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Implementation and verification of a flexible optical tracker Semple, Mark Joseph 2015

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 © Mark Joseph Semple, 2015      IMPLEMENTATION AND VERIFICATION OF A FLEXIBLE OPTICAL TRACKER  by Mark Joseph Semple B.Sc, Mechanical Engineering  Queen’s University, 2012 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) February 2015       ii  Abstract Despite being demonstrably better than conventional surgical techniques with regards to implant alignment and outlier reduction, computer navigation systems have not faced widespread adoption in surgical operating rooms. We believe that one of the reasons for the low uptake stems from the bulky design of the optical tracker assemblies. These trackers must be rigidly fixed to a patient’s bone and they occupy a significant portion of the surgical workspace, which makes them difficult to use. In this thesis we introduce the design for a new optical tracker system, and subsequently we evaluate the tracker’s performance. The novel tracker consists of a set of low-profile flexible pins that can be placed into a rigid body and individually deflect without greatly affecting the pose estimation. By relying on a pin’s stiff axial direction while neglecting lateral deviations, we gain sufficient constraint over the underlying body. We used an unscented Kalman filter based algorithm as a recursive body pose estimator that can account for relative marker displacements. We assessed our tracker’s performance through a series of simulations and experiments inspired by a total knee arthroplasty. We found that the flexible tracker performs comparably to conventional trackers with regards to accuracy and precision, with tracking errors under 0.3mm for typical operating conditions. The tracking error remained below 0.5mm during pin deflections of up to 40mm. Our algorithm ran at computation speeds greater than real-time at 30Hz which means that it would be suitable for use in real-time applications. We conclude that this flexible pin concept provides sufficient accuracy to be used as a replacement for rigid trackers in applications where its lower profile, its reduced invasiveness and its robustness to deflection are desirable characteristics.      iii  Preface This thesis is an original piece of work by the author, Mark Semple. The original concept for the flexible tracker was conceived by Dr Antony Hodgson, who also provided guidance on methodology and revisions for the thesis writing. Early results of this work were presented at the 14th meeting of the International Society for Computer Assisted Orthopaedic Surgery (CAOS) in Milan, in June 2014. An extended technical paper has been submitted for the upcoming 2015 CAOS meeting in Vancouver, in June 2015. A provisional patent has been submitted to protect the intellectual property described in this thesis. The questionnaire documented in Chapter 1 and Appendix A was approved by the UBC Behavioral Research Ethics Board (H13-01360).       iv  Table of Contents  Abstract .............................................................................................................................................. ii Preface ............................................................................................................................................... iii Table of Contents ............................................................................................................................. iv List of Tables ................................................................................................................................... vii List of Figures ................................................................................................................................. viii List of Abbreviations ...................................................................................................................... xiii Acknowledgments .......................................................................................................................... xiv Chapter 1: Introduction and Background ...................................................................................... 1 1.1 Current State of Orthopaedic Navigation ............................................................................. 1 1.1.1 Lack of Evidence Supporting Clinical Benefits of Navigation ............................... 2 1.1.2 Cost of Navigation Systems .................................................................................... 4 1.1.3 Navigation Increases Operating Time ..................................................................... 5 1.2 Human Factors Issues with Navigation Technology ............................................................ 6 1.2.1 Human-Computer Interface Issues .......................................................................... 6 1.2.2 Line of Sight Issue ................................................................................................... 7 1.2.3 Orthopaedic Surgeon Questionnaire ........................................................................ 7 1.3 Rigid Tracker Hardware Design Issues ................................................................................ 9 1.3.1 Tracker Accuracy versus Usability: a Design Contradiction ................................ 11 1.3.2 Invasive Tracker Fixation Pins .............................................................................. 12 1.3.3 Selective Applicability of Rigid Trackers ............................................................. 13 1.4 Defining an Ideal Motion Tracker ...................................................................................... 15 1.5 Proposed Surgical Tracker Redesign .................................................................................. 15 1.6 Summary and Research Goals ............................................................................................ 16 1.7 Thesis Organization ............................................................................................................ 17        v  Chapter 2: Design, Theory, and Implementation ......................................................................... 18 2.1 Flexible Optical Tracker Design Concept ........................................................................... 18 2.1.1 Single-Marker Flexible Pins .................................................................................. 20 2.1.2 Double-Marker Flexible Pin Concept.................................................................... 21 2.1.3 Detailed Design Considerations ............................................................................ 23 2.2 Theoretical Principles of the Flexible Tracker Design ....................................................... 24 2.2.1 Pin Bending Variability ......................................................................................... 28 2.2.2 Flexible Pin Configuration Considerations ........................................................... 31 2.3 Implementation of Design .................................................................................................. 32 2.3.1 Geometrical Formulation ....................................................................................... 33 2.3.2 The Kalman Filter.................................................................................................. 34 2.3.3 Implementation of Unscented Kalman Filter Algorithm ....................................... 35 Chapter 3: Methods ........................................................................................................................ 42 3.1 Simulation Methods ............................................................................................................ 42 3.1.1 Combined Evaluation Metric ................................................................................ 44 3.1.2 Flexible Pin Configuration .................................................................................... 46 3.1.3 Static Simulation Method ...................................................................................... 48 3.1.4 Assumed Motions Simulation Methods ................................................................ 50 3.1.5 Motion Speed Simulations .................................................................................... 51 3.1.6 Process Noise Simulation Methods ....................................................................... 51 3.1.7 Pin Flexion Simulations ........................................................................................ 53 3.1.8 Anchor Digitization Uncertainty ........................................................................... 54 3.2 Experiment Methods ........................................................................................................... 54 3.2.1 Experiment Apparatus ........................................................................................... 55 3.2.2 Experiment Evaluation Metrics ............................................................................. 58 3.2.3 Static Experiment Methods ................................................................................... 59 3.2.4 Freehand Motion Experiment Methods ................................................................. 60 3.2.5 Pin Flexion Experiment Methods .......................................................................... 60 3.2.6 Translation Stage Experiment Methods ................................................................ 61         vi  Chapter 4: Results ........................................................................................................................... 63 4.1 Simulation Results .............................................................................................................. 63 4.1.1 Static Simulation Results ....................................................................................... 64 4.1.2 Assumed Motion Simulation Results .................................................................... 67 4.1.3 Motion Speed Simulation Results ......................................................................... 71 4.1.4 Process Noise Simulation Results ......................................................................... 73 4.1.5 Pin Bending Simulation Results ............................................................................ 77 4.1.6 Anchor Digitization Error Simulation Results ...................................................... 82 4.2 Experiment Results ............................................................................................................. 83 4.2.1 Static Experiment Results ...................................................................................... 84 4.2.2 Freehand Motion Experiment Results ................................................................... 87 4.2.3 Pin Flexion Experiment Results ............................................................................ 89 4.2.4 Translation Stage Experiment Results ................................................................... 97 Chapter 5: Discussion and Conclusion .......................................................................................... 99 5.1 Study Limitations .............................................................................................................. 100 5.2 Evaluating the Flexible Optical Tracker Design ............................................................... 103 5.3 Discussion on Tracking Performance ............................................................................... 104 5.3.1 Simulation Findings............................................................................................. 104 5.3.2 Experiment Findings ........................................................................................... 105 5.4 Discussion on Pin Deflections .......................................................................................... 107 5.4.1 Unpredictable Loading Conditions ...................................................................... 108 5.4.2 Mitigating the Effects of Pin Bending Variability ............................................... 110 5.4.3 Discrepancy in Pin Twist Simulations and Experiments ..................................... 111 5.4.4 Flexion Safeguard ................................................................................................ 111 5.5 Discussion on the Unscented Kalman Filter Algorithm ................................................... 111 5.5.1 Process Noise ...................................................................................................... 113 5.5.2 Rotation Motion Anomaly ................................................................................... 113 5.6 Clinical Considerations ..................................................................................................... 114 5.7 Thesis Contributions and Concluding Remarks ............................................................... 116 Bibliography .................................................................................................................................. 117 Appendices ..................................................................................................................................... 122 Appendix A –   CAOS Questionnaire ................................................................................... 122 Appendix B –   Kalman Filtering .......................................................................................... 134 Appendix C –   Double Marker Pin: Combined Loading...................................................... 136      vii  List of Tables Table 1: Several examples of state-of-the-art rigid trackers used in navigated TKA. The trackers’ configurations and marker types vary, but they all rely on an assumption of zero motion between the marker points and the underlying body. ............................................................. 10 Table 2: A summary of the UKF process noise selections, based on “expected” motion speeds. The pin twist process covariance is only applicable to the double-marker pins. The denominator of “30Hz” in all of the values refers to the optical tracker sampling frequency that we used. .......................................................................................................... 37 Table 3: A summary of the anisotropic marker measurement noise. These values are the basis for our measurement covariance matrix in the UKF, and indicate the relative confidence in the measurement model. ......................................................................................................... 40 Table 4: Descriptions and dimensions of each of the trackers used in the static simulation trials. Each tracker type used only the minimum number of markers to constrain the body. ........... 49 Table 5: Range of bone motion speeds for constant process noise trials. Because the measurement noise is much smaller than the default process noise, we expect that the algorithm will be minimally impacted by the changes of bone speed. .......................................................... 51 Table 6: Scaling factors used to vary the Process Covariance inputs. .................................................. 52 Table 7: Simulation times for each tracker and solver algorithm. The UKF algorithm ran consistently faster than the optimization algorithm. ............................................................... 66 Table 8: Summary of results from the high-precision translation stage experiments. Both the single- and double-marker flexible pin types were translated four times in both the global Y and Z directions. ...................................................................................................... 98        viii  List of Figures Figure 1: A comparison between two simple rigid trackers to demonstrate the effect of marker configuration on target registration error (TRE). In both cases, the measurement jitter is fixed, and the worst-case scenario is shown in red. The TRE δ is markedly greater in case (b) with the markers closer together. .............................................................................. 11 Figure 2: The three main types of long-bone pin fixation, showing cross-section of tubular bone segment. Conventional rigid trackers use bicortical fixation for a secure attachment to the bone segment. (adapted from (Jung 2007)). ..................................................................... 13 Figure 3: Overarching process for the tracker-redesign project. This thesis focuses on the detailed design, implementation, and performance testing stages. The validation of specific usability gains in selected surgical procedures will require future study. .............................. 16 Figure 4: An example of a conventional rigid tracker attached to a femur. At least two bicortical pins are required to fix the tracker to the bone. Any dislocation of the tracker from its calibrated position can result in erroneous readings (typically undetectable by the system). .................................................................................................................................. 19 Figure 5: A demonstration of how the proposed tracker system could conceivably be attached to a femur for use in a navigated surgery. The single-marker pin type requires six pins to fully constrain a body. ............................................................................................................ 19 Figure 6: The single-marker flexible pin design, progression from hand-sketch to experimental prototype. The components of the flexible tracker are the marker, the adaptor, the shaft, and the anchor. ....................................................................................................................... 21 Figure 7: The double-marker flexible pin design, progression from hand-sketch to working experimental prototype. This design is similar to the single-marker design, except now there are two markers spaced apart by a cross-bar component, making the tracker look like a “T”. ............................................................................................................................... 22 Figure 8: An example of how the double-marker flexible pins could be attached to a femur for use in a navigated surgery. The double-marker type only requires three pins to fully constrain the body. ................................................................................................................. 23 Figure 9: The pin’s design enables the markers to move relative to one another, which nullifies the assumption of rigidity. The markers are still partially constrained by the flexible shaft that holds them to the anchor and bone ......................................................................... 25 Figure 10: We model the flexible tracker pin as a slender cantilevered beam. The marker (pin tip) is relatively free to move in the transverse direction compared with the axial direction. ...... 26 Figure 11: A representation of the “bending subspace” concept, which consists of all possible marker positions that can be achieved by deflecting the pin. ................................................. 27     ix  Figure 12: Examples of three distinct loading conditions and their respective bending trajectories (matched by colour). Pin-tip point load (blue) has the greatest contraction between the marker and anchor of the three examples. .............................................................................. 28 Figure 13: The envelope between the two extremes of bending, resulting from simple loading conditions. The blue trajectory shows pin-tip loading (maximum contraction), and the black trajectory shows a rigid-link (no contraction). Our chosen model (red) minimizes the error by being halfway between the two extremes. .......................................................... 30 Figure 14: Parallel pins can constrain rotation, and perpendicular pins can constrain translation, so a combination of both configurations is needed to fully locate the body in space. ................ 31 Figure 15: Geometrical model of flexible tracker system, showing only one single-marker pin. We define a series of Euclidean transformations from the global coordinates of the sensor to the marker point. ................................................................................................................ 33 Figure 16: Sample screenshot from the simulation visualizer for 2D (left) and 3D (right). ..................... 43 Figure 17: The CEM on anatomical femur-based coordinates. The rotation scale factor f calculates the target error due to rotation at either epicondyle................................................................ 45 Figure 18: Configuration of flexible pins used for the 2D Simulations. The long baseline distance between markers creates a high angular resolution over the body. ........................................ 46 Figure 19: The single-marker pin configuration for 3D simulations. The same configuration was used in our experiments. The orthogonality of the pins helps to constrain translations about all directions. The separation between parallel pins constrains rotations about each axis. ........................................................................................................................................ 47 Figure 20: The double-marker pin configuration for 3D simulations. The same configuration was used in our experiments.......................................................................................................... 