA exam had been administered, we then compare it against a baseline RSA exam in order to calculate relative motion between implant and bone.  This is done through a second software package developed at our lab.  Figure 11: Workflow overview of the RSA protocol.  Each RSA exam involved acquiring four radiographs (step #1) followed by RSA software analysis, which included: correcting distortions on radiographs (#2), calibrating the two X-ray views (#3) and reconstructing the 3D position of implant and bone markers (#4).  Lastly, relative motion was calculated between each pair of RSA exams (#5).  3.2 Proposed RSA Protocol As there are several elements that make up a fully functional RSA system, we describe the general approaches we have chosen to take in this section.  The specific details are covered in chapter 2.  As an overview, our proposed protocol leverages two low-cost resources:  (1) C-arm radiographic equipment, and (2) open-source RSA software.  The costs associated with using these resources were discussed in section 1.5. 31  3.2.1 Image Acquisition For image acquisition, a single C-arm (ARCADIS Orbic 3D, Siemens AG, Erlangen, Germany) that was readily available to us was used.   Since the scope of this thesis is limited to phantom studies, we can exploit the fact that the phantom does not move and so can acquire two images by moving the C-arm between exposures, which allows us to perform an RSA with a single C-arm.  Ultimately, a dual C-arm setup will be needed for in-vivo RSA studies.  We therefore briefly explored the spatial constraint issues associated with dual C-arm RSA systems in the previous chapter (Chapter 2).   3.2.2 Distortion Correction Distortion correction was done based on the protocol described by Brainerd et al. (2010) using an open-source software, XROMM (XrayProject-2.2.5, Brown University, Providence, Rhode Island, USA).  The distortion correction device needed to acquire the corrective radiographs is designed and constructed in-house, modelling after the design described by Brainerd.  3.2.3 System Calibration  For our RSA system, the self-calibration approach was chosen to maximize system accuracies, since simultaneous capture of both cage and subject ensures intrinsic and extrinsic parameters of the radiographic setup remains consistent.  This decision meant we were choosing between either a uniplanar or biplanar cage design.    Due to spatial limitations, we found that a uniplanar design would not be effective to use on a C-arm based RSA system.  However, since a biplanar cage will likely yield better accuracies by using interpolative calibration rather than the extrapolative approach needed with a uniplanar cage, we would prefer to use a biplanar cage in any case. In a clinical situation, we will likely wish to use a pre-calibration approach instead of self-calibration to facilitate ease of patient positioning.  For this purpose, a biplanar cage is also the preferred approach. Although a full cage design may provide better system accuracies when compared to a biplanar cage, it poses a challenge during the marker identification process. With a biplanar cage, marker identification is less prone to human error since only two planes of markers are 32  involved in contrast to the 3-4 overlapping layers of a full cage design.  Furthermore, marker occlusion is also relatively easy to prevent in biplanar cages.     For our protocol, we proposed to design and build the biplanar cage in-house, since the size of most commercially available cages were found to be unsuitable for the limited field of view of C-arms.  In addition, we believe that an in-house constructed calibration cage would be much lower in cost.  To produce a fully defined cage, the precise coordinates of each cage marker was found using a Coordinate Measuring Machine (CMM).  We estimated that it was unlikely that the cost of manufacturing a cage together with CMM verification would exceed $2,000, in contrast to a $15,000 commercial cage.1 3.2.4 RSA Software Several software modules are required to create a fully functional RSA system that can measure TKA implant migration.  At the heart of the system is an open-source software package (XrayProject-2.2.5, XROMM, Brown University, Providence, Rhode Island, USA) which provides distortion correction, calibration and point reconstruction capabilities.     Following point reconstruction, further software developments were required in-house to extend the system to calculate and report implant relative motions.  Both XROMM and the in-house programs were developed in MATLAB (Mathworks, Natick, Massachusetts, USA), thus simplifying the data processing workflows.   When calculating DLT parameters using XROMM, we have opted to use the 11-DLT option.  This is done with the understanding that, once successful, one can easily extend our 11-DLT implementation to include additional distortion parameters (e.g., optical and de-centering distortions) using options available within XROMM.  Table 3 summarizes the various software modules involved in our proposed protocol.                                                         1 P. Grundström (personal communication, February 25 2014) 33  CALCULATION STEP MODULE NAME DEVELOPED BY Distortion Correction Distortion Correction (XrayProject-2.2.5) XROMM, Brown University, Providence, Rhode Island, USA Calibration Calibration  (XrayProject-2.2.5) XROMM, Brown University, Providence, Rhode Island, USA 3D Point Reconstruction Digitization  (XrayProject-2.2.5) XROMM, Brown University, Providence, Rhode Island, USA Relative Motion Calculations Relative Motion Calculator v1.4 In-house  Table 3: Multiple software modules were required in developing our RSA protocol.  This table summarizes the modules involved, most of which were available as open-source software, with the rest developed through in-house efforts.  3.3 Image Acquisition As a first step to the protocol, we must capture radiographic images before RSA software analysis can follow.  To later use the software to reduce distortion artefacts and to calibrate the imaging field, two devices must be radiographed in addition to our subject of interest:  the distortion device and the calibration cage.  Since we can capture the calibration cage together with the subject in one shot, only two radiographs are needed per X-ray view, resulting in four radiographs. We will detail in this section the radiographic equipment we used for image acquisition, including relevant system specification and settings.  We also measured the spatial resolution of the system to compare it against guidelines provided in ISO16087, a published standard on RSA systems.   In addition, we also detail image pre-processing steps that were done before radiographs were imported into the open-source RSA software, XROMM (XrayProject-2.2.5, XROMM, Brown University, Providence, Rhode Island, USA).  These steps included enhancing image contrasts and converting file formats.     34  3.3.1 Radiographic Equipment We employed a single C-arm (Siemens ARCADIS Orbic 3D, Siemens AG, Erlangen, Germany) with all shots acquired in digital radiographic mode.  The C-arm is rotated once per RSA examination to capture a single pair of biplanar radiographs (Figure 12).     Figure 12: The isocentric design of the Siemens Arcadis C-arm allows the calibration cage to remain roughly equidistant to the image intensifier after the C-arm is rotated 90-degrees orbitally from one position (left) to the next (right).     Since distortion correction and calibration needs to be done while maintaining the position of the C-arm for each radiographic view (‘A’ and ‘B’), the sequence of image acquisition is crucial to minimize errors.   We propose the following workflow for a single C-arm RSA: (1) Position the C-arm in its ‘A’ position (2) Attach distortion correction device and capture corrective radiograph for view ‘A’ before removing the device (3) Place calibration cage into view in conjunction with the subject (4) Capture radiograph for view ‘A’ (5) Reposition the C-arm in its ‘B’ position (6) Capture radiograph for view ‘B’ (7) Remove calibration cage and subject (8) Attach distortion correction device and capture corrective radiograph for view ‘B’  With this method, the user ends up with four radiographs that can be later used to carry out RSA calculations.  We illustrate the difference between a dual C-arm image acquisition workflow versus a single C-arm workflow in Figure 13.  35    Figure 13: The image acquisition workflow of a dual C-arm (above) involves (a) obtaining distortion correction images of both X-rays, (b) obtain calibration images, and (c) obtain images of the subject.  In contrast, the use of a single C-arm (below) requires a sequence which ensures the cage and subject are not moved between shots from the two X-ray views, and that a unique distortion image is obtained for each X-ray view.  For all image acquisitions, the exposure levels were set on automatic mode, which ranged between 50-61 kV and 0.3-0.5 mA.  The spatial resolution of our C-arm was measured using two radiopaque patterns taped directly onto the surface of the image intensifier (Figure 14).   The first pattern (Type 41, Nuclear Associates Inc., Carle place, NY) provided spatial resolution in terms of line pairs per mm (lp/mm) and the second pattern (Radiographic Ruler, Supertech Inc., Elkhart, IN) in terms of pixels per mm.      36   Figure 14: Two types of test patterns were used to measure the system’s spatial resolution.  A radiographic ruler (top) provides pixels/mm measurements, while the two square patterns (bottom) provide line-pairs/mm measurements.  By taking radiographs of these test patterns, we determined the spatial resolution of our C -arm to be 1.8 lp/mm and 5 pixels/mm.  Using both patterns, we repeated the measurements four times, each with the C-arm configured in a different orientation.  Since electromagnetic interference (EMI) in the environment could cause distortion effects and therefore affect pixel size throughout the image, there was a possibility that spatial resolution measurements could change depending on the degree of distortion present.  At this stage, we had yet established the distortion correction protocol.  Therefore, by varying C-arm orientations, it enabled us to check that spatial resolution measurements were not being affected by distortions.  If spatial resolution measurements were inconsistent between these radiographs, it would mean distortion correction is necessary before spatial resolution could be measured. The angular position of the C-arm was stepped through the following set values to cover a broad range of angular rotations: 0, 90, 135, and 180 degrees.  The resulting spatial resolutions were 1.8 lp/mm and 5 pixels/mm, which were found by observing the sets of lines on the radiographs (Figure 14) and locating the set with the greatest spatial resolution where one can still visually 37  distinguish each unique line within the set.   Results were consistent for all four radiographic measurements.   According to ISO16087:2013, a minimum spatial resolution of 150 dots/inch (or 5.91 dots/mm) is suggested for RSA systems.  Since at no point within our protocol were the images printed, our image resolution was more appropriately expressed in terms of pixels instead of dots.  Assuming equivalency of these two units of measurement, the spatial resolution of our C-arm was slightly worse than the recommended limits, which was expected since most RSA systems in the literature were developed using traditional X-ray machines of higher resolutions.   This difference in resolution is a primary focus of our study:  whether the limited resolution of C-arms is nonetheless sufficient to construct an RSA system capable of measuring implant movement to the needed accuracy.   Other relevant specifications of the C-arm unit are listed in Table 4 below.  The grayscale resolution meets the ISO recommendation of 8 bits.   SPECIFICATIONS DESCRIPTION Make and Model Siemens ARCADIS Orbic 3D, Siemens AG, Erlangen, Germany Spatial Resolution  1.8 lp/mm or 5 pixels/mm Image Size 1024 x 1024 pixels Gray Scale Resolution 24 bit depth  Biplanar or Uniplanar  Biplanar Digital or Analogue Digital   Table 4: Specifications of the single C-arm used in our RSA system to acquire radiographs.  3.3.2 Image Pre-Processing Before any RSA analysis is done using the XROMM software, we first pre-process the images to: (1) Enhance image contrast 38  (2) Shorten long filenames of image files exported by the C-arm (3) Convert files into .tif format   We have found that by enhancing the image contrast, we can improve the quality of calibration by aiding in centroid identification feature of XROMM.   To do so, we used MATLAB’s ‘imadjust’ function to limit the input radiograph’s range of contrast and apply an increase in gamma, which is defined by MATLAB as the shape of the curve relating the input and output images.  If gamma is set equal to one, no changes are made.  If gamma is >1, the output image becomes brighter than the input.  Inversely, if gamma is <1, the output image becomes visibly darker.  Radiographs of the distortion correction device received a different set of enhancement parameters than those of the cage and phantom because the two inherently have different degrees of attenuation.     In appendix B.1, we provide the MATLAB function we developed to apply the contrast enhancement to all images within a given folder.  It also shortens filenames and converts each .bmp image files into .tif format, the latter of which is the preferred format for XROMM.  3.4 Distortion Correction Since our C-arm uses an image intensifier to produce the digital radiographs, it is prone to pincushion distortions as previously explained in section 1.6.2.  In this section, we detail how the distortion correction protocol works in reducing this effect.    To reduce distortions, we used the distortion correction software module of XROMM (XrayProject-2.2.5, XROMM, Brown University, Providence, Rhode Island, USA), which operates on an image of a perforated metal sheet in which the perforations are spaced apart in a consistent pattern.  From this model, the software calculates a corrective transform to reduce distortion effects in all subsequent images taken by the C-arm.  Since the distortion pattern is a function of the alignment of the image intensifier tube with any external magnetic field (including, and normally principally, the earth's magnetic field), the distortion correction protocol must be repeated each time the C-arm or a new piece of equipment that affects the local magnetic field is moved. 39  3.4.1 Distortion Correction Device (Hardware) To acquire a distortion correction image, a perforated steel sheet of 0.91 mm thickness containing 1.59 mm diameter holes (9255T44, McMaster-Carr Robinson, NJ) was used (Figure 15).   Each row of holes were staggered with the next row, creating a perforation pattern where each hole had a center-to-center spacing of 3.18 mm with each of its six immediate holes.    In order to reduce sagging of the sheet metal when it is mounted to the image intensifier (II), the material was cut into a disc shape and mounted to a rigid carbon fiber plate of 3.18 mm thickness (8181K16, McMaster-Carr Robinson, NJ) using four M4 button-head screws.  Carbon fiber was chosen for its stiffness and radiolucency.   Two thumb-triggered latches allow an operator to easily mount and dismount the assembly to/from the II.  The assembly is further constrained spatially by three indexing pins built-into the II, which nest into mirrored cut-outs in the carbon fibre plate.      Figure 15: The distortion correction device is made of a perforated sheet metal that is mounted onto a piece of carbon fiber.  Pictured here is the device before it is mounted onto the image intensifier using two spring-loaded latches.   40  3.4.2 XROMM Distortion Correction Module (Software) The XROMM Distortion Correction Module (XrayProject-2.2.5, XROMM, Brown University, Providence, Rhode Island, USA) is an open-source software that corrects for image distortion (Figure 16) by computing a bilinear transformation that may be applied to all subsequent radiographs to reduce the effects of distortion (Brainerd 2010).   Figure 16: Before (left) and after (right) applying the distortion correction.   When one views the same images in a reduced size (shown at the bottom of each image), t he pincushion distortion effects become more apparent to the naked eye.      This matrix is calculated using a radiographic image of a perforated steel sheet (Lucas 2010). Under the assumption that each circular thru-hole punched into the steel plate was manufactured with perfect concentricity and dimensional accuracy, the algorithm executes the following: (1) Calculates the centroid of each circle identified in the image (2) Calculates an idealized hole spacing using the average distance from the most center hole in the image to its six neighbouring holes (3) Calculates an idealized row alignment using the average angles from the most center hole in the image to its neighbouring six holes (4) Calculates the desired transformation matrix using the cp2tform function of the Image Processing Toolbox in MATLAB (Mathworks, Natick, Massachusetts, USA) and a local weighted mean (LWM) approach   41  After the transform matrix has been obtained, it is stored as a ‘look up table’ (LUT) file and can be called upon to correct for distortion on all subsequent radiographs taken for that specific C-arm position.    3.4.3 Distortion Correction Verification To quantify the level of distortion remaining post-correction, we picked two perforations at random on a given radiograph and drew a line between the perforation centroids.  It is expected that the midpoint of the line should intersect the centroid of a third perforation, if the distortion correction was done perfectly.  We can thus use the closest distance between this third perforation and the line as a method to quantify distortion.   A MATLAB code was written to first segment the post-corrected images and identify the centroid location of each hole within the perforated sheet.  A total of three images were used for this verification, each with the II positioned at a different orientation to create varying conditions of electromagnetic interference (EMI): (1) C-arm positioned with II closest to the ceiling; X-ray source closest to the floor (2) C-arm positioned with II closest to the floor; X-ray source closest to the ceiling (3) C-arm positioned with the II and X-ray source on the same plane, parallel to the floor    Once centroids are identified on the image, two perforations (in yellow) were manually selected to create a straight line (Figure 17).  A total of four lines were plotted on each image:  one horizontal, one vertical and two diagonal lines.  For each line, the perforation that was closest to the mid-point of the line was then identified manually.   By taking the distance between this third perforation’s centroid and the calculated mid-point of each line, we obtain the residual value.  For images with severe distortion, one would expect the perforations along the line to arc away from it, causing high residual values.  Granted, this only works if there were an odd number of perforations between the two chosen perforations – care was taken to ensure this was the case.  42     Figure 17: To quantify the level of distortion which remains after distortion correction, lines are drawn between the centroids of two perforations chosen at random.  If the image is free of distortion, then the midpoint of this line should intersect perfectly with a centroid of a third perforation.  Since we expect some level of distortion to remain, the distance between the midpoint and this third centroid can be defined as the residual.     For the three sample images, we  calculated the residual values for a total of 12 lines and determined the mean residual value was 0.3 pixels.  We then expressed each residual value in terms of microns (Table 5).     SAMPLES OF LINES RESIDUAL [μm] IMAGE 1 IMAGE 2 IMAGE 3 Horizontal Line Sample 69 68 68 Vertical Line Sample 37 81 72 Diagonal Line Sample #1 42 19 55 Diagonal Line Sample #2 51 81 80  Table 5: The residual values after distortion correction, expressed as the distance between midpoint of line drawn and the centroid of the nearest perforation to the midpoint.  Average residual value was 60.0 μm.    The millimeter-to-pixel conversion ratio required to convert pixels into micro millimetres was accomplished by first identifying the centroid closest to the image center on one of the three images.  