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Ankle joint biomechanics applied to computer-assisted total knee replacement 2002

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ANKLE JOINT BIOMECHANICS APPLIED TO COMPUTER-ASSISTED TOTAL KNEE REPLACEMENT by Cameron Shute B.Sc., University of Alberta, 1999 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard: THE UNIVERSITY OF BRITISH COLUMBIA March 2002 © Cameron Shute, 2002 In p resen t i ng this thesis in partial fu l f i lment of the requ i remen ts for an a d v a n c e d d e g r e e at the Univers i ty of Brit ish C o l u m b i a , I agree that the Library shal l m a k e it f reely avai lable fo r re fe rence and s tudy. I fur ther agree that pe rm iss i on fo r ex tens ive c o p y i n g of this thesis fo r scholar ly p u r p o s e s may be granted by the h e a d of m y depa r tmen t o r by his o r her representat ives. It is u n d e r s t o o d that c o p y i n g o r pub l i ca t i on o f this thesis for f inancial gain shall no t be a l l o w e d w i thou t my wr i t ten p e r m i s s i o n . D e p a r t m e n t T h e Univers i ty of Brit ish C o l u m b i a V a n c o u v e r , C a n a d a Da te iViUL U 1007. D E - 6 (2/88) Abstract In total knee arthroplasty, implants are aligned perpendicular to the mechanical axis of the lower limb (the line connecting the centre of the hip to the centre of the ankle). Currently, there is a very precise method for locating the centre of the hip, but not so for the centre of the ankle joint. In this study, I report on the two primary contributions I have made to the problem of locating a meaningful ankle centre for use in the computer-assisted total knee arthroplasty system we are developing at the University of British Columbia. The first contribution I made was by deriving the algorithms for use in fitting mathematical models of the ankle (both ball-in-socket and biaxial) to subject-specific data, and then performing the numerical and physical simulations to identify which algorithms are best suited to each model for use with our computer-assisted surgical system. The second contribution I made was my investigation of locating the ankle centre using these different ankle models mentioned above, in weight-bearing and non-weight-bearing, compared to locating the ankle centre using anatomic digitization. By conducting tests on 12 live subjects to identify the precision of each technique I found that the spherical method of ankle centre localization in the non-weight-bearing was the most reliable method of predicting the location of the ankle centre in weight-bearing. ii Table of Contents Abstract ii Table of Contents Hi List of Tables vii List of Figures viii Terminology xi Glossary of Abbreviations xii Acknowledgements xiii Chapter 1: Importance of locating the Ankle Joint Centre for use in Anatomic Alignment Registration for Computer-Assisted Total Knee Arthroplasty 1 1.0 IMPORTANCE OF ALIGNMENT IN TOTAL K N E E ARTHROPLASTY 1 1.1 TIBIAL COMPONENT ALIGNMENT IN TOTAL K N E E ARTHROPLASTY 3 1.1.1 Classic Alignment Techniques 3 1.1.2 Computer-Assisted Techniques - Surgical Navigation Systems 4 1.2 A N K L E JOINT COMPLEX MODELING 6 1.2.1 Non- Weight Bearing versus Weight Bearing Studies 6 1.2.2 Single Axis and Four-Bar Linkage Models 7 1.2.3 Biaxial Models 7 1.2.4 Ball-in-Socket Model 9 1.2.5 Axis Migration Studies 9 1.3 A N K L E JOINT COMPLEX INVESTIGATION TECHNIQUES 1 0 1.4 RESEARCH QUESTIONS 1 0 1.5 CONTRIBUTIONS OF THESIS 1 0 1.5.1 Optimization Method Formuation and Validation 11 1.5.2 In Vivo Experimental Work 11 1.6 THESIS OVERVIEW 11 1.7 REFERENCES 1 2 iii Chapter 2: Optimal Formulations Of Ankle Joint Models For Use In Computer- Assisted Total Knee Arthroplasty 16 2.0 ABSTRACT 16 2.1 INTRODUCTION 16 2.2 METHODS AND MATERIALS 19 2.2.1 Formulation of AJC Models 19 2.2 A A The Biaxial Model 19 2.2.1.2 The Spherical Model 23 2.2.2 Simulations 25 2.2.2.1 Description of Motion Data Generation Protocols 25 2.2.2.2 Biaxial Model - Evaluation of Optimization Algorithms 25 2.2.2.3 Biaxial Model - Comparison of 8 vs. 12 Parameter Methods 26 2.2.2.4 Spherical Model - Evaluation of Methods & Base Frame Selection 26 2.2.3 Physical Experiments 27 2.2.3.1 Equipment Used 27 2.2.3.2 Biaxial Model 28 2.2.3.3 Spherical Model 28 2.3 RESULTS 28 2.3.1 Biaxial Model 28 2.3.1.1 Simulations 28 2.3.1.2 Physical Experiments 32 2.3.2 Spherical Model 32 2.3.2.1 Simulations 32 2.3.2.2 Physical Experiments 34 2.4 DISCUSSION 34 2.4.1 Biaxial Model: 34 2.4.2 Spherical Model: 36 2.4.3 Conclusions 36 2.5 ACKNOWLEDGEMENTS 37 2.6 REFERENCES 37 iv Chapter 3 : Bias and Repeatability of Motion-Based and Digitized Ankle Joint Centre Estimates Under Both Weight-Bearing and Non-Weight-Bearing Conditions 39 3.0 ABSTRACT 39 RELEVANCE 39 3.1 INTRODUCTION 40 3.1.1 Clinical Application 40 3.1.2 Current Approaches in Computer Assisted TKA 40 3.1.2.1 Anatomic Digitization 41 3.1.2.2 Motion-Based Methods 41 3.1.3 Limitations of Current Approaches 42 3.1.4 Purpose of Study 43 3.2 METHODS AND MATERIALS 43 3.2.1 Measurement Equipment and Subject Information 43 3.2.2 Ankle Digitization Method 45 3.2.3 Motion-Based Methods 45 3.2.3.1 Non-weight-Bearing and Weight-Bearing Testing Protocol 45 3.2.3.2 Spherical Model 46 3.2.3.3 Biaxial Model 46 3.2.4 Data and Statistical Analysis 47 3.3 RESULTS 48 3.3.1 Characterization of the Repeatability of all Methods 48 3.3.2 Bias of Surrogate Estimates with Respect to Weight-Bearing 49 3.3.2.1 Choice of the Reference Weight-Bearing Centre 51 3.3.2.2 Population Variability of the Surrogate Methods 52 3.3.3 Bias of Non-w'eight-Bearing Estimates with Respect to Digitization 53 3.4 DISCUSSION ....54 3.4.1 Interpretation of Results 54 3.4.2 Comparison of our Findings to Other Studies in Literature 54 3.4.3 Strengths and Weaknesses of the Study 56 3.4.4 Conclusions 59 3.5 ACKNOWLEDGEMENTS 60 v 3.6 R E F E R E N C E S 6 0 Chapter 4: Thesis Summary and Direction of Future Work 62 4 . 0 D I S C U S S I O N O F T H E S I S O B J E C T I V E S '. 6 2 4.1 C O N C L U S I O N S 6 2 4 . 2 D I R E C T I O N S F O R F U T U R E W O R K 6 3 4.2.1 Issues to be addressed before going to clinical trials 63 4.2.2 Ideas for future research on ankle centre localization methods 64 4 .3 R E F E R E N C E S 6 5 Appendix: Collection of Published Conference Publications 66 Paper 1: Application of a Biaxial Kinematic Ankle Model to Computer-Assisted Total Knee Arthroplasty. Fifth International Symposium on Computer Methods in Biomechanics and Biomedical Engineering - November 2001 - Rome, Italy 67 Paper 2: Repeatability and Accuracy of Ankle Centre Location Estimates using a Biaxial Joint Model. MICCAI - October 2001 - Utrecht, The Netherlands. Poster Included 74 Paper 3: Investigation of Ankle Centre Location Techniques Applied to Computer- Assisted Total Knee Arthroplasty. Biomechanica IV - September 2001 - Davos, Switzerland 77 Paper 4: Repeatability and Accuracy of Bone Cutting and Ankle Digitization in Computer-Assisted Total Knee Replacement. MICCAI - October 2000 - Pittsburgh, USA. Poster Included 78 vi List of Tables Table 2.1: Svmimary of computational effort in both simulation and experiment 32 Table 3.1: Summary of the bias and population variability of each surrogate method with respect to weight-bearing 53 Table 3.2: Comparison of biaxial model with results from literature 55 Table 3.3: Maximum error of tibial and calcaneal trackers with respect to bone pin mounted markers 56 vii List of Figures Figure 1.1: Definition of frontal plane alignment conditions 1 Figure 1.2: Definitions of the mechanical axes of the lower limb for the femur (left) and tibia (right). Images courtesy Belfast joint group 2 Figure 1.3: Allocation of variance contributors in the registration component of our computer-assisted total knee arthroplasty system 3 Figure 1.4: The centre of the ankle is medial to the midpoint of the malleoli. Image courtesy Belgian orthoweb 4 Figure 1.5: CT models showing the femoral and tibial mechanical axes and the desired resection plane. Image courtesy KneeNav 4 Figure 2.1: Biaxial model of ankle joint complex. Left: exploded view with reference frames attached (from van den Bogert, 1994). Right: physical model with joint axes shown. The triangular bodies carry infrared emitter arrays for position tracking 18 Figure 2.2: Definition of ankle joint centre using biaxial model. The joint centre is the projection of the two axes into a transverse plane when the joint is in a neutral orientation 19 Figure 2.3: Flow chart for solving the 12 parameter formulation of the biaxial ankle joint model 20 Figure 2.4: Parameterizing an axis using four variables. ( P i J V ) defines the pierce point in the yz plane and P3 and P4 represent angles defining the direction the axis points in.... 21 Figure 2.5: Flow chart for solving the 8 parameter formulation of the biaxial ankle joint model 23 Figure 2.6: Schematic illustrating the homogeneous transform method. The frame Fi moves relative to Fo. The point Co is invariant in Fo if it is the centre of a spherical joint 24 Figure 2.7: Physical model of a spherical joint. The marker arrays used in the experiment matched those used with the biaxial physical model 27 viii Figure 2.8: Computational effort for different optimization algorithms used with 8P at different noise levels 29 Figure 2.9: Comparison of repeatability and bias for 12P and 8P in simulation (top and bottom left) and in experiment (bottom right). Estimated means are shown with 95% confidence interval bars in the anteroposterior and mediolateral directions 30 Figure 2.10: Comparison of parameter repeatability 12P and 8P in simulation (top) and in experiment (bottom); error bars represent 95% confidence intervals on estimates 31 Figure 2.11: ML (left) and AP (right) repeatability of spherical methods in simulation (top) and in experiment (bottom); error bars represent 95% confidence intervals on estimates. 33 Figure 2.12: Computational cost of spherical formulations in simulation (left) and in experiment (right) 34 Figure 3.1: Tibial and calcaneal trackers (left), tibial tracker (centre), and calcaneal tracker (right) 44 Figure 3.2: Two Emitter Point probe digitizing lateral malleoli 45 Figure 3.3: Trackers in use for collecting non-weight-bearing data (left) and weight-bearing data (right) 46 Figure 3.4: Definition of ankle joint centre using biaxial model 47 Figure 3.5: Overall Repeatability for all subjects, methods and directions 49 Figure 3.6: Repeatability and bias, subjects a-f, all methods. Error bars indicate standard deviations 50 Figure 3.7: Repeatability and bias, subjects g-1, all methods. Error bars indicate standard deviations 51 Figure 3.8: Bias of the surrogate methods with respect to the weight-bearing reference. Boxes indicate standard deviations, and error bars indicate the 95% confidence intervals predicted with bootstrapping 52 ix Figure 3.9: Mean bias of non-weight-bearing methods relative to digitization for all subjects. Error bars represent the standard deviations of the bias estimates 53 Figure 3.10: Definition of joint orientation angle measurement 55 Figure 3.11: Photo of post experiment dents left in leg 57 Figure 3.12: Typical datasets in non-weight-bearing (left), and weight-bearing (right) 58 x Terminology Accuracy: The ability of a measurement to match the actual value of the quantity being measured. Since we did not have a gold standard to compare our measurements against, accuracy was not measured in this study. Bias: A constant shift or offset between two values (also known as systematic error). In this study I computed the bias between joint centre locations in weight-bearing and joint centre locations in non-weight-bearing. Cost Function: A mathematical formula that describes how well a set of parameters allows a model to fit a set of data. [Satisfies the requirements of an optimization formulation.] Gold Standard: What one considers to be the actual value of the quantity of interest, and thus what we can calculate the bias or accuracy with respect to. Optimization Algorithm: A numerical method for solving complex problems. These algorithms minimize the error associated with a certain solution or set of parameters. In this study, we used optimization algorithms to fit mathematical models of the ankle joint to measured datasets. Optimization Formulation: A method of formulating a problem so that an estimate of the solution can be progressively refined using an optimization algorithm to yield the best possible solution. In this study, I compared several formulations of models of the ankle joint. Precision (= Repeatability): The ability of a measurement to be consistently reproduced. Precision is typically reported as the standard deviation associated with making a measurement; low values imply high precision. Surrogate Method: A method that may be substituted for another method. In this study, I propose that motion-based methods in non-weight-bearing could be used as surrogates for weight-bearing measurements. Variability: The inverse of precision and repeatability. More repeatable is equivalent to less variable. xi Glossary of Abbreviations 12P - 12 Parameter Formulation of the Biaxial Model 8P - 8 Parameter Formulation of the Biaxial Model AJC - Ankle Joint Complex AP - Anteroposterior BM - Biaxial Model CAS - Computer-Assisted Surgical CT - Computerized Tomographic DOF - Degree of Freedom EM - Extramedullary FLOPS - Floating Point Operations HT - Homogeneous Transform IHA - Instantaneous Helical Axis IM - Intramedullary IRED - Infrared Emitting Diode LM - Levenberg-Marquardt ML - Mediolateral NWB - Non-Weight-Bearing RMS - Root Mean Square ROM - Range of Motion RSA - Roentgen Stereophotogrammetry Analysis SD - Standard Deviation ST - Subtalar TC - Talocrural TKA - Total Knee Arthroplasty TR - Trust-Region Reflective Newton WB - Weight-Bearing xii Acknowledgements First thanks go to my family for supporting me in so many ways throughout the course of this degree, and for providing me with a stable foundation to stand on in life. Special thanks go to my partner Sarah who has provided me with day-by-day support and so much understanding and love for so long. Thanks also to Dr. Antony Hodgson for such thoughtful supervision and commitment to my project. I sincerely appreciate all the time you made for me, and for all the thought that went into the discussions we had. Thanks to Dr. Wayne Vogl in the UBC Anatomy Department for assisting me in obtaining cadaveric specimens and the space to conduct experiments with them. Thanks to Rachel MacKay in the UBC Department of Statistics for your help in with the clinical study. Thanks to Dr. Phillip Loewen in the UBC Department of Mathematics for your assistance with optimization theory, and sharp wit in class. Thanks to Kevin Inkpen for the original inspiration for this thesis, and help along the way. Thanks to the surgeons Dr. Robert McGraw, Dr. Bassam Masri, and Dr. Alastair Younger with whom I met with to discuss the project, and who allowed me to observe numerous procedures. Graduate Students and my friends Chris Plaskos, Scott Illsley, Paul McBeth and Willem Astma all provided me with an fantastic lab environment to work within, and assistance with the experiments I conducted. Thanks to many friends for their participation with my study: Becky, Connie, Erica, Hildur, Iman, Michelle, Randy, Patrick and Stephanie. You've all helped science put a foot forward (been waiting to use that pun for 2 years of ankle research). Thanks to all my friends I've had for roommates over the last couple of years: Conor and Rebecca, Tarn and Christine, Colin and Chris and Jen. Thanks for putting up with me being focused on getting the work done, and for all the times when you all were there for me. Special thanks go to all those people involved in providing me with outdoor distractions in the mountains to keep me sane over the last couple of years. I can't begin to describe how important you are all to me. I'll finish off this section with a quote that summarizes what a couple years of living in hectic Vancouver has taught me (kudos to Chris Plaskos for sending it to me in the first place). There is more to life than simply increasing its speed... -Mahatma Gandhi xiii Chapter 1: Importance of locating the Ankle Joint Centre for use in Anatomic Alignment Registration for Computer-Assisted Total Knee Arthroplasty 1.0 Importance of Alignment in Total Knee Arthroplasty The purpose of total knee arthroplasty (TKA) is to provide patients with freedom from pain and improved function for the longest possible time. Degenerative joint disease is the main reason TKA is performed, and limb malalignment in the frontal plane has long been recognized as a primary contributing factor in the development of degenerative arthritis (Johnson et al., 1980; Harrington, 1983). Implant malalignment in the frontal plane can cause increased loading and wear, which can in turn lead to increased risk of early failure or loosening, (Lotke et al., 1977; Denham et al., 1978; Bargren et al., 1983; Tew et al., 1985; Cornell et al., 1986; Hsu et al., 1989; Windsor et al., 1989; Lee et al., 1990; Jeffery et al., 1991). Alignment errors in the frontal plane which cause a patient to stand bow-legged are called varus errors, while those which cause a patient to stand knock-kneed are called valgus errors, as illustrated in Figure 1.1. Varus error Valgus error (a) Varus (b) Neutral (c) Valgus Figure 1.1: Definition of frontal plane alignment conditions Page 1 Varus malalignment has been specifically correlated with early failure, as opposed to valgus and neutral alignment (Ritter et al., 1994). This makes sense when we consider that Hsu found that 75% of the knee joint load passed through the medial compartment when simulating one-legged weight-bearing stance (Hsu et al., 1990). Addition of a varus alignment error to this scenario would cause a further increase in load in the already overloaded medial compartment. The notorious effect of varus alignment on knee biomechanics has also been commented on in a number of other studies (Morrison, 1970; Johnson et al., 1980; Harrington, 1983). Proper implant alignment in TKA is achieved by making bone cuts perpendicular to the mechanical axis of the lower limb (Jeffery et al., 1991; Jessup et al., 1997). The mechanical axis of the femur is defined as the line joining the centre of the hip with the centre of the knee, and the mechanical axis of the tibia is the line connecting the centre of the knee with the centre of the ankle as shown in Figure 1.2 (Kapandji, 1970; Moreland et al., 1987). The primary alignment goal of TKA is to ensure that these two mechanical axes become collinear. Femoral Figure 1.2: Definitions of the mechanical axes of the lower limb for the femur (left) and tibia (right). Images courtesy Belfast joint group. Page 2 In 1998 Inkpen and Hodgson began developing a computer-assisted TKA system to help surgeons achieve proper implant alignment and reduce the alignment variability. Figure 1.3 shows a breakdown of the anticipated contributions to variance from the hip, knee, and ankle registration processes in our system (Inkpen et al., 2000). Since the variance associated with the ankle joint is the largest contributor to variance in our proposed system, my thesis is focused on investigating improvements in the ankle centre location protocol. Figure 1.3: Allocation of variance contributors in the registration component of our computer-assisted total knee arthroplasty system 1.1 Tibial Component Alignment in Total Knee Arthroplasty 1.1.1 Classic Alignment Techniques Traditionally, both intramedullary (IM) or extramedullary (EM) alignment systems have been used to perform alignment of the tibial mechanical axis (Brys et al., 1991; Dennis et al., 1993; Teter et al., 1995; Phillips et al., 1996; Maestro et al., 1998). Different authors claim that both systems are superior than one another, and many authors report that there is no difference. Currently, the most common method for achieving proper coronal plane alignment for tibial alignment is to use an EM guide (Phillips, 1996). By utilizing an EM alignment system, several important complications associated with IM guides are avoided: fat embolization and hypoxia, intraoperative fracture, and the inability of IM rod passage due to bowing of the tibia. When using an EM alignment system it is important to be aware that the centre of the ankle joint is medially offset to the midpoint of the malleoli, and failing to realize this will result in a varus tibial bone cut as shown in Figure 1.4 (Dennis et al., 1993). C o n t r i b u t i o n s o f V a r i a n c e i n C u r r e n t C A S T K A P r o c e d u r e Hip Centre Location 8% Knee Centre Location 22% Ankle Centre Location 70% Page 3 Midpoint between Malleoli Midpoint of Ankle Varus Cut Neutral Cut Figure 1.4: The centre of the ankle is medial to the midpoint of the malleoli. Image courtesy Belgian orthoweb. 