47 Figure 21: The four loading conditions that we simulated. The mid-pin loading case is the implicit bending model for the Kalman filter algorithm. ..................................................................... 53 Figure 22: The prototype flexible tracker pins, with dimensions. The single- and double-marker pin types both have identical anchor and shaft components. ........................................................ 56 Figure 23: The experimental set-up, showing our rigid body, the single-marker flexible pins, and the NDI rigid tracker component............................................................................................ 57 Figure 24: The three distinct modes of deflection of the double-marker pin type: in-plane, out-of-plane, and axial twist. The in-plane mode is the most sensitive, because the resulting marker positions are a function of the cross-bar angle. .......................................................... 61 Figure 25: The test piece was firmly attached to a high-precision linear translation stage, which was securely attached to a work table. The translation stage has a maximum range of 13mm, and a precision of 0.005mm. .................................................................................................. 62 Figure 26: A comparison of simulated 2D static tracking accuracies for the flexible and rigid trackers, each analyzed with an optimization and a Kalman filter algorithm. The two algorithms granted similar combined accuracies for both trackers. ....................................... 64     x  Figure 27: A comparison of simulated 3D static tracking accuracies for the flexible and rigid trackers, each analyzed with an optimization and a UKF algorithm. ..................................... 65 Figure 28: The combined error throughout the simulated 2D translation trial. The 5th, 50th and 95th percentiles of the error are shown as red lines. The error shows no variations compared with the 2D static case on the right in Figure 26. ................................................................... 67 Figure 29: Summary of the CEM data set for the 2D simulated translation trial, compared with the static case. The width to the distribution was added for visual effect. ................................... 68 Figure 30: A summary of our 2D assumed motion simulations. The translation trials show no loss in accuracy compared to the static case. The rotation case, however, shows a significant jump in error compared to static case. The errors in rotation are relatively low, compared with the accuracy requirements for the system. ..................................................................... 69 Figure 31:  Summaries of the two flexible tracker pins’ assumed motion simulations in 3D. Both types of flexible pins demonstrated similar behavior, consistent with the 2D simulations: no increase of error due to translation, and a noticeable jump in error during rotation. ......... 70 Figure 32: A summary of the 2D constant process noise, varying motion speed simulations.  At default process noise values, the translation appears to be insensitive to motion speed, but the rotation case shows errors with increasing speeds. .................................................... 71 Figure 33: Results of the 3D motion speed simulation trials. The top axis shows the single-marker pin, and the bottom axis shows the double-marker pin. Both tracker types demonstrate similar trends at high speeds: minimal effect for translation and greater effect for rotation. .................................................................................................................................. 72 Figure 34: A summary of the 2D constant speed, varying process noise simulations, showing the static, translation, and rotation cases. In each case, the input process noise values were scaled by the factors shown on the X-axis. The hollow points are at the default process noise (ie. a scale factor of 1).  In the static case (left) there is a distinct transition between two plateaus as the process noise is decreased. In translation case (centre) the error begins to grow as process noise is decreased. Rotation (right) shows no effect. ................... 74 Figure 35: A summary of the 3D constant speed, varying process noise simulations for the single-marker flexible pins.  In each case, the input process noise values were scaled by the factors shown on the X-axis. The hollow points are at the default process noise. The trends in this case are the same as the 2D analysis. ................................................................ 75 Figure 36: A summary of the 3D constant speed, varying process noise simulations for the double-marker flexible pins. In each case, the input process noise values were scaled by the factors shown on the X-axis. The hollow points are at the default process noise. The trends in this case are the same as the previous analyses, except that we see a lag behavior in the rotation case at low PCV levels. .................................................................... 76 Figure 37: The results of the 2D pin bending simulations. Four different loading conditions were tested, each with tip displacements to 40mm, in 5mm increments. The assumed bending model was a mid-pin point load. The worst case tracking errors remain under 0.5mm, even with pin deflections up to 40mm. .................................................................................. 78     xi  Figure 38: The results of the 3D pin bending simulations for the single-marker flexible tracker. Four different loading conditions were tested, each with tip displacements to 40mm, in 5mm increments. The assumed bending model was a mid-pin point load. The tracking errors remain submillimetric, even under the worst case loading conditions, up to 30mm of tip deflection. ............................................................................................................................... 79 Figure 39: The results of the 3D pin bending simulations for the double-marker flexible tracker. The top plot shows in-plane bending, and the bottom plot shows out of plane bending. The in-plane case is significantly more sensitive to bending than the out of plane case. ....... 80 Figure 40: Double Marker pin twist simulation. The pin was twisted to a maximum angle of 45º, in increments of 5º. Unlike in the bending cases, there are no different trajectories or loading conditions for the pin twist. The accuracy remains largely unchanged during the twist simulation. ..................................................................................................................... 81 Figure 41: Results of the simulated anchor digitization error trials, showing the 2D and 3D results. In the static case, anchor errors have little impact on tracking accuracy, even up to 10mm of anchor position error. ......................................................................................................... 82 Figure 42: Sample trace of the estimated rigid body position X component in static experiment. There is a brief settling behavior as the algorithm locates the body (lasting well under a second). .................................................................................................................................. 84 Figure 43: The precision combined evaluation metric over the duration of the single-marker pin static experiment trial, showing deviations from the mean pose. ........................................... 85 Figure 44: A comparison between experiment and simulation under static conditions, for all tracker types. The experiment data are reported as precisions, and the simulation data as accuracies – though in the static case the two metrics are likely quite comparable. .............. 86 Figure 45: Sample position trajectory during the single-marker pin freehand motion trial (translation case). The trajectory of the NDI rigid body is super imposed on the axis, and sit indistinguishable from the estimated trajectory curves. .............................................. 87 Figure 46: The discrepancy between the single-marker flexible pins and the NDI rigid tracker during the freehand translation trial, reported as a CEM. After a settling period, the discrepancy remains fairly consistent throughout the trial. .................................................... 88 Figure 47: A comparison between experimental and simulated motion trials. The experiments are reported as discrepancy CEMs, and simulations are reported as accuracy CEMs, so a direct comparison is not entirely valid. .................................................................................. 89 Figure 48: Small, medium, and large out-of-plane deflections of the double-marker flexible tracker pin.  The top row shows a measure of the horizontal component of the deflection, and the bottom row shows the corresponding precision CEM. ..................................................... 90 Figure 49: Small, medium, and large in-plane deflections of the double-marker flexible tracker pin.  The top row shows a measure of the horizontal component of the deflection, and the bottom row shows the corresponding precision CEM............................................................ 91     xii  Figure 50: Small and large axial twist of the double-marker flexible tracker pin. The top row shows detected twist angle, and the bottom row shows the corresponding precision CEM. These results are inconsistent with our simulations, which showed an insensitivity to pin twist. ....................................................................................................................................... 92 Figure 51: Small, medium, and large deflections of the single-marker pin, bent with an approximated mid-pin point load. This loading condition is the implicit bending model for our Kalman filter. The top row shows the detected pin bending trajectory, and the bottom row shows the corresponding precision CEM............................................................ 93 Figure 52: Small, medium, and large deflections of the single-marker pin, bent with an approximated pin-tip point load. We expect greater error for these loading conditions. The top row shows the detected pin bending trajectory, and the bottom row shows the corresponding precision CEM. ............................................................................................... 94 Figure 53: Plot of CEM versus pin deflection for the double-marker in-plane bending case, comparing the experiment and simulation data sets. The experiment data was fit linearly, and agrees with the simulation results (shown as dashed lines). ............................................ 95 Figure 54: A plot of CEM versus pin deflection for the double-marker out-of-plane bending case, comparing the experiment and simulation data sets. The experimental results lay outside of the envelope predicted by simulations. The experimental data was fit linearly. ................ 95 Figure 55: A plot of CEM versus pin deflection for the single-marker pins, comparing experiment with simulation. The blue curves represent pin-tip point load, and the red curves represent mid-pin point loads. The experimental data was fit with a cubic polynomial. The results are consistent in that pin-tip loading creates larger tracking errors than mid-pin loading. ............................................................................................................................. 96 Figure 56: Motion profile for both translation stage trials: in-plane (y) and out-of-plane (z). Each trial consisted of two “there-and-back” translations about the stage’s maximum range. Rest-periods were manually segmented and are shown in red. .............................................. 97 Figure 57: A demonstration of how different bending trajectories can invoke errors in the pose estimation. (Left) Anchor-aligned comparison between two different bends. (Right) The only input to the algorithm is the location of the marker, so bending variability is interpreted by shifting the anchor point. .............................................................................. 108 Figure 58: A demonstration of in-plane bending. Red shows mid-pin loading, blue shows pin tip loading, and black shows a rigid link deflection. The crossbar segment amplifies errors associated with bending variations, leads to considerable tracking errors (~5mm at 40mm deflection for certain loading conditions). ................................................................ 109 Figure 59: An example of a design solution for overcoming errors associated with bending variability. Concentrating the pin’s flexure by adding a compliant joint segment would make pin bending more predictable. .................................................................................... 110       xiii  List of Abbreviations CAOS Computer assisted orthopaedic surgery CAS Computer assisted surgery CEM Combined evaluation matrix CMM Coordinate measurement machine ECD Exact constraint design DOF Degrees of freedom KF Kalman filter NDI Northern Digital Inc. OR Operating room PCV Process covariance TKA Total knee arthroplasty TKR Total knee replacement TRE Target registration error UBC University of British Columbia UKF Unscented Kalman filter         xiv  Acknowledgments I would like to thank my research supervisor, Dr. Antony Hodgson, for his guidance and support in this project, and for being highly accommodating to working across the country. I would also like to thank Dr. Cari Whyne, and everybody at the Orthopaedic Biomechanics Laboratory at Sunnybrook Hospital for hosting me in their lab. Jeff Stanley, and Northern Digital Inc. for lending a Polaris system for my experiments. My fellow students in the Neuromotor Control Lab at UBC: Jake McIvor, Jeremy Kooyman, Andrew Meyer, and Vivian Chung. Your feedback has been a great deal of help. And lastly, the Natural Sciences and Engineering Research Council of Canada for their financial support.  Thesis body    1  Chapter 1  Introduction and Background This chapter presents the background and motivations for this work, which gives context to project objectives. We then provide an overview of the remaining thesis chapters. 1.1 Current State of Orthopaedic Navigation Computer Assisted Orthopaedic Surgery (CAOS, also known as navigation) is a technology that has emerged over the past 15-20 years to address the problem of early failures of joint implants associated with significant malalignment (Bae 2011; Catani 2013; Hodgson 2008). Despite being demonstrably better than conventional techniques with regards to implantation accuracy and reduction in outliers, CAOS has yet to succeed in being widely adopted; less than one third of surgeons use navigation regularly for total knee arthroplasty (TKA) procedures (Friederich 2008). The most commonly cited reasons for the low uptake include a lack of substantial data relating use of CAOS to improved long-term patient outcomes (Heesterbeek 2013), the comparatively high cost of the technology (Laskin 2006), and     2  increased operating time (Sikorski 2003). Although less frequently cited as a deterrent for use of CAOS technologies, several authors have indicated that current CAOS optical tracker hardware is large, cumbersome and challenging to use (Bellemans 2009; Laskin 2006; Lionberger 2007). This sentiment is echoed in testimonials from experienced orthopaedic surgeons that we acquired as a part of a questionnaire (discussed in Section 0). The focus of this project is to implement a novel optical tracking system (based on a design concept proposed by Hodgson (2012)), and to verify the system’s tracking performance through a series of simulations and experiments inspired by a navigated total knee arthroplasty (TKA). In the next three sections we provide an overview of the most commonly cited factors affecting uptake of navigation systems, and then turn our attention to the relatively limited literature on human factors issues in CAOS. 1.1.1 Lack of Evidence Supporting Clinical Benefits of Navigation It is widely accepted that implant alignment accuracy affects long term survival rates. Malaligned components can cause uneven loading patterns, leading to accelerated component wear, and ultimately implant loosening and failure (Wasielewski 1994). Jeffery (1991) demonstrated that failure risk rose significantly for knee implants aligned outside of a ± 3°  window in the coronal plane (varus/valgus). A meta-analysis by Mason (2007) found that mechanical axis errors beyond Jeffery’s ± 3° occurred in 31.8% of non-navigated cases (Mason 2007).     3  CAOS technologies emerged as a promising means of decreasing the number of outliers in joint arthroplasty. Since the technology’s inception, several studies have compared the alignment accuracies achieved between navigated and conventional TKA approaches. In contrasting to the 31.8% of outliers from conventional TKA, Mason (2007) also found that only 9% of navigated knees were outliers. Despite this significant reduction in outliers, there are many who doubt the clinical efficacy of CAOS technology.  “Although it is philosophically difficult to argue against any system that increases accuracy, it is reasonable to question whether this improvement, which is often in the range of 1-1.5 degrees, will have a salutary effect on the long-term implant survival.”     (Laskin 2006)  As correct implant alignment can positively influence its survival, and CAOS can improve alignment, logic would dictate that CAOS can positively impact prosthesis survival. Unfortunately, at this time, there is a lack of published evidence confirming or denying the clinical benefits of CAOS. Advocates defend CAOS on the basis that the technology is still in its infancy, so not enough time has passed to accumulate enough data to achieve statistically significant test results. Skeptics believe that the clinical impact of CAOS is too insignificant ever to be detected statistically (Bellemans 2009; Heesterbeek 2013). Many authors, however, argue that it is still prudent to use CAOS to reduce outliers (Laskin 2006). While the lack of supporting evidence creates a cloud of uncertainty surrounding the     4  necessity of CAOS, other reasons can partially explain the low rates of CAOS uptake: high cost, additional operating time, and usability issues. 1.1.2 Cost of Navigation Systems CAOS is commonly criticized for its comparatively high cost  (Bellemans 2009; Novak 2007; Sikorski 2003). If CAOS can reduce the rate of implant failures, a clear economic benefit arises from having to perform fewer implant revision surgeries, which are typically much more expensive than the primary procedure (Gøthesen 2013; Lavernia 2006). Several studies have analyzed the cost-effectiveness of CAOS technologies, finding that high-volume hospitals would benefit most from the technology and, that to be cost effective, CAOS only needs to marginally improve survival rates (Novak 2007; Slover 2008). Gøtheson (2013) showed that the 10-year implant survival rate would only have to improve by 0.1% (for a Norwegian centre that does 250 TKAs per year) for CAOS technology to be considered cost-effective (Gøthesen 2013). Continued efforts strive to overcome this financial barrier and thus help CAOS proliferate. Some authors advocate concentrating arthoplasty procedures into regional high-volume centres in order to maximize the cost-effectiveness of CAOS systems while achieving better implant alignment results (Slover 2008). Other groups are designing low-cost navigation systems. For example, Claron Technologies (Toronto, ON) market their MicronTracker product as a low-cost optical tracker for navigated interventions, though not explicitly for orthopaedic applications.     5  1.1.3 Navigation Increases Operating Time While CAOS simplifies some surgical tasks, such as component alignment and spatial measurement, the technology increases total operating time because it requires several additional pre- and intra-operative steps (Bäthis 2004; Haritinian 2013; Sikorski 2003). Without sacrificing quality of care, surgeries should be kept as short as possible because operating room (OR) costs are very high (approx. $20 per minute), operating space is limited, and longer surgeries increase patient risk (ie. higher risk of infection and complication from anaesthetic) (Hodgson, 2008). Bäthis (2004) found that once the initial learning curve is overcome, navigated surgeries took just 14 minutes longer than conventional techniques – an increase they claim is “acceptable in clinical practice” (Bäthis 2004). Surgeons who responded to our CAOS Questionnaire (see next section) described the time-increase as “negligible”, and that “surgical time is actually decreased significantly”. There is little consensus regarding the extent or significance of any increased operating time resulting from navigation. Recent innovations in the field aim to optimize surgical workflow and decrease operating time, which when refined could definitely make CAOS faster than conventional approaches. For example, Praxim’s adjustable cutting blocks have been shown to reduce CAOS operating time by 14.8 minutes (compared with conventional cutting blocks) without sacrificing implantation accuracy (Suero 2012).      6  1.2 Human Factors Issues with Navigation Technology We found that issues relating to system design and human-factors are infrequently discussed in the literature. Some authors express concern over the cumbersome nature of existing CAOS systems, suggesting that improvements to the user experience could reduce barriers to adoption and thus help proliferate the technology (Bellemans 2009; Laskin 2006). Indeed, there have been numerous innovations introduced related to streamlining the intraoperative workflow and improving the visibility of the markers. Despite these improvements, there has been little attention paid to the bulky and obtrusive hardware components, which will be the area of focus for this thesis. 