The distance between this centroid and its surrounding six centroids were then measured in 43  MATLAB, and determined to be 15.7 pixels on average.  With the knowledge that the perforations had an average center-to-center spacing of 3.1691 mm (measured with a CMM using 56 randomly selected samples; CMM resolution was 0.0001 mm and accuracy was 0.0039 mm), we thus calculated the conversion rate to be 0.2023 mm/pixels.  We assumed that this conversion rate is constant throughout the plane (parallel to the image intensifier).  Using this conversion factor, we calculated the average residual value to be 60 microns.  Although there are no existing guidelines on limits of distortion errors and how errors should be quantified, the average residual we have quantified here is nearly an order of magnitude better than the clinical magnitude of migration we would like to detect (>500 microns – as discussed in section 1.2).  We thus conclude that the distortion correction protocol would likely be sufficient for our intended purposes.    3.5 System Calibration As mentioned in the overview (section 3.1), a calibration cage device is needed in order to obtain the necessary 2D-to-3D transforms for the RSA system. Several considerations were pivotal during the calibration cage design phase.   The principal decision regarding the cage design is the choice of whether to use a multi-layer or surrounding design (i.e., full cage or biplanar cage).  A uniplanar cage (single-paneled design) was immediately eliminated because of its tendency to produce higher errors along the axis perpendicular to the radiographs (Cai 2008; Yuan 2000; Choo 2003).      The primary difference between a full vs. a biplanar cage is in their possible workflows:  a full cage can only be used in a pre-calibration workflow (where the calibration cage and subject are radiographed separately), while a biplanar cage can be used in both a pre-calibration and self-calibration workflow (where the calibration cage and subject are radiographed simultaneously).  For the purpose of streamlining workflow, our protocol calls for simultaneous imaging of both the cage and the subject of interest.  While this protocol is effective when our subject of interest is a phantom or cadaveric specimen, we recommend calibration images to be obtained separately in clinical practice since wrapping the cage around a patient’s knee in a dual C-arm setup could prove to be a challenge spatially.  Ideally, the C-arms would be fixed in space and could be pre-calibrated before the patient’s arrival.   44  To keep our workflow options open, we chose a biplanar cage design, thus allowing the cage to be radiographed on its own, or simultaneously radiographed with the subject of interest.  In this section, we will detail the cage design process (section 3.5.1) followed by a description of the software used for RSA calibration (section 3.5.2). 3.5.1 Calibration Cage Device (Hardware) As shown in Appendix C.1, a minimum of six calibration markers need to be captured by each radiograph in order to solve for the eleven DLT parameters.  Additional markers will improve the quality of calibration.  For each marker, we also need to know the exact 3D coordinates as identified using CMM (section 3.2.3).  Thus, in designing the biplanar calibration cage, we set out to meet the following two criteria: (1) Be able to see the majority of cage markers on each calibration radiograph, without markers overlapping one another (2) Be able to match each calibration marker seen on radiographs with their corresponding 3D coordinates identified during the cage design phase   To effectively meet these two criteria, we need to first quantify the field of view of the C-arm to determine the appropriate placement area, spacing and pattern of calibration markers.  We will detail in this section, the design process involved in setting cage specifications. Investigating Field of View In order to determine the common volume that could be captured by two C-arms in two images when the C-arm gantry is moved through an arc of 90°, we modeled the C-arm’s field of view. Since the X-rays produced by a C-arm are substantially emitted from a point source and travel in a straight line to the image intensifier, the volume they pass through can be represented as a cone.  Figure 18 shows a representation of the resulting conical fields of view, in conjunction with two C-arms in a biplanar configuration.  In order to design a calibration cage of suitable size with appropriate marker spacing, we needed to approximate the dimensions of this cone.    Since we could not directly measure the effective source position relative to the position of the imaging plane, we devised an alternative method to determine the imaging volume.  A piece of sheet metal with a single 0.9 mm diameter hole was placed parallel to the image intensifier at various 45  distances from it and radiographed in order to quantify the scale (pixels per mm) at different distances.  The change in scale provided sufficient information to determine the conical volume.  A total of thirteen radiographs were obtained, with distances ranging from 15 mm to 679 mm away from the II. The scale (expressed in pixel/mm) was calculated for each of the thirteen radiographs by manually counting the number of pixels representing the diameter of the hole.  Since we know each radiograph had an approximate diameter of 900 pixels, we can determine the diameter of each circular ‘slice’ of the conical imaging volume in millimeters.  These thirteen circles were then drafted in SolidWorks to model a cone, superimposed onto a CAD model of the C-arm (Figure 18).      Figure 18: Through empirical experimentation, we determined the dimensions of the conical imaging volume of our C-arm.  Show here is the CAD model of a biplanar C-arm configuration along with the corresponding imaging volume.  This approach allowed us to determine appropriate marker placements for each panel of the calibration cage (also shown).   Once the conical imaging volume was modelled to be approximately 171 mm wide in diameter at its base with a focal length (i.e., height of the cone) of 977 mm, we had sufficient information to determine the appropriate coverage area for each panel of markers – which varies according to the size of the cage and position of the panel relative to the X-ray source.  If markers are spread out across an excessively large area, matching markers seen on radiographs with their corresponding 3D coordinates becomes difficult.  On the other hand, if markers cover only a small area on the radiograph, we then limit the quality of calibration.    46  Marker Placement As previously mentioned in section 1.6.3, we refer to the panel closest to the X-ray source as the 'control plane', while the one closest to the image intensifier is the 'fiducial plane’.  Once we had approximated the shape and size of the C-arm’s field of view, several two-panel test cages were prototyped and radiographed to check for marker occlusion between the fiducial and control planes.   It was found that if we used 2 mm diameter beads placed on two planes that were 295 mm apart, then marker spacing of 21 mm x 17 mm and 32 mm x 27 mm were well suited for the control and fiducial planes, respectively.  Through visual inspections, we found that these specifications placed sufficient distances between markers which reduced the chance of marker occlusions between calibration markers, anatomical features, and markers on the implant and bone.  The spacing also distributed all 50 calibration markers evenly while covering most of the field of view.   It was also found from the test shots (Figure 19) that any bracing structure should be placed about 20 mm away from the nearest row or column of markers on the control plane, in order to mitigate marker occlusion by mounting screws or the acrylic material itself.     Lastly, we found that while having sufficient markers captured ensures quality of calibration, having an excessive number of markers raised the potential for human error.  During calibration, it was crucial that one can match the 3D coordinates of a given marker with the correct marker seen on each radiographs.  The greater the number of markers that fell beyond the X-ray’s field of view, the more difficult and error-prone this marker identification task became.  We therefore decided to keep the number of markers to a minimum.  In the final biplanar cage design, a total of 50 markers were used.  This consisted of two control panels each with a 4x4 marker pattern and two fiducial panels of 3x3 markers.  In comparison to a commercially available cage, Tilly Medical’s Cage #10 (Tilly Medical Products, Lund, Sweden) is purposed for RSA imaging on knees and uses a comparable pattern of 3x3 for all four panels.  We note that a standard commercial cage can include as few as 8 markers per panel, with some studies using as much as 64 markers per panel (Cai 2008). 47        Figure 19: Several cages were prototyped for the purpose of testing marker placement.  With an early stage two-panel cage prototype (top left) and its resulting radiograph (top right), we found that support structures (shown by arrows) can potentially occlude markers.  We also discovered that having an excessive number of markers extending beyond the C-arm's field of view made it difficult to match each marker with its corresponding 3D coordinates.   The bottom two figures show a later prototype which addressed these two issues by reducing the number of markers and placing sufficient distances between markers and support structure.    Material Selection Another consideration in designing our cage was choosing the panel materials that would make up the cage.   Specifically, we required panels with sufficient thickness for structural rigidity, radio-translucency for imaging, transparency for aligning the subject within the cage using laser guidance, and lastly, very low coefficient of thermal expansion.   Thermal expansion was an important consideration in material selection since temperature fluctuations in a room can potentially expand the cage and change the relative positions of the calibration markers.   For example, if we construct one of our 314 mm long cage panels out of an 48  acetal material which has a high coefficient of thermal expansion (21.6 x 10-5 mm/mm/°C), then the panel will expand by 67.8 μm for each degree Celsius of temperature increase.  Since we are aiming to make measurements accurate to a few tens of microns, this degree of temperature sensitivity would be unacceptable.   Thus, we have opted to use 9.53 mm thick panels of UV-resistant polycarbonate (8707K144, McMaster-Carr, Robinson, NJ) with a maximum coefficient of thermal expansion of 6.8 x 10-5 mm/mm/°C, which will reduce the panel's expansion to 21 μm per °C.  While still not a trivial level of expansion, it will likely be acceptable for our application.  If we needed to reduce the thermal expansion further, we could consider glass-filled variants of this polycarbonate, which can achieve coefficients as low as 2.7 x 10-5 mm/mm/°C;  however, they are difficult to machine and we therefore opted not to use this material unless it later proved necessary.   Manufacturing and Verification As fiducials for the calibration cage, we used 50 stainless steel beads of 2 mm diameter (1598K18, McMaster-Carr, Robinson, NJ) mounted onto four polycarbonate panels using liquid epoxy.  All machining on the cage parts was done using a Computer Numerically Controlled (CNC) machine.   To precisely attach each bead during manufacturing, a shallow tunnel 1.83 mm wide was pre-drilled at each location on the panel through CNC machining (Figure 20).  Epoxy was then injected into the tunnel using a needled syringe before the bead was placed.  Care was taken to ensure no adhesive residue coated the surface of the beads during placement.  With this manufacturing technique, approximately two-thirds of each marker remained exposed to air, enabling access by CMM tool tips.  49   Figure 20: Shown here is the CAD model of a cage panel, with pre-drilled tunnels at each location where markers are placed.  The tunnels are narrower than the 2 mm markers, thus allowing portions of each marker surface to remain exposed.  This exposure facilitates CMM verification at later stages.     After the cage was constructed, the coordinates of each bead were determined using a Coordinate Measuring Machine (CMM).  The CMM system used (Crysta-Apex C 7106 fitted with TP200-PM10MQ probe, Mitutoyo Corporation, Japan) is capable of 0.0001 mm resolution and an accuracy expressed as a maximum permissible error (MPE) of 1.9 + 3*L/1000 μm, where L is the measurement length in millimetres.  For our cage, the maximum distance between any two beads was 305.5953 mm, so the CMM accuracy is estimated to be 3.9 μm.  3.5.2  XROMM Calibration Module (Software) Once radiographs have been obtained, we used the XROMM Calibration Module (XrayProject-2.2.5, XROMM, Brown University, Providence, Rhode Island, USA) to calibrate each X-ray view.   We do so by first importing the ‘frame specs file’ – a .csv file containing the 3D coordinates of all 50 beads on the calibration cage.     The user then identifies each corresponding calibration marker on the radiograph with the assistance of a ‘find marker centroid’ feature, which uses an intensity-based algorithm to snap the user’s mouse click onto the centroid of the nearest marker.  Depending on the quality of the images 50  and size of the markers, this feature may not always be effective in identifying marker centroids.  In this case, the user would need to intervene and identify the correct location of each centroid.     For our protocol, we use the following guidelines: (1) Identify at least 10 markers in total, preferably five from the fiducial plane and five from the control plane (2) Avoid using any markers that overlap with metal or bone artefacts, or any that are bordering the edge of the radiograph such that the marker is partially occluded. (3) After identifying markers, click ‘Computer Coefficient’ and ensure the displayed average residual value is less than 0.4 pixels.  With our calibration cage, a residual value of 0.1 to 0.3 pixels is expected (Figure 21).  (4) Click ‘Error Analysis’ and check the residual plot.  Make sure that residual value is roughly the same for all markers.  Any obvious outliers will indicate potential identification errors (e.g., if a marker was mismatched with its corresponding 3D coordinates).  If there are obvious outliers, check that the correct marker was identified before proceeding.            Once this is done for both radiographs (AP and ML views), the user should have two DLT files, each containing eleven parameters.  The two files can then be merged using XROMM as a single 11x2 matrix, which is later used for point reconstruction.    Figure 21: A screenshot of the XROMM Calibration Module.  After importing the 3D coordinates of all 50 calibration markers, the user identifies each marker's location on the radiograph.  Residual values (in pixels) are then calculated and plotted to provide information on the quality of calibration.  Typically, an average residual of 0.1 to 0.3 pixels is expected (y-axis of plot on right).     51  3.6 Point Reconstruction  Once DLT parameters have been calculated for both radiographic views, the XROMM Digitization Module (XrayProject-2.2.5, XROMM, Brown University, Providence, Rhode Island, USA) is used to reconstruct the location of each marker embedded on the subject.     To do so, the user identifies the location of each bone and implant marker on both radiographs.  Once the user has identified a given marker’s location on one of the radiographs, XROMM assists in identifying the corresponding marker on the second radiograph by displaying a guiding line upon which it expects the markers to lie (Figure 22).   The DLT residual value is calculated for each individual marker.  For our protocol, we recommend a cut-off limit of no more than 0.5 pixels. It should be noted that residual values ranging from 0.03 to 0.40 pixels are often achievable with our RSA system.  An exception to the cut-off limit can be made if the user can only find 3 markers to represent a given rigid body.  In this case, all markers must be used regardless of residual values.  The effect of using markers with high residual values will be reflected in later steps of the protocol, and care must be taken to ensure results are rejected if rigid body errors exceed recommended limits for RSA systems.    Figure 22: Screenshot of the XROMM Digitization Module.  Here,  an implant marker was identified by the user on the ML radiograph (on left).  XROMM then guides the user in identifying the same marker’s location on the AP radiograph (right) by providing a line upon which it expects the marker to lie.   52  3.7 Relative Motion Calculations 3.7.1 Conventions, Definitions and Guidelines For TKA implants, migration is a measure of the movement of each implant component (tibial or femoral) relative to its corresponding bone segment over time.  In other words, it is defined as the measurable relative motion between implant and bone over two RSA exams (referred to as time T1 and time T2).    Following guidelines developed by Valstar (2005) and standards set out in ISO 16087:2013, relative motion of an implant should be reported either in terms of Maximum Total Point Motion (MTPM) or as six components of motion.  These components of motion can be represented as three Euler angles of rotation (Ψ, φ, θ) and three components of translation (dx, dy, dz).  All reported relative motions are normally reported in the implant coordinate system – a three-dimensional Cartesian coordinate system defined by four 1mm markers embedded within the tibial implant.  In addition, each pair of T1-T2 RSA exams is also accompanied by two metrics which indicates the quality of the two RSA exams.  For each rigid body, the condition number (CN) and the rigid body errors (RBE) between the two exams are reported.  The definition of CN and RBE and the cut-off thresholds for both are detailed later in this chapter. Cage Coordinate System Since DLT parameters for both radiographs within the biplanar setup are derived using the calibration cage, any coordinates provided by point reconstructions following calibration steps are also expressed in terms of the cage coordinate system. The origin of the cage coordinate system was defined to be coincident with the first marker on the cage, and the orientation of each axis was set during the CMM-verification of the cage, which was nominally aligned with the four panels on the cage. Figure 23 below shows the position and orientation of this coordinate system relative to the physical cage.   53      Figure 23: Photo of the CMM-verified calibration cage (left), showing the locations of the origin and the orthogonal axes of the cage coordinate system.  Also shown is a visualization of cage marker locations with respect to the cage coordinate system (right).  Markers 1 to 25 are used to calibrate the first X-ray view, markers 26 to 50 for the second view.   Implant Coordinate System Ultimately, RSA results should be reported in terms of an implant coordinate system in order to provide clinical context.  To do so, we first define the implant coordinate system using four markers embedded in the tibial implant (Figure 24).  The first three markers are embedded within the polyethylene liner (rigidly attached to the tibial component), and the last marker is embedded within a polyethylene plug that is inserted distally into the metal tibial prosthesis.  It should be noted that this plug is part of the Zimmer Nexgen PS TKA implant design, and is not a feature that exists in all TKA implants on the market.  The purpose of the stem plug is to allow both the primary and revision versions of the tibial implant to share the same body design to ease manufacturing processes.  For revision surgeries, the plug is removed and replaced by an intramedullary rod which provides additional structural support.  Together, the four markers form a tetrahedron shaped rigid body.  54   Figure 24: A total of four markers were embedded within the polyethylene portions of the tibial component (left).  Three markers were implanted within the liner atop the tibial tray, and a fourth was inserted into the tibial plug.  Using these four markers, the implant coordinate system (right) is defined.  First, the y-axis is defined as the vector drawn from the rigid body centroid (‘Pc’) location to its projection (‘N’) on the plane defined by the first three markers. Then, the z-axis is defined as the vector from ‘N’ to the first marker (‘P1’).  Lastly, the x-axis is calculated as the cross product from y-axis to z-axis.  