1.1.2 Computer-Assisted Techniques - Surgical Navigation Systems Surgical navigation systems can be broken down into two broad categories: those based on preoperative models and those based on intraoperative models (Picard et al., 2000). Preoperative models generally require three-dimensional bone models obtained from computerized tomographic (CT) scans or other expensive imaging techniques, as shown in Figure 1.5, whereas intraoperative model based systems are generally based on limb kinematics and/or anatomic digitization acquired at the time of surgery. The latter systems therefore tend to be much less expensive (Inkpen, 1999b). Figure 1.5: CT models showing the femoral and tibial mechanical axes and the desired resection plane. Image courtesy KneeNav. Page 4 There are many preoperative model surgical navigation systems for clinical use: Musculographics (www.musculographics.com), KneeNav (www.ri.cmu.edu), and Navitrack (www.orthosoft.ca) all make use of three-dimensional bone models from CT scans to aid the surgeon in determining where to make bone cuts with their respective systems. Delp reported the average registration error of the tibia using the Musculographics system, without including bone cutting error, to be 0.4° in the frontal plane, with a maximum error of 0.8° (Delp etal., 1998). Our system (Inkpen, 1999b) falls into the less expensive category of intraoperative surgical navigation systems, along with several other similar systems developed by various groups: Leitner and Picard (Leitner et al., 1997), Krackow (Krackow et al., 1999), and Kunz, Sati and Nolte (Kunz et al., 2001). Inkpen's focus was to investigate the precision and accuracy of this class of computer-assisted systems since many groups had not reported results in a comprehensive manner. A comparative study between the system developed by Leitner and Picard versus the conventional TKA technique showed no statistically significant reduction of variability for tibial alignment for the clinical trial involving 25 surgeries with both systems (Saragaglia et al., 2001). Including bone cutting error, Saragaglia reported the SD associated with tibial frontal plane alignment to be 1.34° for the computer-assisted technique. Kiefer conducted another similar study consisting of 50 surgeries with the same surgical navigation system and reported that 75% of patients had overall mechanical axis alignment errors of less than 2° using the computer-assisted approach compared to only 45% of the 50 subjects who underwent a conventional TKA (Kiefer et al., 2001). The method of determining the centre of the ankle joint varies considerably between these systems, and no "gold standard" has been agreed upon. In Krackow's system, the surgeon digitizes their best estimate of the centre along the anterior portion of the ankle whereas in Kunz's system, the surgeon digitizes the tendon of tibialis anterior. Inkpen investigated digitizing the extremes of the malleoli and then calculating the midpoint of the malleolar axis. Inkpen did this only for the purpose of establishing repeatability in a cadaveric study; he did not actually propose to use the midpoint as the ankle centre estimate in a live procedure. Of all these studies, Inkpen was the only author to report the standard deviation (SD) associated with his chosen method. He reported the SD of using digitization to locate the midpoint of the ankle joint complex (AJC) to be 0.75 mm (0.11°) in the ML direction and 1.12 mm (0.17°) in the AP direction based on a single operator and seven cadaver specimens Page 5 (Inkpen et al., 2000). Leitner's system uses a motion-based method of locating the ankle centre by modeling the joint complex as a spherical joint, but did not report the repeatability associated with the technique. Inkpen investigated the repeatability associated with this method, and reported the SD associated with the spherical model to be 2-3X larger than digitization at 1.49 mm (0.23°) in the mediolateral (ML) direction and 2.72 mm (0.42°) in the anteroposterior (AP) direction; this result was based on two cadaver specimens and he used a non-invasive calcaneal tracker referenced to a bone-pin-mounted tibial frame. He also measured repeatability using a bone pin in the calcaneus instead of the non-invasive tracker and found the SDs to be comparable to digitization: 0.82 mm (0.12°) in the ML direction, and 1.35 mm (0.21°) in the AP direction (Inkpen, 1999b). One of the recommendations in Inkpen's thesis was to investigate the use of alternative models of the ankle joint, and to do so both in weight-bearing (WB) and non-weight-bearing (NWB). 1.2 Ankle Joint Complex Modeling There is not currently a widely accepted model for the AJC. Two issues that researchers have attempted to answer are (1) whether the axes of rotation of the talocrural and subtalar joints are fixed or translating relative to the tibia and talus respectively, and (2) how many meaningful degrees of freedom (DOF) there are in the AJC. Answers to these questions have been hampered because there is not a well established method of obtaining accurate and repeatable data regarding the joint movements (Bottlang et al., 1999; Leardini et al., 1999). Although developments in instrumentation have improved the ability for researchers to obtain and analyze large amounts of data, there has not been the same amount of success in developing useful mathematical models (Leardini et al., 1999). The need for a standardized model and coordinate system for the ankle would not only benefit developers of surgical navigation systems, but the biomechanics community at large (Cole et al., 1993; van den Bogertetal., 1994). 1.2.1 Non-Weight Bearing versus Weight Bearing Studies Before a model is proposed, it is important to realize that ankle joint kinematics are significantly different in a non-weight-bearing (NWB) state than in a weight-bearing (WB) state (Lundberg et al., 1989; Sarrafian, 1993; Leardini et al., 1999). The main differences are in the range of motion and the amount of axis migration (Lundberg et al., 1989; Sarrafian, 1993). The ankle centre in WB is the most relevant datum to use for alignment in TKA since Page 6 it is the point within the AJC through which loads are transferred (Inkpen, 1999b). However, the WB centre is not easily measured intraoperatively since patients are under anesthesia and in the supine position for the duration of the surgery. What surgeons do have at their disposal are several surrogate ankle centre measurement techniques, although the relationship of these centres to the weight-bearing centre has never been established. 1.2.2 Single Axis and Four-Bar Linkage Models Leardini has investigated the kinematics of the AJC in passive flexion on seven cadaveric specimens, and found that under these condition the joint acts a single DOF system (Leardini et al., 1999). They concluded that the tibiocalcaneal and calcaneofibular ligaments in conjuction with articular surfaces guide passive ankle motion, whereas the role of the other ligaments is to limit motion. In passive flexion, he found that the primary ankle motion is at the talocrural joint, and subtalar motion is only generated by external forces much larger than those required to generate dorsi/plantarflexion motion. Leardini also used instantaneous helical axis (IHA) analysis to estimate the location of the talocrural axis, and to investigate migration. All specimens tested showed a continuously moving path of the IHA as the ankle was flexed from dorsiflexion to plantarflexion. The IHAs seemed generally to pass through the distal tips of the malleoli, which agrees with findings from previous studies (Inman, 1976; Lundberg et al., 1989; Singh et al., 1992; Bottlang et al., 1999). Leardini, along with O'Connor, has done work on formulating a 2D four-bar linkage geometric model which describes the kinematics of this single DOF motion (Leardini et al., 1999). This is still early work, and does not incorporate the full three dimensional aspects of ankle joint motion which are important if one wishes to use this model to determine an AJC centre. Since it was an in vitro kinematic study, it also suffers from the criticism that the cadaveric specimens used were extensively dissected, the joints were not loaded, and the motion was generated by experimenters and not by muscle forces. This is a criticism that has also been applied to the early work on ankle motion done by Inman (Inman, 1976). 1.2.3 Biaxial Models Inman's work was done in vitro on cadaver specimens that had most ligaments and soft tissue removed. He used a self-designed and built mechanical apparatus with goniometers to Page 7 determine the location of the joint axes. The underlying assumption that he made was that the talocrural and subtalar axes function as ideal fixed hinge joints (Inman, 1976). The most useful aspect of his research is the large number of specimens he tested (n=46), which provides other researchers with a good idea of the normal range of variation of axis locations. Inman also noted that in general the location of the talocrural axis can be estimated by passing a line through the distal tips of the malleoli. Van den Bogert extended Inman's work by modeling the AJC as a three component system (tibia, talus, calcaneus) and locating the talocrural and subtalar joints using an optimization method (van den Bogert et al., 1994). Van den Bogert argues that using other models like a ball-in-socket model is not correct anatomically, kinematically, or dynamically. His optimization method allows for determining the location of both joint axes using the kinematic model developed originally by Areblad, which requires measures of the positions of only the tibia and calcaneus (Areblad, 1990). This is a significant benefit from a clinical perspective since the less invasive the procedure the better, and currently there is no simple way to non-invasively measure the motion of the talus. Van den Bogert used reflective skin markers and a four camera video system to capture the motion data. The markers were attached to three points on both the shank and the shoe. In total, 14 healthy subjects were tested, and one person was tested eight times in order to assess the repeatability of this procedure. All subjects were tested in a passive NWB state. The repeatability experiment has the most applicability to computer-assisted TKA since we want to know the repeatability of using this model of the ankle to locate the joint centre. The results for the repeatability study showed that the location of the axes could be measured with a SD of less than three degrees for all axes except for the orientation of the subtalar axis in the transverse plane which had a SD of five degrees. The unavoidable limitation of modeling the ankle this way is the assumption that both joints have fixed axes of rotation, and the inability to accurately resolve the position of the subtalar joint reflects this. The important part of this study is that it shows how a motion analysis tool can be used to non-invasively obtain kinematic measures of the ankle joint. Perhaps with improved design of the marker attachment device (Inkpen, 1999b) the measurement variability could be reduced. However, the question of the relevance of a non-weight-bearing measurement to the weight-bearing behaviour remains unanswered. Page 8 1.2.4 Ball-in-Socket Model Leitner and Picard use motion analysis tools to determine the ankle centre for use in computer-aided TKA (Leitner et al., 1997). They originally performed the procedure on several cadaver specimens using an Optotrak™ localizer to track the three-dimensional motion of infrared emitting diodes mounted on bone pins in the pelvis, femur, tibia, and calcaneus. They model the ankle as a ball-in-socket joint, and the centre is estimated as the centre of the sphere that best fits the motion of the calcaneal markers relative to the tibial markers. Their procedure requires a bone pin in the calcaneus which is outside of the normal surgical field in TKA and is obviously undesirable. The precision and accuracy of Leitner's work was not presented in a comprehensive manner, which prompted Inkpen to launch an investigation to quantify the errors associated with these methods (Inkpen, 1999b). Inkpen used the same method as described by Leitner to determine the AJC centre, and he also compared a non-invasive approach using a minimally constrained motion tracking device to Leitner's bone-pin-mounted array. After cadaver testing to determine the repeatability of the two techniques, Inkpen concluded that the ankle was not well modeled as a ball-in-socket joint because the repeatability of the measurements was poor compared with those made at the hip. His suggestions for future work included investigating an improved kinematic model of the ankle, such as the biaxial model suggested by van den Bogert, as well as addressing the weight-bearing question (Inkpen, 1999b). 1.2.5 Axis Migration Studies The idea of axis migration is not new, and it has been studied by many researchers over the years (Barnett et al., 1952; Hicks, 1953; van Langelaan, 1983; Lundberg et al., 1989; Lundberg et al., 1993). The most comprehensive paper of this branch of research is Lundberg's study from 1989. Lundberg used roentgen stereophotogrammetry analysis (RSA) to investigate talocrural joint axis migration in the weight-bearing position of eight healthy volunteers. The subjects were tested bare foot in a weight-bearing state and samples were taken at 10° intervals throughout the range of motion. The method of IHA was used to determine the locations of the joint axes. The results showed that there were large differences in the locations of the axis for different locations within the range of motion of the ankle. They found a rather abrupt change in the axis location near the neutral position between plantarflexion and dorsiflexion which agreed with results from other researchers (Barnett et Page 9 al., 1952). However, when all axes were projected onto the same plane (coronal or sagittal) the intersection of all axes occurred in a relatively small area within the talus. Lundberg speculated that about this centre, located at the midpoint of a line connecting the distal tips of the malleoli, there is more freedom of movement than would be predicted by a single degree of freedom joint model. 1.3 Ankle Joint Complex Investigation Techniques Once a kinematically plausible model for the AJC has been selected, another significant problem arises in determining what equipment to use for tracking bone motion. A number of different measurement devices have been used for such studies, including simple radiographs (Sammarco et al., 1973), roentgen stereophotogrammetry (Lundberg et al., 1989), goniometers (Boone et al., 1979), stereoscopic techniques (Siegler et al., 1988), video (van den Bogert et al., 1994), fluoroscopy (Komistek et al., 2000) and optical tracking device methods (Scott et al., 1991; Inkpen, 1999b; Shute et al., 2001b). Non-invasive methods are obviously desirable, but techniques which rely on skin-mounted markers suffer from errors introduced by motion of these markers relative to the underlying bone (Cappozzo et al., 1996). 1.4 Research Questions There is no gold standard for defining the ankle centre as discussed above. There are several reasons for this as follows: No standard mathematical model of the ankle joint exists, no standard for anatomic joint centre location has been selected, the weight-bearing state of the subject influences the location of the ankle centre, and the relationships between the existing techniques are not treated in the literature. My objective in this study then was to characterize the repeatability and bias associated with motion-based ankle centre localization methods in weight-bearing and non-weight bearing and compare them to direct anatomic digitization techniques. 1.5 Contributions of Thesis This thesis makes two primary contributions to the problem of locating a meaningful ankle centre for use in computer-assisted surgery. First, I derive optimal algorithms for use in fitting mathematical models (both ball-in-socket and biaxial) to subject-specific data; and second, I investigate these methods by conducting tests on live subjects to establish the Page 10 r e p e a t a b i l i t y o f t h e m e a s u r e m e n t t e c h n i q u e s a n d t h e r e l a t i v e p o s i t i o n o f w e i g h t - b e a r i n g a n d n o n - w e i g h t - b e a r i n g c e n t r e s . I n o r d e r t o a c c o m p l i s h t h e s e t a s k s , I c o m p l e t e d t h e t a s k s d e s c r i b e d i n t h e f o l l o w i n g s e c t i o n s : 1 .5.1 O p t i m i z a t i o n M e t h o d F o r m u a t i o n a n d V a l i d a t i o n I n o r d e r t o c o m p a r e t h e f e a s i b i l i t y o f a n k l e j o i n t m o d e l s f o r u s e i n c o m p u t e r - a s s i s t e d s u r g i c a l s y s t e m s , i t i s i m p o r t a n t t o k n o w t h e o p t i m a l m a t h e m a t i c a l f o r m u l a t i o n o f e a c h o f t h e s p h e r i c a l a n d b i a x i a l j o i n t m o d e l s . T h e p r i n c i p a l c r i t e r i a I u s e t o e v a l u a t e p e r f o r m a n c e are t h e v a r i a b i l i t y o f t h e p r e d i c t e d j o i n t c e n t r e u n d e r r e p e a t e d m e a s u r e m e n t s a n d t h e c o m p u t a t i o n a l e f f i c i e n c y o f t h e a l g o r i t h m ( i n b o t h s i m u l a t i o n a n d e x p e r i m e n t ) , as t h i s m a y h a v e s o m e i m p a c t o n t h e p r a c t i c a l i t y f o r i n t r a o p e r a t i v e u s e . 1 . 5 . 2 In Vivo E x p e r i m e n t a l W o r k I c o n d u c t e d a n in vivo e x p e r i m e n t t o c o m p a r e t h e d i g i t i z a t i o n m e t h o d t o t h e s p h e r i c a l a n d b i a x i a l m e t h o d s o f l o c a t i n g t h e a n k l e c e n t r e i n b o t h t h e W B a n d N W B states. T h e r e s u l t s o f t h i s e x p e r i m e n t a d d r e s s t h e q u e s t i o n o f w h e t h e r o r n o t s u r r o g a t e m e a s u r e m e n t t e c h n i q u e s c a n r e l i a b l y p r e d i c t t h e l o c a t i o n o f t h e W B a n k l e c e n t r e . I n o r d e r t o a n s w e r t h i s q u e s t i o n , I c h a r a c t e r i z e t h e r e p e a t a b i l i t y a s s o c i a t e d w i t h e a c h o f t h e i n d i v i d u a l m e t h o d s t e s t e d , a n d t h e n d e t e r m i n e t h e b i a s o f t h e s u r r o g a t e m e a s u r e m e n t t e c h n i q u e s w i t h r e s p e c t t o t h e W B e s t i m a t e s . I a l s o p r e s e n t a c r i t e r i o n b y w h i c h t o j u d g e w h i c h s u r r o g a t e m e a s u r e i s t h e m o s t r e l i a b l e . 1.6 Thesis Overview I h a v e o r g a n i z e d m y t h e s i s i n t h e f o l l o w i n g m a n n e r : Chapter 1. I n t r o d u c t o r y m a t e r i a l , l i t e r a t u r e r e v i e w , p r o b l e m d e f i n i t i o n , a n d p r o j e c t o b j e c t i v e s . Chapter 2. S t u d y #1 - O p t i m a l F o r m u l a t i o n s O f A n k l e J o i n t M o d e l s F o r U s e I n C o m p u t e r - A s s i s t e d T o t a l K n e e A r t h r o p l a s t y . T h e d e r i v a t i o n as w e l l as t h e n u m e r i c a l a n d e x p e r i m e n t a l v a l i d a t i o n o f t h e o p t i m i z a t i o n m e t h o d s I u s e i n C h a p t e r 3 . S u b m i t t e d t o C o m p u t e r M e t h o d s i n B i o m e c h a n i c s a n d B i o m e d i c a l E n g i n e e r i n g . P a g e 11 Chapter 3. Study #2 - Investigations of the Ankle Joint Centre for use in Computer- Assisted Total Knee Arthroplasty. The results of my in vivo experimental work. To be submitted to Clinical Biomechanics. Chapter 4. Discussion of thesis, conclusion, and the direction for further work. Appendix contains conference publications describing aspects of the work presented in the main chapters: Investigation of Ankle Centre Location Techniques Applied to Computer- Assisted Total Knee Arthroplasty. Biomechanica IV - September 2001 - Davos, Switzerland. (Abstract/Presentation). Repeatability and Accuracy of Ankle Centre Location Estimates using a Biaxial Joint Model. MICCAI - October 2001 - Utrecht, The Netherlands. (Short Paper/Poster). Application of a Biaxial Kinematic Ankle Model to Computer-Assisted Total Knee Arthroplasty. Fifth International Symposium on Computer Methods in Biomechanics and Biomedical Engineering - November 2001 - Rome, Italy. (Paper/Presentation). Repeatability and Accuracy of Bone Cutting and Ankle Digitization in Computer-Assisted Total Knee Replacement. MICCAI - October 2000 - Pittsburgh, USA. (Paper/Poster). 1.7 References 1. ) Johnson, F., S. Leitl, et al. (1980). "The distribution of load across the knee. A comparison of static and dynamic measurements." J Bone Joint Surg Br 62(3): 346-9. 2. ) Harrington, I. J. (1983). "Static and dynamic loading patterns in knee joints with deformities." J Bone Joint Surg Am 65(2): 247-59. 3. ) Lotke, P. A. and M. L. Ecker (1977). "Influence of positioning of prosthesis in total knee replacement." J Bone Joint Surg Am 59(1): 77-9. 4. ) Denham, R. A. and R. E. Bishop (1978). "Mechanics of the knee and problems in reconstructive surgery." J Bone Joint Surg Br 60-B(3): 345-52. 5. ) Bargren, J. H., J. D. Blaha, et al. (1983). "Alignment in total knee arthroplasty. Correlated biomechanical and clinical observations." Clin Orthop(173): 178-83. 6. ) Tew, M. and W. Waugh (1985). "Tibiofemoral alignment and the results of knee replacement." J Bone Joint Surg Br 67(4): 551-6. Page 12 7. ) Cornell, C. N., C. S. Ranawat, et al. (1986). "A clinical and radiographic analysis of loosening of total knee arthroplasty components using a bilateral model." J Arthroplasty 1(3): 157-63. 8. ) Hsu, H. P., A. Garg, et al. (1989). "Effect of knee component alignment on tibial load distribution with clinical correlation." Clin Orthop(248): 135-44. 9. ) Windsor, R. E., G. R. Scuderi, et al. (1989). "Mechanisms of failure of the femoral and tibial components in total knee arthroplasty." Clin Orthop(248): 15-9; discussion 19-20. 10. ) Lee, J. G., E. M. Keating, et al. (1990). "Review of the all-polyethylene tibial component in total knee arthroplasty. A minimum seven-year follow-up period." Clin Orthop(260): 87-92. 11. ) Jeffery, R. S., R. W. Morris, et al. (1991). "Coronal alignment after total knee replacement." J Bone Joint Surg Br 73(5): 709-14. 12. ) Ritter, M., P. Faris, et al. (1994). "Postoperative alignment of total knee replacement. Its effect on survival." Clin Orthop;(299): 153-6. 13. ) Hsu, R. W., S. Himeno, et al. (1990). "Normal axial alignment of the lower extremity and load-bearing distribution at the knee." Clin Orthop(255): 215-27. 14. ) Morrison, J. B. (1970). "The mechanics of the knee joint in relation to normal walking." J Biomech 3(1): 51-61. 15. ) Jessup, D. E., R. L. Worland, et al. (1997). "Restoration of limb alignment in total knee arthroplasty: evaluation and methods." J South Orthop Assoc 6(1): 37-47. 16. ) Kapandji, I. A. (1970). The Physiology of the Joints. New York, Churchill Livingstone. 17. ) Moreland, J. R., L. W. Bassett, et al. (1987). "Radiographic analysis of the axial alignment of the lower extremity." J Bone Joint Surg Am 69(5): 745-9. 18. ) Inkpen, K., A. Hodgson, et al. (2000). Repeatability and Accuracy of Bone Cutting and Ankle Digitization in Computer-Assisted Total Knee Replacement. MICCAI 2000, LNCS Vol. 1935, Pittsburgh, USA, Springer-Verlag, 1163-1172. 19. ) Brys, D. A., A. V. Lombardi, Jr., et al. (1991). "A comparison of intramedullary and extramedullary alignment systems for tibial component placement in total knee arthroplasty." Clin Orthop(263): 175-9. 20. ) Dennis, D. A., M. Channer, et al. (1993). "Intramedullary versus extramedullary tibial alignment systems in total knee arthroplasty." J Arthroplasty 8(1): 43-7. 21. ) Teter, K. E., D. Bregman, et al. (1995). "Accuracy of intramedullary versus extramedullary tibial alignment cutting systems in total knee arthroplasty." Clin Orthop(321): 106-10. 22. ) Phillips, A. M., N. J. Goddard, et al. (1996). "Current techniques in total knee replacement: results of a national survey." Ann R Coll Surg Engl 78(6): 515-20. 23. ) Maestro, A., S. F. Harwin, et al. (1998). "Influence of intramedullary versus extramedullary alignment guides on final total knee arthroplasty component position: a radiographic analysis." J Arthroplasty 13(5): 552-8. 24. ) Picard, F., J. Moody, et al. (2000). A Classification Proposal for Computer-Assisted Knee Systems. MICCAI, LNCS Vol. 1935, Pittsburgh, USA, Springer-Verlag, 1145- 51. 