1.2.1 Human-Computer Interface Issues The computer-user interface is difficult to design because it must allow for complex 3D spatial manipulations using only limited user input. Traditional computer controls (eg. a mouse and keyboard) are impractical in the OR largely because of sterility and space issues. As a result, many commercial CAOS systems rely on foot-pedal controls or voice-activated controls, which often further frustrate 3D navigation efforts (De Momi 2008). There have been promising advancements in the area that use intuitive gesture control systems, such as the Leap Motion Controller (Leap Motion, San Francisco). While this issue is a contributing factor to the technology’s learning curve and may deter some surgeons from routinely using navigation, our work will not address the difficulty of use of the computer controls.      7  1.2.2 Line of Sight Issue Optical tracking for navigation requires a direct line of sight between the markers and the camera, which restricts OR staff movement (Schep 2003). Maintaining a line of sight can be difficult in small or crowded ORs, and some surgeons find it impossible to work out an appropriate arrangement (Laskin 2006). Technological innovators have aimed to combat this issue, notably through the introduction of magnetic tracking technologies which do not interfere with non-magnetic objects (such as human body segments). A prominent magnetic navigation technology was the Zimmer Medtronic Axiem system, which was first used in 2005. While the technology was a significant advancement in CAOS usability, the per-case cost exceeded the benefits of its use and it suffered from distortion around metallic objects (Lionberger 2007). Although it eventually fell out of use entirely, Zimmer’s significant investment towards the development of this technology suggests that CAOS ease-of-use is an issue that warrants attention. 1.2.3 Orthopaedic Surgeon Questionnaire While the literature debates a wide range of issues relating to surgical navigation technology, we sought direct feedback from frontline users about undocumented aspects of the technology relating to hindered usage. We conducted a survey of orthopaedic surgeons with experience using CAOS to investigate their perceived advantages and disadvantages associated with the technology. The complete set of questions is in the Appendix. We     8  distributed the survey to 82 surgeons associated with the International Society for Computer Assisted Orthopaedic Surgery and received back 13 responses, which is a sample size too small to conduct any statistically significant analysis. Their written opinions however, provided useful insight into their frustrations with existing navigation technology. One surgeon said:  “I hated navigation when I first started using it as I felt it made the operation much harder. Undoubtedly you get much better at it and eventually these concerns are no longer a problem.”  This highlights the “steep learning curve” barrier to adoption issue that we mentioned earlier. One surgeon said “the problems come from the hardware,” and, “[I] regret that there [has been] no significant technical improvement of the devices since more than 15 years”.  Another surgeon commented “I now have to hold the saw upside down to cut the lateral side of the tibial plateau” as a result of the tracker component attached to the patient’s bone. Amongst the group of responders, we observed that the potential benefits of CAOS are understood and appreciated, but the current form of the technology is more cumbersome and challenging to use than they would like it to be. Our survey supports a number of the contentions with rigid trackers. In particular, there were comments that suggested that the bulkiness of the optical trackers is an obstacle worth addressing for navigation technology.     9  1.3 Rigid Tracker Hardware Design Issues Conventional tracker assemblies are large and rigid by design, as shown in Table 1. There is a fundamental assumption of zero relative motion between the constellation of markers and the bone. The designers intentionally make the tracker bulky to ensure rigidity and achieve adequate angular resolution of measurements. Once attached, the tracker effectively becomes an extension of the bone, and the positions of the markers on the tracker can be directly related to the position and orientation of the segment.  The testimonials from our survey of surgeons suggest that navigation hardware is challenging to use and in need of innovation. In the following sections, we explore some of the literature related to shortcomings of conventional optical tracker hardware, including their contradictory design, the risks associated with rigid bicortical fixation, and their limited range of applicability. Afterwards, flowing from this investigation, we present a novel tracker design concept to address many of the weaknesses of conventional trackers.       10  Brainlab®  A passive tracker with reflective spheres. Tracker attaches to cutting blocks. Marketed as fast, reliable, and easy to use.  ORTHOsoft  Navitracker™  A passive, multifaceted reflective optical tracker. Marketed to have increased accuracy and increased visibility compared with other trackers. Stryker nGenius Tracker                 A refined, state-of-the-art active tracker that attaches to cutting blocks. Marketed as being less invasive, easy to use, and more time-efficient than other trackers. ExactechGPS®  Active LED tracker. Marketed as “ergonomically designed for ease of use”, fast, and highly accurate. Table 1: Several examples of state-of-the-art rigid trackers used in navigated TKA. The trackers’ configurations and marker types vary, but they all rely on an assumption of zero motion between the marker points and the underlying body.     11  1.3.1 Tracker Accuracy versus Usability: a Design Contradiction A tracker’s target registration error (TRE) is a measure of how accurately it can track a projected point, such as a tool tip or anatomical landmark. The TRE is a function of the geometrical configuration of the tracker’s marker points and the distance from the tracker to the target. As a result, a tracker will provide better accuracy if the markers are spread farther apart (West 2004), as demonstrated in Figure 1. This creates a drive towards maximizing the size of optical tracker components. Figure 1: A comparison between two simple rigid trackers to demonstrate the effect of marker configuration on target registration error (TRE). In both cases, the measurement jitter is fixed, and the worst-case scenario is shown in red. The TRE δ is markedly greater in case (b) with the markers closer together.           (a) Widely separated markers(b) Narrowly separated markers    12  While a larger tracker is more accurate in projecting target points, they come at the cost of occupying a greater portion of the operating space. Their large size increases the risk of being knocked from the calibrated position. This is a critical source of systematic error as optical tracking requires having no relative motion between the cluster of markers and the body to which they are attached. Furthermore, the larger the tracker, the greater the lever-arm about the bone-pin interface. This further increases the risk of tracker displacement compared with smaller trackers. Surgeons may be forced to adopt a new surgical approach to avoid interfering with the cumbersome bone tracker. Recalling our questionnaire (Section 1.2.3), one surgeon mentioned having to hold their saw upside down to accommodate the bulky optical tracker hardware. This highlights the tradeoff between the accuracy of larger tracker versus the maneuverability associated with smaller trackers. 1.3.2 Invasive Tracker Fixation Pins In typical navigated orthopaedic surgeries, rigid trackers are attached to the patient’s anatomy by a set of bicortical fixation pins, which are 2-5mm in diameter. The pins must be large enough to provide adequate stiffness and must penetrate both bony cortices to ensure tracker rigidity. The holes that these pins form in the bone create a stress concentration, which can lead to a post-operative stress fracture requiring further surgical intervention (Ossendorf 2006). One study reported a pin site stress fracture incidence of 1.3% at their local centre,     13  which is non-trivial considering the severity of the outcome (Beldame 2010). We note, however, that no other studies have reported this frequency of complication. These pins have also been reported to damage surrounding muscle and soft tissue around the fixation sites (Lionberger, 2007). Many authors recognize the risk associated with the current fixation technique, and agree that unicortical fixation would be preferable for the patient if only the trackers would maintain their fixed orientation (Laskin 2006). 1.3.3 Selective Applicability of Rigid Trackers Existing rigid trackers are generally too large to use on small bones such as the patella, the scaphoid, the radius, the calcaneus, or smaller fracture segments in general. We envision that orthopaedic applications on these bones and others like them could benefit from computer                                   Figure 2: The three main types of long-bone pin fixation, showing cross-section of tubular bone segment. Conventional rigid trackers use bicortical fixation for a secure attachment to the bone segment. (adapted from (Jung 2007)).     14  navigation to reduce reliance on fluoroscopy, in increase control and visualization, and to enable minimally invasive approaches.  Like many orthopaedic procedures, hand and wrist surgeries heavily rely on fluoroscopy to visualize hidden anatomy. These areas are of particular concern, however, because the surgeon must remain in such close proximity to the operative field. As such, they receive a greater dose of radiation exposure to their hands and thyroid (Bahari 2006). This subdiscipline could benefit from computer navigation to reduce the reliance on fluoroscopy exposure, though the bones in the wrist are too small to attach conventional rigid trackers. To overcome the incompatibility between computer navigation and small wrist bones, Smith (2012) developed a wrist-stabilization device to immobilize the patient’s scaphoid relative to a trackable reference frame. An optical tracker can be fixed externally to the device which is coupled to the scaphoid. Their wrist-stabilization system proved to be effective in a cadaveric study, and is moving forward into preclinical trials (Smith 2014), but it is possible that it would be even more desirable to directly track the bone, either independently of or in conjunction with using an immobilization device.  Similarly, a smaller, less intrusive tracker design could enable use of navigation in other situations in which conventional optical trackers would be impossible to use.      15  1.4 Defining an Ideal Motion Tracker Welch (2002) defined an ideal universal motion tracking system as being small, self-contained, complete, accurate, fast, immune to occlusion, robust, tenacious, wireless, and inexpensive. In effect, their ideal tracker would be perfectly suitable to practically every motion tracking application, and also be minimally inhibitive to the user’s actions. Welch acknowledges that no single motion tracking technology can meet the full set of criteria. Different systems offer unique advantages over one another, and the technology by nature is highly application-specific. Generally, though, tracking technology should be more robust, more accurate, less expensive, and less invasive. An additional important concern is to not inhibit the user’s actions. 1.5 Proposed Surgical Tracker Redesign The work described in this thesis builds on a conceptual design for an optical tracker that was first proposed by Hodgson to address the ergonomic shortcomings of conventional rigid tracker designs (2012). This thesis constitutes the detailed design, implementation, and performance testing stages of this larger design project. We aim to demonstrate that the proposed tracker system could be used to track a rigid body to an acceptable level of accuracy.     16   1.6 Summary and Research Goals It is well accepted in the literature that CAOS technology enables surgeons to implant orthopaedic prostheses more accurately and with fewer outliers compared with conventional techniques. Despite this considerable advantage, the technology’s adoption rates remain low. Discussions of this low uptake usually focus on a lack of long-term evidence supporting patient benefits, uncertainty surrounding the cost-efficiency of the systems, and risks associated with increased operating time. Though less frequently discussed in the literature, issues relating to the usability of the technology also act as a barrier to adoption for many surgeons. We introduce the design of a novel optical tracker system originally proposed by Hodgson (2012) as a more ergonomic drop-in replacement for existing trackers. The goal of this thesis project is to develop the tracker concept through stages of detailed design, Idea SystemUser Needs AssessmentOutline Specs Test SpecsDevelopment& User TestingVerificationValidationImplementationDetailed DesignPerformance TestingFigure 3: Overarching process for the tracker-redesign project. This thesis focuses on the detailed design, implementation, and performance testing stages. The validation of specific usability gains in selected surgical procedures will require future study.     17  implementation, and performance verification. We believe that the new tracker concept has the potential to overcome the usability barriers that are preventing large-scale uptake of CAOS technology. 1.7 Thesis Organization Chapter 1 provides an overview of the relevant literature in the field of orthopaedic navigation, as well as the specific objectives of this thesis project. Chapter 2 details the novel tracker design concept, the theoretical principles of operation, and the proposed method of implementation. Chapter 3 explains our methods for testing and validation for the new flexible tracker system: a series of 2D and 3D simulations, and lab-based physical experiments. Chapter 4 presents the results of our simulations and experiments. Chapter 5 concludes with a discussion of our findings, contextualizing the results and impact on the field, as well as considerations for future work.     18  Chapter 2  Design, Theory, and Implementation In this chapter we present the concept for our novel optical tracker design. We explain the theoretical principles of its operation, highlighting the strengths and weaknesses. Lastly we discuss the proposed method of implementation: an unscented Kalman filter (UKF) algorithm. 2.1 Flexible Optical Tracker Design Concept Conventional optical trackers for orthopaedic navigation, as seen in Figure 5, are challenging to use for several reasons discussed in the previous chapter. To address the limitations of these devices, we present a design for a new type of CAOS tracker.     19  The new tracker design, shown in Figure 4, features markers on the ends of separate, inherently flexible pins that can be anchored into a patient’s bone. The new system allows for bending of the pins during use, meaning that a pin can be physically deflected from its neutral position without significantly affecting the system’s tracking accuracy. We explain the theoretical basis for this in the next section. Figure 5: An example of a conventional rigid tracker attached to a femur. At least two bicortical pins are required to fix the tracker to the bone. Any dislocation of the tracker from its calibrated position can result in erroneous readings (typically undetectable by the system). Figure 4: A demonstration of how the proposed tracker system could conceivably be attached to a femur for use in a navigated surgery. The single-marker pin type requires six pins to fully constrain a body.     20   The modular design of the pins allows the tracker to be distributed over a greater area of the patient’s anatomy. The baseline distance between measurements could conceivably be as long as the bone being tracked, which leads to improved angular resolution and accuracy. If an additional marker is used, it can provide an independent check of system accuracy (eg. in the case of detachment of one of the pins). For this project we created and tested two variations of the flexible tracker concept which we will refer to as the “single-marker” and “double-marker” flexible pin types. 2.1.1 Single-Marker Flexible Pins The components of the single-marker flexible pin are labelled in Figure 6, including the marker, the adaptor, the flexible shaft, and the anchor. A more detailed description of each of the components follows in Section 2.1.3.  The single-marker design requires a minimum of six pins to fully constrain the bone in space, as shown in Figure 4.     21  2.1.2 Double-Marker Flexible Pin Concept We also present a double-marker pin type in which two markers are spaced apart by a rigid segment atop the flexible shaft, as shown in Figure 7. Only three double-marker pins are required to fully constrain a body in space.   Computer sketch Solid model PrototypeRough sketchMarkerAdaptorFlexible shaft AnchorFigure 6: The single-marker flexible pin design, progression from hand-sketch to experimental prototype. The components of the flexible tracker are the marker, the adaptor, the shaft, and the anchor.     22    Computer sketchSolid model PrototypeRough sketchMarkerCrossbar/AdaptorFlexible shaft AnchorFigure 7: The double-marker flexible pin design, progression from hand-sketch to working experimental prototype. This design is similar to the single-marker design, except now there are two markers spaced apart by a cross-bar component, making the tracker look like a “T”.      23  Both variants of the flexible tracker will be tested and evaluated in this project. The single-marker pins are smaller, but the double-marker pins require fewer holes in the patient’s bone, so each system has conceivable advantages. We will assess their respective tracking performance before drawing any conclusions about which is the best flexible tracker type for a particular application. 2.1.3 Detailed Design Considerations The original concept for the flexible optical trackers was devised in our lab at UBC by Dr. Antony Hodgson. The technical development in this thesis spans from an abstract conceptualization to a functioning prototype of the tracker. During the development of our prototype, we selected each of the components of our tracker (labelled in Figure 6 and Figure 7).  Figure 8: An example of how the double-marker flexible pins could be attached to a femur for use in a navigated surgery. The double-marker type only requires three pins to fully constrain the body.     24  The anchor component holds the flexible pin to the rigid body. The prototype anchor had to be insertable and removable, and it had to maintain a secure fixation to the test piece. To this end we modified a non-descript wood screw to have a wrench flats to allow the pin to be torqued into the rigid body. The shaft enables the tracker’s flexibility. The shaft must be sufficiently long to breach a patient’s soft-tissue but short enough to avoid crowding the surgical workspace (on the order of 5–15cm). Within that range, we heuristically selected the shaft dimensions to be stiff enough to resist bending due to its own weight, but compliant enough to bend when moderately loaded.  The adaptor component holds the marker to the flexible shaft. We used a short rigid metal rod with a threaded hole to attach the marker’s mounting post to the shaft. For the double-marker pin we had to consider the separation of the two markers, which introduces a trade-off between angular resolution and bulkiness. We chose a marker separation distance similar to the length of the shaft. The marker can be measured by the position sensor. We used the Northern Digital Inc (NDI) reflective spheres. These passive, single-use markers are widely used in the industry. 2.2 Theoretical Principles of the Flexible Tracker Design The assumption of rigidity of the conventional trackers allows for simple Euclidian rigid-body transformations to locate the body and targets (eg. femoral condyles or tool tip). This     25  assumption is fundamentally invalid for our flexible trackers because the markers can move. It is critical to note that the motion that occurs between markers is still partially constrained by the flexible shaft. If we carefully select the locations and orientations of a sufficient number of flexible pins, we can fully constrain the underlying body in space by exploiting the pin’s stiffness in the axial direction and being indifferent to transverse deviations of the markers (ie. the directions of high flexibility). We model each flexible tracker pin as a uniform, slender cantilevered rod (length ≫ diameter), in which the cantilevered-end sits anchored in the bone and the free-end hosts the trackable marker. The lateral stiffness 𝑘𝑙𝑎𝑡   and the axial stiffness 𝑘𝑎𝑥 for such a beam can be expressed as follows, assuming a circular cross section and a point load at the tip. 𝑘𝑙𝑎𝑡 =𝐹𝛿𝑙𝑎𝑡=3 𝐸 𝜋 𝑟44 𝐿3  𝑘𝑎𝑥 =𝐹𝛿𝑎𝑥=𝐸 𝜋 𝑟2𝐿  Figure 9: The pin’s design enables the markers to move relative to one another, which nullifies the assumption of rigidity. The markers are still partially constrained by the flexible shaft that holds them to the anchor and bone     26  If we consider the ratio between axial to lateral stiffness, it simplifies to:  𝑅𝑘 =𝑘𝑎𝑥𝑘𝑙𝑎𝑡=43 ( 𝑟)  Now assuming a pin diameter of 0.80mm and a shaft length of 70mm (a reasonable set of dimensions for the pins), the ratio of axial stiffness to lateral stiffness is found to be: 𝑅𝑘 =43 (70𝑚𝑚0.8𝑚𝑚) ≈ 10 000 This result suggests that the tip of the pin is approximately ten-thousand times more resistant to deviations in the axial direction than in the transverse direction. Therefore, it is very reasonable for us to say that the marker is constrained axially, while being effectively free to move in the transverse direction, as represented in Figure 10.  ≫  AxialTransverseFigure 10: We model the flexible tracker pin as a slender cantilevered beam. The marker (pin tip) is relatively free to move in the transverse direction compared with the axial direction.     27  By extending this concept of stiff axial constraint over greater angles of pin deflection, we can define a “bending subspace” comprised of all possible marker positions for a given pin. In 3D this subspace can be represented as a curved 2D surface resembling a dome, and requiring only two parameters to describe (shown in Figure 11). In general, optical tracking systems measure the Cartesian (𝑋 𝑌 𝑍)  coordinates of a marker in space, granting three independent constraints over that point. For our single-marker pins, however, the flexibility of the shaft gives each marker two degrees of freedom (DOF) with respect to the body, so we lose two degrees of constraint. This leaves us with a single constraint over the underlying body, per pin. According to the principles of Exact Constraint Design (ECD), “the number of points of constraint should be equal to the number of degrees of freedom to be constrained.” (Slocum, 2010). Since rigid bodies have six DOF (three in translation, three in rotation) and each flexible pin provides a single constraint, ECD dictates that we require six single-marker pins to fully locate the body. The two rigidly connected markers that sit atop each double-marker pin collectively provide five degrees of constraint (three constraints per marker, but one lost from redundancy Bending subspaceFigure 11: A representation of the “bending subspace” concept, which consists of all possible marker positions that can be achieved by deflecting the pin.     28  and inability to resolve rotations along their mutual axis). The double-marker pin’s bending subspace is three-dimensional, because in addition to two DOF for transverse bending the pin can also resolve twists in the axial direction. The markers’ five degrees of constraint, minus three DOF from bending, results in two degrees of constraint over the rigid body per pin. We only require three double-marker pins to fully constrain the rigid body’s 6D pose. 2.2.1 Pin Bending Variability Our flexion model is limited by the variability of the pin’s bending trajectory resulting from different loading conditions. The loading conditions affect the distribution of curvature (ie. the shape) of the shaft during bending. Most concerning for our model is the straight distance (ie. the chord) between the anchor and the marker, which is a function of both the loading Mid-pin point load Distributed load Pin-tip point loadFigure 12: Examples of three distinct loading conditions and their respective bending trajectories (matched by colour). Pin-tip point load (blue) has the greatest contraction between the marker and anchor of the three examples.     29  conditions and the magnitude of deflection as shown in Figure 12. We will refer to this characteristic shortening of the marker-anchor distance as the “contraction”, or “chord contraction”. There is a continuum of possible loading conditions, and we have no way of predicting precisely which will occur during use. If we limit ourselves to “simple” loading cases, consisting of combinations of point loads and distributed loads applied perpendicularly to one side of the pin, we can establish limits on the possible contractions.  One limit would be a point-load applied infinitesimally close to the anchor, concentrating the curvature and causing the pin to effectively behave as a hinged rigid link, maintaining a constant separation between the anchor and marker (shown as the black arc in Figure 13). Although this case is practically unrealistic, if we neglect plastic deformation and idealize the geometry, this represents the lower bound on chord contraction. The other limit would be a point-load applied at the tip of the pin, maximally distributing the curvature and causing the greatest contraction between the anchor and marker (shown as a blue curve in Figure 13). The trajectories associated with all other simple loading conditions would fall in the envelope formed between these two limits, though greater chord contraction may be achieved with more complex loading cases.     30  Since our system cannot explicitly detect the applied loading case, we require an a priori model of the assumed bending trajectory for the pins. We chose to model the expected flexion as resulting from a point load applied at the middle of the pin. This loading case creates a chord-contraction precisely halfway between the two extremes described above, thereby minimizing errors associated with our selection (shown as a dashed red line in Figure 13). The impact of our decision was tested through a series of simulations described in the following chapter, in which we tested in a range of different loading conditions. Figure 13: The envelope between the two extremes of bending, resulting from simple loading conditions. The blue trajectory shows pin-tip loading (maximum contraction), and the black trajectory shows a rigid-link (no contraction). Our chosen model (red) minimizes the error by being halfway between the two extremes.     31  2.2.2 Flexible Pin Configuration Considerations Each flexible pin (single-marker type, specifically) imposes a single constraint on the underlying rigid body. Similarly to conventional rigid trackers, the configuration of the markers is critically important in maximizing constraint over the body. Unlike rigid trackers, the configuration of the flexible trackers is more fluid, because they have to be individually inserted into the bone for each case. Therefore it is important to understand how to attach the pins to maximally constrain a body.  Poorly constrainedLacking rotational resolutionPoorly constrained Lacking horizontal resolutionFigure 14: Parallel pins can constrain rotation, and perpendicular pins can constrain translation, so a combination of both configurations is needed to fully locate the body in space.      32  Figure 14 shows two examples of poorly configured pins in a simple 2D case. Parallel pins constrain rotation and axial translation, but are unable to constrain transverse translations. Conversely, a set of perpendicular pins can constrain translations but not rotations. We therefore require a combination of both parallel and perpendicular pins about orthogonal axes to fully constrain a body in space. While we mostly approach this from a geometrical point of view, anatomical and surgical factors would also influence decisions on pin placement. In the next chapter we present examples of the pin configurations that we explored for our simulations and experiments. 2.3 Implementation of Design Although conventional assumptions of rigidity are invalid for the flexible trackers, we are still able to constrain the underlying body by exploiting the pin’s stiff axial constraint and ignoring transverse deviations of markers. While the theoretical foundations may be sound for rigid-body motion tracking, we now consider the practical side of the system’s operation. First we present a geometrical description of the system; followed by an overview of the algorithm that we used to enable our flexible tracker to work in real-time.     33  2.3.1 Geometrical Formulation In this section we establish a geometrical description of our system, using a series of spatial transformations. We modelled the system as a series of Euclidean rigid-body transformations, defined as follows:  (i) 𝑇𝐺𝐵 – Transformation from global (sensor) to body-mounted coordinates. (ii) 𝑇𝐵𝐴 – Transformation from body-mounted to anchor-site coordinates. (iii) 𝑇𝐴𝑃 – Transformation from anchor to pin coordinates. (iv) 𝑑𝑃𝑀 – Position of a marker point in local pin coordinates. 𝑇𝐵𝐴                                 𝑇𝐺𝐵𝑑⃗𝐺𝑀      𝑑⃗𝑃𝑀                         Figure 15: Geometrical model of flexible tracker system, showing only one single-marker pin. We define a series of Euclidean transformations from the global coordinates of the sensor to the marker point.     34  Our aim in tracking is to acquire the transformation between global and body-mounted coordinates 𝑇𝐺𝐵, which ultimately describes the position and orientation of the bone. The marker position measurement 𝑑𝐺𝑀 is equivalent to the string of rigid transformations shown below.  𝑑𝐺𝑀 = 𝑇𝐺𝐵 × 𝑇𝐵𝐴 × 𝑇𝐴𝑃 × 𝑓(𝑇𝐴𝑃) × 𝑑𝑃𝑀 Working backwards along this kinematic chain, we know 𝑑𝑃𝑀 from the designed dimensions of the tracker pin. The term 𝑓(𝑇𝐴𝑃)  accounts for the contraction due to bending, it is a function of the pin’s deflection which is embedded in the transformation from anchor-to-pin coordinates. Transformation 𝑇𝐴𝑃 involves no translation component, but rather it only contains information about the orientation of a pin relative to its neutral position (ie. it accounts for bending). Transformation 𝑇𝐵𝐴 is a constant, which we can acquire during a tracked pin insertion or a registration step once we have established a local body-mounted coordinate frame. Lastly transformation 𝑇𝐺𝐵 describes the position and orientation of the body-mounted coordinates, relative to the camera. 2.3.2 The Kalman Filter The Kalman filter (KF) was developed by Rudolph Kalman in 1960 as a tool for linear filtering and prediction problems, and was instrumental in the success of the 1969 Apollo lunar landing (Kalman 1960; Grewal 2010). The original KF was specifically formulated for     35  linear systems, and various adaptations have been developed to handle non-linear systems, such as the extended Kalman filter (EKF) which linearizes non-linear problems, and more recently the unscented Kalman filter (UKF). The UKF is based on that idea that “it is easier to estimate a Gaussian distribution than to approximate an arbitrary nonlinear function” (Julier 1997). The use of a UKF for motion tracking in navigated surgery is not entirely new. Simpson (2010) used a UKF to compute intraoperative uncertainty associated with navigation systems, and Vaccarella (2012) used a UKF to account for short-term marker occlusions during surgical tracking. They found that for a few frames of occlusion, their algorithm was able to reasonably compensate for a loss of system observations with marginal loss of pose estimation accuracy.  McIvor (2013) successfully implemented a UKF algorithm to calibrate the primary axis of a tracked surgical drill bit to within 1mm and 0.5º. 2.3.3 Implementation of Unscented Kalman Filter Algorithm Here we present our UKF-based algorithm that recursively and optimally combines new marker measurements from our flexible tracker pins with previous states to estimate the bone pose. Further details about the UKF can be found in Appendix B. The State Vector 𝒙  describes the bone’s pose (position and orientation), and the deflection of each of the flexible pins. 𝑥𝑘 = [ 𝑥 𝑦 𝑧 𝜙 𝜃 𝜓 𝑃 ⃗⃗ ⃗⃗ ⋯ 𝑃𝑁⃗⃗ ⃗⃗ ⃗ ] 𝑇     36  Where vector (𝑥 𝑦 𝑧) is the position (cm), (𝜙 𝜃 𝜓) is the orientation (radians), and (𝑃 ⃗⃗ ⃗⃗ ⋯ 𝑃𝑁⃗⃗ ⃗⃗ ⃗) is the deflection state of each pin (cm for deflection, radians for twist). Parameterizing 3D orientation is a basic challenge in all rigid body tracking applications. Common approaches are to use Euler angles, rotation matrices, or unit quaternions. Euler angles and rotation matrices are criticized for suffering singularities at angles of  ±𝜋, but offer convenient rotation matrix manipulations. Unit quaternions avoid singularities by operating in four dimensions, but the four parameters are not linearly independent making it difficult to implement within a UKF (Kraft, 2003). Welch and Bishop (1997) developed a UKF tracking algorithm in which the object’s orientation can be stored as Euler angles, but to avoid discontinuity the orientation was tracked as incremental differences in orientation between time steps (Welch & Bishop, 1997). The absolute orientation of the rigid body can be stored outside of the UFK. This approach was also used successfully by Simpson (2010), and it is how we chose to represent orientation in our UKF algorithm. The Process Model 𝑭(𝒙) is a function that models the physical behaviour of the system from frame to frame. The input to the function is the current system state estimation, and it returns a prediction of the next system state. For a lack of external information on the process, we modelled the process as a constant position, white noise velocity model. This means we predict that the rigid body pose will remain static within the limits of Gaussian process noise.     37  𝑓(𝑥𝑘, 𝑤𝑘) = ?̂?𝑘+ =[      𝑑 + 𝑤𝑑?⃗? + 𝑤𝑅𝑃 ⃗⃗ ⃗⃗ + 𝑤𝑃⋮𝑃𝑁⃗⃗ ⃗⃗ ⃗ + 𝑤𝑃]       Some authors elect to use a constant velocity white noise acceleration model, but we decided to use a simpler model in which we only describe the pose of the body, assuming zero velocity from frame to frame. The impact of our process model selection is evaluated through simulations described in the following chapter. The Process Covariance Matrix 𝑸 describes our confidence in the process model 𝐹(𝑥). Variation to our static process model is embodied as motion. We therefore base our selection of process noise values on expected speeds of the system state. Table 2: A summary of the UKF process noise selections, based on “expected” motion speeds. The pin twist process covariance is only applicable to the double-marker pins. The denominator of “30Hz” in all of the values refers to the optical tracker sampling frequency that we used.   Process Noise Group Value Physical Meaning Translation  (𝑞 )  ( 30 mm  30 Hz ⁄ )  30mm  ⁄  1mm  f  m ⁄  Rotation (𝑞 )  ( 𝜋4      30 Hz ⁄ )  45°  ⁄  1.5°  f  m ⁄  Pin Deflection (𝑞3)  ( 30 mm  30 Hz ⁄ )  30mm  ⁄  1mm  f  m ⁄  Pin Twist (𝑞4)  ( 𝜋4     30 Hz ⁄ )  45°  ⁄  1.5°  f  m ⁄      38  These values are inherently difficult to select because we cannot observe the process directly (Simpson, 2010). Our process noise values were selected heuristically, based on speeds that one could reasonably expect a bone to be moved in a navigated surgery application. We ran a series of simulations (discussed in Section 3.1.6) to explore the sensitivity of these selections. The process noise values are squared and used to construct the square diagonal process covariance matrix, which had the form: 𝑄 =  [                    1.0 0 0 0 0 0 0 ⋯ 00 1.0 0 0 0 0 0 ⋯ 00 0 1.0 0 0 0 0 ⋯ 00 0 0 (𝜋120) 0 0 0 ⋯ 00 0 0 0 (𝜋120) 0 0 ⋯ 00 0 0 0 0 (𝜋120) 0 ⋯ 00 0 0 0 0 0 (𝑞3)   ⋯ 0⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ⋮0 0 0 0 0 0 0 ⋯ (𝑞3)𝑁    ]                  Where the diagonals correspond to the process noise in elements 𝑥, 𝑦 , 𝑧 , 𝜙, 𝜃, 𝜓 and then each of the pin deflection parameters. If we set the process noise to be very small (~ 0), the UKF places a greater weighting on the prediction made by the process model, and becomes very confident that the rigid body     39  will remain static. Jitter and noise will be dampened, producing a smoother estimation of the system state. In a static case, we expect very good performance because the process model matches the true state behaviour. However, when the body actually moves we expect there would be a notable lagging behavior.  Conversely, if we set the process noise to be relatively large (compared to the measurement noise), then the UKF will lose its confidence in the process model, and will base its estimation more heavily on the marker measurements. This would lead to considerably less lag during motion, but also less dampening of the marker jitter. The Measurement Vector 𝒚 contains the marker position measurements at each sampling interval (or frame). Each marker position component has some inherent noise (jitter), which is assumed Gaussian. A marker’s jitter is anisotropic, meaning that measurements are less precise in the sensor’s line of sight direction (𝑧), than the in-plane directions (?⃗? and ?⃗?). 𝑦𝑘 =[        𝑥 ⃗⃗ ⃗⃗ + 𝑣𝑥𝑦 ⃗⃗⃗⃗⃗ + 𝑣𝑦𝑧 ⃗⃗⃗⃗ + 𝑣𝑧⋮𝑧𝑁⃗⃗⃗⃗⃗ + 𝑣𝑧  ]       The Measurement Model 𝑯(𝒙) is where we implement the geometrical model described in subsection 2.3.1. As a function, the measurement model take the state vector 𝑥     40  as an input, and returns a corresponding estimation of the measurement vector ?̂?. It is a transformation between the state space and the measurement space.  𝐻(𝑥) = ?̂? = 𝑇𝐺𝐵 × 𝑇𝐵𝐴 × 𝑇𝐴𝑃 × 𝑓(𝑇𝐴𝑃) × 𝑑𝑃𝑀 The transformations 𝑇𝐺𝐵 and 𝑇𝐴𝑃 are comprised of elements of the state vector 𝑥, and the other elements: 𝑇𝐵𝐴, 𝑓(𝑇𝐴𝑃), and 𝑑𝑃𝑀 are determined prior to operation.  The Measurement Covariance Matrix 𝑅 describes the anticipated jitter for each marker measurement. We determined values for jitter by repeatedly sampling a fixed target with a position sensor and calculating the RMS deviation. The values for measurement noise are summarized in Table 3.  X (in-plane) Y (in-plane) Z (out-of-plane) 0.006 mm 0.006 mm 0.025 mm Table 3: A summary of the anisotropic marker measurement noise. These values are the basis for our measurement covariance matrix in the UKF, and indicate the relative confidence in the measurement model. The measurement noise values are squared, and put into a diagonal matrix as follows:      41  𝑅 =  [              (0.006)  0 0 0 0 0 ⋮0 (0.006)  0 0 0 ⋮ 00 0 (0.025)  0 ⋮ 0 00 0 0 ⋱ 0 0 00 0 ⋮ 0 (0.006)𝑀 0 00 ⋮ 0 0 0 (0.006)𝑀 0⋮ 0 0 0 0 0 (0.025)𝑀  ]                  42  Chapter 3  Methods In the previous chapter we established the theoretical principles behind the flexible tracker system, and a mathematical framework to enable its function in real time. However a critical question remains: is our flexible tracker system actually able to track a rigid body? And if so, can it track to an adequate level of accuracy needed for surgical navigation applications?  This chapter provides an overview of a set of simulations and experiments that we conducted to assess the system’s performance for a range of realistic conditions. 3.1 Simulation Methods Simulations are a fast, inexpensive, and controlled method of iteratively designing and testing a system. An advantage of simulations is that the ground truth is known by design, so the accuracy can be measured directly, precisely and objectively. However, a downside of simulations is that the results are only as good as the model. We created a MATLAB     43  simulator that runs iteratively through two major stages, (1) generating a set of marker measurements based on a model of the flexible pins attached to a rigid body (inspired by a total knee arthroplasty), and (2) feeding the simulated measurements into the UKF data-processing algorithm to estimate the pose of the underlying bone. We can evaluate the performance of the tracker system by comparing the estimated and ground-truth poses.   We ran simulations in both 2D and 3D spaces (see Figure 16). In the simplified 2D case the rigid body has fewer degrees of freedom, requiring fewer flexible pins to constrain it (three as opposed to six). Despite oversimplifying reality, the 2D simulations were useful for developing the key functions of our algorithm in a simpler dimension, with fewer variables.      𝑚         𝑃𝑟               m      m        m                     m      m      m        m                                                Figure 16: Sample screenshot from the simulation visualizer for 2D (left) and 3D (right).     44  We simulated a range of effects on the flexible tracker system to predict its behavior under increasingly realistic conditions. The table below provides a page reference for each of our simulation method descriptions.  Simulation Section Page  Static body  3.1.3 48 Expected motions  3.1.4 50 Motion speed  3.1.5 51 Process noise  3.1.6 51 Pin bending  3.1.7 53 Anchor error  3.1.8 54  3.1.1 Combined Evaluation Metric To quantify the performance of our simulated flexible tracker, we compared the difference between the estimated rigid body pose and the ground-truth pose. Although the accuracy of the 6D pose can be reported as separate position and orientation components, we devised of a single metric that combines the translational error with a scaled factor of the rotational error, which we refer to as the “Combined Evaluation Metric” (CEM). 𝐶𝐸𝑀𝐴 = √(𝑋𝑒𝑠𝑡 − 𝑋𝐺𝑇) + (𝑌𝑒𝑠𝑡 − 𝑌𝐺𝑇) + (𝑍𝑒𝑠𝑡 − 𝑍𝐺𝑇) + ( 𝑓 𝛥𝜃 )      45  The CEM is the root of the sum of the squared errors. The rotation component Δ𝜃 is magnitude of the rotation about the Euler axis that differentiates the two bodies (Euler rotation theorem). The rotation scale-factor 𝑓 is half of a typical transepicondylar distance (~50mm).  Assuming we use anatomically-based coordinates (such as in the centre of the femoral condyles as shown in Figure 17), our CEM estimates the net positional error for a target point on either the medial or lateral side of the distal femur. The subscript ‘A’ designates an accuracy-based metric. In our experiments, however, we do not have a ground-truth, so instead we use variations of the metric: 𝐶𝐸𝑀𝑃 for precision, and 𝐶𝐸𝑀𝐷 for discrepancy (discussed in the following section).  𝑓Δ𝜃𝑌𝑋𝑍Figure 17: The CEM on anatomical femur-based coordinates. The rotation scale factor f calculates the target error due to rotation at either epicondyle.      46  3.1.2 Flexible Pin Configuration Figure 18 shows the configuration of flexible pins that was used for the 2D Simulations. The body is well constrained in both translation and rotation. By situating Pin 3 perpendicularly at the proximal end of the femur model, we gain considerable angular resolution over the bone. The 3D configuration for the single-marker pins is shown in Figure 19. The orthogonality of the pins helps to constrain translations about all directions, and the separation between pairs of parallel pins constrains rotations about each axis. 45°135°180°30,  22, 22,  2132Figure 18: Configuration of flexible pins used for the 2D Simulations. The long baseline distance between markers creates a high angular resolution over the body.     47    Figure 19: The single-marker pin configuration for 3D simulations. The same configuration was used in our experiments. The orthogonality of the pins helps to constrain translations about all directions. The separation between parallel pins constrains rotations about each axis.  