This tetrahedron, however, is not perfectly symmetrical.  To avoid marker occlusion when the implant is radiographed directly in ML view, marker P2 was strategically implanted more anteriorly than marker P3 (Figure 24).   Consequently, we cannot use the line from P2 to P3 to define the ML-axis (i.e., X-axis) orientation of the implant.  Instead, we take an approach of using a combination of the rigid body centroid and the location of P1, which was placed to roughly align with the AP-plane of the implant.  If we had used the P2 to P3 line to define the orientation of the x-axis, we would end up with an implant coordinate frame that is either overly rotated externally or internally when compared with the implant’s inherent symmetry.  The image below illustrates the detailed steps of how the implant coordinate system was defined. To define the coordinate frame of reference, the three markers on the polyethylene liner (P1, P2, and P3) are first used to define a plane.  We then take the cross product of the two vectors defined by P1, P2, and P3 to obtain the normal vector of the plane:  55  Since P1 is on the plane, the equation of the plane defined by the three points is thus:  or  where    The centroid of the rigid body, Pc(u,v,w), is defined by the four markers and calculated simply as:   To find the point closest to Pc that also lies on the plane, N(x0,y0,z0) we first define the parametric equation for the projection line from Pc to the plane:    We then solve for t = t0, which occurs when the point on this line coincides with the plane:   Thus, plugging in the centroid coordinates and the constants from our plane equation, we can obtain the projection point, N(x0,y0,z0).  Using P1, Pc, and N, we derive the following unit vectors which define the orthogonal axes for the implant-based coordinate system: 56      [1] Once the two coordinate systems have been established, relative motion of the implant (specifically, the marker cluster’s centroid) is then calculated and reported in terms of the implant coordinate system’s six degrees of freedom which consist of rotations and translations about/along the three orthogonal axes:    Anterior-posterior (AP) or x-axis  Proximal-distal (PD) or y-axis  Medial-lateral (ML) or z-axis    During phantom verifications, all radiographs were taken with the AP-axis (or x-axis) of the implant coordinate system (ICS) roughly aligned with the y-axis of the cage, while the ML-axis (or x-axis) of the ICS was aligned with the x-axis of the cage coordinate system.   These alignments were done for the sake of consistency, which ensures the AP view of the implant is always calibrated with the first 25 beads of the calibration cage and the ML view of the implant is calibrated with beads #26 to 50.    Translations and Rotations As previously stated, the primary way of expressing relative motion is in terms of translations and rotations in the implant coordinate system.  These six terms of motion are summarized in Table 6 for a left tibial implant.       57  TYPE NOMENCLATURE AXIS OF  ROTATION/ TRANSLATION POSITIVE DIRECTION DENOTES ROTATIONS [DEGREES] ψ X-axis (or ML-axis) Anterior tilt φ Y-axis (or PD-axis) External rotation θ Z-axis (or AP-axis) Valgus rotation TRANSLATIONS [MM] dx X-axis (or ML-axis) Lateral shift dy Y-axis (or PD-axis) Proximal shift dz Z-axis (or AP-axis) Anterior shift  Table 6: Definitions of each of the six components of motion in the implant coordinate system.  Rotations are reported in Euler sequence of XYZ.   Maximum Total Point Motion A second frequently-used way of reporting implant relative motion is as the Maximum Total Point Motion (MTPM), which looks at the marker within a rigid body that experienced the greatest motion between the two RSA exams.  The magnitude of the translation vector of this marker is reported as the MTPM.   While MTPM offers the convenience of interpreting relative motion as a single value rather than six components of motion, it has limitations which stem from its dependence on the markers used.  Thus, fictive markers (also referred to as ‘virtual markers’) are often used in place of physically implanted markers when calculating MTPM (Valstar 2005).  This technique can be employed in different ways to address marker-dependency issues such as:  Inability to cross-compare study subjects with varying implant marker locations  Sensitivity to marker loosening  Sensitivity to marker occlusion       In this paper, we employed the fictive marker approach for the purpose of addressing marker loosening.  As aforementioned, MTPM is defined as the magnitude of displacement of the marker which moved the most between the baseline and followup RSA exams.  Thus, if MTPM was calculated using each marker’s physical location as detected through RSA point reconstructions, a 58  single loosened marker will likely present itself as the marker which experienced the great displacement and thus overestimates MTPM (Figure 25).    Figure 25: Reporting MTPM using physical marker locations is susceptible to the effect of marker loosening.  For example, marker #3 on the implant pictured here has loosened between the two RSA exams.  Thus, its point motion would have been erroneously reported as the MTPM.  It would be erroneous to report this motion as the MTPM since it does not describe the relative motion of the implant, rather, the movement of one marker which loosened.  Thus, by first calculating the rotation and translation transforms of the rigid body, then applying those transforms to marker coordinates identified during the baseline RSA exam, we reduce the effect that a single loosened marker would have on MTPM measures – this is referred to as the ‘fictive marker’ approach (Figure 26) and has been described by Valstar (2005).   59   Figure 26: The fictive marker approach to reporting MTPM works to reduce the effect markers loosening has on MTPM by first finding the rigid body transforms between the two RSA exams.  Note that in the example here, marker #3 was loosened at followup, but its effect was minimized during MTPM calculation.    In addition, we pre-emptively tackled marker occlusion issues with strategic marker placements.  Also, our verification studies did not involve cross-comparing multiple study subjects.  We thus did not need to address the other two issues associated with using MTPM to report relative motions. 60  The use of fictive marker has been presented in the past.  For example, Nilsson and Kärrholm (1993) used six fictive markers at standardized locations to compute MTPM in a way that allowed for comparison across multiple patients.  Similarly, Regnér et al. (2000) used five fictive points to standardize RSA results in their study.  These studies chose fictive point based on landmark locations (e.g., the anterior midpoint on the edge of the tibial polyethylene liner).  Our approach, however, differs in that we have used the physical locations of markers (instead of landmarks) at baseline as the fictive marker locations for the followup exam.  We have chosen this approach since defining fictive markers using landmarks would require the additional process of segmenting the liner’s edge and identifying landmark locations, which would introduce additional sources of error.      3.7.2 Software Development To calculate implant relative motions, the Relative Motion Calculator program was coded in MATLAB (Mathworks, Natick, Massachusetts, USA).   In this section, we will detail how this program was developed by breaking down the calculations as a four step process. The overall architecture of the Relative Motion Calculator is shown in Figure 27.  In this section, we detail the algorithm behind each of these calculation steps: (1) Import Marker Coordinates (2) Transform to Implant Coordinate System (3) Calculate Translations and Rotations  (4) Calculate MTPM   61   Figure 27: The overall architecture of our custom Relative Motion Calculator program.  It uses the coordinate files generated by XROMM’s Digitization Module as the input and then calculated relative motion of the implant in terms of the six components of motion and as the Maximum Total Point Motion (MTPM).  The w hole process is comprised of the four major calculation steps shown.    STEP 1: Import Marker Coordinates The Relative Motion Calculator requires 3D coordinates of each marker on the implant and bone, as generated by XROMM’s point reconstruction software.  Two files are required:  the T1 marker clusters coordinate file and the T2 marker clusters coordinate file.    Within each file, the first four markers listed are automatically assumed by the software to be those embedded in the tibial implant.  The remaining markers are assumed to be those within the bone.  The file loader then extracts the following four marker clusters from these two files and plots each as a rigid body (Figure 28):  62   Xi = n x 3 matrix of bone marker coordinates at T1  Ui = m x 3 matrix of implant marker coordinates at T1  Yi = n x 3 matrix of bone marker coordinates at T2  Vi = m x 3 matrix of implant marker coordinates at T2     Since our definition of the implant coordinate system relies on having coordinates of all four markers within the implant, n must equal 4.  The number of bone markers, however, only needs to meet the minimal requirement for defining a rigid body (i.e., m ≥ 3).    Figure 28: Screenshot of the Relative Motion Calculator showing T1 (left) and T2 (right) marker clusters   STEP 2: Transform to Implant Coordinate System Once the axes of the implant coordinate system have been defined with respect to the cage coordinate system, each T1 marker coordinate set (Xi, Ui) are immediately transformed from the cage coordinate system into the implant coordinate system. We first derive the necessary rotation transform matrix from one system to the other: 63   From section 3.7.1, equation [1]:   Thus, the cage to implant rotation matrix, RCI, becomes:  Next, the translation transform vector from cage to implant coordinate system, TCI, is simply the negative of the T1 implant marker cluster’s centroid location, which is also the implant coordinate system’s origin.   For each T1 marker coordinates, Xi and Ui, we then update the frame of reference from the cage coordinate system into the implant coordinate system by applying the rotation and translation transforms (which can be combined as a homogenous transform).  This results in the updated coordinate locations, Xi’ and Ui’.    64    For the sake of simplicity, we update our definition of Xi and Ui  to use the implant coordinate system instead:       Note that T2 coordinates are not transformed from implant to cage coordinate system, since the coordinates from T2 will be directly registered to those at T1 by matching the bone segment coordinate frames.   STEP 3: Calculate Translations and Rotations To facilitate interpretation of relative motions, we nominally align the implant coordinate frame with key anatomical directions by setting the x-axis to align with the medial-lateral axis, the y-axis with the superior-inferior axis and the z-axis with the anterior-posterior axis.      For both the left and right-hand side of the anatomy, Valstar (2005) recommended that the positive directions of these axes be set to point medially, superiorly and anteriorly.  This convention means that the reference frame on the left side of the body is a left-handed coordinate system. To calculate the six terms of relative motion, we used the Singular Value Decomposition (SVD) method described by Söderkvist & Wedin (1993).  Prior to Söderkvist’s work, Spoor and Veldpaus (1980) first described a method involving the use of eigenvalue decompositions.  According to Söderkvist, the SVD approach involved formulas that were much simpler and thus easier to implement than those of eigenvalue decompositions.  Although ISO guidelines do not offer suggestions as to how rigid body motions should be calculated, SVD has been advocated by Valstar (2002) as the most elegant method to solving rigid body motions.  Brainerd (2010) has also opted for this approach.  We thus are confident in adopting Söderkvist’s SVD method, which we will detail in this section.  We start out by recalling the four matrices from Step 1, which contains the 3D coordinates of the two marker clusters (bone and implant) at time T1 (Xi and Ui) and time T2 (Yi and Vi). We can calculate the component’s relative motion with the following steps (Söderkvist 1993), which we also illustrate in Figure 29: 65  I. Find the rotation matrix, Rref and translation vector, dref which describe movement of the reference segment (bone) from T1 to T2.   II. Use Rref and dref to calculate new coordinates of the moving segment (implant), Vi’ which provides the implant position at T2 as if the reference segment did not move between T1 and T2.   III. Obtain the relative motion rotation matrix, Rmotion and translation vector, dmotion using Ui as the T1 coordinates of the implant and Vi’ as its T2 coordinates.   Figure 29: Our relative motion calculations use a Singular Value Decomposition approach.  Here, the implant is represented by a triangle, and the bone is shown as a square.   We use the bone as the reference segment in order to solve the relative motion of the implant segment between the baseline and followup RSA exams.   The dashed lines indicate the position of implant and bone at T2, after transforming each T2 marker using Rref and dref such that the bone marker clusters at T2 are matched to those from T1 using a least squares approach.  The detailed algorithm to these three steps are described here.    Step I:  We first find the rotation and translation transforms (Rref, dref) of our reference segment (the bone) between time T1 to T2 (Xi, Yi), in order to later match their positions.  We can describe the relationship between Xi and Yi as:     66  We are then faced with the least-squared problem of solving for Rref and dref while minimizing the error between the measured and predicted locations of the bone markers:  Arun (1987) and Hanson (1981) have shown that the above can be solved using the SVD of a matrix C = BTA, where  A and B are column vectors containing the distances between individual markers in each cluster (Xi , Yi) and the centroid of that cluster ( .         We then compute the Singular Value Decomposition of C using MATLAB’s SVD command, resulting in 3x3 matrices P, Q and , where:   According to derivations by Hanson (1981), we can use matrices P and Q to compute Rref by using the following equation:  Once Rref is found, dref can be computed:   Step II: Since we now know how the reference segment (bone) has moved between T2 and T1, we can apply those same transforms to the moving segment (implant) at time T2 (Vi) to simulate the condition where the reference segment did not move at all between T2 and T1.   67     Step III: Now the reference segments are matched between T1 and T2, we can then solve for relative motion of the implant in terms of rotation (Rmotion) and translation (dmotion) by repeating the SVD approach.    where  The translation vector, dmotion, can be decomposed into dx, dy, and dz, providing the magnitudes of relative translation along each of the three axes.  We can also further decompose the rotation matrix, Rmotion, into Euler angles.  Using an XYZ sequence (arbitrary choice, since all angles are expected to be small):   Thus the three Euler angles of relative motion are as follows:        We then have our six terms of relative motion:  ψ, φ, θ, dx dy, dz   STEP 4: Calculating MTPM  After we have obtained the transforms (rotation matrix and the translation vector) which describes relative motion of the implant from T1 to time T2 (Rref and dref, respectively), we then generate 68  coordinates of each fictive implant marker at time T2 by applying these transforms to the T1 marker cluster.  The rationale behind this specific approach to the use of fictive markers is described in the previous section (3.7.1).   We thus have: Ui = m x 3 matrix of physical implant markers coordinates at T1   Vi’’ = m x 3 matrix of fictive implant markers coordinates at T2        The Relative Motion Calculator can then plot the T2 and T1 implant rigid bodies (Vi’’ and Ui) in a superimposed manner to show relative motion (Figure 30).  The MTPM is also calculated by: (1) Finding the absolute distance moved for each of the m markers from T1 to T2 (i.e., between Ui and Vi’’ for i=1,2,3,…n) (2) Finding the maximum of these m number of absolute distances   Figure 30: Screenshot of the Relative Motion Calculator, reporting relative motion results of an implant that was micromanipulated 3mm proximally.  The six components of motion along with the MTPM are provided.    69  3.7.3 Reporting RSA Quality Metrics The ability of an RSA system to accurately measure relative motion is heavily influenced by the placement of the markers and the stability of the markers that define each rigid body.  If a cluster of markers are not well distributed within a rigid body (e.g., if some are placed collinearly), the relative motion measured may become unreliable due to insufficient 3D information provided by the markers.  We can quantify the quality of maker distribution by using condition numbers (CN) – a variable that essentially measures the extent to which markers spread.  The larger the CN, the more the cluster forms a straight line (Söderkvist 1993).   Furthermore, rigid body error (RBE) can also speak to the quality of relative motion measurements.  RBE occurs when a cluster of markers defining a rigid body shows differences in terms of the distances between markers from T1 to T2.  Conceptually, the distance between any two markers on a rigid body should show zero displacement over time.  However, if a marker were to become loose or if there were point-reconstruction errors within the RSA process, then small displacements can be seen.  Since some reconstruction errors are always expected, RBE is rarely zero.       When reporting RSA results, it has been recommended by Valstar (2005) to always provide a condition number and rigid body error to each marker cluster.   The guideline also suggests that CN should be kept under 150 to indicate sufficiently well-distributed markers, and RBE be kept under 0.35 mm to ensure T1 and T2 marker clusters are well matched.   It should be noted that these limits of acceptance are not numerically derived, but are rather based upon past researchers’ experience.  For example, Nilsson & Kärrholm (1993) used 99 as the maximum limit for acceptable CN and 0.370 mm as the limit for RBE.   Condition Number  For a given rigid body defined by a marker cluster of n markers with 3D coordinates, Xi:   70    The SVD solution to A includes a 3x3 diagonal matrix, , which contains the three singular values of  A:  where σ1 ≥ σ2  ≥ σ3   ≥ 0 Geometrically, if one were to draw a line of best fit through the 3D marker cluster Xi, then σ22 + σ32  describes the squared sum of the distance of each marker to this line in millimetres (Söderkvist 1993).  The smaller this value, the closer the markers are to becoming collinear (indicating poor marker distribution). Condition Number (CN) is a metric that is typically defined as the reciprocal of the square root of σ22 + σ32, such that the larger the CN value, the poorer the distribution.  In ISO16087 standard, the CN is defined with a unit of metres-1 by multiplying the CN by 1000:  According to ISO16087, CN should preferably be below 150 in general, and 120 for knee, hip, and shoulder implants. Rigid Body Error RBE is defined as the mean difference between the relative distances of markers in a rigid body between exams (Valstar 2005).  More specifically, we first apply the calculated transforms of relative motion to the bone (Xi) and implant (Ui) marker clusters from the baseline RSA:  71    The resulting marker clusters Xi’ and Ui’ should match closely with clusters coordinates from the followup RSA exam (i.e., Xi’ ≈ Yi and Ui’ ≈ Vi’).  We can thus quantify the mean rigid body error for each segment as follows (ISO16087):       Valstar (2005) has suggested that RBE should not exceed 350 μm.  As examples, Solomon (2010) reported RBE of under 100 μm and Kedgley (2010) reported RBE average of 118 and 206 μm.   3.8 Cost of System Using an existing C-arm available to us at the Centre for Hip Health and Mobility (CHHM) on the Vancouver General Hospital campus, our single C-arm RSA system was constructed for under $1,500 USD.  This valuation includes the cost of constructing the calibration cage and distortion correction device.  In addition, we also did not incur software licensing fees by basing our calculation pipeline on open-source RSA software (XROMM) together with in-house developed relative motion calculation software.      