25. ) Delp, S. L., S. D. Stulberg, et al. (1998). "Computer assisted knee replacement." Clin Orthop(354): 49-56. Page 13 Inkpen, K. (1999b). Precision and Accuracy in Computer-Assisted Total Knee Replacement. M.A.Sc. Thesis. Department of Mechanical Engineering, University of British Columbia: 153. Leitner, F., F. Picard, et al. (1997). Computer-Assisted Knee Surgical Total Replacement. CVRMed-MRCAS'97, LNCS Vol. 1205, Grenoble, France, Springer - Verlag, 629-638. Saragaglia, D., F. Picard, et al. (2001). "Computer-assisted knee arthroplasty: comparison with a conventional procedure. Results of 50 cases in a prospective randomized study." Rev Chir Orthop Reparatrice Appar Mot 87(1): 18-28. Krackow, K. A., M. Bayers-Thering, et al. (1999). "A new technique for determining proper mechanical axis alignment during total knee arthroplasty: progress toward computer-assisted TKA." Orthopedics 22(7): 698-702. Kunz, M., M. Strauss, et al. (2001). A Non-CT Based Total Knee Arthroplasty System Featuring Complete Soft-Tissue Balancing. MICCAI 2001, LNCS Vol. 2208, Utrecht, The Netherlands, Springer-Verlag, 409-415. Kiefer, H., D. Langemeyer, et al. (2001). "Computer-assisted navigation in total-knee replacement." European Journal of Trauma E- Suppl. 1: S128-132. Bottlang, M., J. L. Marsh, et al. (1999). "Articulated external fixation of the ankle: minimizing motion resistance by accurate axis alignment." J Biomech 32(1): 63-70. Leardini, A., J. J. O'Connor, et al. (1999). "A geometric model of the human ankle joint." J Biomech 32(6): 585-91. Cole, G. K., B. M. Nigg, et al. (1993). "Application of the joint coordinate system to three-dimensional joint attitude and movement representation: a standardization proposal." J Biomech Eng 115(4A): 344-9. van den Bogert, A. J., G. D. Smith, et al. (1994). "In vivo determination of the anatomical axes of the ankle joint complex: an optimization approach." J Biomech 27(12): 1477-88. Lundberg, A., O. K. Svensson, et al. (1989). "The axis of rotation of the ankle joint." J Bone Joint Surg Br 71(1): 94-9. Sarrafian, S. K. (1993). "Biomechanics of the subtalar joint complex." Clin Orthop(290): 17-26. Leardini, A., J. J. O'Connor, et al. (1999). "Kinematics of the human ankle complex in passive flexion; a single degree of freedom system." J Biomech 32(2): 111-8. Inman, V. T. (1976). Inman's joints of the ankle. Baltimore, William & Wilkins. Singh, A. K., K. D. Starkweather, et al. (1992). "Kinematics of the ankle: a hinge axis model." Foot Ankle 13(8): 439-46. Areblad, M. (1990). On Modelling of the Human Rearfoot. Thesis No. 238. Linkdping Institute of Technology, Sweden. Barnett, C. H. and J. R. Napier (1952). "Axis of rotation at ankle joint in man: its influence upon the for of talus and mobility of fibula." J Anat 86: 1-9. Hicks, J. H. (1953). "The Mechanics of the foot I. The Joints." J Anat 87: 345-357. van Langelaan, E. J. (1983). "A kinematical analysis of the tarsal joints. An X-ray photogrammetric study." Acta Orthop Scand Suppl 204: 1-269. Lundberg, A. and O. K. Svensson (1993). "The axis of rotation of the talocalcaneal and talonavicular joints." The Foot 3: 65 - 70. Sammarco, G. J., A. H. Burstein, et al. (1973). "Biomechanics of the ankle: a kinematic study." Orthop Clin North Am 4(1): 75-96. Page 14 47. ) Boone, D. C. and S. P. Azen (1979). "Normal range of motion of joints in male subjects." J Bone Joint Surg Am 61(5): 756-9. 48. ) Siegler, S., J. Chen, et al. (1988). "The three-dimensional kinematics and flexibility characteristics of the human ankle and subtalar joints—Part I: Kinematics." J Biomech Eng 110(4): 364-73. 49. ) Komistek, R. D., J. B. Stiehl, et al. (2000). "A determination of ankle kinematics using fluoroscopy." Foot Ankle Int 21(4): 343-50. 50. ) Scott, S. H. and D. A. Winter (1991). "Talocrural and talocalcaneal joint kinematics and kinetics during the stance phase of walking." J Biomech 24(8): 743-52. 51. ) Shute, C. and A. Hodgson (2001b). Application of a Biaxial Kinematic Ankle Model to Computer-Assisted Total Knee Arthroplasty. Fifth International Symposium on Computer Methods in Biomechanics and Biomedical Engineering, Rome, Italy, 52. ) Cappozzo, A., F. Catani, et al. (1996). "Position and orientation in space of bones during movement: experimental artefacts." Clin Biomech (Bristol. Avon) 11(2): 90- 100. Page 15 Submitted to Computer Methods in Biomechanics and Biomedical Engineering Chapter 2: Optimal Formulations Of Ankle Joint Models For Use In Computer-Assisted Total Knee Arthroplasty 2.0 Abstract In computer-assisted total knee arthroplasty, the surgeon must identify the kinematic centre of the hip and ankle joints. The spherical joint model commonly used is not as realistic as a biaxial model proposed more recently. Since there are different ways to formulate each joint model, our purpose in this paper was to identify the optimal formulation of each. In particular, we introduce a new formulation of the biaxial joint model with the minimum possible number of parameters (8 vs. 12). Using both simulations and physical experiments we show that the 8 parameter formulation requires an order of magnitude fewer computations than the 12 parameter formulation while maintaining comparable repeatability of ankle joint centre estimates. Three formulations of the spherical joint model have roughly comparable repeatability of the joint centre estimates, but differ modestly in computational requirements. Two of the three formulations are sensitive to which limb segment we consider to be fixed and which moving. Keywords: ankle joint kinematics, biaxial model, biomechanics, computer-assisted surgery 2.1 Introduction To properly position implants in total knee arthroplasty (TKA) the surgeon aims to identify the mechanical axis, the line passing through the centre of the femoral head and the centre of the ankle joint (Jeffery et al., 1991; Jessup et al., 1997), and to align the implants relative to this line. Such alignment ensures that the loads carried by the prosthetic knee are properly balanced and significantly decreases the likelihood of early failure (Jeffery et al., 1991). The standard surgical technique uses anatomic references to estimate the locations of the joint centres (e.g., lining up the tibial guide with the second ray of the foot (Phillips et al., 1996)), but because of various factors such as the thickness of overlying soft tissues and the use of Esmarch bandages to swath the ankle, it is often difficult to identify these references with good repeatability. Also, individual variations in the relative position of the anatomic centre to the functional centre of the joint add to the overall variability of using anatomic references. Because of this variability, emerging computer-assisted surgical (CAS) techniques are based Page 16 on more direct measurements of the kinematic joint centres. For example, in Krackow's technique (Krackow et al., 1999), a marker array is mounted to the exposed distal end of the femur and the femur is manipulated about the hip centre such that the markers trace out a portion of the surface of a sphere. The locations of the markers are measured using a metrology system (typically an optoelectronic localizer, a video system, or a magnetic tracker). The hip centre is taken to be the centre of the sphere which best fits the marker data. Results for locating the hip centre have been quite promising (Krackow et al., 1999; Inkpen, 1999b), but results for the ankle have been more problematic, with standard deviations 4-7 times higher than for the hip (Leitner et al., 1997; Inkpen, 1999b). In all these techniques, a marker array is either explicitly or implicitly mounted on either side of the joint complex and movements of one segment relative to the other are recorded; for the ankle, arrays are typically mounted on the tibia and the calcaneus. The primary technique used to date on the ankle in CAS systems is based on a spherical joint model, which is not a particularly realistic representation of the ankle joint complex anatomy (AJC - see Figure 2.1 left); this may be the cause of the comparatively poor repeatability. The three primary formulations of the centre-finding optimization routine which can be used with the spherical joint model that we considered were: (i) a 3-parameter model (x,y,z coordinates of the joint centre) in which we minimize the variability of distances of each marker from the centre, (ii) a (3+N)-parameter model in which, in addition to the joint centre coordinates, we explicitly parameterize the radii of the spheres to the N emitters on the marker array (N is typically equal to 3), and (iii) a 3-parameter formulation based on homogeneous transforms in which we minimize the movement in one reference frame of a point rigidly fixed to the other frame. For each of these three formulations, there is also the question of which of the two reference frames (tibial or calcaneal) we should treat as fixed and which as moving. It is currently not clear which of the three formulations is most suitable for use in computer-assisted surgical systems. Page 17 Figure 2.1: Biaxial model of ankle joint complex. Left: exploded view with reference frames attached (from van den Bogert, 1994). Right: physical model with joint axes shown. The triangular bodies carry infrared emitter arrays for position tracking. Anatomically, a more plausible model for this joint complex is the biaxial model introduced by Areblad (Areblad, 1990) and later refined by van den Bogert (van den Bogert et al., 1994). This model represents the talocrural (TC) and subtalar (ST) joints separately as hinge-type joints with fixed axes (see Figure 2.1 (left)). According to this model, axial translations are negligible and the rotation axes are fixed in each bone, not migrating throughout the range of motion. While these assumptions are not strictly accurate (Barnett et al., 1952; Hicks, 1953; van Langelaan, 1983; Lundberg et al., 1989; Lundberg et al., 1993), van den Bogert presented a 12 parameter formulation of this model and showed that it produced good fits to experimental data (RMS marker errors in the 2 mm range), although he did not explicitly compare it with a spherical joint model. He used the model to investigate the locations and directions of the joint axes in a group of 14 subjects, but did not use the model to define a joint centre for use in arthroplasty. Such a centre is more difficult to define for the biaxial model than for the spherical joint model, but a force acting along the line passing through the knee centre and both the TC and ST axes will cause no net moment around either axis (see Figure 2.2); we therefore take the joint centre to be the point on this line lying in a transverse plane at the level of the malleoli when the foot is in a neutral position (note that the no- Page 18 moment line will shift slightly mediolaterally during plantar- or dorsiflexion, so we define the mechanical axis and ankle centre at a specified flexion angle). Van den Bogert's formulation also uses more parameters than are strictly required to represent a biaxial joint model; this use of redundant parameters will likely increase the computational time required to fit the model to data. Figure 2.2: Definition of ankle joint centre using biaxial model. The joint centre is the projection of the two axes into a transverse plane when the joint is in a neutral orientation. In order to compare the performance of the two kinematic models for use in CAS systems, it is important to know the optimal formulation of each. The purpose of this paper, therefore, is to identify the optimal formulation for each of the spherical and biaxial joint models treated separately. Future work will compare these two models to one another and to anatomic references to determine which approach is most appropriate for intraoperative use. The principal criteria we will use for evaluating performance will be the variability of the predicted joint centre under repeated measurements and the computational efficiency of the algorithm, as this may have some impact on the potential for intraoperative use. 2.2 Methods and Materials 2.2.1 Formulation of AJC Models 2.2.1.1 The Biaxial Model 2.2.1.1.1 12 Parameter Formulation of Biaxial Model (12P) Van den Bogert's model is shown in Figure 2.1 (left); it consists of six different coordinate frames whose positions relative to one another are described by 12 parameters. Two free Page 19 variables, a and fi, represent motion about the two axes. In this formulation, coordinate system Ftj is fixed to the tibia and represents the location of the marker array used to track the tibia. Coordinate system Fo is also fixed to the tibia with the xo axis coinciding with the TC axis and the origin of Fo is at the point on the TC axis closest to the ST axis (on the common normal). The transform Atjo is defined by five parameters, P 1 - P 5 , consisting of two rotations to position the xo axis, and three translations to position the origin. The transform A01 represents rotation about the TC axis by the variable a. The transform A c a3 is defined by five parameters, P 6 - P 1 0 , and positions the Z3 axis to coincide with the ST axis, and positions the origin of F3 at the point on the ST axis closest to the TC axis. The transform A 2 3 represents rotation about the ST axis by the variable p. The last two parameters P11-P12 are used to form the transform An which establishes the relationship between the TC and ST axes in the talus by defining the distance between them (common normal length), and the twist about the common normal. Nframes Data 1 Guess Estimate a and p for each measured frame of data using current guess Check if attol Calculate predicted marker location using angle estimates, and define cost as the difference between measured and predicted location Optimization routine alters guess Exit Figure 2.3: Flow chart for solving the 12 parameter formulation of the biaxial ankle joint model. The flowchart in Figure 2.3 outlines the data fitting process for the 12 parameter formulation (12P). The Nframes of data are the measured overall transforms from the tibia to the calcaneus, Atica, obtained as the ankle is manipulated throughout its range of motion, and for each frame of data an estimate for a and p is calculated. These estimates are determined using a guess at the axis locations and the measured transform A tiC a to calculate A 0 3 , from which estimates of a and P can be easily extracted. In contrast to the 8 parameter formulation (8P) we present in the next section, this formulation uses its parameters and the two rotation angles to explicitly Page 20 calculate the overall transform A t iCa and predict the marker locations. The cost function is the sum of squared distances between the predicted (rpredicted) and measured marker locations (rmeasured) over all frames (Nframes) and markers (Nmarkers)(\an den Bogert, 1994): Q\7P ~ Nframes Nmar ker J- I 2> predicted measured ' 1=1 7=1 3 N frames^mar ker A' 2.2.1.1.2 8 Parameter Formulation of Biaxial Model (8P) As mentioned in the introduction, the 12 parameter formulation uses more parameters than are strictly required to represent the locations of the two axes; this redundant representation is likely to lead to computational inefficiencies when solving for the model parameters. We have developed a minimally represented formulation of the biaxial joint model which requires only 8 parameters to fully define the model (8P). Figure 2.4 illustrates our method of parameterizing an axis in three dimensions using only four parameters; we pair two such representations to model a biaxial joint. (Pi,P 2) X Figure 2.4: Parameterizing an axis using four variables. (Pi,Pa) defines the pierce point in theyz plane and P 3 and P4 represent angles defining the direction the axis points in. The TC axis is parameterized in the tibial frame using the following four parameters: Pi and P 2 are the coordinates of the pierce point in the xy plane, and P 3 and P4 are two angles which define the direction of the TC axis. The ST axis is parameterized in the calcaneal frame using a similar set of four parameters: P5 and P6 define the pierce point in the yz plane, and P5 and P6 define the direction of the ST axis. In contrast to 12P, 8P does not contain enough information to represent the overall transform Atica in terms of the parameters and rotation angles, so we cannot use the same cost function Page 21 to fit the model to the data. Instead, we use the measured overall transform A ti c a in conjunction with the parameters alone (i.e., we do not require any estimates of the current rotation angles) to produce an estimate of the relative locations of the two joint axes. Since these joint axes are assumed to be fixed in the talus, they should remain fixed relative to one another for all frames of data. For each frame of data, then, we compute values for the four descriptors of the relative axis positions (dst - position of common normal on ST axis, dtc - position of common normal on TC axis, LCN - length of common normal, 8CN - angle of twist between TC and ST axes about the common normal); the cost function we use in the model- fitting process is a weighted sum (W1.4) of the variances associated with these theoretically invariant descriptors: QZP = J £ -d,)2+(W2(dJ-dLC)2+(W3(LCN'-LCN)2 +(W4(0CN'-GCN)2) Since the four descriptors used in the cost function have different physical dimensions (i.e., lengths and angles), the weighting scheme must incorporate reference dimensions with which to non-dimensionalize the descriptors. In our current implementation, we divide each descriptor by the standard deviation of its predicted value over all frames of a data set; this gives each descriptor roughly equal weight in the overall optimization. If we were particularly interested in optimizing a particular aspect of the final representation (such as mediolateral position of the subtalar axis, for example), we could adjust the weighting scheme accordingly. Figure 2.5 contains a flowchart that shows how 8P is implemented. Once the routine has converged to a set of eight parameters, it is possible to generate explicit estimates of the 12 parameters of van den Bogert's model, which allows the two models to be compared directly. Page 22 Nframes Data Guess For each frame of data calculate: 1. CN Length 2. Twist between axes 3. Position of CN intersection on TC axis 4. Position of CN intersection on ST axis Minimize the variance of each of implicit parameters using a weighting scheme so that each of these implicit parameters is equally important. i Optimization routine alters guess Extract 12 params from 8 Figure 2.5: Flow chart for solving the 8 parameter formulation of the biaxial ankle joint model. 2.2.1.2 The Spherical Model 2.2.1.2.1 3 +N Parameter Formulation The 3+N parameter spherefitting formulation is based on the observation that each marker moves on a spherical surface centred at the spherical joint. Each marker will move on a surface with a different radius, but all spheres will share a common centre (xg, ygl zg). In this formulation, three parameters define the centre of the spherical joint and N parameters explicitly represent the radii of the surfaces associated with each marker (Rg' for i=l.JV). In principle, a single marker is sufficient to locate the joint centre, but since most marker arrays have at least three markers mounted, we can maximize the accuracy of this technique by using all available data. In the simulations and experiments reported here, we use marker arrays with three markers mounted (i.e., N = 3). The cost function for this algorithm is defined as the sum of squared normal distances from each measured marker location (expressed in x, y, z components respectively - rxm, rym, rzm) the to the corresponding sphere surface: Q: 3+N I Nmar ker s Nframes I i=l y=l 2.2.1.2.2 3 Parameter Formulation This formulation is a minimal formulation, somewhat analogous to the 8 parameter biaxial formulation. Instead of explicitly representing the sphere radii as parameters like in the Page 23 previous formulation, we simply minimize the variance of the distances (Rm'J) between each measured marker location and the estimated joint centre: g3= ^ Z (Rj - R ) 2 whereR.' is the mean of Rj for marker / over j=l..Nframes V (=1 7=1 A s was the case with the 8 & 12 parameter biaxial formulations, the parameters of the 3+N parameter formulation are easily determined from the solution to the 3 parameter formulation, so the two formulations can be directly compared. 2.2.1.2.3 Homogeneous Transform Formulation The Homogeneous Transform (HT) method is based on the fact that i f two rigid bodies are connected with a spherical joint, the vector to the joint centre in the moving frame maps to a constant point Co in the fixed frame for all positions. Figure 2.6 illustrates this in two dimensions, and the extension to three dimensions is straightforward. In this method, we seek the three coordinates of a point C i in the moving frame which minimizes the variance of its projections into the fixed frame, Fo, for the Nframes measured transforms between the two frames, TQI ' . The cost function is: Fixed frame Fo Figure 2.6: Schematic illustrating the homogeneous transform method. The frame moves relative to F 0 . The point C 0 is invariant in F 0 if it is the centre of a spherical joint. Page 24 2.2.2 Simulations For both spherical and biaxial models of the ankle joint complex, we performed repeated simulations of acquiring data and applying the fitting procedure in order to assess how repeatable the resulting joint centre estimates were and how much computational effort was involved. All data processing and computation was performed on an PC running Windows 2000 with an AMD Duron 650 MHz processor and 128 Mb of RAM. The Matlab R12 programming environment was used to perform all the computations. 2.2.2.1 Description of Motion Data Generation Protocols For all simulations, we emulated a sequence of ankle movements designed to substantially cover typical ranges of motion. In particular, we emulated a cycle of dorsi/plantarflexion with the ankle in the neutral, inverted and everted positions, followed by a cycle of in/eversion with the ankle in the dorsiflexed, neutral and plantarflexed positions. This combination of movements generates data around the periphery of the joint's range of motion as well as within it. The limits on the range of motion used to generate data were 40° of plantarflexion to 20° of dorsiflexion, and 20° of inversion to 20° of eversion. 