Figure 20: The double-marker pin configuration for 3D simulations. The same configuration was used in our experiments.      48  3.1.3 Static Simulation Method Our first simulation is for the flexible tracker system under perfectly static conditions. This is the simplest conceivable scenario for any motion tracker, making it a logical first-step in assessing our flexible tracker system. The purpose of this simulation is to establish a baseline accuracy for the flexible tracker system as it operates under somewhat ideal conditions. In addition to the UKF algorithm described in Section 2.3.3, we also coded a simple optimization algorithm to locate the bone by minimizing the sum of squared distances between the marker measurements and known tracker geometry. This is a simplified version of an approach used to fit a rigid body pose to marker position measurements (Chiari 2005). We also simulated a conventional rigid tracker design for both the 2D and 3D trials. The geometry for each of the simulated trackers is summarized in Table 4.  While many rigid trackers will feature redundant markers to improve accuracy and robustness to occlusion, we only included the minimum-required number of markers to constrain the body. Our rigid tracker model had two marker in 2D, and three markers in 3D.  During each static simulation trial, the rigid body was kept motionless, the pins remained un-deflected, and the geometrical parameters were assumed known without error. We added a typical level of jitter to the simulated marker position measurements (0.006mm x⃗ ; 0.006mm  ⃗⃗ ; and 0.025mm z⃗ ). The 2D simulations ran for 500 frames (16.67 seconds at 30Hz), and the 3D simulations ran for 300 frames (10 seconds at 30Hz).       49  Rigid Tracker 2D  Simple two marker rigid tracker design for 2D simulations. Flexible Tracker 2D  Three flexible pins are required to constrain the body in a 2D plane Rigid Tracker 3D  Top view of simple three marker rigid tracker design for 3D simulations Flexible Tracker  (Double) 3D  Three of the flexible double-marker “T” pins are required to constrain the body in 3D space. Flexible Tracker  (Single) 3D  Six single-marker flexible pins are required to constrain the body in 3D space.  60𝑚𝑚60𝑚𝑚30𝑚𝑚60𝑚𝑚70𝑚𝑚40𝑚𝑚60𝑚𝑚Table 4: Descriptions and dimensions of each of the trackers used in the static simulation trials. Each tracker type used only the minimum number of markers to constrain the body.      50  In principle, each of our tracker configurations should fully constrain a rigid body in space, therefore we expect that the static simulations will result in very accurate (< 0.1mm CEM) estimations of the body pose. We predict that the flexible tracker system will be more accurate than the rigid tracker design because the markers are modular, and can be placed at opposite ends of the bone creating a greater baseline distance. 3.1.4 Assumed Motions Simulation Methods In addition to tracking a static object, our system must also detect changes of system state and motions of the underlying body. This series of simulations subjected the rigid body to translations and rotations at expected speeds that correspond with the process noise of the UKF. These simulations (and all that follow) focus exclusively on the flexible tracker system and UKF algorithm. The sensitivity of our system to input motion speed will be explored in a later set of simulations. Each assumed motion trial consisted of a constant-rate rigid body motion (for 500 frames in 2D, and 300 frames in 3D).  The body was translated at 30mm/s, and rotated at 45º/s. All degrees of freedom of the rigid body were tested. We expect our algorithm to track the body to a high degree of accuracy (comparable to the static case), because the rates of change are compatible with the UKF process noise.       51  3.1.5 Motion Speed Simulations Expanding on the previous expected motion simulations, in this trial we move the bone over a range of constant rate motion speeds, which are summarized in Table 5. The process noise inputs were kept at default levels for this simulation.  Translation (𝑚𝑚/𝑠) 0 30 60 150 300 3000 Rotation ( °/𝑠) 0 22.5 45 67.5 90 112.5 Table 5: Range of bone motion speeds for constant process noise trials. Because the measurement noise is much smaller than the default process noise, we expect that the algorithm will be minimally impacted by the changes of bone speed. The body was moved at a constant rate for the duration of the trial (500 frames 2D, 300 frames 3D). We expect the system to handle all motion speeds without increase in tracking error because the default measurement noise is much smaller than the process noise, meaning that our system mostly bases its estimations on the incoming measurement information. 3.1.6 Process Noise Simulation Methods The UKF process noise is a measure of our confidence in the process model. It is difficult to select values for process noise because we cannot directly observe the process (Simpson, 2010). The purpose of these simulations is to explore the sensitivity of the flexible tracker system to the selection of process noise values. We chose (1.0)  and (𝜋 120⁄ )  as the default     52  process variance values (noise squared) for translation and rotation respectively, corresponding to a physical meaning of 30mm/s and 45º/s. In this simulation, we analyze an identical set of tracker data using a range of process noise input values (ie. constant speed, varying process noise).  We scaled the default process noise inputs exponentially to show the effect over a large range. The process noise scaling factors are presented in Table 6.   Factor 10−  5 0 10−  4 0 10−  3 0 … 10−   0 10−   0 100 10  10  Approx. Decimal 0.03 0.04 0.05 … 0.63 0.79 1 10 100 Table 6: Scaling factors used to vary the Process Covariance inputs.  The simulated marker data in this trial was identical to the assumed motions simulations. In the static case, we expect the tracking error will decrease with process noise because the process model is to remain static, and a lower process noise gives this prediction more weighting.  In the trials with motion, we expect the tracking error to grow with a decreasing process noise due to a lagging behaviour.  In all trials, we expect the accuracy will eventually stabilize at a certain level of process noise because the estimation will be dominated by the measurements, and the process model will be irrelevant.       53  3.1.7 Pin Flexion Simulations As explained in Section 2.2.1, pin loading condition variability could negatively affect our tracking accuracy. The purpose of these simulations was to assess the severity of the tracking error associated with different pin loading cases. Our current model assumes bending curvature based on a point load in the centre of the pin, shown in red in Figure 21. In each simulation, the tip of one pin was deflected at a constant rate, and then held statically deflected for the remainder of the trial. Each trial lasted for 500 frames in 2D, and 300 frames in 3D. The maximum deflection for each loading case was 40mm transverse, which corresponds with an angle of approximately 40º.    Mid-pin point load Distributed load Pin-tip point loadPin-tip point load withresistive distributed loadFigure 21: The four loading conditions that we simulated. The mid-pin loading case is the implicit bending model for the Kalman filter algorithm.     54  3.1.8 Anchor Digitization Uncertainty Our algorithm requires initial knowledge of the relative positions and orientations of each flexible tracker pin (transformation 𝑇𝐵𝐴), which can be acquired either through a tracked pin insertion, or from a digitization step at the beginning of the procedure. Since both of these methods rely on optical tracking, the anchor location measurement will be prone to some degree of error. The purpose of this simulation trial is to understand the impact of the anchor digitization error on overall tracking error.  The tracking algorithm was given deliberately false random anchor positions for errors levels of 1mm, 2mm, 5mm, and 10mm RMS. We ran a Monte Carlo analysis of 100 randomly generated sets of misplaced anchor coordinates at each noise level, in which every trial was run for 100 frames of tracking. The sets of CEM data for each noise level were analyzed collectively. 3.2 Experiment Methods While simulations are useful for running fast, inexpensive and repeatable tests of a system, the quality of the results is limited by the quality of the model. Therefore to validate our model, we conducted a series of lab-based experiments using a physical set of flexible tracker pins, a mock bone, and a position sensor. The index below summarizes the experiment trials and points to their location within this chapter.      55   Experiment Section Page  Static body  3.2.3 59 Freehand motion 3.2.4 60 Pin flexion 3.2.5 60 Constrained motion  3.2.6 61  3.2.1 Experiment Apparatus For the experimental set-up, we aimed to mimic a navigated total knee arthroplasty in terms of scale, configuration, and equipment. We assembled the critical components of a surgical navigation system including an optical tracking camera, a computer, a tracker, and a rigid body. Optical Tracking Camera We used a Polaris Spectra infrared position sensor from Northern Digital Inc. (NDI; Waterloo, Ontario) to measure the 3D positions of the individual markers. The Polaris was run in passive mode at a sampling rate of 30Hz. We established the jitter for single marker measurement as (0.006 mm x⃗, 0.006 mm  ⃗⃗, 0.025 mm z⃗) by repeatedly sampling a static point.       56  Flexible Tracker Prototype Pins With help from Vancouver-based Implant Mechanix Inc., we created functional prototypes of the single- and double-marker flexible tracker pins. The adaptor component had threaded-holes for NDI passive marker mounting posts, to host the NDI reflective spheres. The shaft was made from a segment of 0.75mm diameter spring stainless steel, 70mm long. We designed the shaft segment such that deflections due to its own weight would be negligible, but little external force would be required to induce a bend. The anchors at the distal end of the pin are made from ordinary wood screws that have been altered with wrench-flats to enable manual insertion into the test piece, shown in Figure 22.  70mm65mm25mmFigure 22: The prototype flexible tracker pins, with dimensions. The single- and double-marker pin types both have identical anchor and shaft components.      57  Rigid Body Test Piece To represent a femur we used a piece of pine wood (45  m × 9  m × 4  m), as shown in Figure 23. The block of wood met our criteria as it was sufficiently rigid and approximately the size of a femur.  Computer Platform We used a standard personal computer (2012 Lenovo Ideapad Y470P running Windows 8.1) to acquire the measurements and conduct the analysis. Unlike a true navigation system, we performed a post-acquisition analysis, meaning that we took measurements first and analyzed them after. The data was recorded using NDI’s Tool Tracking software, saved to disk, and then analyzed with MATLAB 2013a. NDI rigid trackerRigid body test piece (pretend-femur)Single-marker flexible pinsFigure 23: The experimental set-up, showing our rigid body, the single-marker flexible pins, and the NDI rigid tracker component.     58  3.2.2 Experiment Evaluation Metrics In simulation, calculating tracking accuracy is straight forward because we fundamentally know the ground truth pose of the rigid body. In experiments however, we require simultaneous measurements with a gold-standard tracking method. A full 6DOF large-scale tracking system (such as a coordinate measurement machine (CMM)) was cost-prohibitive. To overcome this limitation, in some cases we evaluated the tracker’s precision, and in others we calculated the discrepancy to a rigid tracker’s measurements. In one case we attached the test piece to a high precision linear translation stage, which served as our gold standard for a limited set of experiments (described in Section 3.2.6). Here we present the experiment evaluation methods, and the alternatives to the accuracy-based combined evaluation metric. (i) 𝑪𝑬𝑴 𝑷𝒓𝒆𝒄𝒊𝒔𝒊𝒐𝒏 While our lack of a gold standard is inhibitive to calculating accuracy, we could still determine the tracking precision in the trials where the rigid-body was kept perfectly static. In these experiments (described in the following sections) the rigid body test piece was firmly clamped to a sturdy surface to prevent relative motion between the body and the sensor. We can calculate the mean estimated pose throughout a trial (which we know should be static) and measure how far the estimation deviates from its mean. We refer to this as the Precision Combined Evaluation Metric (𝐶𝐸𝑀𝑃), to make the distinction with accuracy. 𝐶𝐸𝑀𝑃 = √( 𝑋𝑒𝑠𝑡 − ?̅? ) + ( 𝑌𝑒𝑠𝑡 − ?̅? ) + ( 𝑍𝑒𝑠𝑡 − ?̅? ) + ( 𝑓 𝛥𝜃 )      59  (ii) 𝑪𝑬𝑴 𝑫𝒊𝒔𝒄𝒓𝒆𝒑𝒂𝒏𝒄𝒚 For one set of experiments (described in Section 3.2.4) the body undergoes unconstrained large-scale motions, so we can neither calculate accuracy nor precision. Instead, we compare the pose estimation from our algorithm with the measurements made by the NDI rigid tracker (see Figure 23). The NDI rigid tracker itself is not perfect, so we cannot treat it as the gold standard, but it is still a highly accurate measure of the same quantity. Here, we calculate the discrepancy between the two tracker system, and report as the Discrepancy Combined Evaluation Metric (𝐶𝐸𝑀𝐷).  𝐶𝐸𝑀𝐷 = √( 𝑋𝑒𝑠𝑡 − 𝑋𝑅𝐵 ) + ( 𝑌𝑒𝑠𝑡 − 𝑌𝑅𝐵 ) + ( 𝑍𝑒𝑠𝑡 − 𝑍𝑅𝐵 ) + ( 𝑓 𝛥𝜃 )  3.2.3 Static Experiment Methods The first (and simplest) experiment trial had the rigid body in completely static conditions. The test piece was firmly clamped to a sturdy work table and was left undisturbed for the duration of the trial to ensure no relative motion between the body and the Polaris Spectra camera. Static trials were conducted for both the single- and double-marker pin variations, lasting for 10 second each at 30Hz. We expected that our tracker system will detect and maintain a consistent static pose of the rigid body, and we evaluated the performance using the precision based CEM.      60  3.2.4 Freehand Motion Experiment Methods The purpose of the freehand motion experiments was to demonstrate that the flexible tracker system can measure large scale displacements of the rigid body. By “freehand motion”, we mean that the test piece was manually lifted from a surface and moved around the camera volume without any external constraint. We ran two trials for each flexible tracker type; one focused on translations and the other for rotations. Although these experiments are not particularly repeatable, they demonstrate the tracker’s performance under realistic motion conditions within the operating volume. We manually translated and rotated the rigid body along each of the three principal axes in the global frame. For this experiment we evaluated the flexible tracker against the NDI rigid tracker with a discrepancy CEM. We expect the combined discrepancy to remain low (<0.5mm) for all trials.  3.2.5 Pin Flexion Experiment Methods As explained in Section 2.2.1, variability in pin flexion could lead to tracking errors. The purpose of these experimental trials was to observe how well the system handles deflections of its tracker pins in real life. We tested both the single- and double-marker pin types under various pin flexion conditions. For the double-marker pins, we tested each of the three distinct deflection modes: in-plane bending, out of plane bending, and axial twist (shown in Figure 24). For the single-marker pin, we emulated two different loading cases: a mid-pin point load, and a pin-tip point load. Each flexion type was repeated using three different     61  displacements. The test piece was firmly clamped to a worktable to ensure a static pose, and the performance was evaluated as a precision-based CEM. During each trial, the Polaris Spectra sampled measurements for 5 seconds at 30Hz while a pin was manually deflected and returned to neutral. Unless otherwise stated, we always attempted to emulate the expected pin loading case, a mid-pin point load. 3.2.6 Translation Stage Experiment Methods To experimentally validate the accuracy of our system, we simultaneously measured a translation motion with our tracker and with a micrometer-driven translation stage (Parker , shown in Figure 25). The translation stage has a maximum range of 13mm, and the surface can be positioned to within 0.005mm. For these trials, we consider the displacement as measured on the translation stage to be our gold standard of measurement. In-plane flexion Axial twistOut of plane flexionFigure 24: The three distinct modes of deflection of the double-marker pin type: in-plane, out-of-plane, and axial twist. The in-plane mode is the most sensitive, because the resulting marker positions are a function of the cross-bar angle.     62  The translation stage was rigidly attached to a base which we clamped to a workbench. The axes of the translation stage were roughly aligned with the global coordinate axes of the Polaris Spectra. One direction was approximately “cross plane” of the camera (global Y), and the other with the camera’s line of sight (global Z). For each flexible tracker type (single-marker and double-marker) we ran trials for both the Y and Z directions. Each trial consisted of four repetitions of translating the rigid body over the entire range of the micrometer (precisely 13mm), with short pauses between each. Translation StageTest PieceMounting BracketFigure 25: The test piece was firmly attached to a high-precision linear translation stage, which was securely attached to a work table. The translation stage has a maximum range of 13mm, and a precision of 0.005mm.     63  Chapter 4  Results In this chapter we present the results of our simulations and experiments with the flexible tracker system. The results follow the same sequence as in the Methods chapter. 4.1 Simulation Results  This section contains the results of our simulations. The index below lists the simulations, and provides a page reference for each result.  Simulation Section Page Static body  4.1.1 64 Expected motions  4.1.2 67 Motion speed  4.1.3 71 Process noise  4.1.4 73 Pin bending  4.1.5 77 Anchor error  4.1.6 82     64  4.1.1 Static Simulation Results We conducted static simulations to establish baseline accuracy levels under the most basic of motion-tracking conditions. The performance of each tracker was calculated as an accuracy-based CEM between the ground truth pose and the estimated pose. The results of the 2D static simulations are presented in Figure 26, followed by the 3D static simulations in Figure 27. All data points are represented as the median (50th percentile), and 90% confidence interval (5th – 95th percentiles of data). The 2D simulations were run for 500 frames, and the 3D simulations were run for 300 frames. 00.010.020.030.040.050.060.070.08CEMA(mm)Optimization UKF Optimization UKFRigid Tracker Flexible TrackerFigure 26: A comparison of simulated 2D static tracking accuracies for the flexible and rigid trackers, each analyzed with an optimization and a Kalman filter algorithm. The two algorithms granted similar combined accuracies for both trackers.     65  The results of the static simulations show that the optimization and the UKF algorithms provided similar tracking accuracies for both tracker types. The 2D and single-marker 3D static cases were more accurate than the rigid trackers, but only by 0.01 – 0.02mm. For our given application, this small of a difference in tracking accuracy is clinically inconsequential. This is discussed in greater detail in Chapter 5.     00.010.020.030.040.05Optimization UKF Optimization UKF Optimization UKFRigid Tracker Double-Marker Flexible Single-Marker FlexibleCEMA(mm)Figure 27: A comparison of simulated 3D static tracking accuracies for the flexible and rigid trackers, each analyzed with an optimization and a UKF algorithm.     66  In addition to finding the tracking error, we also recorded the computation time for each simulation trial, which is summarized in Table 7.    Tracker  Trial Duration UKF  Optimization Rigid Tracker 2D 500 frames (16.7 s) 0.4 s 15.2 s Flexible Tracker 2D 500 frames (16.7 s) 1.8 s 13.5 s Rigid Tracker 3D 300 frames (10.0 s) 0.8 s 36.7 s Double Marker 3D 300 frames (10.0 s) 9.2 s 25.4 s Single Marker 3D 300 frames (10.0 s) 9.5 s 21.6 s Table 7: Simulation times for each tracker and solver algorithm. The UKF algorithm ran consistently faster than the optimization algorithm.  In general the UKF ran consistently faster than the optimization algorithm for all tracker types. These simulations were run on a personal computer, but all of the optimization cases in 3D were slower than real-time.    67  4.1.2 Assumed Motion Simulation Results We conducted the assumed motion simulations to verify that the flexible tracker can measure simple large scale motions of the underlying bone. Tracking motions is a basic function for any motion tracking technology, and we expect our system to handle these conditions with minimal error. First, we present a sample of the data processing procedure. The disagreement between the estimated rigid body pose and the ground truth pose is calculated at every time step as an accuracy CEM value, shown in Figure 28.  