72   Chapter 4   System Verification 4.1 Introduction In this chapter, we present verification results for our Roentgen Stereophotogrammetric Analysis (RSA) system.  The goal of this verification process was to estimate the accuracy, precision and inter-rater reliability of this system using a realistic phantom model.   To verify our system’s accuracy and precision, we follow the general approach of Valstar (2005) and the ISO16087:2013 guidelines.  We use zero-motion repeated measurements to quantify precision and a series of phantom studies in which accurately-known displacements are applied to quantify accuracy.   Since some aspects of the RSA protocol depend on human intervention, it is necessary to assess intra- and inter-rater reliability, which we do through asking multiple analysts to repeat measurements on the same images.   To enable us to produce accurately-known relative displacements between phantom components, we built a precision micromanipulator platform to which the tracked components could be attached.  This approach has been well described and accepted in the literature (Laende 2009; Kedgley 2009; Onsten 2001).  We note that the verification studies reported here on phantoms cannot and do not replace the repeated examinations that are recommended to estimate precision for clinical systems (Valstar 73  2005), as clinical evaluations are required to ensure that the system is precise in the presence of conditions that only exist in the clinical setting.  If our system ends up being used clinically, it will be necessary to conduct repeated examinations at that time.   4.2 Materials To verify the RSA system we have developed in Chapter 3, we needed to replicate clinical scenarios where a tibial TKA component experiences submillimetre-scale motion relative to the bone segment in which it is implanted.  In these scenarios, the surgeon would typically have already implanted tantalum markers into both the implant and bone during surgery.  To simulate these measurement conditions, we constructed a phantom using a tibial TKA component and an artificial tibial bone replica (Figure 31).  The two parts are then moved relative to one other using translation and rotation micromanipulators, thus allowing us to induce migration in a controlled and precise manner.  The resulting micromanipulator-based phantom therefore serves as our ‘gold standard’ for the true motion between the components.       Additional calculations were required beyond the protocol described in Chapter 3 in order to combine components of motion into single magnitudes (one for rotation, one for translation)1.  By doing so, it enabled us to compare relative motions detected by our system against the gold standard values.  The details of these additional calculations are described later in the Methods section (4.3.1).  To house the phantom, a transparent box was constructed using 3/8”-thick polycarbonate (8574K56, McMaster-Carr Robinson, NJ) rigidly assembled with epoxy.  The artificial bone (Sawbones #1117-42 Tibia, Pacific Research Laboratories, Inc., Vashon Island, WA) was mounted to the base inside the box, while the implant was mounted to the roof of the box through one of the two micromanipulators (one for rotation, the other for translation).  This effectively suspended the implant above the bone and allowed controlled application of translational and rotational migration                                                    1 The three components of translation (along x, y, z axes) are combined into a single magnitude using the 3D Pythagorean Theorem, while the three components of rotation are combined by converting Euler angles into a quaternion value.  Since we cannot perfectly align the axis of motion of the micromanipulator with the implant coordinate system, we must assume that any RSA-detected motion (regardless of direction) reflects micromanipulator motion.  Thus, combined motions are used in place of motion along a specific axis. 74  of the implant relative to a stationary bone.     Figure 31: The phantom box (right) fitted with the translational micromanipulator and bone ‘A’.  Bone ‘B’ (middle) had some bone removed, primarily on the posterior aspects, to accommodate extreme rotations (>5°) of the implant during ML and AP rotational accuracy tests.  A rotational micromanipulator (left) is also used to induce rotational motions. To generate relative motions along all six degrees of freedom (6-DOF), two micromanipulators were used1: (1) A 2-axis translational micromanipulator  (M4022M, Parker Daedal, Irwin, PA) capable of 13 mm of travel in 10 μm increments, accurate to 1.39 μm (2) A 1-axis rotational micromanipulator (M-UTR90, Newport, Irvine, CA) capable of 4° of travel in 1 arcminute (0.0166°) increments, accurate to 4 arcseconds                                                     1 According to the manufacturer, the rotational micromanipulator has a sensitivity of 4 arcseconds, which is defined as the travel associated with 1° turn of the manual control dial.  The manufacturer assumes that most users have sufficient dexterity to make 1° adjustments with the dial.  In our study, we quote this sensitivity value provided by the manufacturer as the accuracy of the micromanipulator.  For the sake of consistency, we also defined the accuracy of the translational micromanipulator as the resulting displacement for every 1° turn of the dial, which would be 0.5 mm travel per turn / 360°= 1.39 μm. 75  Since these micromanipulators only offer limited degrees of freedom, repositioning was required to move the implant in all 6 nominal DOFs.   Additional metal brackets and pre-drilled mounting holes on the polycarbonate box (shown in Figure 31) provided the ability to reposition each micromanipulator to align nominally with each desired axis of motion. The tibial TKA implant (Zimmer Nexgen Posterior Stabilizing implant, size 5, Zimmer, Warshaw) was mounted to the translational or rotational micromanipulator using a combination of custom made polycarbonate brackets and nylon screws.   Three 1 mm-diameter stainless steel beads were embedded into the polyethylene liner of the implant through pre-drilled tunnels and permanently attached using liquid adhesives.  A fourth marker was placed into the stem plug of the tibial component.  Due to the limited number of stem plugs available on hand, we replicated the plug using a 3D printer (Makerbot, New York City, NY) and recreated the plug’s screw-threads using a standard tap kit.   The replicated stem plug was printed using polylactic acid (PLA) and laid in a honeycomb pattern that was aligned axially to the plug.  Together, these four markers define the implant coordinate system (see section 3.7.1 for details). The artificial tibial bone remained stationary relative to the phantom box at all times and was mounted to the bottom of the box using screws.  Two bones were constructed: one used in translational accuracy studies and precision studies (bone ‘A’), the other (bone ‘B’) was used in a rotational accuracy study only.  The reason for a separate bone for rotational verifications was to avoid implant-to-bone collision during verification of extreme rotation angles (>5°).  In bone ‘B’, a portion of the posterior-lateral wall of the tibia was removed to allow a greater range of rotational implant motion.  A total of sixteen 1 mm-diameter stainless steel beads were embedded into bone ‘A’ (Figure 32).   Bone ‘B’ had 2 fewer beads due to the reduced volume of bone available.  The locations of the other 14 beads were roughly similar in both bones.  76   Figure 32: The AP and ML radiographs of the phantom, showing locations of the markers embedded in the implant (double circle) and artificial bone (single circle).  Note that although 16 beads were implanted in the bone, on average, 8.7 bone markers were used between each RSA-pair during our verifications.  The average number of markers suggested in the literature for each bone structure is 6-9 (Valstar 2005).   For the sake of consistency, the AP axis of the implant coordinate system was always nominally aligned with the y-axis of the cage coordinate system.  Similarly, the ML-axis was nominally aligned with the x-axis (Figure 33).   Figure 33: Experiment setup for testing system accuracy when the implant was translated axially in the PD and ML directions.  The same setup was also used during zero motion exams to assess precisions.  The phantom box was placed in the middle of the calibration cage, with the axial directions nominally aligned between the cage and implant coordinate systems.    77  4.3 Methods 4.3.1 Relative Motion Calculations Our Relative Motion Calculator software reports relative motion in terms of Euler angles (in XYZ order) and translation along the three axes, each with respect to a coordinate system defined by the four markers embedded in the implant.    To verify the system’s ability to resolve relative motion in terms of each of the six cardinal DOFs (three translations and three rotations), we used the micromanipulators to generate known displacements. That is, the readout of the micromanipulator was considered to be our gold standard measurement.  Since the micromanipulators’ motion axes were only nominally aligned with the implant’s coordinate frame, we compared total translational or rotational displacements by combining the translations and rotations reported by our system rather than comparing only the nominally corresponding motion components. For RSA-reported components of translation (dx, dy, dz), the combined magnitude of translation becomes:     [2]   For RSA-reported components of rotation (ϕ, ψ, θ), one can derive a single combined magnitude of rotation by converting the Euler angles into quaternions (Diebel 2006).   Starting with the quaternion matrix:    Where ) ;  ) ; ) ; 78   ;  ) ; ) ; Normalizing the quaternion in case of any deviation from unity:       Extracting the angle of rotation in degrees provides the combined rotation:   [3]   By comparing the RSA combined rotations and translations against the values generated by the micromanipulators, we can compute accuracy estimates for each of the six types of cardinal movements (three translational and three rotations).   An example calculation of the quaternion approach to combining the Euler rotation angles is provided in Appendix B.6. 4.3.2 Precision We define precision as one standard deviation of any motion detected with RSA during 12 repeated exams where no motion was induced between the implant and bone;  this is commonly referred to as a ‘zero-motion’ condition in the literature (Valstar 2005).   Between each exam, the phantom was removed from the X-ray’s field of view and repositioned back into view to mimic clinical conditions where it is impossible for the patient to be repositioned in exactly the same position within the imaging field at two time-separated RSA exams. Since the relative position between implant and bone should remain unchanged regardless of where the phantom is placed, any detected relative motion between any given two RSA exams 79  should reflect system precision.  The 12 RSA exams yielded a total possible combination of 12*(12-1)/2 = 66 RSA-to-RSA pairs where relative motion could be calculated (Table 7). In ISO16087:2013, it was recommended for precision that relative motions detected for each of the six components of motion should be checked for fit to a normal distribution.  If they are not normally distributed, the sample standard deviation is reported as the precision, and the sample mean is reported as the bias.  If the relative motions are indeed normally distributed, then the 95% confidence interval is reported.  If the mean exceeds the standard deviation, it should also be reported as a bias.  These reporting methods were set in the interest of standardizing results amongst various RSA studies in the literature.  4.3.3 Accuracy Accuracy was calculated as the mean difference in the detected motion between the RSA system and actual motion as induced by micromanipulators.   Each of the 6 DOFs was tested individually. For translations, a total of 8 RSA exams were performed for movements along each of the three axes.  Between each exam, the implant was displaced along the axis from a reference zero position to 7 additional positions:  0.05, 0.15, 0.20, 0.50, 1.00, 2.00 and 3.00 mm.  For rotations, 7 RSA exams were administered per axis, with the implant displaced to 6 different angles from a reference zero position:  1/6, 1/3, 1, 2, 5 and 10°.  The axes of rotation were set by the position of the micromanipulator, which was positioned proximal to the implant in order to avoid occlusions.  For the purpose of this experiment, since we are comparing the magnitude of the combined rotations (i.e., the quaternion) against the magnitude of the displaced rotation on the micromanipulator, it did not matter that the axis of rotation did not intersect the centroid of the implant coordinate system.      The ranges tested were those used by Laende (2009), where the maximum translation of 3 mm was based upon the upper limit of tibial component motion observed clinically.  For example, at 2-year follow-up, Ryd et al. (1983) saw a maximum migration of 2.7 mm in translation and 12.6° in rotation for six patients, while Albrektsson et al. (1990) reported a mean migration of 1.3 to 2.4 mm for 21 patients.  In more recent studies, Molt et al. (2014) reported a maximum mean translation and rotation of 0.18 mm and 0.23° for 47 patients, respectively.  Thus, setting a test range for translation up to 3 mm was considered appropriate for the system’s intended applications.  For rotations, the maximum degree of motion reported clinically varied greatly in the literature.  Through a systematic review of over 20,000 TKAs (Pijls 2012), it was suggested that any migration in terms 80  of MTPM beyond 0.6 mm at one-year follow-up is considered to be at high risk of revision (i.e., revision rate of over 5% at 10 years).  Since >10° of rotation about any axis would likely equate to an MTPM of more than 0.6 mm (dependent upon size of implant and marker locations), testing accuracies against rotations beyond 10° was considered excessive.  Thus, we chose to follow Laende’s methodology of limiting the test range for rotations to a maximum of 10° per axis.   Since migration is the change in relative position between implant and bone from one examination to the next, it can only be calculated when one has pairs of RSA data (one taken at time T1, one at time T2).  With the 8 RSA exams we obtained per axis with induced translations between them, we have 8*(8-1)/2 = 28 possible combinations of RSA-to-RSA pairs where we have known gold standard migrations to compare against (Table 7).  Similarly, the 7 RSA exams per axes administered for rotations yields 7*(7-1)/2 = 21 combinations of RSA-to-RSA pairs.    EXPERIMENT TYPE(S) # OF RSA EXAMS, r DISPLACEMENTS TESTED COMBINATIONS OF RSA PAIRS, n (n = r (r-1)/2) Precision  (Zero-motion) 12 none 66 Translational Accuracy  and Inter-Rater Reliability 8 per axis 0, 0.05, 0.15, 0.20, 0.50, 1.00, 2.00, 3.00 mm 28 per axis Rotational Accuracy and Inter-Rater Reliability 7 per axis 0, 1/6, 1/3, 1, 2, 5, 10° 21 per axis  Table 7: Summary of images acquired for each corresponding experiment used to verify accuracies, precisions, and inter-rater reliability of the RSA system  81  4.3.4 Inter-Rater Reliability Since the calibration and point reconstruction steps of this RSA protocol require significant user input, we used an inter-rater reliability test to assess how sensitive accuracy and precision results are to inter-rater effects.  Specifically, human intervention is required whenever the centroid recognition feature of XROMM is not identifying centroid locations of beads on radiographs accurately.  This occurs if beads overlap with artefacts caused by other objects, creating non-circular shaped silhouettes or circular silhouettes which lack sufficient image contrast.   In either case, the centroid recognition feature will not work optimally.  The user can use XROMM’s plot of residuals to easily identify problematic centroids, but must then decide what an acceptable residual is and what is not for a given image.  While our protocol includes cut-off limits for residual means during calibration and point reconstruction, our experience show that different users’ perception of centroid locations can result in differences in residual means as great as 0.1 pixels for the same set of radiographs, which is equivalent to ~20 μm for our C-arm model.  With cut-off limits of 0.4 to 0.5 pixels (~80 to 100 μm), this can be the difference between a set of usable versus unusable RSA results.  It should be noted that this limit of acceptance was based upon XROMM’s recommended limits of 0.4 to 0.8 pixels when using the software for tracking animal kinematics.  Thus, for this inter-rater reliability test, we recruited a volunteer rater to repeat the calibration and point reconstruction steps on all 45 sets of RSA images taken for the aforementioned accuracy experiment.  The volunteer’s results were then compared to the first rater’s results.  Prior to starting the experiment, the volunteer (a first year undergraduate student) received:  3 hours of RSA image acquisition training using a C-Arm,  2 hours of demonstration and instruction on the XROMM software, and  5 hours of self-training on sample RSA images (12 sets) using the XROMM software Once completed, we computed the agreement between the two raters’ RSA-identified relative motions by calculating the intra-class correlation coefficients (ICCs) with a two-way random effects model, single measures and absolute agreement approach (McGraw 1996).  This ICC value indicates the extent to which two raters would be consistent in their RSA relative motion measurements.  The higher the value, the less variability exists between findings of the two raters.   The ICC was calculated for MPTM and motion along each of the 6 DOFs, using all 21*3+28*3=147 possible combinations of the image pairs obtained during the accuracy study.  All 82  relative motions data used were reported in the implant coordinate system.  Table 7 provides a summary of all RSA paired images acquired in this study to evaluate system accuracies, precisions, and inter-rater reliability. Although only two raters were available to assess interrater reliability, an ICC approach was chosen over Pearson’s product-moment correlation coefficient since we are also interested in any systematic bias.  If one rater systematically rates greater relative motions along a specific degree of freedom, Pearson’s coefficient would not be able to detect this bias (Fawcett 2013).   Each of the seven ICCs and their corresponding 95% confidence intervals were calculated using the .icc command in STATA 13.0 (STATA, College Station, TX, USA) using a two-way random-effects model (detailed in Appendix D).  In addition, we carried out one-sided F-tests on the ICCs to test the null hypotheses that each ICC is equal to zero, with α set at 0.05.  Since we have seven ICCs, a Bonferroni correction for multiple comparison was used to adjust the significance cut-off to 0.05/7 = 0.007.    4.4 Results 4.4.1 Precision For precision data, the 66 possible pairs of RSA exams were analyzed for relative motions.  Relative motion data was reported in terms of MTPM and the 6 components of motion.  Note that MTPM is an unsigned value, which resulted in an asymmetric distribution, while each of the components of motion exhibited a normal distribution.   As per ISO guidelines, we define precision as one standard deviation of detected relative motions (Table 8).  We thus report that the range of precision of the RSA system is between 16 to 27 μm for translations and 0.041 to 0.059° for rotations. We also provide box plots of the results to show the distribution of precision results (Figure 34).   83   TRANSLATIONS [μm] ROTATIONS [°] MTPM [μm] ML-AXIS PD-AXIS AP-AXIS ML-AXIS PD-AXIS AP-AXIS STANDARD DEVIATION 24 16 27 0.050 0.059 0.041 31 MEAN -3 -2 2 0.010 0.017 -0.009 73 ±95%  CONFIDENCE INTERVAL 47 32 52 0.097 0.115 0.079 61  Table 8: Precision results from the 66 paired-combinations of zero motion RSA exams, with precision defined as one standard deviation of any detected relative motions – which is ideally zero since no displacements were induced.   84   Figure 34: Box plots of precision results.  Each box encloses the interquartile range (IQR), with the upper and lower edges marking the 25 th percentile (q1) and the 75th percentile (q3).  The central line of each box represents the median.  