2.2.2.2 Biaxial Model - Evaluation of Optimization Algorithms Because the biaxial formulations are more complex than the spherefitting ones, there is more potential for the choice of optimization method to have a significant effect on the model's utility, particularly for computer-assisted surgery. We therefore compared the performance of two different algorithms that are well-suited to solving the kinds of nonlinear least squares optimization problems found in both formulations of the biaxial model; these optimization methods are the Levenberg-Marquardt (LM) method (Levenberg, 1944; Marquardt, 1963; More, 1977) and the trust-region reflective Newton (TR) method (Coleman et al., 1994; Coleman et al., 1996) We expected that the trust-region method would be more robust than the LM algorithm as it explicitly controls both step size and search direction, but that this robustness may come at some cost in computational effort. We applied both optimization methods (LM and TR) in conjunction with both 8 and 12 parameter formulations to three datasets consisting of 24 frames of data (4 equally spaced points along each of the 6 movements described in §2.2.2.1); each of the three datasets had a Page 25 different level of isotropic measurement noise added (0.2mm, 0.5mm, and 1.0mm SD). The data sets were generated using marker geometry similar to that used in a typical arthroplasty setup, and we initialized both optimization algorithms with the same initial guess for the axis locations. We evaluated each algorithm using robustness (ability to converge to a relative termination tolerance on the parameter values of 10"6 with a maximum of 400 iterations) and computational effort as performance measures. 2.2.2.3 Biaxial Model - Comparison of 8 vs. 12 Parameter Methods Based on the results from the comparison of optimization methods described above (§2.2.2.2), we exclusively used the trust-region reflective Newton optimization algorithm for this simulation. A total of 15 simulated datasets containing 120 frames each (20 equally spaced points along each of the 6 movements described in §2.2.2.1) were generated and white noise was added to the measurements. Again, we used three levels of isotropic white noise with standard deviations equal to 0.1mm, 0.5mm, and 1.0mm, respectively. The same initial estimate of the joint axis location was used for both formulations of the biaxial optimization problem. As described in the introduction, we defined the centre of the ankle joint complex to be the point in the transverse plane pierced by a line from the knee centre passing through both identified joint axes; a force oriented along this line will cause no net moment about either joint axis. Since this point can be identified for both the 8 and 12 parameter formulations, we can directly compare the performance of the two formulations. In addition, we can use the results of the 8 parameter model to estimate the values for the parameters of the 12 parameter model, so the variability of these parameter estimates can also be directly compared. The results of this simulation will highlight differences between the two formulations in terms of parameter repeatability and computational cost. 2.2.2.4 Spherical Model - Evaluation of Methods & Base Frame Selection We used the same marker geometry and data generation protocols described above (§2.2.2.2) for testing the three spherical joint formulations. In addition, we tested each of the three formulations in two ways: first, by using the tibia as the fixed frame and, second, by using the calcaneus as the fixed frame. We generated a single clean set of data containing 90 frames, and then superimposed white noise (SD = 0.2 mm) to create 30 sets of noisy data to which the methods were applied. The trust-region reflective Newton optimization algorithm Page 26 was used for all spherefitting simulations. In contrast to the biaxial model, we do not expect to see significant differences in computational performance between the different model formulations, but the results of these simulations will enable us to compare the model formulations on the basis of repeatability of the joint centre estimate. We will also see if the choice of the fixed base frame (i.e., tibial or calcaneal) has a significant effect. 2.2.3 Physical Experiments To confirm the predictions of the simulations, we built physical models of both spherical and biaxial joints, instrumented them with marker arrays substantially as described in the simulation section, generated data by moving the joints through ranges of motion roughly equivalent to the ranges of real joints, and analyzed the results using formulations described in §2.2.1.1 and §2.2.1.2. 2.2.3.1 Equipment Used Marker position data for the physical model experiments was obtained using an Image Guided Technologies Flashpoint 5000 optoelectronic metrology system to track infrared emitting diode (IRED) marker triads. Figures 2.1 (right) and 2.7 show the biaxial and spherical physical models that were constructed for testing. The expected SD of the noise associated with the Flashpoint system has been reported to be in the range of 0.15 - 0.42 mm (Chassat et al., 1998; Li et al., 1999), which agrees with a value of 0.27 mm that we obtained in preliminary testing. Figure 2.7: Physical model of a spherical joint. The marker arrays used in the experiment matched those used with the biaxial physical model. Page 27 2.2.3.2 Biaxial Model The model was put through the motions described in §2.2.2.1 and an average of 160 (± 20 SD) frames of kinematic data were collected for 30 trials. We used two methods to obtain initial estimates of the locations of the joint axes: (i) digitizing the tips of the hinge pin of each axis, and (ii) using a finite helical axis identification technique based on measured homogeneous transformations across each joint (Spoor et al., 1980). In all other respects, the data analysis was identical to that described for the simulations in §2.2.2.3. 2.2.3.3 Spherical Model We exercised the spherical model in the same way described above for the biaxial model (§2.2.3.2) and used the three algorithms outlined in §2.2.1.2 to find the sphere centres which best fit the data. In addition, we analyzed the data twice - once with the tibial frame fixed and the second time with the calcaneal frame fixed. This analysis is substantially similar to that described for the simulations in §2.2.2.4. 2.3 Results On all plots presented in this section, the error bars represent the 95% (a=0.05) confidence intervals predicted using the t statistic for mean values, and the %2 statistic for standard deviation values. To test the difference between mean values, we used the t test assuming unequal variances (a=0.05) and the F test to test the difference between variances (a=0.05). 2.3.1 Biaxial Model 2.3.1.1 Simulations 2.3.1.1.1 Comparison of Optimization Algorithms There were distinct differences in performance for the two optimization algorithms, and these differences were dependent on the biaxial model formulation used. For 12P, the Levenburg- Marquardt method (LM) failed to converge for all of the three noise levels (in the sense that the algorithm was not able to reach the relative termination tolerance of 10"6), whereas the trust-region reflective Newton method (TR) converged in all cases; this suggests that the LM method is not as robust as the TR method. Page 28 In contrast, for 8P, both optimization algorithms converged for all three noise values tested. In addition, the TR method was more computationally efficient than the LM method (see Figure 2.8), typically requiring only a third the number of floating point operations. The most striking result, however, (discussed further in §2.3.1.1.3) is that both optimization methods required an order of magnitude fewer computations for 8P than 12P. I/I Q. O 5.0 4.5 4.0 3.5 3.0 g1 2.0 5 1.5 -\ 1.0 0.5 H 0.0 U L M • TR 0.2mm 1.0mm 0.5mm Noise Level, (mm) Figure 2.8: Computational effort for different optimization algorithms used with 8P at different noise levels. In summary, then, the trust-region reflective Newton algorithm appears to be both more robust and more computationally efficient for both biaxial model formulations; we therefore used this algorithm exclusively for the remaining analyses described in this paper. 2.3.1.1.2 Comparison of Repeatability and Bias On average, for the three noise levels tested and for the two directions of interest (anteroposterior (AP) and mediolateral (ML)), the joint centre estimates produced by 8P were 17% less variable than the corresponding estimates produced by 12P as the raw data shows in Figure 2.9 (range of SD ratios: 0.89-1.60), although this difference between formulations was not statistically significant (p>0.04). Similarly, the 12 equivalent parameters computed from 8P had variabilities which were typically not significantly different (see Figure 2.10 top). Page 29 o 8P • 12p • mean 8P • mean 12P 0.1 mm SD Noise - 2 - 1 0 1 ML Position, mm o •M w O a, 5! - 2 - 1 0 1 2 ML Position, mm 34 s 33 6 ef 32 o •a o 31 CL, CL, < 30 Physical Experiment . 2 - 1 0 1 2 - 1 3 - 1 2 - 1 1 - 1 0 - 9 ML Position, mm ML Position, mm Figure 2.9: Comparison of repeatability and bias for 12P and 8P in simulation (top and bottom left) and in experiment (bottom right). Estimated means are shown with 95% confidence interval bars in the anteroposterior and mediolateral directions.. In contrast, the bias in the joint centre location associated with each technique was significantly different for the 0.5mm and 1.0mm noise level in the AP direction and increased with noise level (p=0.015 and pO.OOl). For all trials the centre estimate for both methods were medial and posterior to the known "true" centre located at the origin for the three simulation plots in Figure 2.9. Page 30 0.40 § 1 u fe 0.35 0.30 • 0.25 0.20 0.15 0.10 0.05 0.00 -I— p 12 Parameter • 8 Parameter S 3 P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 Parameter 0.60 0.50 S3 a ••3 0.40 CQ 0.30 Q <Z> u 0.20 0 u a § 0.10 OH 0.00 0 12 Parameter • 8 Parameter ft P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 Parameter Figure 2.10: Comparison of parameter repeatability 12P and 8P in simulation (top) and in experiment (bottom); error bars represent 95% confidence intervals on estimates. 2.3.1.1.3 Comparison of Computational Effort As shown in Table 2.1, 8P typically required roughly an order of magnitude fewer floating point operations than 12P when evaluated on identical datasets (p<0.001). Optimizing a single dataset took roughly 20 seconds for 8P and two and a half minutes for 12P, so one Page 31 important practical consequence of this result is that 8P may be more easily implemented in the operating room where computational delays are unacceptable.' 2.3.1.2 Physical Experiments 2.3.1.2.1 Comparison of Repeatability and Bias Slightly in contrast to the simulations, we found marginally improved repeatability of the ankle joint centre location in both the ML (0.36 mm SD vs. 0.42 mm SD) and AP (0.18 mm SD vs. 0.21 mm SD) directions for 8P vs. 12P (see Figure 2.9 - bottom right). There was no significant difference in the location of estimates the ankle joint centre between 8P and 12P (p>0.4). As was found in simulations, the repeatability of the 12 equivalent parameters as computed from the parameters of 8P was not typically different than 12P except for parameter 11 which is the common normal length, (p<0.01) (see Figure 2.10 bottom). 2.3.1.2.2 Comparison of Computational Effort The results for the physical experiment are comparable to those predicted by the simulations, and 8P again required just over an order of magnitude fewer floating point operations (FLOPS) than 12P to converge, as summarized in Table 2.1. Table 2.1: Summary of computational effort in both simulation and experiment. 8 Parameter 12 Parameter (Megaflops) (Megaflops) Simulation (0.1 mm noise) 5.5 80.1 Simulation (0.5 mm noise) 5.1 57.8 Simulation (1.0 mm noise) 5.7 64.5 Physical Experiment 7.1 104.1 2.3.2 Spherical Model 2.3.2.1 Simulations The repeatability of the joint centre found by each of the three methods is shown for both the ML and AP directions in Figure 2.11. There were insignificant differences in repeatability 1 The computations were implemented in Matlab, so we might expect a speedup of up to perhaps 100 times if we implemented the optimization algorithm as compiled code. Even so, the 12 parameter formulation would still require a significant delay in the operating room. Page 32 between all three methods (3 and 3 +N parameter methods and the homogeneous transform method (HT)) (p>0.05). There were no significant differences in the centre locations between methods and no significant bias associated with any of the three methods. The choice of base frame had a large effect on both repeatability and residual error. The residual for all three methods using the tibia as the base frame was significantly higher at 0.49 ± 0.03 mm, compared to 0.28 ± 0.01 mm using the calcaneus as the base frame. Similarly, using the calcaneal base frame improved joint centre estimate repeatability significantly for all methods (p<0.02), although this effect was most marked for the 3 and 3 +N parameter methods (see Figure 2.11). Formulation ] Tibia Fixed (TJ Calcaneus Fixed Formulation Q oo 0 40 0 35 0 30 0.25 0.20 0.15 0.10 0.05 0.00 040 035 0 3 0 , 0.25 Q 0 2 0 OO 0 15 010 0.05 000 3 3 Param 6 Param Formulation 3 Param 6 Param Formulation Transform Figure 2.11: M L (left) and A P (right) repeatability of spherical methods in simulation (top) and in experiment (bottom); error bars represent 95% confidence intervals on estimates. The computational costs associated with each of the three methods found in the simulation are illustrated in Figure 2.12 for both base frame choices. The 3 parameter formulation was found to be the most computationally efficient, followed in turn by the homogeneous transform method and the 3+vV parameter method. The 3 parameter formulation required roughly half the number of floating point operations as the H T method, although in practice these differences in computational effort would be negligible since the entire computation takes only a second or two. A s expected, there was no significant effect of base frame choice on computational efficiency. Page 33 P Tibia Fixed P Calcaneus Fixed 3 Pa r am 6 Pa r am Transform 3 P a , a m 6 P a r a r n Transform Formulation Formulation Figure 2.12: Computational cost of spherical formulations in simulation (left) and in experiment (right). 2.3.2.2 Physical Experiments Overall, the experimental results supported the simulation predictions. There were no significant differences in repeatability between the 3 and 3+N parameter methods; however, in contrast to the simulations, the homogeneous transform method was 13 - 70% less variable in all directions (all statistically significant except for the M L direction when using the calcaneal baseframe (p>0.2) - see Figure 2.11). There was no significant difference in the bias of the joint centre location amongst the three methods. A s was the case in simulation, choosing the calcaneal frame as the fixed reference frame resulted in the lowest residual errors (0.11 ± 0.01 mm vs. 0.49 ± 0.06 mm for the tibial base frame for all methods). The choice of base frame had the largest effect on results using the 3 and 3+vV parameter methods, whereas it had little impact on results using the homogeneous transform method shown in Figure 2.11. We again found that the 3 parameter formulation was the most computationally efficient, followed in turn by the homogeneous transform method and the  2+N parameter method. The choice of base frame again had no significant effect on computational cost (see Figure 2.12). 2.4 Discussion 2.4.1 B i a x i a l M o d e l : We have presented a new formulation (designated 8P) for the biaxial joint model introduced by Areblad (Areblad, 1990) and refined by van den Bogert (van den Bogert, 1994) which uses the minimum possible number of parameters, namely eight. The optimization problem we solve to identify our parameter values is based on minimizing the variance of four quantities predicted by the model to be invariant; this approach is in contrast to the Page 34 optimization problem used with the 12 parameter formulation (12P) in which the cost function is based on a sum of squared distances between predicted and measured marker locations. In our first simulation study, we compared two well-known optimization algorithms for non- linear least squares problems and demonstrated that the best one to use for either model formulation on grounds of computational efficiency (roughly a 70% reduction of effort) and robustness was the trust-region reflective Newton method. For the particular test problem we tried, both algorithms converged for 8P, but the Levenberg-Marquardt method failed to converge for 12P. We therefore recommend using the trust-region reflective Newton method for either formulation. 8P produced slightly less repeatable joint centre estimates from the same artificially generated data set than did 12P (17% increase in variability, as measured by the standard deviation of the joint centre positions). In the physical experiment we found the variability was 10-20%) lower for 8P than 12P. The reasons for this difference between simulation and experiment are unclear, although the differences in both noise level and isotropy between the simulation and the physical measurements may play an important role here. The simulation predicted that the joint centres predicted by both formulations tended to be medial and posterior to known "true" centre; we could not confirm this experimentally as we did not have an independent estimate of the joint centre position to the accuracy necessary to test this prediction, but we found no significant difference in the experimental joint centre estimates between 8P and 12P. 8P also required less computational effort in both simulations and the physical experiment - typically an order of magnitude fewer floating point operations were required to fit the parameters to the data. Overall then, we recommend using 8P since there were relatively minor differences in repeatability compared to 12P and a large savings of computational cost. It should be noted that the robostness of 8P has not yet been validated on human data where the assumptions of the biaxial model may not be strictly accurate. Page 35 2.4.2 Spherical Model: The three spherical joint models we examined all produced similar repeatability results in simulation, although the homogeneous transform method produced somewhat less variable joint centre estimates in the experiment. All methods also produced substantially unbiased joint centre estimates. We did find a strong reduction in variability for the 3 and 3+vV parameter methods when the calcaneal reference frame was treated as the fixed frame rather than the tibial frame; the homogeneous transform method seemed to be insensitive to this choice. This asymmetry in variability with choice of base frame likely has to do with changes in the cost function. For the 3 and 3+N parameter methods, the cost function represents the distance between the marker locations and a sphere passing through their midst. When the base frame is changed, the markers lie on spheres of significantly different radii (typically, the tibial markers are 4-5 times more distant from the joint centre than the calcaneal markers). In contrast, with the HT method, the cost function is based on the assumed invariance of a point in the moving frame; this cost function is little changed by a change in base frame. In a practical sense, there were comparatively minor differences in computational efficiency between the three methods, but in both simulation and experiment, the 3 parameter method was most efficient, with the homogeneous transform and 3+7V methods requiring 2X and 20X more FLOPS, respectively. In general, this was roughly 5% of the number of FLOPS required to solve the biaxial joint model. There was essentially no effect of choice of base frame on computational efficiency. Overall then, we recommend using either (i) the 3 parameter method with the calcaneal reference frame treated as the fixed frame if computational efficiency is an important consideration or (ii) the homogeneous transform method with either reference frame treated as the fixed frame if one wishes to use the technique which seems empirically to produce the most repeatable joint.centre estimates. 2.4.3 Conclusions We have identified the optimal formulations of two models of the ankle joint complex being considered for applications in computer-assisted total knee arthroplasty. The spherical model has been the most commonly used (Leitner et al., 1997; Inkpen, 1999b) because of its ease of formulation and use, but we feel that the biaxial model is more representative of the underlying anatomy of the joint and therefore ought to produce more reliable results. This Page 36 study has not addressed which joint model best represents the human ankle joint complex, which model is the most appropriate for use in a computer-assisted surgical system, or what the relationship is between the joint centre defined using either method and the anatomical joint centre, and we are currently exploring this question in more detail. 2.5 Acknowledgements We thank Dr. Philip Loewen (UBC Department of Mathematics) for his assistance on optimization theory. 2.6 References 1. ) Jeffery, R. S., R. W. Morris, et al. (1991). "Coronal alignment after total knee replacement." J Bone Joint Surg Br 73(5): 709-14. 2. ) Jessup, D. E., R. L. Worland, et al. (1997). "Restoration of limb alignment in total knee arthroplasty: evaluation and methods." J South Orthop Assoc 6(1): 37-47. 3. ) Phillips, A. M., N. J. Goddard, et al. (1996). "Current techniques in total knee replacement: results of a national survey." Ann R Coll Surg Engl 78(6): 515-20. 4. ) Krackow, K. A., M. Bayers-Thering, et al. (1999). "A new technique for determining proper mechanical axis alignment during total knee arthroplasty: progress toward computer-assisted TKA." Orthopedics 22(7): 698-702. 5. ) Inkpen, K. (1999b). Precision and Accuracy in Computer-Assisted Total Knee Replacement. M.A.Sc. Thesis. Department of Mechanical Engineering, University of British Columbia: 153. 6. ) Leitner, F., F. Picard, et al. (1997). Computer-Assisted Knee Surgical Total Replacement. CVRMed-MRCAS'97, LNCS Vol. 1205, Grenoble, France, Springer - Verlag, 629-638. 7. ) Areblad, M. (1990). On Modelling of the Human Rearfoot. Thesis No. 238. Linkoping Institute of Technology, Sweden. 8. ) van den Bogert, A. J., G. D. Smith, et al. (1994). "In vivo determination of the anatomical axes of the ankle joint complex: an optimization approach." J Biomech 27(12): 1477-88. 9. ) Barnett, C. H. and J. R. Napier (1952). "Axis of rotation at ankle joint in man: its influence upon the for of talus and mobility of fibula." J Anat 86: 1-9. 10. ) Hicks, J. H. (1953). "The Mechanics of the foot I. The Joints." J Anat 87: 345-357. 11. ) van Langelaan, E. J. (1983). "A kinematical analysis of the tarsal joints. An X-ray photogrammetric study." Acta Orthop Scand Suppl 204: 1-269. 