0 50 100 150 200 250 300 350 400 450 50000.0050.010.0150.020.0250.030.0350.040.0450.05Time (frames)CEMA(mm)Figure 28: The combined error throughout the simulated 2D translation trial. The 5th, 50th and 95th percentiles of the error are shown as red lines. The error shows no variations compared with the 2D static case on the right in Figure 26.       68  From our collection of CEM data points, we extract the 5th, 50th, and 95th percentiles (ie. the median ± 2 standard deviations). Figure 29 presents the same data as a point with a set of upper and lower bounds, which is how the data sets will be presented from here on. This analysis was run for each of our assumed motion simulations involving translations and rotations. Figure 30 shows the results of the 2D assumed motion simulations. Static X Translation00.010.020.030.040.05CEMA(mm)Figure 29: Summary of the CEM data set for the 2D simulated translation trial, compared with the static case. The width to the distribution was added for visual effect.     69  Translation shows no increase in error from the static case. The rotation set, however, shows a significant jump in error compared to static.  The error returned to static levels once the rotation motion had ended. This rotation error is explored in greater detail in later simulations. The same behaviour was seen with the 3D simulations, presented in Figure 31. 00.010.020.030.040.05Static X Translation Y Translation RotationCEMA(mm)Figure 30: A summary of our 2D assumed motion simulations. The translation trials show no loss in accuracy compared to the static case. The rotation case, however, shows a significant jump in error compared to static case. The errors in rotation are relatively low, compared with the accuracy requirements for the system.     70  00.010.020.030.040.050.060.070.080.090.1X Y Z X Y ZStaticTranslation (30 mm/s) Rotation (45o/s)CEMA(mm)00.010.020.030.04. 5. 60.070.080.09X Y Z X Y ZStaticTranslation (30 mm/s) Rotation (45o/s)CEMA(mm)Figure 31:  Summaries of the two flexible tracker pins’ assumed motion simulations in 3D. Both types of flexible pins demonstrated similar behavior, consistent with the 2D simulations: no increase of error due to translation, and a noticeable jump in error during rotation.      71  4.1.3 Motion Speed Simulation Results Figure 32 and Figure 33 show the results of the simulations in which we moved the bone over a range of speeds in both translation and rotation, while keeping the process noise constant. Each data point represents an individual simulation trial with constant-rate motion at the indicated speed. The translation case appears insensitive to motion speed, with no increase in error even up to 3m/s. Rotation, however, seems more sensitive to motion speed, but the magnitude of these errors remains objectively low, at under 0.5mm CEMA for rotations of 112.5º /s.  00.050.10.150.20 30 60 150 300 3000 0 22.5 45 67.5 90 112.5Translation (mm/s) Rotation (o/s)CEMA(mm)Figure 32: A summary of the 2D constant process noise, varying motion speed simulations.  At default process noise values, the translation appears to be insensitive to motion speed, but the rotation case shows errors with increasing speeds.      72    00.10.20.30.40.50 30 60 150 300 3000 0 22.5 45 67.5 90 112.5CEMA(mm)Translation (mm/s) Rotation ( /s)o00.050.10.150.20.250.30.350 30 60 150 300 3000 0 22.5 45 67.5 90 112.5CEMA(mm)Translation (mm/s) Rotation ( /s)oFigure 33: Results of the 3D motion speed simulation trials. The top axis shows the single-marker pin, and the bottom axis shows the double-marker pin. Both tracker types demonstrate similar trends at high speeds: minimal effect for translation and greater effect for rotation.      73  4.1.4 Process Noise Simulation Results The purpose of the process covariance simulations was to explore the sensitivity of the UKF process model to variations in process noise inputs. We re-analyzed the same set of simulated marker data as in our static and assumed-motion simulations, but with a range of process noise input levels.  Results of the 2D process covariance simulations are presented in Figure 34, and the results of the 3D process covariance simulations are in Figure 35 and Figure 36 for the single and double-marker pins respectively. The static cases used a different set of process noise scale factors to exemplify the underlying trend.  In all static cases, as we decreased the process noise from the default levels (white point on each plot) we see a transition between two steady levels of tracking error. This transition demonstrates the UKF shifting its trust between the two models: the measurements and the process prediction. Since our process model is correct in the static case, a lower process noise places a greater weighting on a true model, and we see improved performance. However in motion, we see considerable lagging behaviour. Rotation only begins to show lagging behaviour in the double-marker trial, but we believe that with a lower process noise it would arise in all cases.       74   00.010.020.030.040.0510-710-510-310-11 10 100 10-3/210-110-1/21 10 100 10-3/210-110-1/2Static Translation RotationCEMA(mm)1 10 100Figure 34: A summary of the 2D constant speed, varying process noise simulations, showing the static, translation, and rotation cases. In each case, the input process noise values were scaled by the factors shown on the X-axis. The hollow points are at the default process noise (ie. a scale factor of 1).  In the static case (left) there is a distinct transition between two plateaus as the process noise is decreased. In translation case (centre) the error begins to grow as process noise is decreased. Rotation (right) shows no effect.     75   00.010.020.030.040.050.060.070.08Translation RotationStaticCEMA(mm)1 10 10010− 10−  ⁄10−3  ⁄1 10 10010− 10−  ⁄10−3  ⁄1 10 10010− 10−510−310− Figure 35: A summary of the 3D constant speed, varying process noise simulations for the single-marker flexible pins.  In each case, the input process noise values were scaled by the factors shown on the X-axis. The hollow points are at the default process noise. The trends in this case are the same as the 2D analysis.     76   00.020.040.060.080.100.120.140.16Translation Rotation1 10 10010− 10−  ⁄10−3  ⁄1 10 10010− 10−  ⁄10−3  ⁄CEMA(mm)1 10 10010− Static10−510−310− Figure 36: A summary of the 3D constant speed, varying process noise simulations for the double-marker flexible pins. In each case, the input process noise values were scaled by the factors shown on the X-axis. The hollow points are at the default process noise. The trends in this case are the same as the previous analyses, except that we see a lag behavior in the rotation case at low PCV levels.      77  4.1.5 Pin Bending Simulation Results The following series of plots (starting with Figure 37) present the results of the pin bending simulations. In these trials, the rigid body was kept static while one pin was deflected from the neutral position to a set amplitude and held for the remainder of the trial. Four different loading conditions were tested: a mid-pin point load, a distributed load, a pin-tip point load, and a pin-tip point load with a counteractive distributed load (simulate soft tissue resistance). Each condition was tested for tip deflections up to 40mm, in increments of 5mm. Our algorithm’s assumed loading condition was a mid-pin point load for all trials. The simulations were run in 2D, and for both tracker types in 3D. The 3D double-marker pins have two modes of bending (in-plane and out of plane), as well as an axial twist.      78     00.10.20.30.40.55 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40Static Mid-Pin Pt. Load Distributed Load Pin-Tip Pt. Load Pt. Load w. Skin ResistanceTip Displacement (mm)CEMA(mm)Figure 37: The results of the 2D pin bending simulations. Four different loading conditions were tested, each with tip displacements to 40mm, in 5mm increments. The assumed bending model was a mid-pin point load. The worst case tracking errors remain under 0.5mm, even with pin deflections up to 40mm.      79  00.511.525 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40Static Mid-Pin Pt. Load Distributed Load Pin-Tip Pt. Load Pt. Load w. Skin ResistanceCEMA(mm)Tip Displacement (mm)Figure 38: The results of the 3D pin bending simulations for the single-marker flexible tracker. Four different loading conditions were tested, each with tip displacements to 40mm, in 5mm increments. The assumed bending model was a mid-pin point load. The tracking errors remain submillimetric, even under the worst case loading conditions, up to 30mm of tip deflection.     80    01234565 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40Static Mid-Pin Pt. Load Distributed Load Pin-Tip Pt. Load Pt. Load w. Skin ResistanceTip Displacement (mm)CEMA(mm)5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 400123456Static Mid-Pin Pt. Load Distributed Load Pin-Tip Pt. Load Pt. Load w. Skin ResistanceTip Displacement (mm)CEMA(mm)Figure 39: The results of the 3D pin bending simulations for the double-marker flexible tracker. The top plot shows in-plane bending, and the bottom plot shows out of plane bending. The in-plane case is significantly more sensitive to bending than the out of plane case.     81  In general, all of our simulations demonstrated similar trends. Intuitively, we found that the outcome tracking error is proportional to the extent of deflection. In all cases, the mid-pin point load condition resulted in no increase of error from static, which makes sense as this condition represents the nominal bending model. The 3D single marker simulation showed submillimetric tracking errors for up to 30mm of tip deflection in the least-compatible loading case. The double-marker pins are significantly more sensitive to in-plane bending than out of plane bending, where in the worst case tracking errors rose to greater than 5mm. This is due to the nature of the double-marker pin design (discussed in more detail in Section 5.4).  00.010.020.030.045 10 15 20 25 30 35 40 45StaticCEMA(mm)Pin Twist (  )oFigure 40: Double Marker pin twist simulation. The pin was twisted to a maximum angle of 45º, in increments of 5º. Unlike in the bending cases, there are no different trajectories or loading conditions for the pin twist. The accuracy remains largely unchanged during the twist simulation.     82  4.1.6 Anchor Digitization Error Simulation Results The flexible tracker system requires the relative positions of each of the pins’ anchor points. This information could conceivably be acquired through a tracked pin insertion tool, or through a digitization step after the pins have been installed. Each of these methods introduce an additional source of error, which could be difficult to detect during use. Therefore, we want know how sensitive our system is to poorly located anchor points. The results of all of these simulations are summarized in Figure 41.  Each point represents a set of Monte Carlo simulations in which random noise of a specified magnitude was added to each of the anchor positions.  00.010.020.030.040.050.060 1 2 5 10 0 1 2 5 10 0 1 2 5 102D Flexible Tracker 3D Double-Marker 3D Single-MarkerAnchor Digitization Error, RMS (mm)CEMA(mm)Figure 41: Results of the simulated anchor digitization error trials, showing the 2D and 3D results. In the static case, anchor errors have little impact on tracking accuracy, even up to 10mm of anchor position error.     83  4.2 Experiment Results In the following section, we present the results of our experiment trials. We used a Northern Digital Inc. (NDI) Polaris Spectra positon sensor and a prototype set of flexible tracker pins to validate the system’s real world performance for a range of different conditions. The index below gives an overview of our experiments, and page references for specific results.  Experiment Section Page Static body  4.2.1 84 Freehand motion 4.2.2 87 Pin flexion 4.2.3 88 Constrained motion  4.2.4 97     84  4.2.1 Static Experiment Results In our static experiment the test piece was kept motionless and the system state was left undisturbed. As such, we expected our algorithm to estimate a static bone pose. In this trial the rigid body was kept fixed relative to the sensor, and we calculate the precision combined evaluation metric. First we present a sample of the data processing steps, beginning with the estimated rigid body coordinate shown in Figure 42. This figure shows only the X-coordinate of the rigid body pose, from the double-marker pin static trial.  The vertical axis has been adjusted to exemplify the stability of the estimation. We can see a short settling behaviour at 0 50 100 150 200 250 300103.0103.2103.4103.6103.8104.0104.2104.4104.6104.8105.0Time (frames)Position (mm)Settling Figure 42: Sample trace of the estimated rigid body position X component in static experiment. There is a brief settling behavior as the algorithm locates the body (lasting well under a second).      85  the beginning of the trial (frames 0 - 10) as the UKF corrects its initial estimation of the position coordinate. The position estimation remains invariable (± 0.015mm) for the remainder of the trial. The mean pose was calculated from post settling until the end of the trial, and this was used to calculate the combined precision metric. The CEMP is shown over time in Figure 43 for the single-marker pin static experiment, along with the 5th, 50th, and 95th percentiles of the data as red lines. 0 50 100 150 200 250 30000.010.020.030.040.050.060.070.080.090.1CEMP(mm)Time (frames)Figure 43: The precision combined evaluation metric over the duration of the single-marker pin static experiment trial, showing deviations from the mean pose.       86  Figure 44 presents the experimental static tracking precisions of our two flexible trackers and a rigid tracker. The plot also shows the corresponding simulation results (reported as accuracy metrics). The double-marker flexible pins and the rigid tracker were consistent between simulation and experiment results. The single-marker pins showed a notable increase in error from simulation to experiment, though only by about 0.02mm. 00.010.020.030.040.050.060.070.08Exp. Sim.Double-Marker Pins Single-Marker Pins Rigid TrackerExp. Sim. Exp. Sim.CEM (mm)Figure 44: A comparison between experiment and simulation under static conditions, for all tracker types. The experiment data are reported as precisions, and the simulation data as accuracies – though in the static case the two metrics are likely quite comparable.     87  4.2.2 Freehand Motion Experiment Results In the freehand motion experiments, we lifted the rigid body test piece from its initial static position and moved it around within the sensor’s tracking volume. We ran two trials for each flexible pin type, focusing on translations and rotations respectively. A sample set of translation trajectories is shown in Figure 45, from the single-marker pin freehand translation trial. The NDI rigid body data is superimposed on (and indistinguishable from) the flexible tracker’s estimated motion. 0 50 100 150 200 250 300-150-100-50050100150200250Time (frames)Displacement (mm)XYZNDI RBLifted test pieceLowered test piece Figure 45: Sample position trajectory during the single-marker pin freehand motion trial (translation case). The trajectory of the NDI rigid body is super imposed on the axis, and sit indistinguishable from the estimated trajectory curves.     88  Figure 47 shows a comparison between simulated and experimental motion trials. This comparison is not entirely valid, because the simulations are reported as accuracy CEMs, and the experiments as discrepancy CEMs. Figure 47 summarizes the discrepancy metrics for all of the freehand motion trials, and compares them with corresponding simulation results. This likely contributes to the greater spread of the CEM for the experiment points. The single-marker pin rotation trial (although on average was around 0.2mm CEMD), reached combined pose discrepancies up to 0.75mm, which is not negligible. This error in rotation is likely related to the rotation error in the motion-speed simulations (Section 4.1.3).  0 50 100 150 200 25000.20.40.60.811.21.41.61.82CEMD(mm)Time (frames)Figure 46: The discrepancy between the single-marker flexible pins and the NDI rigid tracker during the freehand translation trial, reported as a CEM. After a settling period, the discrepancy remains fairly consistent throughout the trial.     89  4.2.3 Pin Flexion Experiment Results We ran the pin flexion experiments to validate our pin bending model. The rigid body was kept static, and a single pin was bent while the position sensor recorded marker coordinates. We first present the experimental bending data over time, comparing the pin bending trajectory with the disturbance of the rigid body pose from static. We also present plots of CEM against pin displacement to compare our experiments against our simulations.   00.10.20.30.40.50.60.70.80.9CEM (mm)Translation RotationDouble-Marker PinsSingle-Marker PinsExperiment Simulation Experiment SimulationTranslation Rotation Translation Rotation Translation RotationFigure 47: A comparison between experimental and simulated motion trials. The experiments are reported as discrepancy CEMs, and simulations are reported as accuracy CEMs, so a direct comparison is not entirely valid.     90  Double-marker pins, out-of-plane bending  20 40 60 80 100 120 1400510152025303540Time (frames)20 40 60 80 100 120 1400510152025303540Time (frames)20 40 60 80 100 120 1400510152025303540Time (frames)Time (frames) Time (frames)20 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.91Displacement (mm)Time (frames)CEMP(mm)Small Deflection Medium Deflection Large DeflectionFigure 48: Small, medium, and large out-of-plane deflections of the double-marker flexible tracker pin.  The top row shows a measure of the horizontal component of the deflection, and the bottom row shows the corresponding precision CEM.     91  Double-marker pins, in-plane bending  20 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.91CEMP(mm)Displacement (mm)Time (frames) Time (frames) Time (frames)Small Deflection Medium Deflection Large Deflection20 40 60 80 100 120 14005101520253020 40 60 80 100 120 14005101520253020 40 60 80 100 120 140051015202530Time (frames) Time (frames) Time (frames)Figure 49: Small, medium, and large in-plane deflections of the double-marker flexible tracker pin.  The top row shows a measure of the horizontal component of the deflection, and the bottom row shows the corresponding precision CEM.     92  Double-marker pins, axial twist  0 20 40 60 80 100 120 140-50-40-30-20-1001020304050Displacement (degrees)Time (frames)0 20 40 60 80 100 120 140-50-40-30-20-1001020304050Time (frames)0 20 40 60 80 100 120 1400.10.20.30.40.50.60.70.80.91CEMP(mm)Time (frames)0 20 40 60 80 100 120 1400.10.20.30.40.50.60.70.80.91Time (frames)Small Twist Large TwistFigure 50: Small and large axial twist of the double-marker flexible tracker pin. The top row shows detected twist angle, and the bottom row shows the corresponding precision CEM. These results are inconsistent with our simulations, which showed an insensitivity to pin twist.     93  Single-marker pins, approximated mid-pin loading  20 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 1400510152025303540455020 40 60 80 100 120 1400510152025303540455020 40 60 80 100 120 14005101520253035404550Time (frames) Time (frames) Time (frames)Time (frames) Time (frames) Time (frames)CEMP(mm)Displacement (mm)Small Deflection Medium Deflection Large DeflectionFigure 51: Small, medium, and large deflections of the single-marker pin, bent with an approximated mid-pin point load. This loading condition is the implicit bending model for our Kalman filter. The top row shows the detected pin bending trajectory, and the bottom row shows the corresponding precision CEM.     94  Single-marker pins, approximated pin-tip loading 20 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 14000.10.20.30.40.50.60.70.80.9120 40 60 80 100 120 1400510152025303540455020 40 60 80 100 120 1400510152025303540455020 40 60 80 100 120 14005101520253035404550Time (frames) Time (frames) Time (frames)Time (frames) Time (frames) Time (frames)CEMP(mm)Displacement (mm)Small Deflection Medium Deflection Large DeflectionFigure 52: Small, medium, and large deflections of the single-marker pin, bent with an approximated pin-tip point load. We expect greater error for these loading conditions. The top row shows the detected pin bending trajectory, and the bottom row shows the corresponding precision CEM.      95   0 5 10 15 20 25 3000.511.522.53Pin Deflection (mm)CEM  (mm)Large def’lMed. def’lSmall def’lSim. Tip PLSim. Mid PLFigure 53: Plot of CEM versus pin deflection for the double-marker in-plane bending case, comparing the experiment and simulation data sets. The experiment data was fit linearly, and agrees with the simulation results (shown as dashed lines). 0 5 10 15 20 25 30 35 4000.10.20.30.40.50.60.7Large def’lMed. def’lSmall def’lSim. Tip PLSim. Mid PLPin Deflection (mm)CEM  (mm)Figure 54: A plot of CEM versus pin deflection for the double-marker out-of-plane bending case, comparing the experiment and simulation data sets. The experimental results lay outside of the envelope predicted by simulations. The experimental data was fit linearly.     96     0 5 10 15 20 25 30 35 40 45 5000.20.40.60.811.21.41.61.8Exp. Mid PLExp. Tip PLSim. Tip PLSim. Mid PLPin Deflection (mm)CEM  (mm)Figure 55: A plot of CEM versus pin deflection for the single-marker pins, comparing experiment with simulation. The blue curves represent pin-tip point load, and the red curves represent mid-pin point loads. The experimental data was fit with a cubic polynomial. The results are consistent in that pin-tip loading creates larger tracking errors than mid-pin loading.       