Each set of whiskers corresponds to the limits of what we have defined as outliers; with upper limit set at q3 + 1.5*(IQR) and lower limit as q1 – 1.5*(IQR).  Outliers are shown as crosses.  Each bar indicates the mean error, while error bars indicate one standard deviation of the error. All reported relative motions are in the implant coordinate system.  Note that MTPM values are unsigned (unlike the six components of motion), thus its plotted bias can only be positive. 85  4.4.2 Accuracy For accuracy data, all possible RSA-to-RSA combinations within each individual degree of freedom were analyzed for relative motion, including:  28 combinations for each translational axis, and   21 combinations for each rotational axis    For evaluating MTPM accuracy, only the combinations where the implant was translated were used.  It was not possible to use results from rotational tests, since we do not know what each degree of displacement would be in terms of equivalent MTPM.   The results are reported in Table 9, with accuracy defined as the mean construction error as per ISO guidelines.  The range of our system’s accuracy is thus between -39 to 11 μm for translations, and -0.025 to 0.029° for rotations.  MPTM accuracy was 20 μm with a standard deviation of 34 μm.  A Bland-Altman plot for MTPM (Figure 35) and the 6 components of motion are plotted (Figure 36).    TRANSLATIONS [μm] ROTATIONS [°] MTPM [μm] (n=84) ML-AXIS (n=28) PD-AXIS (n=28) AP-AXIS (n=28) ML-AXIS (n=21) PD-AXIS (n=21) AP-AXIS (n=21) MEAN ERROR 5 -39 11 -0.010 -0.025 0.029 20 STANDARD DEVIATION OF ERROR 22 35 30 0.057 0.052 0.067 34  Table 9: Accuracy results, reported as the mean difference between RSA-detected relative motions and actual migrations induced using a micromanipulator (the ‘gold standard’).        86       Figure 35: Bland-Altman plot of the accuracy test results for implant relative motion expressed in terms of MTPM.  Results show slight bias with the zero line positioned beyond the 95% CI boundaries of the mean.  87   Figure 36: Bland-Altman plot of the accuracy test results for implant relative motion along the six degrees of freedom.  Results show low degrees of error overall, but slight bias on PD translation, PD rotation and AP translation.  88  4.4.3 Inter-Rater Reliability  We found a high degree of agreement between the relative motion measures produced by the two raters, with ICCs ranging from 0.999 to 1.000 for the seven relative motion variables tested (MTPM and the six components of motion).   Table 10 reports all seven ICCs along with 95% confidence intervals as calculated in STATA.  In addition, the seven F-tests all rejected the null hypothesis that each ICC is equal to zero (p-value of less than 0.007 - adjusted for multiple comparisons).   We thus have statistical evidence to reject the null hypothesis that the two raters did not produce correlated results.  This finding, supported by the high ICC values, suggests that different raters can produce relative motion measurements that are in close agreement with one another, regardless of whether we are measuring MTPM or translation/rotation along specific anatomical axes. Furthermore, it was interesting to note that the difference between the two raters ranged between -70 to 80 μm in MTPM, averaging at 1 μm.  The largest difference occurred when the two raters measured a micromanipulator translation of 2.80 mm.    TRANSLATIONS ROTATIONS MTPM ML-AXIS PD-AXIS AP-AXIS ML-AXIS PD-AXIS AP-AXIS ICC 1.000 1.000 1.000 1.000 1.000 1.000 1.000 95% CI LOWER LIMIT OF ICC 1.000 0.999 1.000 1.000 1.000 1.000 1.000 95% CI UPPER LIMIT OF ICC 1.000 1.000 1.000 1.000 1.000 1.000 1.000  Table 10: The ICC results between two raters tested using a two-way random effects model, single measures and absolute agreement.  Significance cut -off was set at p<0.007 (after multiple comparison correction) for each ICC.  Results suggest that the two raters produced highly correlated relative motion measures.     89  4.5 Discussion The purpose of the studies reported in this chapter was to assess the precision, accuracy and inter-rater reliability of our proposed RSA process.  In this section, we discuss the outcomes. 4.5.1 System Performance Precision, as assessed by zero-motion tests, resulted with an average deviation of 73 μm in MTPM (maximum of 164 μm).  Accuracy in translation was under 39 μm, with a maximum standard deviation of 35 μm, while accuracy in rotation was under 0.029° with a maximum standard deviation of 0.067°.  In clinical applications, we would like to see MTPM under 50 μm in zero-motion tests, such that the accuracy is at least an order of magnitude better than what we are seeking to detect when assessing migration in TKA implants (i.e., >500 μm)1.  Therefore, our system appears to be capable of meeting this criterion.  There are currently no studies, to our knowledge, which discuss clinically acceptable migration limits for patients in terms of individual components of motion (e.g., a migration limit in the anterior tilt direction).  Furthermore, ISO standards and guidelines by Valstar (2005) on RSA systems do not outline minimal requirements on system precision and accuracy.  Thus, for rotation and translation measures, we aim to match existing RSA systems’ accuracies and precisions reported in the literature. Although overall accuracies and precisions were satisfactory in comparison to other studies in the literature, we noted slight bias on the PD-axis for both translation and rotation during precision tests (mean error of -39 μm and -0.025°).  To a lesser degree, bias was also observed on the measurements for AP translation (mean error = 11 μm) and MTPM (mean error = 20 μm).  For these four measures, the Bland-Altman plot had the horizontal zero-line positioned beyond the boundaries of the 95% confidence interval, indicating the presence of a small amount of bias.  For AP translation and MTPM, the bias was in the direction of overestimating the true relative motion, while PD rotation and translation had biases that underestimated the relative motion. Clinically, overestimating migration is less concerning than underestimating, since the ultimate purpose of an RSA system is to assist in identifying patients at risk of early implant failure.                                                     1 Base on a systematic review which suggested that MTPM migrations of 0.5 to 1.6mm highlighted patients at risk of higher revision rates (Pijls 2012) 90  Thus, the slight bias on AP translation and MTPM is of relatively little concern.  It is recommended that future use of the system considers correcting for bias on the PD axis of translation and rotation for phantom based studies.  We speculate that the PD-axial biases may have been caused by collision issues within our verification phantom, which will be discussed later in the limitations section (5.3). For clinical studies, precision should be reassessed for individual patients using double-examinations as recommended by Valstar (2005) and ISO guidelines. In terms of inter-rater reliability tests, the ICC values and corresponding 95% confidence intervals showed a high degree of correlation between the two raters (ICC > 0.999).  While the first rater was the author, who had close knowledge of the system, the second rater had only 10 hours of training and did not have prior training in the operation of X-ray equipment or in image analysis.  This suggests that most individuals can be competently trained to use the system to evaluate TKA implant relative motions.  4.5.2 Comparison with Other Studies The results presented in this study1 are comparable to those reported in the literature.  Existing RSA systems have reported translational and rotational accuracies in the range of  -48 to 277 μm and -0.45 to 0.15°, respectively (Laende 2009; Kedgley 2009; Onsten 2001; Seehaus 2013).  Similarly, these studies also used phantoms to evaluate system precisions and have reported results ranging from 12 to 181 μm for translation and 0.01 to 0.4° for rotation.  Our system was able to achieve comparable results. Table 11 provides a detailed comparison between our results and those reported in the literature.     Specifically, Kedgley (2009) was the only other C-arm based RSA study that we found in the literature which used similar definitions of accuracy and precision.  Comparing against Kedgley’s system, we were able to achieve higher levels of translational accuracy and very similar precisions.  Similar to our study, Kedgley defined accuracy as the mean difference between RSA-identified vs. gold standard measurements.  However, Kedgley was measuring inter-marker distances while we measured rigid body motions.                                                       1 The range of our system’s accuracy was between -39 to 11 μm for translations, and -0.03 to 0.03° for rotations; precision was between 16 to 27 μm for translations and 0.04 to 0.06° for rotations. 91   RSA EQUIPMENT ACCURACIES PRECISIONS RSA SYSTEMS X-RAY SETUP* CAGE† SOFTWARE‡ TRANSLATION [μm] ROTATION  [°] TRANSLATION [μm] ROTATION  [°] Our System C-Arm (Radiographic) Custom-Built XrayProject 2.2.5 (marker-based) -39 to 11 a -0.025 to 0.029 a 16 to 27 d 0.041 to 0.059 d Kedgley, 2009 C-Arm (Fluoroscopic) Custom-Built Custom-Built (marker-based) 90 to 277 b - 16 to 23 d - Seehaus, 2013 Conventional Medis Medical MBRSA 2.0 Beta (model-based mode) -48 to 48 a -0.449 to 0.078 a 12 to 181 d 0.015 to 0.354 d Onsten, 2001 Conventional RSA Biomedical XRAY90 & Kinema (marker-based) 45 to 121 c - 24 to 104 d - Laende, 2009 Conventional Tilly Medical RSA-CMS 4.3 (feature & marker-based) 18 to 75 c 0.061 to 0.153 c 17 to 44 d 0.014 to 0.049 d Trozzi, 2008 Conventional Tilly Medical MBRSA  (marker-based mode) - - 24 to 157 e  0.079 to 0.246 e Kaptein, 2007 Conventional Medis Medical MBRSA 3.02  (marker-based mode) - - 27 to 83 e 0.140 to 0.246 e a Accuracy calculated as the mean difference between measured and actual displacements induced using micromanipulators b Accuracy calculated as the mean difference between RSA and CMM measured marker-to-marker distances c Accuracy calculated as the 95% prediction interval from regression analyses between measured and actual displacements induced using micromanipulators d Precision calculated as one standard deviation of reconstruction error in repeated measures in phantom tests e Precision calculated as one standard deviation of reconstruction error in repeated measures in clinical double examinations  * 'Conventional' refers to any X-Ray setup involving the use of stand-alone Roentgen tubes and radiographic film/cassettes † Cage manufacturers:  Medis Medical Imaging Systems BV (Leiden, Netherlands), RSA Biomedical AB (Umeå, Sweden), Tilly Medical Products AB (Lund, Sweden) ‡ Software providers: XrayProject (XROMM, Brown University, Providence, RI), MBRSA (Medis Specials BV, Leiden, the Netherlands), XRAY90 and Kinema (RSA Biomedical Innovations AB, Umeå, Sweden), RSA-CMS (MEDIS medical imaging systems BV, Leiden, The Netherlands)   Table 11: Summary of accuracy and precision results obtained in this study in comparison with studies in the literature  92  4.5.3 Clinical Relevance   To obtain clinically relevant RSA results, it has been suggested in a systematic review (Pijls 2012) that tibial components with migrations beyond 500 μm in MTPM during the first postoperative year are considered at risk of having higher revision rates (i.e., 10-year survival rate of >5%).  Therefore, since our RSA system has demonstrated precision in the range of 70 μm, which is nearly an order of magnitude better than this clinical criterion, it should be capable of being used to diagnose motions of this magnitude.   4.5.4 Limitations Our verification studies to date have been limited to phantom studies as a first step, further in vivo trials would be required to assess whether soft tissues, patient motion and other clinical factors (e.g., room temperature fluctuations, radiologist training, etc.) will significantly reduce the system’s accuracy and precision.  While we expect that accuracies presented in this paper will closely reflect system capabilities clinically, we recognize that precision levels may be patient-dependent and will thus require reassessment for each patient when measuring in vivo relative motions.  These patient-based factors include the locations and quantity of implanted markers, the levels of marker occlusion during image acquisition, and the degree of soft tissue artefacts captured on images.  Another limitation to this study is that we did not quantify the accuracies and precisions of the micromanipulators we used to generate our ‘gold standard’ motions.  In principle, the accuracy of these devices14 were much better than the movements we needed to measure, so it is unlikely that these devices would contribute significant error to the overall errors in our system.  However, it is possible that if there were any laxities in the kinematic chain supporting the model components, this might present as biases that could depend on the orientation of the micromanipulators.  For the translational tests, we would expect little influence on precision because the magnitude and direction of external forces would remain roughly constant during the test, and this was confirmed by our finding that the precisions in the three coordinate directions varied a relatively small amount between directions - between 16 and 27 μm.                                                    14 Translational micromanipulator was accurate to 1.39 μm.  Rotational micromanipulator was accurate to 4 arcseconds. 93  In terms of accuracy, we did observe a larger bias of -39 μm (i.e., an underestimation) along the PD axis in contrast to the 5 to 11 μm in the other two directions.  This PD axial bias had an apparent trend towards increasing the degree of underestimation with distance moved.  To further investigate this, we ran a linear regression analysis on the Bland-Altman plot of PD translation tests, and used a T-test to challenge the null hypothesis that the slope of this regression line equals zero.   We found a regression slope of -0.025 with p-value < 0.0001, thus providing the statistical evidence to reject the null hypothesis that the slope was zero.  This suggested the presence of a non-constant bias.  We speculate that potential collisions between the implant and bone during PD translation tests may have contributed to this behaviour, since the underestimation of translations was accompanied by an overestimation of rotations (mean error of 0.15° during PD translation tests).  Since the bone was opaque, collisions between bone and the inferior portion of tibial implant on the scale of microns were difficult to detect with the naked eye.  The mean error of rotations during AP and ML translation tests were much lower (0.04 to 0.06°). For rotational tests, we expect that the forces acting on the micromanipulator could vary as the implant is rotated since the direction of gravity relative to the implant changes.  However, there were no visible correlation between the changes in RSA detection error and the changes in length of the moment arm for any configuration of the micromanipulator, so we do not believe this potential issue had any detectable effect.   Since the overall size of most of the effects discussed here are relatively small (with the possible exception of the 39 μm bias in the PD direction), we did not feel that it was necessary to independently measure the micromanipulator movements.  However, it would certainly be possible (though somewhat costly and time-consuming) to use a Coordinate Measuring Machine with micron-level accuracy to investigate any deviations between the distances indicated by the micromanipulators and the 'true' distances moved.   4.5.5 Recommendations In summary, we have demonstrated that, in the context of the phantom studies performed here, our RSA system has resolution and accuracy comparable to the best of previously reported systems and is therefore likely to be suitable for use in investigating the clinical questions for which it has been designed. 94  Some minor questions remain that may be considered worthy of followup investigation.  The most pressing question is the source of the 39 μm bias in the PD direction, though even at the extreme of a 3 mm displacement, the discrepancy between the translation reported by RSA and by the micromanipulator readout is only 100 μm, or about 3% of the total displacement range (or a maximum of 10% if we consider the bias of ~50 μm at 0.5 mm of displacement to be the figure of interest), so reducing this discrepancy is unlikely to have any significant impact on clinical decisions.  If one wished to pursue this issue, it would likely be necessary to use a higher-resolution and higher-accuracy device such as a CMM to make the necessary measurements.  It might also be useful to build a combined and integrated 6DOF micromanipulation device combining a 3-DOF rotation stage and a 3-DOF translational stage to facilitate the device characterization since the need to disassemble the current system between tests in the different directions prevents us from evaluating combined movements.           95   Chapter 5   Summary and Conclusions 5.1 Summary of Findings Our primary research goal in this thesis was to develop a fully functional RSA system capable of measuring implant relative motion of a tibial TKA component with respect to bone.  The scope of our development was limited to the use of a single X-ray source (a C-arm unit) and verifying the developed RSA protocol through phantom-based experiments.  Future developments will focus on implementing the protocol for clinical applications by incorporating a dual C-arm setup and executing clinical trials. We began our protocol development by first evaluating the spatial feasibility of using a dual C-arm RSA setup in clinical practice (chapter 2).  This evaluation provided preliminary insights on the eventual clinical feasibility of our proposed RSA protocol.  Our findings suggest that to capture the desired RSA images of a TKA implant in-vivo using dual C-arms, it was easiest to stand the patient on an elevated platform and position one of the two C-arm gantries below the platform while the other gantry remained above the platform (referred to as the ‘AB’ or ‘above-below’ configuration).  We concluded that for the two specific C-arm models that were available to us, this configuration offered sufficient space for the patient to enter the imaging field (>11.5” wide) and maintain an upright stance once within the field (> 20” wide). 96  In addition, we recommend that a pre-calibration approach is taken when using a dual C-arm setup clinically.  By doing so, we maintained sufficient room for the patient to maneuver within the imaging field, and it also reduced the risk of damaging the calibration cage during examinations.  Consequently, to prevent the C-arms from being touched after calibration shots have been acquired, we suggest constructing barriers to surround the imaging field. After the system was developed using a single C-arm (chapter 3), we verified our system through a series of phantom experiments (chapter 4).  Our system achieved translational and rotational accuracies ranging between -39 to 11 μm and -0.025 to 0.029°, comparable to systems reported in the literature which varied from -48 to 277 μm and -0.449 to 0.153°.   Perhaps more importantly, our system precisions ranged between 16 to 27 μm and 0.041 to 0.059°, matching those reported in the literature (16 to 81 μm; 0.014 to 0.354°) obtained through similar phantom-based verification methodologies.  Although clinical precision should be verified using double examinations for individual patients since many factors can affect precision (e.g., number of markers, varying levels of soft-tissue artefact, and quality of marker distribution), these phantom-based verifications show results sufficiently promising to warrant clinical trials.   Aside from precise and accuracy verifications, we also recognized that the analysis steps within our RSA protocol involve some level of judgment by the person using the software.  The resulting quality of RSA reconstruction is thus person-dependent.  Our inter-rater reliability tests showed a high degree of correlation between two raters (ICC > 0.999), thus suggesting that a very small portion of the measured variability was attributed to the difference in raters.   It should be noted that we may not conclude from ICC results whether the inter-rater differences (in terms of millimeters) were small enough as to not significantly impact system accuracy and precision in the clinical context.  