12. ) Lundberg, A., O. K. Svensson, et al. (1989). "The axis of rotation of the ankle joint." J Bone Joint Surg Br 71(1): 94-9. 13. ) Lundberg, A. and O. K. Svensson (1993). "The axis of rotation of the talocalcaneal and talonavicular joints." The Foot 3: 65 - 70. 14. ) Levenberg, K. (1944). "A Method for the Solution of Certain Problems in Least Squares." Quarterly Applied Math 2: 164-168. 15. ) Marquardt, D. (1963). "An Algorithm for Least Squares Estimation of Nonlinear Parameters." SIAM Journal Applied Math 11: 431-441. Page 37 16. ) More, J. J. (1977). The Levenberg-Marquardt Algorithm: Implementation and Theory. Numerical Analysis. G. A. Watson, Springer Verlag. LNM Vol. 630: 105 - 116. 17. ) Coleman, T. F. and Y. Li (1994). "On the Convergence of Reflective Newton Methods for Large-Scale Nonlinear Minimization Subject to Bounds." Mathematical Programming 67(2): 189 - 224. 18. ) Coleman, T. F. and Y. Li (1996). "An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds." SIAM Journal on Optimization 6: 418 - 445. 19. ) Chassat, F. and S. Lavallee (1998). Experimental Protocol of Accuracy Evaluation of 6-D Localizers for Computer-Integrated Surgery: Application to Four Optical Localizers. MICCAI'98, LNCS Vol. 1496, Cambridge, USA, Springer-Verlag, 421- 430. 20. ) Li, Q., L. Zamorano, et al. (1999). "Effect of optical digitizer selection on the application accuracy of a surgical localization system - a quantitative comparison between the OPTOTRAK and flashpoint tracking systems." Computer Aided Surgery 4(6): 314-321. 21. ) Spoor, C. W. and F. E. Veldpaus (1980). "Rigid body motion calculated from spatial co-ordinates of markers." J Biomech 13(4): 391-3. Page 38 To be submitted to Clinical Biomechanics Chapter 3: Bias and Repeatability of Motion-Based and Digitized Ankle Joint Centre Estimates Under Both Weight- Bearing and Non-Weight-Bearing Conditions 3.0 Abstract Objective. To compare the precision of several methods of estimating the ankle joint centre for use in the registration component of computer-assisted total knee arthroplasty, and to comment on the evaluation criteria for these methods. Design. Three surrogate methods of ankle centre estimation were compared to direct localization of the ankle centre in weight-bearing. Background. Previous studies have focused largely upon the precision of individual techniques rather than examining the intersubject variability of the techniques relative to one another. The effect of weight-bearing versus non-weight-bearing on the location of the ankle centre has not ever been addressed. Methods. The position and orientation of the tibia with respect to the calcaneus was measured on 12 subjects as the foot was moved through a representative range of motion in both non-weight- bearing and weight-bearing. The surrogate methods evaluated for prediction of the weight- bearing ankle centre are as follows: spherical ankle model method, biaxial hinge ankle model method, and direct anatomic digitization. Results. We found the spherical ankle model to be the most repeatable method for estimating the centre of the ankle joint. Conclusion. We found that a lateral and posterior shift of 2.7 mm and 6.9 mm respectively should be applied to the centre obtained from the non-weight-bearing spherical method to best estimate the weight-bearing centre. Relevance As more surgical navigation systems are being developed for use in total knee arthroplasty it is important for developers, and clinicians alike, to recognize the issues and variability associated with each method of ankle centre localization. Page 39 Keywords: ankle joint complex, kinematics, weight-bearing, in vivo, biaxial model, biomechanics, computer-assisted surgery, registration. 3.1 Introduction 3.1.1 Clinical Application It has long been recognized that achieving proper frontal plane implant alignment in total knee arthroplasty (TKA) is critical to minimizing the risk of short term failure. In fact varus/valgus alignment errors of as little as 3° have been strongly correlated to increased risk of early loosening and implant failure (Jeffery et al., 1991). To properly align implants, the surgeon must intraoperatively define the mechanical axis of the lower limb, which passes through the centre of the hip, knee, and ankle joints. Implants are then oriented perpendicular to this axis to ensure the equal balancing of loads between the medial and lateral compartments of the knee. Ideally, the joint centres should be measured for every patient and should represent the location in each joint through which loads are transferred during locomotion. Clinically, a weight-bearing (WB) measurement is very difficult to obtain since the patient is supine and under anesthesia, so surgeons use several surrogate measures to estimate the joint centres. We have been developing a computer-assisted TKA system with the aim of assisting surgeons by improving the precision of the registration process. The focus of this study was to investigate several surrogate methods for estimating the centre of the ankle joint complex (AJC) and to examine how reliably these estimates can predict the WB centre of the ankle. 3.1.2 Current Approaches in Computer Assisted TKA Although no "gold standard" for the ankle joint centre in computer-assisted TKA currently exists, there are two main approaches used: anatomical digitization and motion-based methods. There are differences in both repeatability and bias between these two estimation techniques. Joint centre errors in the mediolateral (ML) direction give rise to varus/valgus angular errors, whereas anteroposterior (AP) errors correspond to flexion/extension errors. If one chooses a centre that is medial to the "true" centre, a valgus deformity will result, whereas lateral errors give rise to varus deformities. Similarly, anterior errors result in flexion errors, and posterior errors produce extension errors. It is most critical to minimize the varus/valgus errors, since they directly affect the load balance between the compartments of the knee. The results we quote in the following sections will appear in the following form: joint centre error (Sj0in, in mm) followed Page 40 by the equivalent angular error (stheta in °). The angular error in defining the mechanical axis is proportional to the joint centre location errors and inversely proportional to the length of the mechanical axis in the tibia. To be conservative, we have used the tibial mechanical axis length th (L = 375 mm) for a 5 percentile female (mean of Japanese and Swedish populations; ~1.5 m tall) for computing all angular errors (i.e. 1° ~ 6.5 mm). The relationship of angular error to joint centre location error is computed as etheta ~ arc tan (Sjoint IL)- 3.1.2.1 Anatomic Digitization The anatomical ankle centre has been loosely defined by locating points on the extremes of the medial and lateral malleoli, and then choosing some point along the line connecting these points. The surgical technique manual for the DePuy P.F.C.® Sigma system recommends a 3 mm medial offset to the midpoint of the malleolar axis to account for the prominence of the lateral malleolus (DePuy, 2000). The following methods of defining the anatomical ankle centre for computer- assisted TKA systems have been used: Krackow digitized the surgeons' best estimate of the centre along the anterior portion of the ankle (Krackow et al., 1999), Kunz digitized the ligament of tibialis anterior (Kunz et al., 2001), and Inkpen used the midpoint of the malleolar axis (Inkpen et al., 1999a; Inkpen, 1999b), although only for the purpose of establishing repeatability in a cadaveric study; he did not actually propose to use the midpoint as the ankle centre estimate in a live procedure. Of all these studies, Inkpen was the only author to report the standard deviation (SD) associated with his chosen method. He reported the SD of using digitization to locate the midpoint of the AJC to be 0.75 mm (0.11°) in the ML direction and 1.12 mm (0.17°) in the AP direction based on a single operator and seven cadaver specimens (Inkpen et al., 2000). 3.1.2.2 Motion-Based Methods Motion-based estimates in computer-assisted TKA are obtained by performing the following four steps: (1) choosing an appropriate mathematical model of the ankle joint, (2) measuring a subject move the ankle through a representative range of motion (ROM), (3) applying an optimization technique to optimally fit the mathematical model to the measured data, and lastly (4) computing the effective joint centre from the model parameters (this computation depends on the type of model chosen). Traditionally, ball-in-socket, or spherical, models of the AJC have been used in computer-assisted TKA systems (Leitner et al., 1997; Inkpen, 1999b). Inkpen quantified the SD associated with the ball-in-socket model to be 1.49 mm (0.23°) in the ML Page 41 direction, and 2.72 mm (0.42°) in the AP direction based on two cadaver specimens using a non- invasive calcaneal tracker referenced to a bone-pin-mounted tibial frame. When he used a bone pin in the calcaneus instead of the tracker he found the SDs to be 0.82 mm (0.12°) in the ML direction, and 1.35 mm (0.21°) in the AP direction (Inkpen, 1999b). An alternative to the spherical model, the biaxial model, was originally proposed by Areblad (Areblad, 1990), and further developed by van den Bogert (van den Bogert et al., 1994). It is a two degree of freedom (DOF) kinematic model in which all ankle motion is assumed to arise from rotation about the talocrural (TC) and subtalar (ST) joints. The model relates the motion of three rigid bodies: the tibia, the talus, and the calcaneus. The TC joint allows articulation between the tibia and the talus about a fixed hinge, and the ST joint allows articulation between the talus and the calcaneus about another fixed hinge. The optimal locations of the joint axes are determined based solely on measurements of the location of the calcaneus relative to the tibia throughout the ROM. A pilot study we conducted with five subjects indicated that the biaxial model showed promise for use in computer-assisted surgery since we obtained SDs of 1.14 mm (0.17°) in the ML direction, and 0.92 mm (0.14°) in the AP direction (Shute et al., 2001c). 3.1.3 Limitations of Current Approaches There have been no reported studies that discuss the relationship between any of the ankle joint centre estimates discussed above and a functional weight-bearing (WB) centre. The WB studies in the literature have generally focused on issues such as axis migration and the range of motion (ROM) of the ankle (Lundberg et al., 1989; Tinley et al., 1996). As discussed previously, it is difficult to obtain an intraoperative measure of the WB kinematics, so we must rely on surrogate measures such as a non-weight-bearing (NWB) motion-based estimate or a digitized estimate to predict the location of the WB centre. There has also been a lack of recognition of the intersubject bias, and variability of this bias, between centre localization techniques. Although Inkpen did not focus on the bias variability, he did note that in general the motion-based AJC centre estimates were medial and anterior relative to the digitized centre estimates (Inkpen, 1999b). We investigated the variability due to bias in a pilot project and reported that the anatomical digitization method would exhibit 8-9X the variability of the NWB motion-based methods when this effect is taken into account (Shute et al., 2001b). So, since we would ideally like a subject-specific measure of a functional WB ankle Page 42 joint centre, it is important to include the intersubject variability of the bias between this and the surrogate measurement technique when calculating the overall precision of a technique. Other limitations of previous studies include their reliance on cadavers and use of suboptimal formulations of the joint centre finding routines. Cadavers lack muscle tone (trackers might move more with muscle contraction/release) and the absence of elasticity in the skin causes dimpling, which can artificially improve the repeatability of digitization. Most published studies do not report in detail how they formulate the spherefitting optimizations, but we have found that the choice of algorithm has a significant effect on the repeatability of the centre estimates (Shute et al., 200Id). In particular, we found that a transform-based method using data from a marker triad was significantly less variable than the single marker technique used by Inkpen (and possibly others). 3.1.4 Purpose of Study In order to better understand weight-bearing ankle joint kinematics and to overcome the limitations of the previous studies mentioned, we conducted an in vivo experiment to compare the digitization method to the spherical and biaxial methods of locating the ankle centre in both the WB and NWB states. Specifically, we addressed the following research question: which of the tested surrogate measurement techniques can most reliably predict the location of the WB ankle centre across our tested population? In order to answer this question, we characterized the repeatability associated with each of the individual methods tested, determined the bias of the NWB methods and digitization with respect to the WB estimates, and investigated the relationships between surrogate measurement techniques. 3.2 Methods and Materials 3.2.1 Measurement Equipment and Subject Information Altogether, 12 healthy subjects aged 25 ± 3 years weighing 70 ± 9 kg were tested (6 male age: 27 ± 3 years, weight: 77 ± 5 kg; 6 female age: 23 ± 3 years, weight: 62 ± 5 kg). No subject had ever suffered fractures of the ankle or displayed excessive ankle joint laxity. The location of the calcaneus relative to the tibia was required to be measured in order use the motion-based methods. To achieve these measurements we designed non-invasive bone tracking devices for both the tibia and calcaneus as depicted in Figure 3.1. These devices were designed Page 43 using a minimum constraint design approach to minimize relative motion between the tracker and the underlying bone, and the contact patches were generally placed on areas of the tibia and calcaneus where the overlying tissues are comparatively thin. Figure 3.1: Tibial and calcaneal trackers (left), tibial tracker (centre), and calcaneal tracker (right). For each subject the tibial tracker was attached around the midshank and the calcaneal tracker was attached to the heel. Triads of infrared emitting diodes (equilateral triangles 120 mm on a side) were mounted to each tracker to establish a reference frame. The geometry of these references was in accordance with the design suggestion by Cappozzo that the root mean square (RMS) distance of the markers from their centroid should be greater than ten times the SD of the marker position error (Cappozzo et al., 1997). The spatial locations of the triads were measured using a Flashpoint 5000 localizer (Image Guided Technologies, Boulder, CO, USA). The SD of the noise associated with the Flashpoint system has been reported to be in the range of 0.15 - 0.42 mm (Chassat et al., 1998; Li et al., 1999), which agrees with the value of 0.27 mm we obtained in preliminary testing. In order to relate our results to anatomical reference planes, a nominal body reference frame fixed to the calcaneus was defined. While the subject was seated with the foot flat on the floor the head of the fibula was vertically aligned with the lateral malleolus, and the tibial tuberosity was positioned in line with the midline of the foot (the line connecting the centre of the heel and the centre of the second digit) when viewed from above. Once the subject was in position, the following three points were digitized to allow calculation of the reference frame: the centre of the heel (origin), the centre of the second digit (AP direction), and lastly a point on the ground that defined the transverse plane. This is similar to the reference position used previously by van den Bogert (van den Bogert et al., 1994). Page 44 3.2.2 Ankle Digitization Method A single operator performed 15 digitization trials on each subject using the conventional 135 mm digitizing probe shown in Figure 3.2. A single trial consisted of the operator digitizing what they considered to be the extremes of both medial and lateral malleoli. We report the midpoint of this malleolar axis as the digitization estimate for the centre of the AJC. Figure 3.2: Two Emitter Point probe digitizing lateral malleoli 3.2.3 Motion-Based Methods 3.2.3.1 Non-weight-Bearing and Weight-Bearing Testing Protocol To fit either the spherical or the biaxial models to a subject we required measurements of the calcaneus with respect to the tibia throughout a representative ROM for the ankle. We followed the method outlined by van den Bogert in which subjects performed the following movements: dorsi/plantarflexion with the ankle in the neutral, inverted and everted positions, in/eversion with the ankle in the dorsiflexed, neutral and plantarflexed positions, and lastly a cycle of circumduction. The same dataset was used to fit both models, and to extract the centre estimates for each model. Each subject performed 15 trials of these motions in both NWB as well as in WB. The NWB measurements were taken with the subject seated with their right foot elevated off the ground as shown in Figure 3.3 (left). The WB measurements were more difficult to obtain since the calcaneal tracker on the heel of the subject would shift if loads were applied. Subjects stood on the ball of their foot as shown in Figure 3.3 (right), and went through the motions mentioned above. In this way the calcaneal tracker did not encounter external forces that could have shifted it relative to the bone between trials. Since there is no significant difference in the ROM for the right or left foot in normal subjects (Stefanyshyn et al., 1994; Moseley et al., 2001) we tested the right foot for all subjects. Page 45 Figure 3.3: Trackers in use for collecting non-weight-bearing data (left) and weight-bearing data (right) 3.2.3.2 Spherical Model Extracting the centre of the spherical model is trivial since a single joint centre is what defines the model. The algorithm we use to perform the spherefitting was the homogeneous transform (HT) method which is based on the fact that i f two rigid bodies are connected with a spherical joint, the vector to the joint centre in the moving frame maps to a constant point in the fixed frame for all positions. We derived this and validated it in a previous study (Shute et al., 200Id). 3.2.3.3 Biaxial Model Our joint centre definition for the biaxial model is the intersection point of the T C and ST axes that one would see in a transverse projection looking down from the knee centre shown in Figure 3.4 (Shute et al., 200Id). The rational for this choice is based on what we are trying to achieve in alignment with respect to force transfer. This centre is the o;dy point that a force oriented through the A J C towards the knee could pass through and generate no net moments about either the T C joint or ST joint. Figure 3.4: Definition of ankle joint centre using biaxial model. Page 46 The method was tested using Flashpoint on a wooden model with two hinges to get an idea of the repeatability one could expect for a true biaxial system with markers rigidly attached. A set of 30 trials were performed, and the repeatability of locating the centre was: 0.42mm (0.06°) SD in the ML direction and 0.21mm (0.03°) SD in the AP direction. 3.2.4 Data and Statistical Analysis '" We are interested in knowing both the bias of the three surrogate measures relative to the weight- bearing centres and the repeatability of each of the surrogates. In our calculations, we consider mediolateral and anteroposterior directions independently, and on the plots we show these directions as follows: medial to the left, lateral to the right, anterior upwards, and posterior downwards. Let Wj be our mean estimate of the weight-bearing centre for subject i (assumed to be unbiased), _ 1 N' computed as wi = — ^ w{j, where M A is the y'th estimate of the weight-bearing centre for N, 7=1 ' " subject / and Nt represents the number of centre measurements made per subject (15 in our experiments). Similarly, we define xy and 5y = x:j - wi to be the y'th estimates of the surrogate centre and bias, respectively, for subject /. Our experiments therefore provide us with Ns sets of N, estimates of the bias of a surrogate method, where yVs. is the number of subjects studied (12 in our case). We estimate this bias by averaging the full set of 180 bias measurements: _ j N.< N , 8 = SS^y • Although m e igg bias measurements are not independent (since each set of 15 from each subject are highly correlated), the variance of the whole set (hereafter referred to as the population variance) is a good estimate of the repeatability associated with the bias estimate 1 N< 1 N l itself. We define the population variance as si = V — V(c> -8)2, which is a nearly unbiased estimate if the intrasubject measurement variabilities for both the weight-bearing and surrogate centre estimates are low relative to the variability of the bias across subjects, as is typically the case. The 95% confidence interval for 8 is computed as ±1.96^. I^N~s . Because the population variance is due to the three different factors just described, it is difficult to develop an analytical expression for the confidence interval associated with ss itself. We Page 47 therefore applied a bootstrapping approach to estimate the confidence intervals (DiCiccio et al., 1996). In particular, we simulated new experiments using a two-stage process: first, we randomly resampled (with replacement) the original set of 12 subject indices; second, for each subject index in the resampled set, we resampled with replacement both the weight-bearing and surrogate centre measurements corresponding to that subject index. We then computed the population standard deviation for the simulated experiment. We repeated this simulation process a large number of times (-3000) until the 95% confidence limits appeared to be stable to the second significant digit. As a check, we also computed analytically the confidence limits accounting only for the intersubject bias variability (by assuming a %2 distribution with Ns -1 degrees of freedom). We expect that our bootstrapped confidence intervals will be slightly wider than the analytical intervals because they account for two additional sources of variability, although bootstrapped intervals computed from small data sets do tend to underestimate the upper limit of the interval because the tail ends of the probability distribution are typically not represented in the original data set. Our results were in accord with this expectation. We also use the bootstrapping approach to test the null hypotheses that the non-weight-bearing centre estimates have lower variability than the digitized estimate. In this case, we simply resample all surrogate measures in the second stage of the simulation process and compute the difference between the NWB and digitized population standard deviations as the bootstrapped variables of interest. We report the one-sided p value and regard a probability of less than 0.05 as significant. Finally, we also assess the repeatability of each method considered independently, without reference to a "gold standard". In these cases, we used F tests (a=0.05) to assess differences in variability between pairs of methods. 