97  The results of our flexion experiments largely agree with the simulations. The double-marker twist trial was one notable anomaly; our simulations predicted an insensitivity to pin twisting, but our experiments showed a notable spike of error during twist, as high as 1mm for a 45º twist. These discrepancies are discussed in greater detail in Section 5.4. 4.2.4 Translation Stage Experiment Results In this experiment we attached the rigid body test piece to a high-precision linear translation stage. Figure 56 shows the in-plane (Y) and out of plane (Z) translation trajectories for single marker pins. 0 500 1000 1500 2000 2500 3000 3500 400002468101214Time (frames)Position (mm)0 500 1000 1500 2000 2500 3000 3500 400002468101214Time (frames)In Plane (Y) Out of Plane (Z)Figure 56: Motion profile for both translation stage trials: in-plane (y) and out-of-plane (z). Each trial consisted of two “there-and-back” translations about the stage’s maximum range. Rest-periods were manually segmented and are shown in red.      98   In general, the translation errors fell between 0.02mm and 0.23mm. The Z-direction translations were more accurate, having a mean translation error of only 0.04mm for both the single and double-marker pin types. The Y-direction translations were less accurate, with mean translation errors of 0.10 and 0.12mm. The following table summarizes the results of our high-precision translation stage trials, for both tracker types and both translation directions.  Tracker Type Translation Direction Mean accuracy (range) Double Marker Tracker Cross Plane (Y) 0.10mm   (0.07 to 0.11) Camera Direction (Z) 0.04mm   (0.02 to 0.06) Single Marker Tracker Cross Plane (Y) 0.12mm   (0.04 to 0.23) Camera Direction (Z) 0.04mm   (0.03 to 0.05) Table 8: Summary of results from the high-precision translation stage experiments. Both the single- and double-marker flexible pin types were translated four times in both the global Y and Z directions.   For both types of tracker pins, we see greater accuracy in the Z-direction. At first this seems unexpected because the Polaris Spectra suffers the poorest measurement quality along that axis (line-of-sight direction). However, the experimental apparatus was configured such that the Z-direction is the most constrained. For our single-marker tracker, three pins are oriented in the global Z axis, and only 1 pin was oriented in the Y direction.    99  Chapter 5  Discussion and Conclusion In this chapter we discuss the significance of our results in the context of orthopaedic navigation, we highlight some limitations of our analysis, and we make recommendations for future tasks. The purpose of this thesis was to demonstrate the feasibility of estimating a rigid body pose using a novel flexible optical tracker system. We ran simulations and experiments inspired by a total knee arthroplasty and found that our flexible pin system could track a body with high levels of accuracy and precision under typical operating conditions. In static and translation cases, our simulations predicted tracking errors less than 0.04mm. In rotation, the tracking errors scaled proportionally with speed, but remained below 0.30mm for rotations up to 90º/s and settled back to static levels after motion. The tracker’s estimated body pose deviated less than 1mm for simulated unexpected bending conditions up to 30mm of lateral deflection. We observed comparable trends in our experiments, but for a lack of gold standard we generally reported precision metrics instead of accuracy.     100  We encountered some noteworthy anomalies in the double-marker pin bending trials. Our simulations predicted that the in-plane bending case is highly sensitive to variability in loading conditions, with CEM errors up to 5mm for lateral deflections of 30mm under different loading conditions. Additionally, our simulations predicted that the double-marker pin is insensitive to axial twist, however in the experiments we found observed pose deviations of about 1mm for 45º pin twists. We will explore these topics in greater detail later in the chapter, but first we discuss general limitations of our study. 5.1 Study Limitations As explained in Section 2.2, our algorithm’s implicit pin bending model was based on a mid-pin point load condition. However, in making this decision, we neglected to consider: (1) lateral bending stiffness is not a constant under all loading conditions; marker deflections are easier to achieve with loads applied nearer to the marker, (2) all loading conditions are not equally as probable, for example the pin may be knocked more frequently at the tip than at the base, and (3) the upper limit we set on bending contraction is not the most extreme case possible, since greater bending contractions could be achieved with a counteracting force such as soft tissue resistance. To address these limitations, we could reconsider our implicit bending model to better reflect realistic bending cases, or we could alter the design of the hardware to give better control over how the pins deflect (discussed later in this chapter).     101  Another limitation to our findings about pin bending comes from the limited scope of the bending simulations and experiments. Our tests involved deflecting one pin at a time, but not combinations of multiple pins bending in different ways. With this information, we could perhaps create a more detailed, probabilistic UKF measurement model, which would improve the system’s performance/robustness. We did, however, establish a basic understanding of how our tracker behaves under a variety of bending cases and loading conditions, on which further analysis could be based. We estimate that the tracking errors from multiple pins bending would compound; in a best case scenario they may cancel out, but otherwise they may proportionately scale the tracking error.  Our study was limited in part by our basis for selecting our configuration of flexible pins. Our configuration was chosen heuristically based on our understanding of constraint theory (Section 3.1.2), and although it proved sufficiently accurate, we did not explicitly demonstrate that our chosen pin layout was optimal. Simon (1995) presents a novel technique for optimally selecting constraint points for a given geometry, which we believe could be adapted for this work to determine ideal pin locations. In addition to geometrical issues, we must also consider the pin configuration limits that would be imposed by the surgical environment. Possible anchor sites could be restricted by access to anatomical structures, the patient’s pose, and the OR layout (eg. position sensor, staff). These factors would vary by application and surgeon preference, so it is difficult to describe how to select optimal pin configuration for a given surgical application. Such a     102  process would need to take into account a set of surgical constraints (incision sites, bone access, patient pose) in order to determine where to best attach the pins for maximum geometric constraint. Our simulated and experimental configurations did not result from an explicit optimization process, so our resulting performance measures are likely somewhat understated. We explored the impact of anchor digitization error on the overall tracking accuracy with a series of simulations (Section 4.1.6). The proposed methods for determining relative anchor positions rely on optical tracking, which will inherently introduce some error in these estimates. A limitation of our anchor uncertainty simulations is that we only considered the static case. It is likely that a combination of anchor errors with pin bending would result in slightly larger overall position estimation errors. However, since both of these errors are relatively small, we believe that the overall system would remain acceptably accurate. Our experiments were limited in several ways. First, in contrast to simulation, we had no independent ‘gold standard’ for assessing the true position of the tracked object. Optical tracking is generally regarded as a very accurate motion tracking technology, and to validate our optical tracker with a more accurate technology (such as a coordinate measurement machine (CMM)) would be prohibitively expensive. Therefore in experiments we generally restricted ourselves to metrics of measurement precision and discrepancy. In one case, we assessed the system accuracy by attaching the test piece to a high precision linear translation stage (Section 3.2.6), but limited our assessment to relatively short translations, neglecting     103  rotations. Additionally, the constrained translations were at approximately 0.5mm per second, which is slower than we might expect motions to occur during use. A further limitation of our experiments comes from unrepeatable and imprecise pin bending trials (Section 3.2.5). The loading conditions, the magnitude and the direction of bending were based on our crude approximations. Ideally we would have had some sort of rig that could reliably and repeatedly impose controlled deflections to the pins. This limitation affects our precise characterization of the system’s behaviour, but does not detract from the observed performance. 5.2 Evaluating the Flexible Optical Tracker Design In this project we tested a novel optical tracker system under typical operating conditions through a series of simulations and experiments. This work constitutes a proof of concept of the flexible tracker in which we verify the system’s tracking performance. In the introduction chapter (Section 1.5), we laid out a set of requirements for the new tracker system. In the following sections we will readdress and evaluate these requirements by drawing on evidence from the results of our simulations and experiments. First we analyze the basic tracking performance by comparing precisions and accuracies to the literature. Second, we expand our discussion of pin bending to explain its effect on tracking error, and we propose bending error mitigation strategies. Third, we discuss the UKF algorithm. Lastly, we cover some clinical considerations for the design, followed by a conclusion.     104  5.3 Discussion on Tracking Performance An important requirement for our flexible tracker system is that the tracking performance must be comparable to, or better than conventional rigid systems. Before discussing our results, we briefly explain the distinction between accuracy and precision. Accuracy is the degree of closeness of an estimated quantity and the true value. Precision is the degree to which repeated estimations under unchanged conditions show the same results (Leardini 2013). There are several factors that determine the precision of optical tracking technologies, including a finite number of pixels in the sensor, imperfect optics, and triangulation errors (Khadem 2000).  5.3.1 Simulation Findings Our static simulations (Section 4.1.1) predicted that the flexible tracker can estimate the body’s pose to within 0.015 and 0.009mm of the true position (mean CEM for the double- and single-marker pin types respectively). The simulated rigid tracker design had a mean accuracy of 0.018mm CEM. There is no clinical significance to such minor discrepancies of tracker accuracy. For all intents and purposes, both the rigid and flexible trackers perform comparably well. As an example, consider an arc formed by rotating 50mm by 3º (our CEM scale factor rotated by the threshold of what is considered an alignment outlier (Jeffery 1991)). The resulting arc length is 2.62mm, or 2618μm. By comparison, our simulated tracker demonstrated combined errors less than 0.1mm for typical operating conditions,     105  which constitutes less than 5% of the 2.62mm arc length. The trackers were all comparably precise with mean combined precisions ranging from 0.01 to 0.03mm under simulated static conditions. Typical motions in orthopaedic surgery are infrequent and fairly modest; we estimate limb segment velocities to be on order of cm per second in translation, and tens of degrees per second in rotation based on our observations of arthroplasty procedures. Our simulations predicted that the flexible tracker’s accuracy is insensitive to translations at least as fast as 3000mm/s. In rotation, however, we observed that tracking errors grew proportionally with motion speed, to about 0.30mm mean combined error at 90º/s. We discuss this rotation error in more detail in Section 5.5. The rotation error returned to static levels as the rotation motion ceased. This rotation error is of little practical concern because the errors remain acceptably low (0.30mm) at considerable rotation speeds (90º/s). Furthermore, surgical actions will generally only occur under static conditions. 5.3.2 Experiment Findings The results of our experiments were generally consistent with our simulations. However, a direct comparison is unwarranted because our simulations reported system accuracy, whereas our experiments mostly reported system precisions and discrepancies. We only measured system accuracy in one case in which the body was attached to a high precision translation stage.     106  From our static experiments, we calculated mean precisions of 0.015 and 0.034mm CEM for the double- and single-marker flexible pins respectively. For the rigid tracker, we calculated a mean combined precision of 0.018mm. Much like in simulations, these quantities are comparable and represent very small deviations from the mean for the application. These precision findings are consistent with those reported by Khadem (2000). Their study compared the precision of five different optical sensors by repeatedly measuring a static rigid body at different locations in the measurement volume. For the original Polaris sensor in passive mode, they reported rigid body positional precisions ranging from 0.050 – 0.180mm, varying as a function of measurement depth (Z-position). The Polaris Spectra sensor that we used (launched in 2006) has a slightly better tracking accuracy than the older Polaris (by 0.10mm RMS), which may explain discrepancy between our results. In our freehand experiment motion trials, we measured the discrepancy between the flexible and rigid trackers. We found a mean discrepancy of 0.19mm CEM for the single marker pin, and 0.16mm CEM for the double marker pin. The discrepancy metric was notably larger in magnitude than our other precision and accuracy metrics, with peak deviations as high as 0.75mm for the single-marker pins in rotation. Although this could be related to the rotation speed error that we observed in simulations, the discrepancy metric is fundamentally larger than the other metrics we used because the two trackers’ individual errors compound. Because we have no independent gold standard, we cannot determine which of the rigid and flexible tracker designs contribute most to the overall discrepancy measure.     107  In one trial we simultaneously measured the body’s translation with a high-precision translation stage and a micrometer. The mean translation errors were 0.12mm( ⃗⃗) and 0.04mm(z⃗) for the single-marker tracker, and 0.10mm( ⃗⃗) and 0.04mm(z⃗) for the double-marker tracker, based on four translations per direction per tracker. These findings are consistent with values reported by Wiles (2004), who found mean translation accuracies of 0.185mm for a simple 3 marker passive rigid body with the original Polaris sensor. The accuracy improvements in the Polaris Spectra model may again explain some of the discrepancy between our findings and those of the earlier study. 5.4 Discussion on Pin Deflections A key aspect of our tracker design is that it allows for modest deflections of individual pins without significantly impacting the system accuracy. For a detailed explanation of our bending methods, see Section 2.2.  This section explains source of bending error, strategies to mitigate this error, and other practical considerations. Conventional rigid optical trackers risk losing registration if knocked, which could severely affect the surgical outcome if undetected (Leardini 2013). We demonstrated in experiment that the (single-marker) flexible tracker could be deflected by 45mm without disturbing the estimated pose by any more than 0.5mm. Pin bending error stems mostly from variability in pin loading conditions.     108  5.4.1 Unpredictable Loading Conditions An important source of error in our method comes from the variability in loading conditions, which we describe in detail in Section 2.2.1. Our flexion simulations (Section 4.1.5) demonstrated that tracking error is proportional to the magnitude of a deflection, and the amount of disagreement between the applied bend and modelled bend. Figure 57 (left) shows a pin-tip loading case aligned with our modelled loading case. The markers’ locations are the only conduit between the physical and virtual spaces. As such, pin bending discrepancy is interpreted as a flexion plus a displacement of the anchor point, which affects the pose estimation as shown in Figure 57 (right).   Figure 57: A demonstration of how different bending trajectories can invoke errors in the pose estimation. (Left) Anchor-aligned comparison between two different bends. (Right) The only input to the algorithm is the location of the marker, so bending variability is interpreted by shifting the anchor point.     109  The double-marker pins exhibit similar behaviour in bending, but with greater complexity due to the crossbar at the end of the shaft. The double-marker pin is most sensitive to in-plane bending, with simulated tracking errors of 5mm for 40mm deflections (Figure 39 on page 80). In this case, the discrepancy between applied and assumed bends is amplified by the marker separation. The crossbar’s orientation is dictated by the shaft’s tip angle, which varies with loading conditions. Figure 58 compares marker positions for three different in-plane loading conditions.  6Figure 58: A demonstration of in-plane bending. Red shows mid-pin loading, blue shows pin tip loading, and black shows a rigid link deflection. The crossbar segment amplifies errors associated with bending variations, leads to considerable tracking errors (~5mm at 40mm deflection for certain loading conditions).      110  5.4.2 Mitigating the Effects of Pin Bending Variability From our flexion simulations, we observed that the system becomes less sensitive to marker deflections when the applied load more closely matches our implicit model. We could substantially reduce these errors by specifically designing the pin hardware to reduce bending variability. For example, we could design the pins to have a short compliant segment near the shaft’s anchor to absorb deflections (as shown in Figure 59). The compliance could come from a flexible polymer segment, or from adding notches to the metal shaft to promote localized flexure. This concept will require further study.  Slender uniform shaft Non-uniform shaftFigure 59: An example of a design solution for overcoming errors associated with bending variability. Concentrating the pin’s flexure by adding a compliant joint segment would make pin bending more predictable.     111  5.4.3 Discrepancy in Pin Twist Simulations and Experiments Our simulations predicted that the double-marker tracker would be insensitive to axial twist effects, but our twist experiments (Section 4.2.3) showed pose deviations up to 1mm CEM for 45º twists. We lack an explanation for this discrepancy, but we speculate that it may result from systematic errors or from an oversight in our simulation model. This topic merits further investigation. In its current form, the pin’s torsional stiffness is greater than its lateral stiffness (derived in in Appendix C), meaning that tracker will be displaced more by bending than by twisting under most loading conditions. We could increase torsional stiffness by adjusting the tracker’s dimensions, further reducing twist effects. 5.4.4 Flexion Safeguard As a practical consideration, our system could alert the user if a pin is being overflexed. The UKF explicitly estimates the deflection state of each pin. Aspects such as the threshold for overbending, or the optimal way to display errors to the surgeon should be studied through more comprehensive user testing. 5.5 Discussion on the Unscented Kalman Filter Algorithm We used an unscented Kalman filter algorithm to enable the flexible tracker system to function at real-time speeds. Detailed information about our UKF can be found in Section 2.3.3, and in Appendix B. In this section, we compare our UKF with the optimization-based     112  algorithm, we discuss the significance of the process noise simulations, and we explain the rotation motion anomaly. We investigated two major approaches to estimating the location of the rigid body given the marker measurements:  (1) an optimization algorithm and (2) a Kalman-filter-based observer approach.  The optimization algorithm works by trying to solve an optimization problem at each time step in which the system state consists of the position of the rigid body, together with the deflections of all the pins. The cost function is the squared distance between the predicted and measured marker locations. We integrated MATLAB’s standard fminunc function into a recursive loop, solving for the body pose at each time step. While this approach worked well in our simulations, it proved to be computationally inefficient compared with the Kalman filter approach and had occasional problems with local (false) minima. The observer-based approach recursively updates a state estimate based on a weighted consideration of the previous state and information from new measurements. We implemented a nonlinear version of an observer known as an unscented Kalman filter (UKF) that enabled us to take into account various nonlinearities in the relationships between the state and the predicted marker locations. In practice, the UKF was robust, computationally efficient and accurate, so we stopped using the optimization approach after completing the static simulations; all subsequent work in this thesis was based on the UKF.     113  5.5.1 Process Noise Our process noise simulations (Section 4.1.3) changes in the tracker’s behaviour for different levels of process noise. The results were largely as expected: decreasing the process noise levels increases the UKF’s certainty that the body will remain static, which enables a smoother pose estimation in the static case but creates a lag effect in motions. At the nominal process noise levels, however, the static process model is almost ignored entirely, and the state estimation is mostly based on the measurements and measurement model. We found that this worked well across most testing conditions, and we recommend keeping the current values for process noise. 