We can however, provide an idea as to the magnitude of this difference by calculating the absolute differences between each of the 147 relative motion measurements as judged by the two raters, which was 22 μm on average for MTPM measures.  This difference is small relative to the migration we intend to detect clinically (>500 μm), thus suggesting that inter-rater differences would have minor impact on system performance.   97  5.2 Novelty  Studies in the literature have reported dual C-arms RSA systems purposed for animal or human kinematic studies (Brainerd 2010; Kedgley 2010).  Other RSA systems developed for measuring relative motion of TKA implants are also commercially available (Laende 2009; Kaptein 2007; Nilsson 1993; Seehaus 2013).  However, to our knowledge, there exists no RSA study which leverages C-arm technology to measure relative motion of TKA implants.  As discussed in section 1.5, the use of C-arm technology has the potential to lower the cost of RSA systems in clinical settings where C-arms are normally routinely used.  Furthermore, we have also found no studies which take advantage of open-source RSA software for this purpose.   The novelty of our study is therefore the use of a widely-available imaging resource (i.e., C-arms) in combination with open-source software to develop a low-cost RSA system capable of achieving sufficient accuracy and precision to resolve relative motion of TKA implants for the kinds of studies normally performed using purpose-built RSA systems.    5.3 Limitations The scope of our study was limited to phantom-based verifications, for which it was sufficient to use a single C-Arm for image acquisition.  The RSA protocol developed here can be used in future phantom and cadaveric studies, but cannot be immediately applied to in-vivo studies without additional system development and verification.   In particular, we have not yet demonstrated simultaneous image acquisitions on in-vivo subjects using two C-arms.  This would be the logical next step. In terms of methodological limitations, we did not independently measure the precision and accuracy of the micromanipulators used, but simply accepted the manufacturer's stated accuracy and precision estimates instead.  This may have introduced some small errors into our results, but, for the reasons presented in chapter 4, we believe that these are comparatively negligible. With regard to our system’s intended application to assessing TKA relative motion, we limited our initial study to measuring motions in the tibial component only.  We anticipate that measuring femoral component motion will involve substantially similar issues and our existing 98  MATLAB software can be easily reprogrammed to include this feature.  We chose to initially focus on a single component since studies have shown that when implants were revised, the largest percentage of cases (~35%) involved exchanging all components (Leta 2015; Bozic 2010).  In addition, a multicenter study of over 60 thousand TKA revisions by Bozic (2010) showed that tibial component revisions (9.5%) were more prevalent than that of exclusive femoral (4%) and patellar (5.2%) revisions.  Similar findings have also been echoed by Windsor  (1989) and Ghomrawi (2009).   5.4 Future Directions We intend to further develop the RSA protocol detailed in this thesis for future clinical use.  To aid in this goal, we briefly discuss future developments in this section.   The immediate priority beyond this thesis will be to extend our marker-based system to include model-based or feature-based capabilities (section 5.4.1).  The alternative development path is to continue with the marker-based approached (Figure 37).  Either path will require developing the protocol to use a dual C-arm X-ray configuration as previously discussed in chapter 2.  Once developments are completed, we will execute a series of phantom and in-vivo verifications on the modified system (section 5.4.2).  If system precision and accuracy are unsatisfactory, we will explore potential system improvement opportunities (section 5.4.3) before reassessing system performance.  Once satisfactory results are obtained, we will have an RSA system suitable for clinical studies on TKA patients. Beyond the major milestone of a verified clinical RSA system, our continuous improvement initiatives on the system will focus on optimizing the workflow through automated software (section 5.4.3).  Furthermore, we will also explore other applications of RSA, such as measuring polyethylene wear in TKA patients (section 5.4.4).  The overall plan for future developments is summarized in a flowchart below (Figure 37).  99   Figure 37:  Flowchart of future developments  5.4.1 Model-Based and Feature-Based RSA In our RSA system, we embedded markers within the implant in order to track the precise location of the implant.  This approach is commonly referred to as ‘marker-based’ RSA and involves making modifications to the implant.   In clinical context, these modifications are often costly and may require additional regulatory approval, and the standard implants currently used are not so modified (Valstar 2001).  In more recent years, many researchers have worked on developing an alternative ‘model-based’ approach which eliminates the need for embedding markers within the implant. With model-based RSA, only the bone segments have markers embedded, while the implant is left unaltered.  Using a 3D computer model of the implant, its pose can be determined by matching the outline of the implant on 2D radiographs with the 3D model through optimization algorithms.   The downside of this approach is that its accuracy depends on specific implant geometries and is generally less accurate (Seehaus 2013).  However, some studies have concluded that it is possible to obtain results remain within acceptable limits for clinical applications (Kaptein 2007; Hurschler 2009).  In Kaptein’s study (2007), the authors reported comparable results between the two approaches, reporting precisions (as one standard deviation of zero motion exams) of 0.06 mm and 0.20° for marker-based RSA and 0.11 mm and 0.23° for model-based RSA. 100  Furthermore, other studies have shown that the 3D model of implants should be reversed engineered (e.g., by CT or laser/optical scanning) each time, instead of using CAD models provided by the manufacturer (Kaptein 2003; Seehaus 2013). A study by Valstar (2001) concluded that CAD models provided by the manufacturer resulted in unacceptable accuracy for clinical use, speculating that there may have been dimensional tolerance issues introduced through casting and polishing processes during manufacturing.   It is also worthwhile to note a third RSA approach:  feature-based RSA (Amiri 2012; Baldursson 1979; Valstar 1997).  Instead of matching 3D models to 2D radiographs, feature-based RSA requires the user to segment geometric features of the implant (e.g. the hemispherical heads of pegs, straight/circular/curved edges of flat surfaces).  Using these features, the software then determines implant pose.  This approach has been used to determine relative motion in total hip implants (Baldursson 1979; Valstar 1997).  However, the extent of feature-based RSA application on TKA implants has been limited to kinematic studies.  In Amiri’s study (2012), the kinematic relationship between the tibial and femoral TKA components was determined with an accuracy of 1.03 mm and 1.45°, along with precision of 0.26 mm and 0.57°.  Additionally, the authors also compared these findings against a model-based RSA approach to resolving implant kinematics, resulting in comparable accuracy (1.32 mm and 0.95°) and precision (0.44 mm and 0.42°).  Although feature-based RSA was promising, there currently exists insufficient evidence to suggest that it would provide sufficient accuracy and precision in TKA relative motion studies.  Our future developments will thus focus on first achieving a clinical model-based RSA system, before exploring feature-based RSA options.  We contrast the requirements of each RSA approach in Table 12.          101    Marker Based Model Based  Feature Based Pre-op Requirements Implant modification by manufacturer 3D Model of Implant using CT or laser/optical scanning A list of elementary geometric features of the implant and the relationships between features  Intra-op Requirements Bone marker insertion Bone marker insertion Bone marker insertion  Table 12: A comparison of protocol requirements for the three RSA approaches that may be used to measure TKA implant relative motions.  While marker-based and model-based RSA have been well studied, feature-based RSA remains a relatively unexplored technique for TKA relative motion measurements.     It is our vision that we will extend our RSA system to include model-based capabilities by continually leveraging XROMM’S open-source software developments, which is discussed later in this chapter (section 5.4.3).   Consequently, we will also need to assess various methods of acquiring sufficiently accurate 3D models of the implant, which will need to be developed into a clinically feasible process – one that is effective both in terms of time and cost. 5.4.2 Additional Verifications The next phase following dual C-arm model-based RSA developments will be to assess performance of the new system. Our primary research objective will be to assess whether the system will be suitable for clinical use, as such, we will be resolving relative motion in both the tibial and femoral components through a two-part study: (1) Accuracy verification using a 6-DOF micromanipulator phantom (2) Precision verification  using double examination on TKA patients   While phantom-based verification results for accuracy can be used to infer system performance under clinical conditions, precision must be quantified through in-vivo double examinations (Valstar 2005).  For accuracy verification, we will follow the same methodologies 102  described in this thesis that was used to test our marker-based system.  However, a new phantom will need to be constructed to consist of a complete set of TKA implant.  The phantom should be capable of inducing migration in all 6 DOFs for each individual component (tibial and femoral) relative to a set of artificial bones.  We will repeat the methods described in section 4.3.3 on each component.  For clinical precision, we will verify using a sample size of at least 15 patients (as recommended by ISO16087).  During surgery, 6-9 well-distributed tantalum markers (Valstar 2005) will be injected into the proximal segment of each patient’s tibia.  Each patient will receive double examination under weight-bearing conditions 6 months after surgery.  We suspect that several factors may impact system performance under these conditions, such as soft tissue artefacts (varies from patient to patient), potential for marker loosening and patient movement during image acquisition. It should be noted that while this will provide users with an idea of system precision, re-evaluation of precision will be required with each new RSA study by carrying out double examinations on at least 25% of the cases being evaluated (ISO16087).   We expect accuracy to be comparable to our marker-based RSA results reported in this thesis (-39 to 11 μm; -0.025 to 0.029°), which has been shown to be possible with model-based RSA (Seehaus 2013).   For precision, we expect to achieve ranges similar to other model-based systems.     As a reference, Trozzi (2008) reported clinical precision of 100 to 229 μm and 0.239 to 0.589° and Kaptein (2007) reported similar results of 58 to 161 μm and 0.159 to 0.293°.  Both results maintain the submillimetre precision required to monitor implant migration in TKA patients.  Recall from section 4.5.3, any migration (measured as MTPM one year after surgery) beyond 0.5 mm is considered at risk, and any migration beyond 1.6 mm is considered unacceptable (Pijls 2012).    5.4.3 System Improvement Opportunities During the development of our system, we noted several areas of potential improvement that may result in faster processing time and/or greater precision and accuracy.   In this section, we highlight some of these opportunities that could be explored through additional experiments.    Open-Source Software Our RSA protocol has been developed to work with the marker-based workflow of XrayProject v2.2.5 (released May 2014) developed by XROMM.  In recent years, XROMM has also developed a 103  model-based RSA workflow (XROMM Autoscoper Beta, released October 2014) which uses CT models of bones as an input and can calculate the relative motion between two bone segments.   As previously mentioned, model-based RSA tracks position of the implant using its 3D model, while the bone is tracked using implanted markers.  In contrast, marker-based RSA tracks both bone and implant using implanted markers.  Thus, to work towards developing a functional model-based RSA system, the first priority is to develop an RSA system that is capable of tracking rigid bodies using markers only (i.e., a marker-based system) – which is the scope of this thesis.  Our next step will be to eliminate the need to attach markers to implants by integrating XROMM Autoscoper’s model-based approach.  In addition, XMALab 1.2.16 was recently released (October 2015) by Brown University, which offers a more streamlined and automated method for performing RSA analysis.  In particular, we recognize the value in using XMALab to automate calibration procedures within our protocol.  This is a feature that we will also explore to improve the efficiency of our protocol’s workflow. In summary, we recommend that future developments to include incorporating XROMM Autoscoper and XMALab tools into our existing RSA protocol.  These developments will need to be followed by additional verifications to ensure that the precision and accuracy of our system remain within acceptable limits. Although we have yet investigated these alternative open-source software packages, it is worthwhile to mention that Brown University continues to develop and refine RSA software that is free to researchers globally.  We can be leverage this access in future developments to reduce processing time associated with our developed RSA system. Distortion Correction Mesh In principle, the perforated steel sheet should be as thin as possible without introducing excessive flexibility or radiolucency.   Figure 38 below illustrates how a thick sheet may create a radiographic image where the center-to-center distances between perforations may begin to increase for those that are furthest away from the X-ray source (i.e., those that are closest to the perimeter of the image intensifier).  Since XROMM’s algorithm corrects for distortion using distances between centroid pairs, it is important that the sheet thickness is kept to a minimum without becoming so thin that flexing of the sheet can introduce additional errors.   104    Figure 38: To illustrate the effect material thickness has on the quality of distortion correction, a cross-sectional view of the perforated sheet metal is shown here as rectangles and the projection center of the X-ray beams is represented by the circle.  When beams (dashed lines) pass through each perforation, a thick sheet material will produce radiographs where the centroid representation of perforation ‘B’ poorly reflects its true location.  Since the distortion correction algorithm relies on distances between perforation centroids, this effect can reduce the quality of correction.     At 0.91 mm thick, the sheet metal used in our device was felt to be appropriately thin; we also confirmed this by estimating the potential impact of this thickness may have on accuracy.  In section 3.5.1, we approximated the dimensions of the X-ray beam as a cone with a base radius of 85.5 mm and height of 977 mm.  Thus, we know rays nearing the perimeter of the conical beam would have an angle of approximately tan-1(85.5/977) = 5° from beam center.  A hole nearing the edge of the plate would thus have a shadow diameter (distance ‘B’ in Figure 38) that is approximately 80 μm (= 85.5/977 * 0.91 mm) less than the diameter of the actual hole.  Half of this error can then be used to estimate the maximum centroid offset of the shadow, which becomes 40 μm.  While this is not entirely negligible, it is a factor of 10 less than the minimum 500 micron migration we would like to detect in TKA implants.           A B 105  To improve upon this, we would recommend exploring the use of thinner plate materials.  For example, a 0.5 mm thick sheet can reduce this offset error to approximately 22 microns.    However, the use of thinner plates would introduce additional flexibility.  While we have addressed the issue of flexibility in our system by mounting the plate against a sheet of carbon fiber using four screws, it may also be worthwhile to investigate the option of mounting the plate atop a carbon fiber sheet using adhesives. Calibration Cage When training new users on the RSA system, we noticed that it was intuitive for most to lift the calibration cage by pinching the side panels with their hands.  This approach effectively puts the user’s fingers in contact with cage markers, which may cause marker loosening and compromise calibration quality, and with a portion of the cage that should not be subject to distorting forces.    We therefore recommend that should the cage ever require re-verification of marker coordinates using CMM, it would be worthwhile to use that opportunity to permanently add an additional protective polycarbonate panel to the top of the cage.  This added panel should include a circular cut-out in the middle (similar to the base panel) such that the imaging subject can still be placed at the center of the cage.  In addition, explicit lift points should be designed and added to the cages as affordances to ensure that the cage can be lifted without introducing unwanted stresses into the cage. 5.4.4 Measuring Polyethylene Wear Beyond implant relative motions, we are also interested in using the RSA system to measure wear on the polyethylene liner.  It is well recognized that most implant failures occur due to aseptic loosening, instability, infection and polyethylene wear (Sharkey 2014; Paxton 2011; Bozic 2015; Dalury 2013; Schroer 2013).  However, the primary mechanism of failure varies greatly from study to study.  Thus, we are interested in the two mechanisms that RSA can quantify:  implant relative motion and wear.  In a study of over 800 TKA revisions across six institutions, Schroer et al. (2013) found that for early failures (>2 years), the top failure mechanisms were distributed amongst loosening, instability and infection  (each between 19-25%).  However, for failures occurring in the mid-term (2-15 years), loosening was a larger factor (39%).  For implants that failed later (>15 years of service), polyethylene wear was the reason behind nearly half of all revisions.    106     A similar contrast between early and late mechanisms of TKA failures was also observed by Sharkey (2002) and Dalury (2013) for revisions performed between 1997-2011.  In Sharkey’s study, polyethylene wear attributed to 44% of failures in the late revision group (>2 year). Interestingly, when the authors repeated the study at the same institution 10 years later, failure rates due to wear dropped for both late (4.3%) and early (2%) revision groups (Sharkey 2014).     As it stands, there is little consensus on whether polyethylene wear remains a major mechanism of failure in TKAs (Figure 39).  While some suggests that rates have reduced in recent years due to biomaterial advancements (Chakravarty 2015; Sharkey 2014), the ability to quantify wear in-vivo remains to be of interest.   By providing the means to measure wear progression in patients before failure occurs (van Ijsseldijk 2011), RSA is especially valuable for studies focused on late failure mechanisms.    Figure 39: Historically, the primary mechanisms of failure for TKAs are typically polyethylene wear, loosening, instability and infection.  In this chart, we summarize percentages of revisions that were attributed to these failure mechanisms as reported by six large-scaled studies.  The patient data included in these studies varied between 200 to 300,000 revision surgeries performed between 1997 and 2012.  107  5.5 Conclusions We have developed a marker-based RSA protocol which leverages two low-cost resources: (1) C-Arm for image acquisition (2) Open-source software (XROMM) for RSA analysis This system is suitable for phantom and cadaveric studies, and can be used to quantify relative motion of the tibial component in a set of TKA prosthesis.   In addition, the system was also developed in adherence to ISO 16087:2013(E).      