3.3 Results 3.3.1 Characterization of the Repeatability of all Methods The SDs averaged over all subjects for each method are illustrated in Figure 3.5. In the ML direction digitization was found to be the most repeatable of all the other methods (pmax - 0.003). The spherical method was significantly more repeatable than the biaxial method in the ML direction in both NWB and WB (p<0.01). Page 48 Grouped Repeatability 2.5 £ 2.0 £ i • ML Direction > • AP Direction Q 1.0 0.0 Digitization Spherical NWB Biaxiall2NWB Spherical WB Biaxiall2WB Method Figure 3.5: Overall Repeatability for all subjects, methods and directions Results were different for the A P direction where digitization was found to be the least repeatable measure when compared to both the N W B and W B results ( p m a x = 0.02). There was no significant difference in the repeatability of either the spherical or biaxial method in the A P direction in N W B , however, in W B the spherical method was significantly more repeatable (pO .Ol ) . 3.3.2 Bias of Surrogate Estimates with Respect to Weight-Bearing Figure 3.6 shows the bias and repeatability associated with each method for all subjects individually. We can see that by only examining repeatability as shown above in Figure 3.5 we lose the context of how the different methods relate to one another in terms of bias which we can clearly see dominates the variability on an intersubject basis. Page 49 AP Direction, (mm) U l cn AP Direction, (mm) cn CM o AP Direction, (mm) 3 3 CM CM o CM CM cn o 3 3 AP Direction, (mm) AP Direction, (mm) CM O CO o AP Direction, (mm) 3 3 o O CM 3 3 11 4 - Figure 3.6: Repeatability and bias, subjects a-f, all methods. Error bars indicate standard deviations. Page 50 A P Direction, (mm) A P Direction, (mm) A P Direction (mm) A P Direction, (mm) A P Direction, (mm) A P Direction, (mm) O 3 cn 3 + 3 3 fi Figure 3.7: Repeatability and bias, subjects g-1, all methods. Error bars indicate standard deviations. 3.3.2.1 Choice of the Reference Weight-Bearing Centre In §3.3.1 we found that in WB the spherical model was significantly more repeatable than the biaxial model in all directions. Furthermore, we found that there was no significant difference in Page 51 the absolute position of the biaxial WB estimate when compared to the spherical WB estimate (mean differences across all subjects: 0.6 mm (0.09°) ML, 0.7 mm (0.11°) AP). In light of this we chose to use the more repeatable spherical WB centre estimate as our WB reference to which we compare the surrogate methods. The relationships of each surrogate method to the WB reference are shown in Figure 3.8. Comparison of Surrogate Methods to the Weight-Bearing Reference 10 E E, c g o Hi b i_ o •c Q) w o CL o 0) -*—1 c < 2.7 mm Medial 6.9 mm Anterior A DIGITIZATION • BM-NWB o SP-NWB 0.8 mm Lateral 5.4 mm Anterior 6.5 mm Lateral 1.1 mm Posterior -5 0 5 Medial Lateral Direction, (mm) 10 Figure 3.8: Bias of the surrogate methods with respect to the weight-bearing reference. Boxes indicate standard deviations, and error bars indicate the 95% confidence intervals predicted with bootstrapping. 3.3.2.2 Population Variability of the Surrogate Methods The mean bias values, or correction values, for all three surrogate methods (digitization, NWB spherical method, and NWB biaxial method) are shown in Figure 3.8. These corrections were then applied for each subject, and the population variability of each method was calculated using the methodology described in §3.2.4. We found that the spherical NWB centre estimate technique was significantly more repeatable than both the biaxial and digitization methods in the ML direction (p<0.05). There was no significant difference in the ML variability between the digitization and NWB biaxial method. Page 52 In the AP direction the spherical method was significantly more repeatable than the digitization method (p>0.05), but not the biaxial method (p>0.05). There was no significant difference in repeatability between the biaxial method and digitization in the AP direction (p>0.05). Table 3.1 contains a summary of the bias for each method as well as the population variability associated with the method relative to the WB centre. Table 3.1: Summary of the bias and population variability of each surrogate method with respect to weight-bearing Surrogate Bias in M L Bias in A P M L Standard Deviation A P Standard Deviation Method Direction Direction (95% confidence limits) (95% confidence limits) (mm) (mm) (mm) (mm) Spherical 2.7 Medial 6.9 Anterior 3.7 (2.5-4.3) 3.5 (2.2-4.3) Biaxial 0.8 Lateral 5.4 Anterior 5.5 (3.3-6.6) 3.9 (2.4 - 4.7) Digitization 6.5 Lateral 1.1 Posterior 5.1 (2.9-6.1) 4.8 (3.0 - 5.7) 3.3.3 Bias of Non-weight-Bearing Estimates with Respect to Digitization Both the NWB centre estimates were medial and anterior to the digitization estimates, as shown in Figure 3.9. This was consistent with what we have found in the past as well with Inkpen's work (Inkpen, 1999b; Shute et al., 2001b). Overall, the centres estimated with the spherical method were 3.4 mm medial (0.52°), and 1.5 mm anterior (0.23°) to the centres estimated using the biaxial method. The spherical model centre estimates were furthest from the digitized centre at 9.2 mm medial and 8.0 mm anterior. 10 I al c o ' b 1 o in o 0. CD -4-« c < Comparison of the NWB Methods to Digitization 9.2 mm Medial 8.0 mm Anterior 0 - • B M - N W B O S P - N W B 5.7 mm Medial 6.4 mm Anterior -12 -10 - 8 . - 6 -4 Medial Lateral Direction, (mm) Figure 3.9: Mean bias of non-weight-bearing methods relative to digitization for all subjects. Error bars Page 53 represent the standard deviations of the bias estimates. 3.4 Discussion 3.4.1 Interpretation of Results We conducted an in vivo experiment to compare anatomic estimates of the ankle joint centre to those estimated using spherical and biaxial models in both the NWB and WB states. In this study we sought first to characterize the repeatability of all five methods, second to examine the bias and associated population variability for the three surrogate methods relative to a WB reference, and lastly to determine the most repeatable surrogate method. In some of our previous studies, the precision (or repeatability) of a method was treated as the primary criterion for assessing the validity of the centre location technique used in computer- assisted TKA (Inkpen et al., 1999a; Inkpen, 1999b; Inkpen et al., 2000; Shute et al., 2001c). However, after examining the bias between techniques shown in Figure 3.6, we can see that the variability of each of the methods is considerably smaller than the population variability of the bias of one surrogate technique relative to another. To further emphasize this point, the surrogate method with the largest variability (biaxial method NWB SD = 0.81 mm (0.12°) ML) represents less than 5% of the population variability associated with the bias of the most precise surrogate method (spherical method NWB with 2.7 mm lateral (0.41°) correction, SD = 3.74 mm (0.57°) ML). We now regard the key criterion for assessing which technique is most suited for use in our surgical system to be the population variability associated of the surrogate measure used for predicting the WB centre of the ankle joint. We found that of all the tested surrogate methods, the spherical NWB method was the most precise for estimating the WB centre, with a reduction in ML variability of nearly 50% over the next most precise technique: digitization. 3.4.2 Comparison of our Findings to Other Studies in Literature Table 3.2 contrasts the results for the biaxial method to the results obtained previously (Inman, 1976; van den Bogert et al., 1994). An illustration of how the joint axes orientations are measured is shown in Figure 3.10. The joint axes directions in NWB that were most consistent between the three studies are the talocrural inclination and deviation as well as the subtalar inclination. The subtalar deviation orientation, which was the most variable orientation in all studies, showed the least agreement. Van den Bogert noted the variability of this, and used an 11 Page 54 parameter model that fixed the ST deviation at 0°, but this only marginally improved the repeatability of the other joint axis orientations. Table 3.2: Comparison of biaxial model with results from literature Shute van den Bogert Inman (2002) (1994) (1976) 12 subjects 14 Subjects 46 Subjects Joint Axis Orientation N W B in vivo WB in vivo in vivo in vitro Subtalar Inclination 44.8 ± 6.0° 52.9 ±9.0° 35.3 ±4.8° 42 ± 9 ° Subtalar Deviation 36.3 ±21.3° 18.7 ±26.0° 18.0 ± 16.2° 23 ±11° Talocrural Inclination 0.4 ±6.4° -2.0 ± 14.7° 4.6 ± 7.4° 8 ± 4 ° Talocrural Deviation -5.3 ± 9.2° -7.1 ± 10.7° l"± 15.1° 6 ± 7 ° Sagittal Plane Transverse Plane Figure 3.10: Definition of joint orientation angle measurement According to our study, the recommended 3 mm medial offset suggested by one surgical instrumentation manufacturer is in the proper direction but of too small a magnitude (DePuy, 2000). The average offset we found for the W B centre relative to the digitized centre was 6.5 mm medial and 1.1 mm anterior. Inkpen reported the medial bias of the N W B spherical method with respect to digitization to be in the range of 2.2 - 3.8 mm, and the anterior bias to be 1.0 - 14.1 mm based on two cadaver specimens (Inkpen, 1999b). Comparing this to our study we again found the bias to be in the same directions, but of different magnitudes (medial bias 9.2 mm, anterior bias 8.0 mm). Since Inkpen's results were obtained from cadaver specimens with limited R O M it is not surprising that we obtained different results when testing live subjects. Page 55 The homogeneous transform (HT) algorithm we developed for spherefitting was more repeatable than the single marker algorithm used previously. We used the single marker algorithm for spherefitting in a pilot project, and obtained a SD of 1.84 mm (0.28°) in the ML direction (Shute et al., 2001b). We performed the analysis on the same data using the HT algorithm instead and obtained a SD of 0.98 mm (0.15°) in the ML direction. The variance has effectively been reduced 3.5X by ensuring that the optimal algorithm was used for spherefitting. 3.4.3 Strengths and Weaknesses of the Study Skin movement artefact, the movement of skin-mounted markers relative to the underlying bone, is considered by some to be the most important error to account for in human movement analysis (Cappozzo et al., 1996). Attempts at quantifying these types of errors for the AJC have compared bone pin movement (Reinschmidt et al., 1997a; Reinschmidt et al., 1997b) or tantalum beads implanted in bone (Tranberg et al., 1998) to externally mounted markers. These studies report that measurements from externally mounted markers typically exceed the true bone motion. We conducted a small study on a cadaveric specimen comparing our bone tracking devices to markers on bone pins, and found that they represented bone motion well; the maximum errors over 15 motion trials are summarized in Table 3.3 (Shute et al., 2001a), and all were smaller than the upper limit of 4.3 mm reported by Tranberg (Tranberg et al., 1998). Although the maximum errors were in the millimeter range, it should be noted that we obtained sub-millimeter repeatabilities for both the spherical and biaxial methods when using these devices. In contrast to typical skin-mounted markers which move with the skin as it slides over the underlying bone, our external tracking device was specifically designed to maintain apposition to the bone through a set of normal constraints. As Figure 3.11 illustrates, all contact points for the tibial tracker made good contact with the desired areas on the subjects tested. Table 3.3: Maximum error of tibial and calcaneal trackers with respect to bone pin mounted markers. Non-Invasive Tracker Medial Lateral Anterior Posterior Proximal Distal Tibial Tracker 2.15 mm 1.30 mm 0.63 mm Calcaneal Tracker 0.89 mm 1.96 mm 2.65 mm Page 56 AH 5 contacts points made good contact with the tibia Figure 3.11: Photo of post experiment dents left in leg. We tested 12 healthy young normal subjects which clearly do not represent the population of typical TKA candidates. Since this is a preclinical study comparing NWB and WB ankle centres, we first wanted to determine if we could distinguish any differences between models on healthy normal subjects. Neither the validity nor the repeatability of these results has been verified on obese subjects where the thickness of subcutaneous tissue could be much greater, and could therefore cause more relative motion between the tracker and the underlying bone. However, it should be noted that in our clinical TKA system we will only require a calcaneal tracker because the tibia will be exposed during the surgery and we will be free to mount a marker directly to the bone. Another limitation of the study was the number of subjects tested. To establish a population correction between NWB and WB for either motion-based method with tighter confidence bounds, more subjects must be tested. The WB centre estimates for both motion-based methods were the most variable of the study. This is not surprising due to the limited range of motion of the subtalar (ST) joint when in WB, as can be seen in Figure 3.12 which illustrates typical NWB and WB datasets. Sarrafian found that under vertical loading the ST joint is in a close-packed position with maximum talar head surface in contact with the calcaneus and navicular bones (Sarrafian, 1993). The guiding ligaments are under maximum tension and the posterior talocalcaneal surfaces are interlocked laterally. This restricts the motion about the ST joint, which explains the small range of inversion-eversion motion. Under loaded conditions, therefore, the dominant motion is about the talocrural joint. Page 57 Figure 3.12: Typical datasets in non-weight-bearing (left), and weight-bearing (right). The WB datasets were generated by loading the joint via the ball of the foot instead of via the heel. This is a limitation due to the non-invasive tracker we chose to use in our study. Even so, the loads were still transferred through the AJC, and would be similar to the loads present near the toe off portion of the gait cycle. There are several limitations associated with both the biaxial and spherical models implemented in this study. The most common criticism of the biaxial method is that there is evidence that the joint axes migrate throughout the ROM of the ankle (Barnett et al., 1952; Hicks, 1953; van Langelaan, 1983; Lundberg et al., 1989; Lundberg et al., 1993); however, in both van den Bogert's and our study, small fitting errors were obtained (<2mm RMS) with good repeatability (0.81 mm SD ML), so the model captures much of the kinematic behaviour of the joint. Although the centre estimates obtained with the spherical model of the ankle were more repeatable than those found with the biaxial model, the spherical model is anatomically Page 58 unrealistic. For researchers interested primarily in using joint models for realistically modeling force and torque transmission through the joint complex, the spherical model is a poor choice since the true AJC can transmit external/internal rotation torques whereas the spherical model offers no constraint to such torques. This may lead to overestimation of muscular forces required to keep the joint in equilibrium (van den Bogert et al., 1994). An additional source of variability associated with the digitization method that was not addressed by this study is the variability across different operators. We conducted a small experiment with four operators performing 30 digitization trials each on a single cadaver, and we obtained interoperator range of mean digitization values of 1.3 mm (0.20°) in the ML direction and 3.0 mm (0.46°) in the AP direction for the point probe used in this study. This interoperator variability would make the raw variability of digitization roughly comparable to, or perhaps slightly larger than, the variability associated with the NWB centre methods. Because the optimization technique uses data obtained throughout the whole range of motion of the joint, we expect the motion-based centre methods to be considerably more robust to interoperator effects than digitization, so the relative advantage of the spherical NWB model would likely stand. Finally, we also introduce intersubject variability by the process of defining anatomical reference frames. The distance between the centres found by the different approaches is not affected by this choice, but it does alter the relative angular orientation of the AP and ML axes from subject to subject. The SD associated with repeatedly locating the centre of the heel has been reported to be 2.5 mm (Robinson et al., 2001) which would correspond to a rotational variability of 0.6° SD based on a mean foot length for the tested subjects of 248 mm. This would contribute an additional variability of <0.1 mm SD in the relative positions of the surrogate centres across subjects; this amount is entirely negligible compared with standard deviations in the bias across subjects of 3-5 mm. 3.4.4 Conclusions We recommend that instead of evaluating each method used for estimating the ankle centre solely on the precision of that method, the population variability associated with the bias relative to the WB centre should be the primary consideration. Based on this criterion, the recommended surrogate method for estimating the WB ankle joint centre is the NWB spherical method (correction: 2.7 mm lateral (0.41°), 6.9 mm posterior (1.05°)), which will introduce a variability of ~4 mm SD (0.6°) in both the AP and ML directions. Page 59 3.5 Acknowledgements We thank Rachel J. MacKay (UBC Department of Statistics) for her assistance with the statistical theory used to evaluate the surrogate measures. We also thank Dr. Alastair Younger (UBC Department of Orthopaedics) for valuable discussions regarding the ankle joint complex and Drs. Robert McGraw and Bassam Masri (UBC Department of Orthopaedics) for opportunities to observe numerous total knee replacement procedures and discussions on surgical technique. 3.6 References 1. ) Jeffery, R. S., R. W. Morris, et al. (1991). "Coronal alignment after total knee replacement." J Bone Joint Surg Br 73(5): 709-14. 2. ) DePuy Orthopaedics Inc (2000). P.F.C. Sigma Knee System: Revision Surgical Technique. 52. 3. ) Krackow, K. A., M. Bayers-Thering, et al. (1999). "A new technique for determining proper mechanical axis alignment during total knee arthroplasty: progress toward computer-assisted TKA." Orthopedics 22(7): 698-702. 4. ) Kunz, M., M. Strauss, et al. (2001). A Non-CT Based Total Knee Arthroplasty System Featuring Complete Soft-Tissue Balancing. MICCAI 2001, LNCS Vol. 2208, Utrecht, The Netherlands, Springer-Verlag, 409-415. 5. ) Inkpen, K. and A. Hodgson (1999a). Accuracy and repeatability of joint centre location in computer-assisted knee surgery. MICCAI'99, LNCS Vol. 1679, Cambridge, UK, Springer-Verlag, 1072 - 1079. 6. ) Inkpen, K. (1999b). Precision and Accuracy in Computer-Assisted Total Knee Replacement. M.A.Sc. Thesis. Department of Mechanical Engineering, University of British Columbia: 153. 7. ) Inkpen, K., A. Hodgson, et al. (2000). Repeatability and Accuracy of Bone Cutting and Ankle Digitization in Computer-Assisted Total Knee Replacement. MICCAI 2000, LNCS Vol. 1935, Pittsburgh, USA, Springer-Verlag, 1163-1172. 8. ) Leitner, F., F. Picard, et al. (1997). Computer-Assisted Knee Surgical Total Replacement. CVRMed-MRCAS'97, LNCS Vol. 1205, Grenoble, France, Springer - Verlag, 629-638. 9. ) Areblad, M. (1990). On Modelling of the Human Rearfoot. Thesis No. 238. Linkdping Institute of Technology, Sweden. 10. ) van den Bogert, A. J., G. D. Smith, et al. (1994). "In vivo determination of the anatomical axes of the ankle joint complex: an optimization approach." J Biomech 27(12): 1477-88. 11. ) Shute, C. and A. Hodgson (2001c). Repeatability and Accuracy of Ankle Centre Location Estimates using a Biaxial Joint Model. MICCAI 2001, LNCS Vol. 2208, Utrecht, The Netherlands, Springer-Verlag, 1166-7. 12. ) Lundberg, A., O. K. Svensson, et al. (1989). "The axis of rotation of the ankle joint." J Bone Joint Surg Br 71(1): 94-9. 13. ) Tinley, P., T. Barker, et al. (1996). Assessment of the position and motion of the axis of the subtalar and ankle joints of the left feet of a male and female sample. Australasian Biomechanics Conference, 1, Sydney, Australia, 36-37. Page 60 14. ) Shute, C. and A. Hodgson (2001b). Application of a Biaxial Kinematic Ankle Model to Computer-Assisted Total Knee Arthroplasty. Fifth International Symposium on Computer Methods in Biomechanics and Biomedical Engineering, Rome, Italy, 15. ) Shute, C. and A. Hodgson (2001d). "Optimal Formulations Of Ankle Joint Models For Use In Computer-Assisted Total Knee Arthroplasty." Computer Methods in Biomechanics and Biomedical Engineering (Submitted). 16. ) Cappozzo, A., A. Cappello, et al. (1997). "Surface-marker cluster design criteria for 3-D bone movement reconstruction." IEEE Trans Biomed Eng 44(12): 1165-74. 17. ) Chassat, F. and S. Lavallee (1998). Experimental Protocol of Accuracy Evaluation of 6-D Localizers for Computer-Integrated Surgery: Application to Four Optical Localizers. MICCAi'98, LNCS Vol. 1496, Cambridge, USA, Springer-Verlag, 421-430. 18. ) Li, Q., L. Zamorano, et al. (1999). "Effect of optical digitizer selection on the application accuracy of a surgical localization system - a quantitative comparison between the OPTOTRAK and flashpoint tracking systems." Computer Aided Surgery 4(6): 314-321. 19. ) Stefanyshyn, D. J. and J. R. Engsberg (1994). "Right to left differences in the ankle joint complex range of motion." Med Sci Sports Exerc 26(5): 551-5. 20. ) Moseley, A. M., J. Crosbie, et al. (2001). "Normative data for passive ankle plantarflexion—dorsiflexion flexibility." Clin Biomech (Bristol, Avon) 16(6): 514-21. 21. ) DiCiccio, T. and B. Efron (1996). "Bootstrap confidence intervals." Statistical Science 11(3): 189-212. 22. ) Inman, V. T. (1976). Inman's joints of the ankle. Baltimore, William & Wilkins. 