5.5.2 Rotation Motion Anomaly While the UKF generally worked reliably and accurately, we encountered situations in which it gave unexpected results. For example, our rotation simulations showed that tracking error rises proportionately with speed. We believe that this behaviour results from how the UKF handles nonlinear functions. Rotation is the non-linear component of the measurement model, and evidently we see greater errors arise as we increase the rate of the rotation. Small changes of orientation (slow rotations) can be approximated well linearly. However, the nonlinearity of rotation dominates as wider sigma points (ie. greater rotation speeds) are used in the UKF’s unscented transform to calculate a weighted average of estimated marker coordinates.     114  Our motion speed simulations predicted that the estimated pose would deviate by only 0.3mm for rotations of 90º per second – a speed faster than we would expect to see in a typical TKA.  Furthermore, no surgical actions would be made while the bone is in motion. 5.6 Clinical Considerations This thesis served to verify the feasibility of tracking a rigid body using flexible optical tracker pins. This was a lab-based proof of concept stage in a larger design project aimed at creating a more intuitive and user-friendly optical tracker for orthopedic navigation systems. In future technical development of the tracker system, we anticipate running user-studies with cadaveric specimens to assess the system’s ergonomics and clinical workflow. Many of our system’s design requirements were established to improve upon the perceived shortcomings of conventional rigid trackers; it should not further increase operating time or worsen the line of sight issue, and it should enable unicortical tracker fixation. Although a complete user validation is beyond the scope of this thesis, we can at least speculate about how a flexible tracker system could achieve these requirements. First, the flexible tracker system should not significantly increase operating time. Many authors criticize navigation technology for taking too much time, thereby increasing risk for patients (Bäthis 2004; Haritinian 2013; Sikorski 2003). Once the flexible tracker is installed on the bone, the navigated surgery would proceed much the same as with a rigid tracker and would not affect execution of existing steps such as the bone registration or the     115  joint centre calibration. The greatest difference between the tracker types (regarding time) would be in the tracker insertion and removal steps. Future iterations of the flexible tracker system could require an optically-tracked “pin injector” tool for quickly inserting flexible pins into the bone while recording each of their relative positions and orientations. We believe that this is entirely feasible, based on numerous existing designs that enable rapid bone penetration, such as the Pyng FAST system, the EZ IO system, the Bone Injection Gun, and others, including pneumatic nail-gun-like technologies that have been shown to be superior to drilling (Franssen 2009).  Second, the flexible tracker system should not worsen the line of sight issue that hinders optical tracking systems. This issue is a common frustration for surgeons using optical navigation, especially in a crowded OR (Craven 2005). We have no compelling reason to believe that the flexible tracker would worsen the line of sight issue. The measurement volume remains the same for both types of optical trackers, covering from approximately the distal tibia to the proximal femur in a TKA. While the configuration of markers on the rigid trackers is more compact, the surgeon requires a fairly large field of view to be able to view their tracked tools. This would be a topic for a future user study in a mock clinical environment. Lastly, the flexible pin system should enable unicortical fixation of the trackers to minimize the invasiveness and patient risk (Laskin 2006; Ossendorf 2006). Conventional trackers require bicortical fixation to retain stability on the bone (Jung 2007). Although the     116  flexible trackers require more insertion sites than conventional trackers, conceivably the anchor could be very small and allow for unicortical fixation. Flexible trackers are well suited for unicortical fixation because the flexible shaft would not transmit any significant moment to the anchor due to a deflecting force applied at the tip. The anchor mechanism presented in this work was designed for use with wood, and would be unsuitable for surgery. For future versions of the anchor, we could look to a number of existing design solutions for short-term unicortical bone penetration and fixation, such as the Unitek Temporary Anchorage Device System (TAD), or the Pyng FAST infusion system. 5.7 Thesis Contributions and Concluding Remarks Through a series of simulations and experiments, we have demonstrated the feasibility of tracking a bone segment with a set of flexible pins and an optical sensor. The design of our tracker addresses many of the human-factors issues associated with conventional rigid trackers for computer assisted orthopaedic surgery, which are generally large and cumbersome. 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IEEE Transactions on Medical Imaging, 23(5), 533–545. Wiles, A. D., Thompson, D. G., & Frantz, D. D. (2004). Accuracy assessment and interpretation for optical tracking systems. In R. L. Galloway, Jr. (Ed.), Medical Imaging 2004: Visualization, Image-Guided Procedures, and Display (Vol. 5367, pp. 421–432).        122  Appendices Appendix A –   CAOS Questionnaire We sent a questionnaire to 82 orthopaedic surgeons around the world who have experience using CAOS technology, on May 17 2014. The survey was approved by the UBC Behavioral Research Ethics Board (H13-01360). We received 13 responses. This appendix presents the survey questions and a summary of the numerical results. Any self-identifying responses have not been included in this section to for the privacy of our responders. Part 1 – Your Experience as an Orthopaedic Surgeon Q1. Surgical Experience Indicate your level of surgical experience       Resident 2 14% Fellow 0 0% Attending 5 36% Other 4 29%     123  Q2. Area of Expertise Computer assisted, minimally invasive hip and Knee arthroplasty Computer Navigation, hip resurfacing TKA THA Osteotomies around the knee Sport Traumatology Knee and hip arthroplasty and lower limb surgery Hip and Knee Replacement Lower limb arthroplasty. Total Hip and Knee Replacement Surgery Computer and Robotically Assisted Orthopaedic Surgery joints replacement cas Lower Limb Arthroplasty I often use a computer assisted in Knee Joint replacement Hip-,Knee-and Shoulder replacement TKR computer assisted THR computer assisted MAKO robotic PKR Orthopaedics, Spine, Hip.  Q3. Experience with CAOS Briefly describe your level of experience using CAOS systems (ie. you use it whenever possible, often, occasionally, rarely, once, etc.). Which procedures do you most often use CAOS with?  Often for hip resurfacing. Conducted PhD in area. Have used it for 10 years on all knees and most hips. I was part of the development team for DePuy Ci System (BrainLab). I also developed Robodoc. CAOS used 100% Total Knee Replacement 20% revision TKR and 100% HTO 80% UniKnee All of my Total Knees have been performed using CAOS since 2005 soon to reach 2000 in number. I have only used CAOS in the lab for THR not in clinical use. As above 350 cases/yr Previous user for knees - very rare now. Not used except for TKR i use CAOS since 2000 in evrey knee joint replacement. also short experience in hip joint replacement     124   … TKA UniKA Osteotomies around the knee More than 400 cases per year in my department Using Stryker Nav since 2003 (knee) and 2004 (hip), approximately 250 TKA and 175 THA PER YEAR. Also 25 - 30 unicompartmental knees per year, and frequent use in revision TKA and THA I always use computer navigation in total knee arthroplasty in cases of osteoarthritis in combination with extraarticular femoral deformities I've been using cas for knee and hip since 1999 I have been a consultant for 6 years and having been using computer navigation for my primary knees. I do about 150-200 knees a year using this system Spine: very often Hip:always (except revision) Knee: always (except revision) HTO: always Q4. Type of CAOS System Used Stryker Stryker/MAKO BrainLab orthopilot brainlab Mako orthokey orthosoft CT based Intra-op 3D imaging (C arm) Image free +++ BBraun (Aesculap) Columbus Knee replacement Galileo PI Aarau,Switzerland) now smith and nephew PiGalileo Medacta GMK Nav Omni-Praxim iBlock OrthoPilot (BBraun/ Aesculap) Navio PFS (BBT) BrainLAB VectorVision (IR) Stryker Have observed Orthopilot but never used. Stryker Orthopilot device (BBraun-Aesculap Company) Stryker Nav. Have limited experience with other systems.       125  Part 2 – Your Experience with CAOS Systems The following statements are common concerns dealing with CAOS systems.  Please rate each concern on a scale from 1 to 7, based on your experience using CAOS.   1   A significant impact 4   A moderate impact 7   A negligible impact Q5. Line of Sight Optically-based CAOS systems require a constant line-of-sight between the camera and the bone-mounted markers.  To what degree do you find this interferes with your workflow?  Additional Comments: We used for years active trackers and I would have put 7. Since we used passive trackers I would mark 6 Not an issue Stryker is unique with its active tracker system. I adapted Slightly, but my entire routine is based around using it. Occassionally new assistants have to be told to get out the way.       126  Q6. Size of CAOS Marker Frames Commercially available optical marker frames can be on the order of 10-20 cm in size, or even larger, depending on the application. To what extent do you find that the size of the marker frames interferes with your ability to perform surgical actions?  Additional Comments: We recently changed from the active to the passive trackers which are bigger. I now have to hold the saw upside down to cut the lateral side of the tibial plateau. Used to them already. Markers are too big. I think it could be possible to have smaller ones with the same accuracy. Q7. Invasiveness CAOS systems sometimes require additional incisions to be made for the marker frames to be fixed to the patient's bones. How much of a concern is it to you when additional incisions have to be made for this purpose?     127   Additional Comments: If we have smaller markers it will be possible to insert the markers in the surgical approach... I'm used to using it. I have had I believe three infected tibial tracker sites out of many hundreds of knees. 2 stab wounds in Tibia for TKR and Femoral array is incorporated in my TKR incision. After thousands of procedures, only one known pin site infection and two broken pin tips. No fractures although a known risk. Incisions are on the order of 3mm and are of no concern to the patients. Q8. Time Consuming CAOS systems have been reported to extend the duration of surgical procedures. In your experience, what effect does using a CAOS system have on the length of an operation?        128  Additional Comments: I really think it saves time in the long run. We reported our data at _________. If one uses "light" softwares the increasing time is about 5 mn that is negligible when an intervention is about 1 hour... Adds 10 min to hips and 20 to knees. I'm so used to using the navigation that I would now be slower if I didn't use it. A typical primary knee using navigation takes me 50-55 minutes from skin incision to dressings on After A short learning curve, the Surgical Time actually decreased significantly  Q9. Reliability Surgical delays have been reported due to malfunctions or miscalibrations of CAOS systems.  In approximately what proportion of procedures do you find that you have to revert to a conventional method (without CAOS), because the CAOS technology is too unreliable or finnicky?     Additional Comments: Almost never. We had a couple of computer crashes in the last year after the company updated the software. Fortunately we have four machines in our unit and brought in a different one and re-registered the navigation. Maybe 1 in 400 cases I have abandoned CAOS less than once per 1000 cases. I have had to abort 3 cases out of a consecutive series of almost 2000. I went to conventional surgery and boy that is when you really miss the benefits of CAOS. You know what they say you don't know how much you love something or someone until you don't have it. I found that I lost all the finesse and data feedback I was used to helping me make decisions.      129  Part 3 – Your Opinion on CAOS Systems Q10. Perceived Benefits Based on your own experience, understanding and judgement of CAOS technology, please rate how compelling each of the following assertions is to you, for adopting CAOS technology in the operating room. You may pick from four options by which to rate each statement: Strong   “this is a strong reason to adopt CAOS” Moderate   “this is a moderate reason to adopt CAOS” Weak   “this is a weak reason to adopt CAOS” N/A   “I do not believe this is a benefit of CAOS”  Using CAOS systems can allow for better accuracy than conventional surgical methods.  CAOS systems allow surgeons to perform more consistently.   Strong 12 86% Moderate 0 0% Weak 1 7% N/A 0 0% Strong 11 79% Moderate 0 0% Weak 2 14% N/A 0 0%     130   Using CAOS systems leads to better functional outcomes for the patient.   CAOS systems are useful in training younger surgeons correct technique.    CAOS systems provide reliable, objective documentation of an operation (in the face of potential malpractice litigation).    Strong 3 21% Moderate 3 21% Weak 6 43% N/A 1 7% Strong 5 36% Moderate 4 29% Weak 1 7% N/A 3 21% Strong 7 50% Moderate 1 7% Weak 4 29% N/A 1 7%     131   CAOS technology allows minimally invasive procedures to be done more easily.   Additional Comments: Using CAOS in TKR since 2005 has made me a better surgeon assuring my patients of an excellent result in the OR. CAS-TKR is both a measured resection bone cutting surgery and ligament and soft tissue balancing procedure using CAS with virtual planning and robotics and real time instant validations assures a good result. This is what ________ has written about so long ago. CAOS brings TKR into the next millennium. I like being able to check my own or trainees cuts. I am horrified when using non navigated knee revision products how much I can move some cutting blocks and being unable to verify the accuracy of what I have done. I have had experience of using the navigation to ward off a malpractise issue. A surgeon elsewhere reviewed a patient of mine and said my tibial component was in varus based on a short x-ray. I was able to print off the navigation report which showed that the knee was exactly on the mechanical axis (as did our long leg x-ray). I believe the Australian registry now demonstrates better survivorship of navigated knees in the under 65 age group. Q11.  Perceived Concerns The following are common criticisms of CAOS technology.  Based on your experience, understanding, and judgement of CAOS technology, please rate how compelling each of the following statements is, to you, for NOT using CAOS in the operating room. You may pick from four options, by which to rate each statement:  Strong   “this is a strong reason to avoid CAOS” Moderate   “this is a moderate reason to avoid CAOS” Weak   “this is a weak reason to avoid CAOS” N/A   “I do not believe CAOS suffers from this problem” Strong 4 29% Moderate 5 36% Weak 3 21% N/A 1 7%        132   Using CAOS systems may increase risk of pin-site bone fracture.   Using CAOS systems can be disruptive to OR workflow.   CAOS systems are too expensive.     Strong 1 7% Moderate 1 7% Weak 10 71% N/A 1 7% Strong 0 0% Moderate 3 21% Weak 8 57% N/A 2 14% Strong 2 14% Moderate 4 29% Weak 5 36% N/A 2 14%     133  CAOS systems take too long to set up.   Optical marker frames are too large to use on smaller bones.    Additional Comments: The above concerns are probably in fact true when you first start using navigation. I hated navigation when I first started using it as I felt it made the operation much harder (I only persisted with navigation because my clinical lead wouldn't give me any knees to do unless I used it). Undoubtedly you get much better at it and eventually these concerns are no longer a problem. Too bulky when using at the same time, arthroscopy and image intensifier... Different systems offer different advantages and disadvantages in regards to these issues. If a CAOS system is disruptive this is because it is flawed or ergonomically unfisnished this is correct because the problems come from the hardware I regret that there is no significant technical improvement of the devices since more than 15 years...   Strong 0 0% Moderate 3 21% Weak 8 57% N/A 2 14% Strong 2 14% Moderate 4 29% Weak 3 21% N/A 4 29%     134  Appendix B –   Kalman Filtering This section gives more detailed overview of our Kalman Filter implementation, and provides the code. For a great introduction to the Kalman Filter, see “Poor Man’s Explanation of Kalman Filtering, or, How I Stopped Worrying and Learned to Love Matrix Inversion”, by Du Plessis (1967). This work was done in MATLAB 2013a, and used a public Kinematics toolbox package.  Below we present three MATLAB functions for the UKF algorithm: the UKF body, the unscented transform, and sigma-point calculator. [UKF.m]  function [x,P] = ukf(fstate,x,P,hmeas,y,Q,R,data)   ns = numel(x);                              %number of states nm = numel(y);                              %number of measurements alpha = 1e-3;                               %default, tunable ki = 0;                                     %default, tunable beta = 2;                                   %default, tunable lambda = alpha^2*(ns+ki)-ns;                %scaling factor c = ns + lambda;                            %scaling factor Wm = [lambda/c 0.5/c + zeros(1,2*ns)];      %weights for means Wc = Wm; Wc(1) = Wc(1)+(1-alpha^2+beta);             %weights for covariance c = sqrt(c);   X=sigmas(x,real(P),c);                      %sigma points around x  [xest,Xsig,Px,delX] = ut(fstate,X,Wm,Wc,ns,Q,data); % unscent trans. proc. [yest,Ysig,Py,delY] = ut(hmeas,Xsig,Wm,Wc,nm,R,data); %unscent trans. meas.   P12 = delX * diag(Wc) * delY';              %transformed cross-covariance K   = P12 * inv(Py); x   = xest + K *(y - yest);                 %state update P   = Px - K * P12';                        %covariance update  end      135  [UT.m]  [sigmas.m]    function X = sigmas(x,P,c)  %Sigma points around reference point %Inputs: %       x: reference point %       P: covariance %       c: coefficient %Output: %       X: Sigma points   A = c*chol(P)'; Y = x(:,ones(1,numel(x))); X = [x Y+A Y-A];  function [yest,Y,Py,delY] = ut(f,X,Wm,Wc,n,R,data) %Unscented Transformation  nsig = size(X,2);    % number of sigma points % yest = zeros(n,1); Y = zeros(n,nsig);  for k=1:nsig     Y(:,k) = f(X(:,k),data); end  yest = mean([Y(:,1)'; mean(Y(:,2:end)')])'; delY = Y - yest(:,ones(1,nsig)); Py = delY*diag(Wc)*delY' + R;   end      136  Appendix C –   Double Marker Pin: Combined Loading This section contains a derivation of the ratio between lateral bending and twisting, as a result of a combined load on the double-marker pin. The rigid crossbar segment allows for axial twisting to be resolved, and provides a moment arm. One option would be to re-select tracker pin dimensions to maximize bending relative to twisting. Let us consider the worst case scenario for both twisting and bend: a point load applied at one marker.  The equation for twist displacement of the pin θ is shown below, where T is the applied torque, L is the shaft length, J is the polar moment of inertia, and G is the modulus of rigidity.  𝜃 =𝑇  𝐽 𝐺 Combined Load Bend Twist    137  The corresponding marker displacement resulting from the pin twist δtwist can be calculated as follows, where s is the marker’s radial distance from the shaft, and F is the applied force at that point. The polar moment of inertial for a circular beam is (   𝜋 𝑟4).   𝑡𝑤𝑖𝑠𝑡  =  𝑇   𝑠𝐽 𝐺=2 𝐹   𝑠 𝜋 𝐺 𝑟4 The torsional stiffness (N/m) for the marker point can be expressed as follows:  𝑘𝑡𝑤𝑖𝑠𝑡  =  𝐹 𝑡𝑤𝑖𝑠𝑡 =  𝜋 𝐺 𝑟42 𝑠    In section 2.2, we found the lateral bending stiffness (N/m) to be: 𝑘𝑙𝑎𝑡  =  𝐹 𝑙𝑎𝑡 =  3 𝐸 𝜋 𝑟44  3 Finding the ratio of lateral stiffness to bending stiffness for our point load, simplifies to a ratio of key dimensions and material constants, where ν is poisons ratio of the material (0.3 for stainless steel).  𝑅𝑘 =𝑘𝑙𝑎𝑡𝑘𝑡𝑤𝑖𝑠𝑡=(3 𝐸 𝜋 𝑟44  3)(𝜋 𝐺 𝑟42 𝑠   )=  3 𝑠  (1 + 𝜈)       138  If we apply our pin’s dimensions (𝑠 = 32.5𝑚𝑚;  = 70.0𝑚𝑚), we find a ratio of 0.84. This means the pin is slightly more prone to lateral bending than twisting. But this ratio is fairly close to one. To shift the pin design towards bending over twisting, we could increase length of pin, or decrease the crossbar length.  

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