The accuracy and precision of the system were quantified through a series of phantom verifications, with results comparable to those reported in the literature.  In addition, the achieved accuracy and precision were also sufficient for identifying excessive migrations (>0.5 mm of MTPM) in TKA patients for the purpose of predicting early implant failure.  However, future clinical verifications will be required to ensure the same level of accuracy and precision is retained when the system is further developed for use in clinical conditions. Through inter-rater reliability tests, we are also confident that new users can ascertain similar levels of accuracy and precision after receiving 5 hours of instructional training followed by 5 hours of practice time. Using an existing C-arm available to us, we constructed a single C-arm RSA system with a calibration cage and distortion correction device that were designed and built in-house.  The combined cost of these two devices were under $1,500 USD.  Of the past studies we used in our comparisons, most were built with traditional X-ray equipment and/or commercially-purchased RSA software.  Based on our enquiries, commissioning a commercial RSA system can cost as much as $750,000 USD.  If a hospital were to implement a dual C-arm setup, we estimate that the total system cost would be a few thousand dollars, in addition to any costs associated with acquiring access to C-arms (often already available in hospitals)15 – this price would be sufficient to cover the cost an imaging platform, calibration cage, distortion correction device, computer equipment, and a mechanism for simultaneous image acquisition.  In addition, as demonstrated in section 1.5, the use of in-house developed software in conjunction with open-source software could potentially reduce                                                    15 ($1,500 for calibration cage and distortion correction device) + ($1,000 for platform) + ($1,500 for computer equipment) + ($2,000 for development and construction of a simultaneous image acquisition mechanism) = $6,000 108  image analysis cost from $600 to $120 per patient.  Even if the cost of two moderate C-arms is included (2x$100,000), this approach would yield roughly a 72% reduction in system cost and 80% reduction in per-patient analysis cost.  Therefore, we have shown that one can reduce this cost drastically by leveraging existing C-Arms and open-source software.   To our knowledge, most surgically-capable hospitals in North American already have C-Arm access.   In terms of limitations, our system was constructed with a single C-arm that is rotated once per exam in order to obtain the required pair of biplanar images.  While this radiographic setup is satisfactory for phantom and cadaveric studies, it is not immediately suitable for use in the clinical environment.  Since patient motion will introduce errors, future development of the protocol will need to address the use of two C-Arms and devise methods to ensure the two radiographs are acquired simultaneously.  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Accuracy analysis for RSA: a computer simulation study on 3D marker reconstruction. Journal of Biomechanics, 33(4), pp.493–498., doi:10.1016/S0021-9290(99)00182-7   115   Appendix A   Summary of Protocol This appendix provides a tutorial on the operation of the RSA system we have developed, as such, the content is only applicable to our specific model of C-arm (ARCADIS Orbic 3D, Siemens AG, Erlangen, Germany), along with the calibration and distortion devices we have built.  The workflow, however, can be universally applied to other custom-built RSA system.    A.1 Image Acquisition To acquire the necessary images for RSA analysis, you will need:  Calibration cage  Calibration cage support platform  Image intensifier positioner plates ‘A’ and ‘B’   Distortion correction device  The Siemens Orbic Arcadis 3D C-Arm  A phantom / cadaveric specimens you wish to image   Detailed in this section is a step-by-step procedure for image acquisition. 116  STEP 1 Position C-arm to -45° orbital position with the orbital plane parallel to the floor (top image on left). This C-arm position is hence forth referred to as view ‘A’ or ‘X-ray #1’.  Place calibration cage together with its support platform at the center of the imaging field.    Roughly align the two panels of the cage that are engraved as ‘X-RAY #1 - FIDUCIAL’ and ‘X-RAY #1 - CONTROL’ with the orientation of the X-ray beam.  Position the fiducial panel closer to the image intensifier.   Note that cage marker numbers 1 through 25 are embedded into these two cage panels.   STEP 2 Now we will fine-tune the cage alignment. Align the cage to the image intensifier by turning on the C-arm’s laser aim crosshairs.  Make sure it passes both the control and fiducial panels at the center of each panel.   Blue crosshairs on the control panel and green tape on the fiducial panel indicate location of panel centers.  117  STEP 3 While maintaining laser alignment, use the image intensifier positioner ‘A’ (IIP-A) to measure a precise gap of 19.2cm between the image intensifier and the outer face of the fiducial panel (top image on left).    Take a fluoroscopic shot to make sure no occlusion occurs between markers on the two panels. Note that the cage has radiopaque letterings ‘26’ and ‘42’ taped onto the upper corners of the control panel which may be visible on the images.  It indicates that you are viewing marker numbers 26-41 (control panel) and 42-50 (fiducial panel).  STEP 4 Remove the IIP-A plate and carefully and slowly rotate the C-arm orbitally by 90°, you should now be in the +45° orbital position.  Using the laser aim, inspect C-arm alignment with the two panels engraved as ‘X-RAY #2’.  If needed, use image intensifier positioner ‘B’ (IIP-B) to adjust the cage position.   If view ‘A’ was properly aligned, this should not be necessary.  Check alignment with a fluoroscopic shot.  118  STEP 5 Remove the cage while keeping its support platform in place.  The position of the platform will help you return the cage into position later on.    STEP 6 Attach the distortion correction device onto the image intensifier using the two finger-triggered latches.   These latches should hold the device securely onto the image intensifier.  Make sure the three protruding indexing pins on the image intensifier line up with red markings on the device. Obtain your distortion correction radiograph for view ‘B’.  119  STEP 7 Remove the distortion correction device carefully without disturbing the position of the image intensifier.   Return the cage back into position using the support platform as a reference. Make sure cage orientation is correct: the laser aim should be now aligned with the two panels engraved as ‘X-RAY #2’ with the ‘X-RAY #2 - FIDUCIAL’ panel closer to the image intensifier. Place the subject you wish to image (e.g., a phantom) inside the cage.  Obtain your radiograph for view ‘B’. Note that the cage has radiopaque letterings ‘1’ and ‘17’ taped onto the upper corners of the control panel which may be visible on the images.  It indicates that you are viewing marker numbers 1-16 (control panel) and 17-25 (fiducial panel).  120  STEP 8 We will now obtain the two radiographs needed for view ‘A’.  Rotate the C-arm back to view ‘A’ position (-45° orbital) and obtain radiograph ‘A’.  Remove both the subject of interest and cage.  Reattach the distortion correction device onto the image intensifier to obtain the distortion correction radiograph for view ‘A’.  You should now have four radiographs needed for RSA:  (1) distortion correction image of view ‘A’ (2) distortion correction image of view ‘B’ (3) subject together with cage from view ‘A’ (4) subject together with cage from view ‘B’       121  A.2 Point Reconstruction with XROMM Once you have obtained the four radiographs from a single RSA exam, run the code provided in appendix B.1 to increase image contrast and convert the file formats from ‘.bmp’ into ‘.tif’.   Next, download the XROMM software from Brown University’s online portal (http://www. http://www.xromm.org/).  In this protocol, we have adopted XROMM’s XrayProject version 2.2.5.  Since Brown University provides online tutorials on the operation of XrayProject-2.2.5, we will not cover this in detail, but rather, provide a general overview of the steps involved within this appendix.  You will need the following files before proceeding with XrayProject: (1) ‘CMMframespec.csv’: A table containing 3D coordinates of the 50 calibration cage markers, with respect to cage coordinate system.  These coordinates were obtained during CMM verification (section 3.5.1).  (2) ‘UNDISTORT_A.tif’ and ‘UNDISTORT_B.tif’:  Radiographs of the distortion correction device for view ‘A’ and ‘B’.  (3) ‘XRAY_A.tif’ and ‘XRAY_B.tif’: Radiographs of the subject of interest together with the calibration cage from view ‘A’ and ‘B’.  Once you have these five files, you may proceed with RSA point reconstruction using XrayProject (detailed below). STEP 1 Using the distortion correction module in XROMM XrayProject-2.2.5, generate the two undistortion transforms, ‘UNDTFORM_A.mat’ and ‘UNDTFORM_B.mat’ (one for each X-ray view).  Next, apply the generated transforms to XRAY_A.tif’ and ‘XRAY_B.tif’ files, ensure the correct transform is used for each corresponding view. You should end up with undistorted images ‘XRAY_A_UNDIST.tif’ and ‘XRAY_B_UNDIST.tif’.  122  STEP 2 Run the calibration module in XROMM XrayProject-2.2.5 and load the un-distorted images from Step 1.  Follow XROMM’s online instructions to generate the 11-DLT coefficient files for each of your X-ray views.  XrayProject will save them as ‘A_DLTcoefs.csv’ and ‘B_DLTcoefs.csv’.  When using the calibration module to generate the 11-DLTs, refer to section 3.5.2 for calibration rules.  For our protocol, we suggest users to use at least 10 calibration markers (five per panel), and aim for an average residual value of less than 0.4 pixels.  With our calibration cage, DLT residual values as low as 0.1 to 0.3 pixels is expected. Next, use the calibration module to merge the two DLT files, saving it as ‘Merged_DLTcoefs.csv’. STEP 3 Run the digitization module in XROMM XrayProject-2.2.5 and load the following files:  ‘XRAY_A.tif’ and ‘XRAY_B.tif’  ‘UNDTFORM_A.mat’ and ‘UNDTFORM_B.mat’  ‘Merged_DLTcoefs.csv’  Note that unlike the calibration module, the digitization module requires image files that have not be undistorted.    Ensure each marker is reconstructed in accordance to the definitions outlined in section 3.6.  It is imperative that the first four markers are the implant markers as defined in this section. For point reconstruction, we specify the cut-off limit for residual value at 0.5 pixels for each marker. On average, residual values ranging from 0.03 to 0.4 pixels are achievable.   STEP 4 Save the 3D coordinates of your subject markers as ‘Subj_xyzpt.csv’. 123  A.3 Reporting Relative Motions To calculate relative motion, you will need RSA exams of the same subject examined at two different times (referred to as time T1 and T2).  Once you have completed the procedures outlined in appendix A.1 and A.2 for each of these two RSA exams, you may then proceed to quantify the relative motion between implant and bone from T1 to T2.  We do so by using the Relative Motion Calculator software we have developed in-house.  Start by running the ‘RelativeMotionCalculator.m’ file within Relative Motion Calculator v1.4 folder.  Follow the instructions below to calculate relative motions between two RSA exams. STEP 1 Load the RSA reconstructed subject markers from T1 and T2 by browsing to the two ‘Subj_xyzpts.csv’ files generated using XROMM.  Click on the ‘STEP 1: IMPORT DATA’ button to load files into program.  STEP 2 Check to ensure all points have been imported correctly for both T1 and T2 by first inspecting the table of imported markers (on left), then inspecting the graphical plot of the implant and bone marker 124  clusters.  Marker labels should be consistent between T1 and T2 for all markers.  Implant markers should also follow the definitions displayed on the right side of the window.  Click on ‘STEP 2: CONFIRM DEFINITION OF MARKERS’ to continue. STEP 3  You should now see relative motion results displayed on a panel to your right, check the ‘DATA QUALITY’ section of this panel to make sure CN is less than 150 and RBE is less than 0.35 mm.  If any value exceeds these limits, do not use the relative motion results and repeat your RSA exams.   On the ‘RELATIVE MOTIONS’ section of the right panel, you will find relative motions reported as follows for a left tibial implant: (1) Rotation about ML-axis of implant, with positive value denoting anterior tilt (2) Rotation about PD-axis of implant, with positive value denoting external rotation  (3) Rotation about AP-axis of implant, with positive value denoting varus tilt (4) Translation about ML-axis of implant, with positive value denoting lateral shift (5) Translation about PD-axis of implant, with positive value denoting proximal shift (6) Translation about AP-axis of implant, with positive value denoting anterior shift (7) Maximum Total Point Motion (MTPM), reported as the magnitude of movement of the implant rigid body’s centroid.  Fictive marker locations are used to calculate MTPM (white 125  text), but MTPM based on actual marker locations are also provided (gray text).    A graphical representation of the implant rigid body at time T1 and T2 is also provided, with the two superimposed atop one-another to show relative motion.  Below the plot, we also provide a summary of the displayed relative motions and data quality results in a table format for quick copying and pasting.  The number of markers that were used for relative motion analysis is also provided within this table. Note that the codes to ‘RelativeMotionCalculator.m’ is not provided in this thesis since it cannot operate without its corresponding ‘RelativeMotionCalculator.fig’ file which generates the graphical user interfaces shown in this section.  However, a code-only version for calculating relative motion between multiple pairs of RSA datasets is provided in appendix B.7 (‘RelativeMotionCalculatorBatchScript.m’).  Appendix B.2 through to B.6 contains supplementary functions which are needed to run this particular non-graphical version of Relative Motion Calculator v1.4.       126   Appendix B   MATLAB Programs B.1 Batch Image Pre-Processing function []=ApplyConstrastFilter()   % DESCRIPTION ---------------------------------------------------------------------------------  % This function pre-processes radiographs to increase image contrast.  % It also reduces length of file names and converts files into .tif formats. % Two contrast settings are available:   %  (A) cage/phantom filter settings, or %  (B) distortion mesh filter settings % ---------------------------------------------------------------------------------------------   % Ask user for which settings to use prompt= 'Use "a" or "b" settings (a= for phantom/cage, b = for distortion mesh)?'; filtertype= inputdlg(prompt) filtertype = char(filtertype)  % Ask user which directory contains the BMPs they would like convert into TIF dirpath = uigetdir('C:\Users\Vivian\Google Drive\Project RSA\_RSA Trials'); cd(dirpath);  files = dir('*.tif');  for k = 1:numel(files) %for each tif in directory:     input_name = files(k).name     [path, name, extension] = fileparts(input_name)     output_name = fullfile(path, [name '.tif'])     I= imread(input_name);      % For option (A): cage/phantom 127      if filtertype == 'a'          I=rgb2gray(I); %convert to grayscale         Ia=imadjust(I,[.2 .5],[0 1] ); imshow(Ia);         output_name = fullfile(path, [name 'filterA.tif'])     end     % For option (B): distortion mesh     if filtertype == 'b'          I=rgb2gray(I);  %convert to grayscale         Ia=imadjust(I,[0 .6],[0 1],35);imshow(Ia);         output_name = fullfile(path, [name 'filterB.tif'])     end  % Output resulting image files imwrite(Ia, output_name); end   % Open file directory containing resulting image files system(sprintf('explorer.exe "%s\"', dirpath))      B.2 Relative Motion and Rigid Body Error function [psi, phi, theta, R_r,d_r, PtMotions_f,PtMotions_p, MTPM_f, MTPM_p,Vi_prime,RBE_bone,RBE_implant] = RelativeMotionWithMPTM(Xi,Yi,Ui,Vi)   % DESCRIPTION -------------------------------------------------------------------------------------- % This script calculates the relative motion of a rigid body of interest (segment B) from time T1 to T2,  % by using a reference rigid body (segment A), which also happens to be moving.  % The relative motion results are reported as: % (1) Euler angles % (2) Rotation and translation transforms  % (3) Maxium Total Point Motion (MTPM) using both fictive and actual marker locations % -----------------------------------------------------------------------------------------------------------   % VARIABLES ------------------------------------------------------------------------------------------ % INPUTS:     % Xi = coordinates of segment A (bone) markers at time T1      % Yi = coordinates of segment A (bone) markers at time T2     % Ui = coordinates of segment B (implant) markers at time T1      % Vi = coordinates of segment B (implant) markers at time T2           % CALCULATED:     % R_ref = the 3x3 rotation matrix to transform Xi to Yi     % d_ref = the 3x1 translation vector to transform Xi to Yi     % Ui_RFT1C_AM = T1 coordinates of implant markers transformed to T2 orientation and location     % Xi_RFT1C_AM = T1 coordinates of bone markers transformed to T2 orientation and location  128           % OUTPUTS:       % psi, phi, theta = the three Euler angles of rotation     % R_r = the 3x3 rotation matrix to transform Ui to Vi_prime, example:  R = [0 -1 0; 1 0 0; 0 0 1]     % d_r = the 3x1 translation vector to transform Ui to Vi_prime) example:  d = [2; 5; 4]     % PtMotions_f = a list of how much each implant marker moved from T1 to T2, using fictive marker locations     % PtMotions_f = a list of how much each implant marker moved from T1 to T2, using physical marker locations     % MTPM_f = MTPM using fictive marker locations     % MTPM_p = MTPM using physical marker locations          % Vi_prime = coordinates of segment B markers, at T2, if segment A did not move          % -------------------------------------------------------------------------------------------------------------   % Begin by updating all coordinates from cage coordinate system to implant coordinate system [Xi Ui coord_R coord_T axes_RFC]=ChangeOriginToImplantCentroid(Xi, Ui)         % STEP 1:  Yi = R_ref * Xi + d_ref  |   find R_ref, d_ref     X_bar = mean(Xi);     Y_bar = mean(Yi);     A=bsxfun(@minus,Xi,X_bar);      % A = Xi-[X_bar;X_bar;X_bar]     B=bsxfun(@minus,Yi,Y_bar);      % B = Yi-[Y_bar;Y_bar;Y_bar]     C=B'*A     [P,Gamma,Q]=svd(C)  % P*Gamma*Q'= SVD of C     Q_t=Q'  % Transpose matrix     if Gamma(2,2) == 0  % Is solution unique?  It is if Gamma(2,2) is not equal to zero         error('Gamma(2,2) = zero, solution not unique')     else          R_ref=P*diag([1,1,det(P*Q_t)])*Q_t % Rotation matrix         d_ref=Y_bar'-R_ref*X_bar' % Translation vector     end      % STEP 2:  Vi_prime = R_ref * Vi + (R_ref)'*d_ref   |   find Vi_prime     [rowcountVi, ~] = size(Vi) ;     R_refT=(R_ref)'     for row = 1:rowcountVi,          Vi_prime(row,:)= (R_refT*Vi(row,:)'- R_refT*d_ref)';     end      Vi_prime      % STEP 3:  Vi_prime = R_r * Ui + d_r    |   find R_r, d_r     U_bar = mean(Ui);     Vi_prime_bar = mean(Vi_prime);     A=bsxfun(@minus,Ui,U_bar); % A = Xi-[X_bar;X_bar;X_bar]:     B=bsxfun(@minus,Vi_prime,Vi_prime_bar); % B = Yi-[Y_bar;Y_bar;Y_bar]     C=B'*A;     [P,Gamma,Q]=svd(C); % P*Gamma*Q'= SVD of C       Q_t=Q'  % Transpose matrix     if Gamma(2,2) == 0  % Is solution unique?  