23. ) Cappozzo, A., F. Catani, et al. (1996). "Position and orientation in space of bones during movement: experimental artefacts." Clin Biomech (Bristol, Avon) 11(2): 90-100. 24. ) Reinschmidt, C, A. J. van Den Bogert, et al. (1997a). "Tibiocalcaneal motion during walking: external vs. skeletal markers." Gait and Posture 6: 98-109. 25. ) Reinschmidt, C , A. J. van Den Bogert, et al. (1997b). "Tibiocalcaneal motion during running, measured with external and bone markers." Clin Biomech (Bristol, Avon) 12(1): 8-16. 26. ) Tranberg, R. and D. Karlsson (1998). "The relative skin movement of the foot: a 2-D roentgen photogrammetry study." Clin Biomech (Bristol, Avon) 13(1): 71-76. 27. ) Shute, C. and A. Hodgson (2001a). "Investigation of Ankle Centre Location Techniques Applied to Computer-Assisted Total Knee Arthroplasty." J Biomech 34((S1)): 25. 28. ) Sarrafian, S. K. (1993). "Biomechanics of the subtalar joint complex." Clin Orthop(290): 17-26. 29. ) Barnett, C. H. and J. R. Napier (1952). "Axis of rotation at ankle joint in man: its influence upon the for of talus and mobility of fibula." J Anat 86: 1-9. 30. ) Hicks, J. H. (1953). "The Mechanics of the foot I. The Joints." JAnat 87: 345-357. 31. ) van Langelaan, E. J. (1983). "A kinematical analysis of the tarsal joints. An X-ray photogrammetric study." Acta Orthop Scand Suppl 204: 1-269. 32. ) Lundberg, A. and O. K. Svensson (1993). "The axis of rotation of the talocalcaneal and talonavicular joints." The Foot 3: 65 - 70. 33. ) Robinson, I., R. Dyson, et al. (2001). "Reliability of clinical and radiographic measurement of rearfoot alignment in a patient population." The Foot 11(1): 2-9. Page 61 Chapter 4 : Thesis Summary and Direction of Future Work 4.0 Discussion of Thesis Objectives I stated in my introduction that I had hoped to make contributions to the problem of locating a meaningful ankle centre for use in computer-assisted surgery in two ways. First by deriving the optimal algorithms for use in fitting different ankle models to subject data, and second by performing an in vivo experiment to validate these methods. I identified the optimal formulation for each of the spherical and biaxial joint models treated separately in chapter 2. I found that the homogeneous transform method was the most reliable method of the tested spherefitting methods. Although I found that the 8 parameter formulation of the biaxial model worked well in simulation and in experiment on an ideal biaxial joint, I did not find the same performance when I applied the method to the data obtained for chapter 3. I found that the 8 parameter formulation had much larger variability than the 12 parameter formulation, and for that reason I used the 12 parameter formulation of the biaxial model for the study in chapter 3. It is possible that the reason that I obtained such poor performance of the 8 parameter formulation could be due to any rolling or sliding occurring with the ankle joint that the model can not account for. I compared the digitization method to the spherical and biaxial methods of locating the ankle centre in both the weight-bearing and non-weight-bearing states on 12 live subjects. I characterized how reliably surrogate measurement techniques could predict the location of the ankle centre in a weight-bearing state, and found that the spherical method in non-weight- bearing was most reliable. A novel evaluation criterion for choosing the most reliable ankle centre location technique was developed, and presented for the first time by looking at the variability associated with the bias relative to the ankle centre in weight-bearing. 4.1 Conclusions I have identified the optimal formulations of two models of the ankle joint complex being considered for applications in computer-assisted total knee arthroplasty. The homogeneous transform method was found to be the most repeatable algorithm for use with the spherical Page 62 model. Although in simulation and under ideal physical circumstances the 8 parameter formulation of the biaxial model produced moderately improved repeatability over the 12 parameter formulation, this did not hold true when applied to the measurements of real ankle motion. I conclude that the 12 parameter formulation of the biaxial model is preferred over the 8 parameter formulation since it is both more repeatable and more robust. I recommend that instead of evaluating each method used for estimating the ankle centre solely on the precision of the individual methods, that the variability associated with the bias relative to the weight-bearing centre also be considered. Applying this, I found that the most reliable surrogate method for estimating the ankle joint centre in weight-bearing is the non- weight-bearing spherical method with a lateral and posterior correction of 2.7 mm and 6.9 mm respectively. 4.2 Directions for Future Work 4.2.1 Issues to be addressed before going to clinical trials • In order to optimize the design of the calcaneal tracking device and validate it on live subjects, roentgen stereophotogrammetry (Tranberg et al., 1998) or bone pins (Reinschmidt et al., 1997a) could be used to quantify the amount of skin movement artefact. Test subjects could be chosen to represent a typical population of total knee arthroplasty candidates (elderly, and perhaps overweight). The tracker should also be designed so that it is relatively comfortable to wear and easy to adjust between subjects before it is used clinically. • The optimal location for the marker triad relative to the calcaneus could be determined prior to clinical trials to ensure the entire setup of the tracker is ideal. • Recoding of the fitting algorithms in a language with a highly optimized compiler would be recommended (an increase in computational speed of perhaps 100 times would be found if the computations were performed using an optimized coding language such as C or Fortran instead of using Matlab). The integration of all the other system software components (i.e. hip location, knee location, soft tissue balancing) into a single easy-to-use software application is required. Consultation with the surgeons is highly recommended once the graphical user interface Page 63 development stage begins in order to provide them with the information and tools they require. 4.2.2 Ideas for future research on ankle centre localization methods • It may be possible to combine information from several surrogate measurement techniques in order to predict the weight-bearing ankle centre more reliably. For example the digitization method has good mediolateral repeatability, but very poor anteroposterior repeatability compared to any of the motion-based methods. A neural network approach may be one way of identifying the optimal way of using several methods together. • A larger group of subjects could be tested in order to determine bias values for surrogate measurement techniques with tighter confidence bounds. Perhaps it would be possible to obtain the data from Tinley's study which involved 120 subjects (Tinley et al., 1996) to perform the analyses discussed in this thesis to achieve this. • The use of force plates could be used to determine where the force vector is located and how it is oriented in weight-bearing. This would further our understanding of force transfer through the joints, and provide more insight on where the line of force through the ankle is positioned compared to the ankle centre in weight-bearing which I investigated. • To establish interoperator variability bounds, digitization with the point probe could be performed on the same subject across different operators. One could also test the difference between untrained operators and expert surgeons as Inkpen did for his robust digitizing probe (Inkpen et al., 2000). • The variability associated with the methods discussed in this thesis for ankle centre localization could be compared to a study aimed at quantifying the repeatability of using an extramedullary rod for alignment in total knee arthroplasty. I used digitization as a surrogate method of anatomic ankle centre localization rather than the extramedullary alignment tool which might be less repeatable. Page 64 4.3 References 1. ) Tranberg, R. and D. Karlsson (1998). "The relative skin movement of the foot: a 2-D roentgen photogrammetry study." Clin Biomech (Bristol, Avon) 13(1): 71-76. 2. ) Reinschmidt, C , A. J. van Den Bogert, et al. (1997a). "Tibiocalcaneal motion during walking: external vs. skeletal markers." Gait and Posture 6: 98-109. 3. ) Tinley, P., T. Barker, et al. (1996). Assessment of the position and motion of the axis of the subtalar and ankle joints of the left feet of a male and female sample. Australasian Biomechanics Conference, 1, Sydney, Australia, 36-37. 4. ) Inkpen, K., A. Hodgson, et al. (2000). Repeatability and Accuracy of Bone Cutting and Ankle Digitization in Computer-Assisted Total Knee Replacement. MICCAI 2000, LNCS Vol. 1935, Pittsburgh, USA, Springer-Verlag, 1163-1172. Page 65 Appendix: Collection of Conference Publications The papers contained within this appendix are in their conference publication format and in the following order. I attended all conferences and presented either posters or lectures. Paper 1: Application of a Biaxial Kinematic Ankle Model to Computer-Assisted Total Knee Arthroplasty. Fifth International Symposium on Computer Methods in Biomechanics and Biomedical Engineering - November 2001 - Rome, Italy. Paper 2: Repeatability and Accuracy of Ankle Centre Location Estimates using a Biaxial Joint Model. MICCAI - October 2001 - Utrecht, The Netherlands. Poster Included. Paper 3: Investigation of Ankle Centre Location Techniques Applied to Computer-Assisted Total Knee Arthroplasty. Biomechanica IV - September 2001 - Davos, Switzerland. Paper 4: Repeatability and Accuracy of Bone Cutting and Ankle Digitization in Computer- Assisted Total Knee Replacement. MICCAI - October 2000 - Pittsburgh, USA. Poster Included. Page 66 APPLICATION OF A BIAXIAL KINEMATIC ANKLE MODEL TO COMPUTER-ASSISTED TOTAL KNEE ARTHROPLASTY CA. Shute1, A.J. Hodgson2 1. ABSTRACT One of the major causes of early failure in total knee replacement (TKR) surgery is improper implant alignment in the frontal plane. Varus/valgus alignment errors of as little as 3° have been implicated in poor outcomes, so to reduce the variability of implant placement we are developing a computer-assisted TKR procedure to assist surgeons in defining the mechanical axis to which the implants are aligned. To define the mechanical axis, which passes through the hip, knee and ankle centres, we use an optoelectronic motion capture system in conjunction with several optimization methods to fit biomechanical joint models which appropriately represent patient kinematics. The focus of this study was to evaluate the precision and bias of using a biaxial model to find the centre of the ankle joint complex (AJC) compared to a spherical ankle joint model and direct anatomic digitization. Five healthy male subjects performed 30 repeated trials of each centre identification technique in a simulated TKR setting. To improve computational efficiency we derived an alternative 8 parameter formulation (8P) of the biaxial model, and evaluated it on a physical model against a previously reported 12 parameter formulation (12P). The biaxial model was significantly more repeatable than the spherical model, but was slightly less repeatable than digitization (1.14 mm SD biaxial, 1.84 mm SD spherical, 0.70 mm SD digitization), although the digitization result should also have added it to the variability due to differences between the kinematic and anatomic joint centres. Both motion-based centre location techniques were located medially (7-11 mm) and anteriorly (1-3 mm) relative to the digitized centre. The mechanical model experiment showed that 8P was 15-fold more computationally efficient (7.1±1.1 megaflops) than 12P (104±25 megaflops). Keywords: Ankle Joint Model, Biomechanics, Optimization 2. INTRODUCTION M.A.Sc Candidate, Mechanical Engineering, University of British Columbia, Vancouver, BC, Canada Assistant Professor, Mechanical Engineering, University of British Columbia, Vancouver, BC, Canada Page 67 We are developing a low cost computer-assisted total knee replacement system to improve alignment accuracy and soft tissue balancing. The system does not require computed tomography scans or extra incisions that are remote to the operating site. Varus/valgus alignment error of 3° has been strongly correlated with aseptic loosening [Jeffery 1991]. In a meta-analysis of 10 published studies involving 1373 knees, Inkpen estimated the SD of varus/valgus alignment to be 2.6° on the overall limb and 1.9° each for the distal femoral and tibial plateau cuts [Inkpen 1999a]. This clearly indicates that it is necessary to reduce the SD of the overall procedure to ensure that implant alignment errors remain within this critical 3° window. Currently there is no universally accepted method of defining the centre of the AJC, but there are two main approaches: anatomical (digitization) and kinematic (motion-based) determination. Published results indicate that digitization methods are generally more repeatable than motion- based studies [Inkpen 1999a, 1999b, Leitner 1997], although these studies only evaluate an anatomically implausible ball-in-socket model of the AJC and the digitized centre has no direct effect on the loads experienced by the joint. The anatomical ankle centre has been loosely defined by locating points on the extremes of the medial and lateral malleoli, and then choosing some point along the line connecting these points (generally somewhat medial to the midpoint). Inkpen chose to examine the repeatabitlity of locating the midpoint of the malleolar axis [Inkpen 1999a], Krackow et al. use a digitized unit vector that represents the surgeons' best estimate of the centre along the anterior portion of the ankle [Krackow 1999], and Kunz et al. digitize the ligament of tibialis anterior [Kunz 2001]. Inkpen most recently reported the SD associated with a digitizing probe to be 0.75 mm in the ML direction and 1.12 mm in the AP direction when used on a cadaver [Inkpen 2000]. Although important, repeatability is not the only factor affecting alignment; the intersubject variability associated with the bias of the biomechanical centre relative to the digitized centre should be included when calculating the overall precision of the digitization method. Inkpen found the SD associated with the spherical model to be 3.99 mm in the ML direction, and 2.29 mm in the AP direction for a single cadaver specimen using bone pin mounted markers [Inkpen 1999b]. A two degree of freedom (DOF) kinematic model of the ankle which is more realistic than the spherical model has been developed recently [van den Bogert 1994]. The biaxial model (BM) represents the talocrural (TC) and subtalar (ST) joints as fixed hinge joints. The model relates the motion of three rigid bodies: the shank (tibia and fibula), the talus, and the calcaneus. The TC joint is assumed to be fixed in the shank and the talus, and the ST joint is fixed in the talus and the calcaneus. In this way the talus acts as an intermediate body, and the optimal locations of the joint axes are determined based on measurements of the location of the calcaneus relative to the shank. In this paper, we compare the three methods of AJC location (anatomic digitization, spherical method, biaxial method) in terms of the repeatability and relative position of the estimated joint centres in an in vivo setting. 3. METHODS Page 68 3.1 Measurement techniques and subjects Five healthy male subjects were tested (mean age: 24; mean weight: 80 kg). Two external reference frames were clamped around the midshank and the heel, and triads of infrared emitting diodes (equilateral triangles 120 mm on a side) were mounted to each reference frame. The spatial locations of the triads were measured using a Flashpoint 5000 localizer (Image Guided Technologies, Boulder, CO, USA). The ankle centre was digitized with a standard 135 mm point probe supplied by IGT. 3.2 Non-invasive bone trackers In keeping with our desire to avoid incisions remote from the operative site in the TKR system we are developing, we designed non-invasive bone tracking devices for both the tibia and calcaneus. These devices were designed using a minimum constraint design philosophy to minimize relative motion between the tracker and the underlying bone and the contact patches were generally placed on areas of the tibia and calcaneus where the overlying tissues are comparatively thin. 3.3 Ankle digitization method Subjects were positioned as they would be if they were undergoing a TKR procedure, and a single operator performed 30 repeated trials on the five live subjects using a conventional 135 mm point probe. The operator then repeatedly and alternately digitized what they considered to be the extremes of both medial and lateral malleoli. 3.4 Motion-based methods The centre of a ball-in-socket joint is well-defined, but is perhaps not so obvious for the biaxial model. We have chosen to define the joint centre location for the BM as the point at which the TC and ST axes intersect when projected onto a transverse plane at the level of the malleoli (shown pictorially in Figure 1). The rational for this choice is that this is the only point through which a vertical force could pass and generate no net moment about either joint. Page 69 •4 Medial Biaxial Model A J C Definition T C Axis ST Axis Lateral Figure 1. Definition of an AJC centre for the biaxial model We follow the method outlined by van den Bogert for determining the optimal TC and ST joint axes, which consists of tracking the tibia and the calcaneus throughout a representative range of motion. Subjects performed 30 trials of ankle movement throughout the entire range of motion. We then used a non-linear least squares optimization algorithm to find the optimal parameters of the 12 parameter model (12P). The same motion data was also used to determine the joint centre using the sphere-fitting method. A simple validation experiment of the BM optimization method was performed using the Flashpoint system and a wooden model with two hinges to quantify the repeatability we could expect for a true biaxial system with markers rigidly attached. A set of 30 trials was performed, and the repeatability of locating the BM centre was 0.41 mm SD in the ML direction and 0.32 mm SD in the AP direction. 3.5 Improved Biaxial Model Cost Function Formulation The 12P formulation uses more parameters than are strictly required to represent the locations of the two axes in space. The redundancy in this representation is likely to lead to computational inefficiencies when solving for the model parameters. We developed a minimally represented formulation of the biaxial joint model that requires only 8 parameters to fully define the model. For each joint axis, two parameters define a position of the axis in an arbitrary plane, and another two parameters orient the axis in space. Since the TC axis and the ST axis are assumed to be fixed in the talus, they should remain fixed relative to one another for all frames of data. The cost function we define for 8P is a weighted sum of the variances of the four theoretically invariant descriptors used to define the TC axis and ST axis relationship. 3.6 Data Analysis We report the average SD of each method, with 95% confidence limits based on the x2 distribution to give an overall indication of the repeatability of each method. The bias of the Page 70 motion-based methods are computed relative to the point probe centre with the 95% confidence limits. 4 R E S U L T S The digitization method had significantly higher precision than both motion-based method in the frontal plane (P<0.001 for both; see Figure 2). This result makes intuitive sense since most of the variability associated with digitizing the malleoli is in the sagittal plane. The biaxial method was significantly less variable than the spherical method (P<0.001). 2.5 c •2 1.5 2 '> u Q 1 l— (3 T3 I 0.5 Digitization Biaxial Method Spherical Method Figure 2. Overall Repeatability of each A J C centre location method Since only five subjects were tested, we do not have enough data yet to say that the centre estimates from the motion-based method are significantly different from those estimates determined using direct digitization. However, all A J C centres found using the motion-based methods were biased medially (7-10 mm) and slightly anteriorly (1-3 mm) (see Figure 3). These results agree well with Inkpen's studies that report on A J C centre bias [Inkpen 1999a]. Page 71 c o o Ul c •— o u c «< • Biaxial • Spherical A Point Probe | -50 -30 -10 10 30 Medial Lateral Direction Figure 3. A J C b i a s o f m o t i o n - b a s e d m e t h o d s r e l a t i v e t o p o i n t p r o b e c e n t r e 8 P s h o w e d s i g n i f i c a n t l y i m p r o v e d e f f i c i e n c y o v e r t h e 1 2 P f o r t h e w o o d e n m o d e l e x p e r i m e n t . F o r t h e 3 0 t r i a l s t e s t e d , 8 P r e q u i r e d o n a v e r a g e 7 . 1 ± 1 . 1 m e g a f l o p s t o c o n v e r g e c o m p a r e d t o 1 0 4 ± 2 5 m e g a f l o p s f o r 1 2 P , a 1 5 - f o l d r e d u c t i o n i n c o m p u t a t i o n e f f o r t . 5 D I S C U S S I O N A s w a s s t r e s s e d i n t h e i n t r o d u c t i o n , w e a r e m o s t i n t e r e s t e d i n r e d u c i n g v a r u s / v a l g u s e r r o r s i n o r d e r p o t e n t i a l l y t o i m p r o v e o u t c o m e s o f c o m p u t e r - a s s i s t e d T K R s . T h e t r e n d s t h a t w e h a v e f o u n d h e r e a r e c o n s i s t e n t w i t h p r e v i o u s s t u d i e s , b u t t h i s i s t h e first t i m e t h a t t h e p r e c i s i o n o f l o c a t i n g t h e A J C c e n t r e w i t h t h e B M h a s b e e n r e p o r t e d . A l t h o u g h t h e r e p e a t a b i l i t y o f the d i g i t i z e d j o i n t c e n t r e i s s i g n i f i c a n t l y b e t t e r t h a n t h e m o t i o n - b a s e d m e t h o d s , t h e c e n t r e s o d e f i n e d b e a r s n o d i r e c t r e l a t i o n s h i p t o the l o a d v e c t o r t h r o u g h t h e a n k l e j o i n t ; i n c o n t r a s t , t h e t h e m o t i o n - b a s e d e s t i m a t e s a r e b o t h d i r e c t m e a s u r e s o f t h e s u b j e c t - s p e c i f i c k i n e m a t i c b e h a v i o u r o f t h e A J C . T h e d i g i t i z e d c e n t r e i s e s s e n t i a l l y j u s t a c h a r a c t e r i s t i c p o i n t w h i c h w e a r e a b l e to l o c a t e p r e c i s e l y . A l t h o u g h w e h a v e a n i n s u f f i c i e n t n u m b e r o f s u b j e c t s t o d r a w a firm c o n c l u s i o n , i f w e a d d e d the i n t e r s u b j e c t v a r i a t i o n s b e t w e e n t h e a n a t o m i c a n d m o t i o n - b a s e d j o i n t c e n t r e e s t i m a t e s w h i c h w e f o u n d a m o n g s t o u r five s u b j e c t s i n t o t h e c a l c u l a t i o n o f o v e r a l l r e p e a t a b i l i t y , w e e s t i m a t e t h a t t h e a n a t o m i c a l d i g i t i z a t i o n m e t h o d w o u l d e x h i b i t 8 - 9 X t h e v a r i a b i l i t y o f t h e m o t i o n - b a s e d m e t h o d s . F i n a l l y , i t i s i n t e r e s t i n g t o n o t e t h a t t h e d i g i t i z e d c e n t r e i s l a t e r a l t o t h e k i n e m a t i c A J C b y a n a m o u n t w h i c h i s m o r e t h a n the 3-5 m m t y p i c a l l y r e c o m m e n d e d as a c o r r e c t i o n i n t h e m a n u a l s f o r m o s t T K R i n s t r u m e n t a t i o n sets. T h i s w o u l d r e s u l t i n a v a r u s e r r o r w h i c h , a c c o r d i n g t o c l i n i c a l e x p e r i e n c e , i s a l w a y s t o b e a v o i d e d . M o r e s u b j e c t s n e e d t o b e t e s t e d i n o r d e r t o p l a c e m o r e P a g e 7 2 reliable bounds on the relationship between the digitized AJC centre and a biomechanically relevant centre. 