It is if Gamma(2,2) is not equal to zero         error('Gamma(2,2) = zero, solution not unique') 129      else          R_r=P*diag([1,1,det(P*Q_t)])*Q_t % Rotation matrix         d_r=Vi_prime_bar'-R_r*U_bar' % Translation vector     end      % Find phi: s2=R_r(1,3) phi=asind(R_r(1,3)) %  Find psi:    -c2s1/c2c1=-s1/c1= - tan(1) = R_r(2,3)/R_r(3,3) psi=atand(-R_r(2,3)/R_r(3,3)) % Find theta:   -c2s3/c2c3=-tan(3) = R_r(1,2)/R_r(1,1) theta=atand(-R_r(1,2)/R_r(1,1))          Ui_RFT1C_AM = TransformRB(Ui,R_r,d_r); % To calculate MTPM with fictive markers, use calculated Ui coordinates (Ui_RFT1C_AM) in place of Vi_prime coordinates [rowcountU, ~] = size(Ui) ; % count number of markers provided for T1 implant markers [rowcountX, ~] = size(Xi) ; % count number of markers provided for T1 bone markers   % MTPM using fictive implant marker locations, 'MTPM_f' for row = 1:rowcountU     PairPtsA = [Ui_RFT1C_AM(row,:);Ui(row,:)];      PtMotions_f(row,:)  = abs(pdist(PairPtsA,'euclidean') ) ;  % the difference in distance as absolute value of each PairPts between T1 and T2 end PtMotions_f % List all point motions  MTPM_f = max(PtMotions_f) % The maximum total point motion using fictive marker locations   % MTPM using physical implant marker locations, 'MTPM_p' for row = 1:rowcountU     PairPtsB = [Vi_prime(row,:);Ui(row,:)];     PtMotions_p(row,:)  = abs(pdist(PairPtsB,'euclidean') ) ;  % the difference in distance as absolute value of each PairPts between T1 and T2 end PtMotions_p % List all point motions  MTPM_p = max(PtMotions_p)  % The maximum total point motion using physical marker locations   % RBE calculations Xi_RFT1C_AM = TransformRB(Xi,R_ref,d_ref);   for row = 1:rowcountU     PairPtsRBE_implant = [Ui_RFT1C_AM(row,:);Vi_prime(row,:)];      pdist_implant(row,:)  = abs(pdist(PairPtsRBE_implant,'euclidean') ) ;  % the difference in distance as absolute value of each PairPts between T1 and T2     RBE_implant = sqrt((1/size(pdist_implant, 1))*sum(pdist_implant.^2)) end   for row = 1:rowcountX     PairPtsRBE_bone = [Xi_RFT1C_AM(row,:);Yi(row,:)];  130      pdist_bone(row,:)  = abs(pdist(PairPtsRBE_bone,'euclidean') ) ;  % the difference in distance as absolute value of each PairPts between T1 and T2     RBE_bone = sqrt((1/size(pdist_bone, 1))*sum(pdist_bone.^2)) end       B.3 Coordinate System Conversion function [Xi Ui coord_R coord_T axes_RFC]=ChangeOriginToImplantCentroid(Xi_old, Ui_old)   % DESCRIPTION --------------------------------------------------------------------------------- % This function defines the implant coordinate system (ICS) using the four markers of the tibial implant. % It also sets the origin of ICS as the centroid of the rigid body defined by these four markers.  Lastly, it updates  % all 3D coordinates to use ICS and provides the transforms between cage coordinate system to ICS. % -----------------------------------------------------------------------------------------------------    % VARIABLES ------------------------------------------------------------------------------------- % INPUTS:     % Xi_old = tibial bone marker coordinates     % Ui_old = tibial implant marker coordinates in the following order: [P1 ; P2 ; P3 ; P4]         % P1 = most anterior point on tibial tray         % P2 = most medial point on tibial tray         % P3 = most lateral point on tibial tray         % P4 = most inferior point on tibial stem           % example Ui_old = [23 28 0;0 0 0;0 56 0;0 28 -55]   % OUTPUTS:     % Xi = new tibial bone marker coordinates based on implant coordinate system     % Ui = new tibial implant marker coordinates based on implant coordinate system     % coord_R = rotation-only transform, from cage coordinate system to ICS     % coord_T = translatio-only transform, from cage coordinate system to ICS     % axes_RFC = unit vector of the ICS axes % -----------------------------------------------------------------------------------------------------     % Find centroid of the tibial implant Pcentroid=mean(Ui_old)   %Find equation of the plane defined by P1, P2, P3 P1 = Ui_old(1,:) P2 = Ui_old(2,:) P3 = Ui_old(3,:) normal=cross(P1-P2,P1-P3)  A = normal(1,1) B = normal(1,2) C = normal(1,3) 131  D =( A*P1(1,1)+B*P1(1,2)+C*P1(1,3)) planefunction=[normal(1,1) ; normal(1,2) ; normal(1,3) ;D]  % equation of the plane  is  0 = Ax + By + Cz  + D   % Find closet point on the plane, from the centroid v = (D - sum(Pcentroid.*normal)) / sum(normal.*normal) x = Pcentroid+ v * normal  % 'x' is the closet point located on plane, from centroid   % Set the new coordinate system  z_axis = (P1 - x)/norm(P1 - x)   % new z-axis, defined as point 'x' to most anterior marker, P1 y_axis=(x-Pcentroid)/norm(x-Pcentroid)  %  new y-axis, defined as unit vector from centroid to 'x' x_axis= cross(y_axis,z_axis) % new x-axis, defined as the cross product of the above two  axes_RFC = [x_axis;y_axis; z_axis] % new unit vectors of ICS axes     % Find rotation transform (coord_R) from cage coordinate system to ICS x_old=[1 0 0]; y_old=[0 1 0]; z_old=[0 0 1]; coord_R =[dot(x_axis,x_old) dot(x_axis,y_old)  dot(x_axis,z_old) ;   dot(y_axis,x_old) dot(y_axis,y_old) dot(y_axis,z_old); dot(z_axis,x_old) dot(z_axis,y_old)  dot(z_axis,z_old)]   % Find translation transform (coord_T) from cage coordinate system to ICS coord_T = -Pcentroid' % set as negative, in order to shift translation in the other direction such that the centroid is at 0,0,0   % Apply transformation from one coordinate system to the other: Ui= TransformRB(Ui_old,[1 0 0;0 1 0; 0 0 1] ,coord_T); % first, apply only translation Ui= TransformRB(Ui,coord_R ,[0 0 0]');% now add rotation Xi= TransformRB(Xi_old,[1 0 0;0 1 0; 0 0 1] ,coord_T);  % first, apply only translation Xi= TransformRB(Xi,coord_R ,[0 0 0]'); % now add rotation    B.4 Rigid Body Transforms function Yi = TransformRB(Xi,R,d)   % DESCRIPTION ---------------------------------------------------------------------------------  % This function translates and rotates a rigid body of markers (Xi) when given  % a rotation matrix (R) and translation vector (d) % -----------------------------------------------------------------------------------------------------    % VARIABLES -------------------------------------------------------------------------------------  % INPUTS:     % Xi = initial 3D coordinates of markers, example:  Xi = [10 2 5 ; 12 5 0 ; 4 20 2 ; 14 1 30 ; 15 20 7]; 132      % R = the 3x3 Rotation Matrix (to transform Xi to Yi), example: R = [0 -1 0; 1 0 0; 0 0 1]     % d = the 3x1 Translation Vector (to transform Xi to Yi), example:  d = [2; 5; 4]   % OUTPUTS:     %Yi = final 3D coordinates of markers after rotations and translations are applied % -----------------------------------------------------------------------------------------------------    % Count number of markers provided: [rowcountX, columncountX] = size(Xi)    % Execute rotations and translations for row=1:rowcountX,         Yi_1(row,:)=(R*Xi(row,:)')'         Yi(row,:)= Yi_1(row,:)+d'                end Yi;    B.5 Condition Number  function [kappa_abs,kappa_r]=CN(Xi)   % DESCRIPTION --------------------------------------------------------------------------------- % This script calculates the Condition Number (CN) of a set of markers representing a rigid body % -------------------------------------------------------------------------------------------- ---------    % VARIABLES -------------------------------------------------------------------------------------  % INPUTS:  % Xi = a set of pt cloud coordinates in 3D, for example:   %          Xi =[25 25 25; -25 -25 -25;-25 -25 25;25 -25 -25;-25 25 25;25 -25 25;25 25 -25;-25 25 -25];      % OUTPUTS:     % kappa_abs = absolutes CN of the set of markers     % kappa_r = relative CN of the set of markers (normalized against volume of boundary) % -----------------------------------------------------------------------------------------------------   % STEP 1: Find mean of Xi X_bar = mean(Xi);   % STEP 2: Calculate residuals, A = Xi-[X_bar;X_bar;X_bar] A=bsxfun(@minus,Xi,X_bar);   % STEP 3: Calculate the CN [P_A,Gamma_A,Q_A]=svd(A); Gamma_A % (a) absolute CN: 133  kappa_abs = 1/ sqrt(Gamma_A(2,2)^2+Gamma_A(3,3)^2 ) % (b) relative CN: kappa_r = sqrt(Gamma_A(1,1)^2+Gamma_A(2,2)^2+Gamma_A(3,3)^2) / sqrt(Gamma_A(2,2)^2+Gamma_A(3,3)^2)     B.6 Euler to Quaternion Conversion  function [Mu_deg]=Euler2Quart(psi,phi,theta)   % DESCRIPTION ---------------------------------------------------------------------------------  % This script converts XYZ Euler angles into a single Quaternion magnitude % ---------------------------------------------------------------------------------------- -------------   % VARIABLES -------------------------------------------------------------------------------------  % INPUTS:     % psi = rotation in degrees about x-axis     % phi = rotation in degrees about y-axis     % theta = rotation in degrees about z-axis                  % OUTPUTS:       % Mu_deg = converted quaternion rotation in degrees          % -----------------------------------------------------------------------------------------------------       % Convert degree to radians psi=psi*pi/180;   phi=phi*pi/180;   theta=theta*pi/180;   % Calculate quaternion terms  c1=cos(psi./2); c2=cos(phi./2); c3=cos(theta./2);  s1=sin(psi./2); s2=sin(phi./2); s3=sin(theta./2);  Q=[c3.*c2.*c1 - s3.*s2.*s1,c3.*c2.*s1 + s3.*c1.*s2,c3.*c1.*s2 - s3.*c2.*s1,c3.*s2.*s1 + c2.*c1.*s3]   % Normalize quaternion in case of deviation from unity   Qnorms=sqrt(sum(Q.*Q,2))  Q=[Q(:,1)./Qnorms,Q(:,2)./Qnorms,Q(:,3)./Qnorms,Q(:,4)./Qnorms]   % Calculate combined angle of rotation  Mu_rad=2*asin(sqrt(sum(Q(:,2:4).*Q(:,2:4),2)));  Mu_deg=Mu_rad*180/pi    134  B.7 Relative Motion Calculator Batched Version function [RESULTS]=RelativeMotionCalculatorBatchScript(ALLPATHS) % DESCRIPTION ----------------------------------------------------------------------------------------------  % This scripts runs the Relative Motion Calculator for n pairs of T1-T2 RSA Exams. % ------------------------------------------------------------------------------------------------------------------    % VARIABLES ------------------------------------------------------------------------------------------------- % INPUTS:  %   ALLPATHS =  a nx2 CELL ARRAY containing the directory paths to each of the .csv files %               containing the 3D coordinates of marker on implant and bone.  The array should  %               have one pair of paths per row: 1st column pointing to T1 data, and column #2  %               pointing to T2 data.  Migration will be calculated row by %               row. The first 4 markers in these files must be the implant markers.   %        %               example:   %               ALLPATHS = {'C:\Users\Vivian\RSA 01\xyzpts.csv' 'C:\Users\Vivian\RSA 02\xyzpts.csv' %               'C:\Users\Vivian\RSA 01\xyzpts.csv' 'C:\Users\Vivian\RSA 03\xyzpts.csv' %               'C:\Users\Vivian\RSA 01\xyzpts.csv' 'C:\Users\Vivian\RSA 04\xyzpts.csv'}   % OUTPUTS: %   RESULTS =   a nx14 cell array containing the following relative motion information: %               (1)     Anterior Rotation [deg] %               (2)     External Rotation [deg] %               (3)     Varus Rotation [deg] %               (4)     Lateral Translation [mm] %               (5)     Proximal Translation [mm] %               (6)     Anterior Translation [mm] %               (7)     MTPM [mm] %               (8)     Condition Number of Implant %               (9)     Rigid Body Error of Implant %               (10)    Condition Number of Bone %               (11)    Rigid Body Error of Bone %               (12)    Number of Bone Markers Used  %               (13)    Magnitude of Translation [mm] -- calculated with 3D Pythagoras Theorem %               (14)    Magniutide of Rotation [deg] -- calculated as a Quaternion % ----------------------------------------------------------------------------------------------------------------- -    % Import each path: [r_,c_]=size(ALLPATHS); for rowtracker = 1:r_ PathMatrix=ALLPATHS(rowtracker,:);   % check paths to be valid length(PathMatrix); for k=1:length(PathMatrix) 135      CurPath=PathMatrix{k};     filetype=exist(CurPath);     if  filetype~=2% && filetype~=7 % filetype 2 = file; 7 = folder.  If neither holds true, warn user and stop script         msgbox(cat(2, {'Path not found:'}, CurPath), 'Warnings');         return % stop script if a path is not valid     end end PathT1=char(PathMatrix(:,1)); PathT2=char(PathMatrix(:,2));   T1_IMPORT = dlmread(PathT1,',',1,0); T2_IMPORT = dlmread(PathT2,',',1,0);   % reformat XROMM's T1_IMPORT into matrix form xyzcount =0; rowcount =1; [~,n1] = size(T1_IMPORT) % length of matrix for  k=1:n1     xyzcount = xyzcount+1;     T1_IMPORT_Formatted(rowcount,xyzcount) = T1_IMPORT(1,k) ;     if xyzcount ==3         xyzcount =0;     end     if xyzcount ==0         rowcount=rowcount+1;     end end T1_IMPORT=T1_IMPORT_Formatted; T1_IMPORT_Formatted=T1_IMPORT_Formatted(1:16,:);  % reformat XROMM's T2_IMPORT into matrix form xyzcount =0; rowcount =1; [~,n2] = size(T2_IMPORT); % length of matrix for  k=1:n2     xyzcount = xyzcount+1;     T2_IMPORT_Formatted(rowcount,xyzcount) = T2_IMPORT(1,k) ;     if xyzcount ==3         xyzcount =0;     end     if xyzcount ==0         rowcount=rowcount+1;     end end T2_IMPORT=T2_IMPORT_Formatted; T2_IMPORT_Formatted=T2_IMPORT_Formatted(1:16,:);      % REMOVE any markers that is missing from either T1 or T2 data,  136  % each marker must be present from both datasets to do relative motion calculations n1; %rowcountofT1 n2; %rowcountofT2 T1_IMPORT(all(T1_IMPORT==0,2),:)=NaN; % set rows with all ZEROS as NaN (i.e., treat them the same way) T2_IMPORT(all(T2_IMPORT==0,2),:)=NaN; % set rows with all ZEROS as NaN (i.e., treat them the same way) n = max(n1,n2) % which ever dataset has most entries  Missing_T1 = all(isnan(T1_IMPORT),2); % 1 = empty Missing_T1 = Missing_T1(1:16,:); Missing_T2 = all(isnan(T2_IMPORT),2); % 1 = empty Missing_T2 = Missing_T2(1:16,:); RowsToRemove = Missing_T1 + Missing_T2; %Find ones that are missing in either RowsToRemove( RowsToRemove>0 )=1; %Turn into binary RowsToRemove = logical(~RowsToRemove); T1_IMPORT =T1_IMPORT(RowsToRemove,:); T2_IMPORT =T2_IMPORT(RowsToRemove,:);   % For both files, the first four pts are presummed to be implant markers T1_IMPLANT = T1_IMPORT(1:4,1:3); T1_BONE = T1_IMPORT(5:end,1:3); T2_IMPLANT = T2_IMPORT(1:4,1:3); T2_BONE = T2_IMPORT(5:end,1:3);   % Calculate condition number [kappa_abs_bone,~]=CN(T1_BONE); [kappa_abs_implant,~]=CN(T1_IMPLANT);   % Calculate relative motions in Euler angles, translation and MTPM based on fictive marksers (also calc RBE): [psi, phi, theta,R_r, d_r, PtMotions_f,PtMotions_p, MTPM_f, MTPM_p,Vi_prime,RBE_bone,RBE_implant] = RelativeMotionWithMPTM(T1_BONE,T2_BONE ,T1_IMPLANT, T2_IMPLANT );   % Display aboves results in a summary table the user can copy directly [BoneMarkersUsed,~] = size(T1_BONE); RESULTSmatrix(rowtracker+1,:) = [psi, phi, theta,d_r(1,:),d_r(2,:),d_r(3,:), MTPM_f,kappa_abs_implant,RBE_implant,kappa_abs_bone,RBE_bone,BoneMarkersUsed, sqrt(d_r(1,:)^2+d_r(2,:)^2+d_r(3,:)^2),Euler2Quart(psi,phi,theta)];      end;      % Create table headers rowlabels= {'ANT ROT','EXT ROT','VAR ROT','LAT TRANS', 'PROX TRANS','ANT TRANS','MTPM','CN IMPL','RBE IMPL','CN BONE','RBE BONE','NUM OF BONE MARKERS USED','3D PYTH','QUARTERNION'}; RESULTS=num2cell(RESULTSmatrix);   % Convert results to cell array RESULTS(1,:) = rowlabels % Paste headers 137   Appendix C   RSA Algorithms C.1 Calibration In this section, the 11-parameter Direct Linear Transform (DLT) equations used for calibration are derived.  These derivations are the work presented by Kwon (1998), with a modification that within the image coordinate system’s reference frame, our projection center is defined in the direction away from the focal point (i.e., the X-ray projection center).   This deviation stems from the fact that the X-ray imaging requires the imaged object placed in between the X-ray’s projection center and the image plane, whilst a camera system would have the projection center (or foci) placed behind in front of the image plane and the object behind the image plane.  We start by defining variables shown on image below.  138   uvwP = [uo,vo,0]O = [x,y,z]I = [u,v,0]N=[uo,vo,-d]dx yz      If we define A as a vector from point N to point O, and B as the vector from point N to I, then: B=cA where c is a scaling constant A transform matrix can be used to relate the image-plane reference frame to the object reference frame:   [4]    where   [5] We can then say that,    [6] 139  Since the image coordinate system expresses measurements in terms of pixels, while the object coordinate system uses millimetres, it is necessary to introduce a unit conversion factor, λ for both the u-axis and v-axis.  Where:   [7]  For our specific C-arm, λ has been pre-determined from DICOM information to be 0.209961 [mm/pixel].  Although λu and λv should be identical in theory, due to optimization errors, the 11-DLT solution will indirectly calculate these two values individually, with typically very small differences between the two constants.    One can then rearrange the previous equation to shown that,       [8]  This can be represented as a pair of 11-parameter equations:      [9]  140  Equation [9] can then be rearranged in matrix form and used to solve for the 11 parameters (L1, L2, …, L11) with a least squared approach.  Since each calibration marker yields two equations, at least six markers are required to solve for the eleven unknowns. It should be noted that each of the eleven parameters contains intrinsic information about the X-ray system, which may be expressed as follows:             [10] 141    Additionally, parameters L1 to L11 can also be used to calculate the transform matrix, TI|o.    [11]  Since TI|o is essentially a rotation matrix which transforms coordinate reference frames between that of the object to image, by definition, the three row vectors of TI|o then represents the unit vectors of axis u, v, w in object reference frame.   [12]  The above equation will be used to define image orientation.  Specifically, these unit vectors are useful in creating a visualization of the relationship between the two reference frames.  In addition, the location of the projection center may be found be rearranging equation [9]:        142    [13]  C.2 Point Reconstruction In this section, we provide the equations used to reconstruct 3D coordinates of object points using their corresponding 2D coordinates on each of the two radiographs.  We refer to this as the ‘point reconstruction’ step, which is made possible once the 11-DLT parameters have been calculated for both X-ray views.      We do so by first rearranging equation [9] for each of the radiographs to remove the denominator as follows:       [14]  Since we have two radiographic views, we have four equations total, which can be expressed as a single linear algebraic equation (equation [15]).  For radiograph ‘A’, we used (uA,vA) to denote its 2D coordinates and L1A-L11A to denote its 11-DLT parameters.  Likewise, radiograph ‘B’ follows the denotations of (uB,vB) and L1B-L11B for its 2D coordinates and DLT parameters, respectively.        [15] We can then solve this equation with a least squared approach using the ‘linsolve’ command in MATLAB, which essentially does the following operation:      [16] 143  The resulting matrix, X = [x,y,z]T, thus contains the 3D coordinates of the object points that were observed on both radiographs ‘A’ and ‘B’.  We refer to this calculation as the 3D ‘point reconstruction’ step. 144   Appendix D   ICC Calculations To calculate ICCs using STATA, we first created .dta table files for each of the seven variables (MTPM and the 6 DOFs).  Each table contained three columns:  rating, target, and judge.   The ‘judge’ refers to the person carrying out the relative motion measurement, and is identified by a discrete number.  In our case, we had judges ‘1’ and ‘2’.  Similarly, the ‘target’ is also identified by a discrete number (1 to 147), referring to the independent observation that our two judges were measuring.  As mentioned in section 4.3.4, there were a total of 147 targets used in the ICC calculations.  Each of these targets were judged twice (one by each judge), with the measured relative motion entered as a continuous variable under the ‘rating’ column.  Thus, there were 147*2=294 entries within each of the 3-column table.   As an example, for AP translation, a table was first created (‘APTrans.dta’) before we ran the following commands in STATA: . use "C:\Users\Vivian\Google Drive\Project RSA\Statistical Analysis\ICC\APTrans.dta" . icc rating target judge   Which returns the following results:  Intraclass correlations Two-way random-effects model Absolute agreement Random effects: target           Number of targets =       147 Random effects: judge            Number of raters  =         2 145  --------------------------------------------------------------          rating      |        ICC          [95% Conf. Interval] ------------------+-------------------------------------------        Individual |   .9999444      .999923    .9999599         Average   |   .9999722      .9999615  .9999799 -------------------------------------------------------------- F test that  ICC=0.00: F(146.0, 146.0) = 36356.33        Prob > F = 0.000 Note: ICCs estimate correlations between individual measurements and between average measurements made on the same target.  

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