6 CONCLUSIONS The motion-based estimates were all medial to anatomically determined centres, and using anatomical centres without recognition of this fact could lead to varus implant alignment errors. The ML variability of the biaxial method is approximately half of that of the spherical method. The variability in relative position between anatomical and kinematic AJC centres dominates the anatomical technique, and we suggest using the motion-based methods in order to achieve the most repeatable measure of a subject-specific kinematically meaningful AJC centre. The results of the study suggest that the biaxial method is best for in terms of frontal plane repeatability, which is the most crucial direction for minimizing implant errors. 7 REFERENCES 1. Jeffery RS, Morris RW, Denham RA: Coronal Alignment After Total Knee Replacement. Journal of Bone and Joint Surgery (Br. Ed.) 73-B, 1991, pp. 709-714. 2. Inkpen KB: Precision and Accuracy in Computer-Assisted Total Knee Replacement. Master's thesis, Dept. of Mechanical Engineering, University of British Columbia, Vancouver BC Canada. 1999a. 3. Inkpen KB, Hodgson AJ: Accuracy and Repeatability of Joint Centre Location in Computer- Assisted Knee Surgery. In: Taylor C, Colchester A (Eds): MICCAI 1999. LNCS Vol. 1679, Springer-Verlag, 1999b, pp. 1072-1079. 4. Leitner F, Picard F, Minfelde R, Schulz H-J, Cinquin P, Saragaglia D: Computer Assisted Knee Surgical Total Replacement. In: Troccaz J, Grimson E, Mosges R (Eds): CVRMed- MRCAS '97. LNCS Vol. 1205, Springer-Verlag, 1997, pp. 627-638. 5. Krackow KA, Bayers-Thering M, Phillips MJ: A New Technique for Determining Proper Mechanical Axis Alignment During Total Knee Arthroplasty: Progress Toward Computer- Assisted TKA. Orthopedics 22(7), 1999, pp.698-702. 6. Inkpen KB, Hodgson AJ, Plaskos C, Shute CA, McGraw RW: Repeatability and Accuracy of Bone Cutting and Ankle Digitization in Computer-Assisted Total Knee Replacement. In: Delp S, DiGioia A, Jaramaz B (Eds): MICCAI 2000. LNCS Vol. 1935, Springer-Verlag, 2000, pp.1163-1172. 7. Kunz M, Strauss M, Langlotz F, Deuretzbacher G, Ruther W, Nolte LP: A Non-CT Based Total Knee Arthroplasty System Featuring Complete Soft-Tissue Balancing. In: Niessen W, Viergever M (Eds): MICCAI 2001. LNCS Vol. 2208, Springer-Verlag, 2001, pp.409-415. 8. van den Bogert AJ, Smith GD, Nigg BM: In Vivo Determination of the Anatomical Axes of the Ankle Joint Complex: an Optimization Approach. Journal of Biomechanics 27(12), 1994. pp 1477-1488. Page 73 Repeatability and Accuracy of Ankle Centre Location Estimates using a Biaxial Joint Model Cameron A. Shute & Antony J. Hodgson Department of Mechanical Engineering University of British Columbia, Vancouver, BC, Canada a h o d g s o n @ m e c h . u b c . c a Abstract. In conventional total knee replacement (TKR) surgery, a significant fraction of implants have varus/valgus alignment errors large enough to reduce the lifespan of the implant, so we are developing a more accurate computer-assisted procedure aimed at reducing the standard deviation (SD) of the implant procedure. In this study we introduce a new method of locating the ankle joint centre (AJC) using a biaxial model (BM), and determine the accuracy and repeatability of this protocol compared to a digitization method and a sphere-fitting method used in a current computer-assisted procedure. Repeated in vivo measurements performed by a single operator were obtained from five normal subjects (450 measurements) using the three methods of A J C location. Based on these experiments we estimate the varus/valgus SD of defining the tibial mechanical axis in the frontal plane for the tested population to be 0.28° for the spherical model, 0 .17° for the biaxial model, and 0.11° for the conventional digitizing point probe. The mean joint centre locations found by the motion-based models are significantly medial and anterior to the point probe centre. 1 Introduction We are developing computer-assisted total knee replacement tooling that eliminates intramedullary rods and improves alignment accuracy without introducing additional imaging requirements (such as preoperative computed tomography scans) or invasive procedures (such as bone pins remote to the operating site). Currently there is no universally accepted method of defining the AJC, but there are two main approaches: anatomical determination (digitization) and biomechanical determination (motion based). This work investigates the feasibility of using a biaxial model to define a biomechanically meaningful AJC by comparing the centre to those centres obtaine with a spherical model and by direct digitization. 2 Methods Digitization measurements were obtained using a 135 mm point probe, and calcaneal motion measurements were recorded with a optoelectronic three-emitter reference frame mounted to a calcaneal tracker (Flashpoint 5000 localizer). A l l measurements were relative to a three-emitter triangular local reference frame (120 mm on a side) mounted rigidly to a tibial tracker. Study design: A simulated computer-assisted T K R setting was constructed, and a single operator performed 30 repeated trials on five subjects (male, mean age 24, mean weight 80 kg) in vivo digitizing what was considered to be the extremes of both medial and lateral malleoli. Subjects then performed 30 trials of the following movements: dorsi/plantarflexion with the ankle in the neutral, inverted and everted positions, in/eversion with the ankle in the dorsiflexed, neutral and plantarflexed positions, and finally circumduction. The measured data was then used to optimize the 12 parameter kinematic model developed previously [Bogert 1994]. The same motion data was then used to determine the AJC using a sphere-fitting method. 3 Results and Discussion Table 1 Summarizes the results of the experiment, and as was expected the digitizing point probe method had significantly higher precision than both motion based method in the frontal plane (P=0.002 Biaxial, P=0.0003 Spherical). The B M was significantly less variable than the spherical method (0.17° SD vs 0.28° SD, P=0.004). Page 74 Table 1. Repeatability averages of methods in the frontal plane Medial Lateral Direction (Frontal Plane - Varus/Valgus) AJC Location Bias normalized to Range Method SD (mm) SD (°) point probe (mm) (mm) Point Probe 0.69 0.11 N/A 2.75 Biaxial Method 1.14 0.17 7.24 (medial) 5.00 Sphere Method 1.84 0.28 10.80 (medial) 7.23 It is the varus/valgus errors that occur in the frontal plane that we are most interested in reducing to improve outcomes of computer-assisted TKRs. This is the first time that the precision of locating an AJC with the B M has been reported on. Although the repeatability of locating the AJC with the point probe is significantly better than the motion based methods, it carries with it no meaningful biomechanical information. The point probe AJC is essentially just some characteristic point that we are able to locate precisely. Since only five subjects were tested, we do not have enough data yet to say that the biases of the motion based studies are statistically significantly different than the point probe centre locations. However, all AJCs found using the motion based methods were biased medially (7-10 mm) and slightly anteriorly (1-3 mm). It should be noted that by choosing an AJC lateral to the true AJC would results in a varus error which, according to clinical experience, is always to be avoided. References 1. van den Bogert AJ, Smith GD, Nigg B M : In Vivo Determination of the Anatomical Axes of the Ankle Joint Complex: an Optimization Approach. Journal of Biomechanics 27(12), 1994. pp 1477-1488. Page 75 Repeatability and Accuracy of Ankle Centre Location Estimates using a Biaxial Joint Model Cameron A . Shute & Antony J. Hodgson DcparuricM of Mechanical Engineering University ol British Columbia, Vancouver, BC, Canada Email: ahodgson®mech.ubc.ca Motivation In conventional total knee replacement surgery, a significant fraction of implants have varus/valgus alignment emus large enough to reduce the lifespan of ihe implant, so wc arc developing a more accurate computer-assisted procedure aimed at reducing the standard deviation (SD) of the implantation procedure. This work compares the performance of a more physically realistic biaxial model of the ankle joint complex (van den Bogert, 1994) both to a spherical model currently used in CAS systems and to direct digitization of the malleoli. Study Design Overall Goal Characterize Repeatability and Bias \—i of AJC centre estimation techniques j Cen t re I^)cali 011 Teclinitjues r.:;.i;:: , Vlol ion Based A n a t o n u i a l l y Based » i A B i a x i a l Spherical i Digi t i sa t ion ] [Bone Tracker Design [Tibial and Calcaneal Trackers! | Optoelectronic marker J ; tracking system [ [In Vivo Experiment Ankle Joint Centre Dclinition I'm- Biaxial Model Biaxial Model: 2 DOF Motion occurs about j j Y/J-4 two fixed hinge joints: talocrural and submit Joint Centre Definition: 'Pie point at which -A the TC and ST axes intersect when viewed from above with foot in the stance position. A force through this point towards the knee centre generates no net moments about either joint Biaxial Model AJC Definition Transverse projection of axes viewed from the knee centre Non-Invasive Bone Tracking Devices Tibial Tracker Calcaneal Tracker \ Tibial tracker Distributed set of contact points Elasticized straps maintain contact Calcaneal tracker Flexible aluminum heel cup - Applies constant force on calcaneus IRED marker triads define ref. frames Results Ankle Centre Estimates - Raw Data all Methods, all Subjects Subject A Subject B Subject C -4- M L uxi<r (mm) Subject I) .a I A. tj MI, axis (mm) Subject E m^e * M l J K I S itiiinl I Point Probe Estimates I Biaxial Estimates I Spherical Estimates Datasets for each subject normalized by individual point probe means M L axis (mm) Repeatabil i ty of Methods for ai l Subjects Medial Lateral Repeatability Anterior Posterior Repeatability . n i l ^ : M S 1 . . Point Biaxial Spherical Probe Model Model O.MIIMII l.tomm l.glmm Point Biaxial Spherical Probe Model Model I . S M M O.WIIIIU 0.49 nun Bias of Estin ates Relative to Point Probe Centre Mean Joint Centres of Methods Bias of Motion Based Methods Relative to Point Probe Centre Biaxial Spherical Direction Method Method ofBias (mm) (llJIll) Medial 7.2 10.8 Anterior 2.7 1.1 compared to c H„,,„„,J, = direct I 40 < 20 0 •20 | - 4 0 o -60 r • Biaxial • Spherical a Point Probe 50 30 Medial 10 30 I -ateral In Vivo Exper imenta l Protocol 5 Normal Male Subjects Tested - (mean age: 24 / mean weight: 80kg) Motion Based Centre Estimation 30 repeated trials of subject generated motion Optimization used to fit model parameters (for both biaxial and spherical models) Anatomically Based Centre Estimation Digitize exuemes of medial and lalei.il malleoli and midpoint used as centre estimate Conclusions 1. Motion based centres are medial to Anatomically determined centres Using anatomic ccntics leads to varus ericas 2. MI. variability of the Biaxial method is ~V? that of the Spherical method 3. Variability in bias dominates anatomical technique • Variance of motion based methods ~ 11 13% thai of anatomic 4. Biaxial method is tht most repeatable method for defining frontal plane alignment reference (which is most crucial for minimizing varus/valgus implantation errors) Page 76 INVESTIGATION OF ANKLE CENTRE LOCATION TECHNIQUES APPLIED TO COMPUTER-ASSISTED TOTAL KNEE ARTHROPLASTY C A . Shute, A.J. Hodgson Department of Mechanical Engineering, University of British Columbia, Vancouver, BC, Canada INTRODUCTION: Implants in total knee replacement (TKR) surgery are aligned relative to the mechanical axis of the lower limb which is defined by the hip, knee, and ankle centres. We are developing a computer-assisted TKR system, and currently our ankle joint centre (AJC) location protocol is the largest contributor of error1. In conventional surgery, the AJC is defined anatomically as the midpoint between the malleoli, but the relation between this point and an AJC based on ankle kinematics has not been established. In this study we examine the precision and bias of AJC estimates obtained from digitisation and motion-based techniques. M E T H O D S : An In Vivo experiment was performed on 5 healthy males (mean age 24, mean weight 80 kg). Three methods of estimating the AJC were investigated: point probe digitisation, biaxial modeling2 and ball-in-socket modeling. Thirty repeated trials of each AJC estimation method were performed on the subjects. The digitisation method consisted of digitising the medial and lateral malleoli, and then calculating the AJC as the midpoint. The motion-based techniques involved recording tibial and calcaneal motion data as the ankle was articulated throughout the range of motion then applying optimisation methods to determine the AJCs that best represent the data for the chosen model. Non-invasive bone tracking devices shown in Figure 1 were developed to track the motion of tibia and calcaneous. f - A Fig. 1: Non-invasive bone trackers. Tibial tracker (left) and calcaneal tracker(right). We also performed a cadaveric experiment to determine how accurately the non-invasive bone trackers represent true bone motion (by comparing tracker markers to bone pins), and to determine the location of the digitised AJC with respect to the tibial plafond (four operators, 30 trials each). For all trials a Flashpoint 5000 optical localizer was utilized and is capable of measuring the position of its active LED markers within 0.5 mm. Table 1.Results of In Vivo study, Precision of AJC method in Frontal Plane - Varus/Valgus Errors. AJC Location Method Standard Deviation Medial Bias W R T to Point Probe Range Point Probe Biaxial Method Sphere Method 0.69 mm 1.14 mm 1.84 mm N / A 7.24 mm 10.80 mm 2.75 mm 5.00 mm 7.23 mm Table 2. Results of In Vitro study, Range of tibial and calcaneal non-invasive tracker motion with respect to bone pin mounted markers. Non-Invasive Tracker Medial Lateral Anterior Posterior Proximal Distal Tibial tracker Calcaneal Tracker 2.15 mm 0.89 mm 1.30 mm 1.96 mm 0.63 mm 2.65 mm The mean location of the digitized AJC was found to be 0.55 mm lateral to and 4.20 mm posterior of the centroid of area of the tibial plafond. DISCUSSION & C O N C L U S I O N S : The most striking result of this study is the bias associated with each AJC location method. The motion-based centres determined from biomechanics are both located significantly medial to the centre determined using the traditional anatomical techniques. The consequences of choosing an AJC lateral to the true centre will result in a varus implant alignment error. The study also shows that non-invasive bone tracking devices can be used to accurately represent underlying bone motion. R E F E R E N C E S : 1 Inkpen KB, Hodgson AJ, Plaskos C, Shute CA, McGraw RW: Repeatability and Accuracy of Bone Cutting and Ankle Digitization in Computer-Assisted Total Knee Replacement. (2000) MICCAI: 1163-1172. 2 van den Bogert AJ, Smith GD, Nigg BM: In Vivo Determination of the Anatomical Axes of the Ankle Joint Complex: an Optimization Approach. (1994) J Biomech 27:1477-1488. ACKNOWLEDGEMENTS: Thanks to Dr. Wayne Vogl, UBC Department of Anatomy for your support and assistance with this research. R E S U L T S : Page 77 REPEATABILITY AND ACCURACY OF BONE CUTTING AND ANKLE DIGITIZATION IN COMPUTER-ASSISTED TOTAL KNEE REPLACEMENTS Kevin B. Inkpen1, Antony J. Hodgson1, Christopher Plaskos1, Cameron A. Shute1 and Robert W. McGraw2 Departments of 'Mechanical Engineering and 2Orthopaedics University of British Columbia, Vancouver, BC, Canada Email: ahodgson@mech.ubc.ca Abstract - In conventional total knee replacement (TKR) surgery, a significant fraction of implants have varus/valgus alignment errors large enough to reduce the lifespan of the implant, so we are developing a more accurate computer-assisted procedure aimed at reducing the standard deviation (SD) of the implant's frontal alignment to under 1°. In this study we measured the contributions to overall alignment error of two steps in our proposed procedure: ankle digitization and manual bone cutting. We introduce a new digitizing probe that quickly and robustly locates the midpoint between the ankle malleoli. Based on repeated measurements on 8 cadavers by 6 operators (318 measurements), we estimate that the new probe introduces only 0.15° (SD) of variability into the definition of the tibial mechanical axis in the frontal plane. We also measured the accuracy and repeatability with which surgeons can implement a bone cut using conventional cutting guides to see if conventional cutting techniques are sufficiently accurate. A total of 53 tibial plateau, distal and anterior/posterior (A/P) femoral cuts approximated primary and revision TKR resections. In 20 test cuts on cadaver bone made by 2 expert TKR surgeons, we found a SD of 0.37° (bias of 0.29°) in the varus/valgus difference between the guide orientation and the implant orientation before cementing, but in 23 additional cuts performed by four less- experienced surgeons, the SD was 0.83° (bias of 0.31°). Ten A/P femoral cuts showed similar trends. We conclude that, in the hands of an experienced surgeon, our current technique (based on our previously-reported non-invasive hip centre locating technique [Inkpen 1999b], robust ankle digitization and manual cutting using computer-guided cutting guides) can approach our target alignment variability goal of a SD less than I °. 1 Introduction We are developing computer-assisted total knee replacement (TKR) tooling that eliminates intramedullary rods and improves alignment accuracy without introducing additional imaging requirements (such as preoperative computed tomography scans) or invasive procedures (such as bone pins remote to the operating site). Varus/valgus alignment error o f 3° has been strongly correlated with aseptic loosening [Jeffery 1991]. In a meta-analysis of 10 published studies involving 1373 knees, we estimate the SD of varus/valgus alignment to be 2.6° on the overall limb and 1.9° each for the distal femoral and tibial plateau cuts [Inkpen 1999a]. To ensure that upwards of 95% of all computer-assisted procedure result in overall alignment lying within this critical 3° window, it is necessary to reduce the SD of the overall procedure to approximately 1°. In the computer assisted T K R technique developed by Leitner et al [Leitner 1997], the mechanical axis is defined intraoperatively using a 3D optoelectronic localizer. They find the hip centre by tracking a marker at the distal femur in a reference frame rigidly attached to the pelvis (using a bone pin) as the femur is moved through its range of motion, taking the centre of the best fitting sphere to the femoral path as the hip centre. Krakow et al [Krackow 1999] use a similar technique for the hip but eliminate the bone pin at the pelvis by simply holding the patient steady during the femoral motion. In early testing we have achieved precise and accurate hip centre location using a new non-invasive hip tracker and, although more testing on a variety o f patients is ongoing, we estimate that the femoral mechanical axis can be located intraoperatively with a SD of 0.05° [Inkpen 1999a]. Krakow et al use a conventional point probe to digitize the medial and lateral malleoli, and the ankle centre point is then defined as the intersection of this malleolar axis and a.digitized unit vector representing the surgeons' estimate of the best centre along the anterior portion of the ankle. Leitner, despite the complexity of the ankle joint complex, uses the same sphere-fitting algorithm applied to the hip to find the ankle centre. They track a bone pin marker inserted in the calcaneus in a coordinate frame pinned to the tibia. In an attempt to validate this determination of a kinematic ankle centre, Leitner compared the sphere fitting method with direct digitization on a plastic skeleton leg. Although not directly comparable to a human study, digitizing the Page 78 mechanical centres of the skeleton showed better repeatability over the sphere fitting method (0.29° vs. 0.49° SD). We also have found poor repeatability with the motion tracking technique even when a bone pin marker is used in the calcaneus [Inkpen 1999a, Inkpen 1999b]. Even though digitization is more repeatable than motion-tracking techniques at the ankle, the contribution of variance is still large in comparison with that found at the hip. Using a conventional point probe, a user who is asked to digitize an ankle malleolus could select any point within an area of radius -3-5 mm and still consider the point to be 'on' the malleolus. Although a careful operator may be very precise using this method, we would not expect it to be robust across different operators and patients. It is also not clear that a motion tracking and sphere fitting method without realistic weight bearing load gives a biomechanically correct ankle centre. A passively manipulated ankle acts about a single talocrural joint axis [Leardini 1999], although inversion/eversion torques generate motion at the subtalar joint, so the ankle complex can act more like a biaxial joint [van den Bogert 1994]. Given the practical problems of simulating and tracking weight bearing ankle motion intraoperatively, we propose using a robust digitizing probe to precisely locate the malleoli, allowing the trauitional ankle centre estimate of the midpoint between the malleoli to be used in the computer-assisted technique. Finally, once the patient's mechanical axis is registered, cutting planes for the desired limb alignment can be calculated and conventional cutting guides fitted with markers can be manually adjusted to the correct pose [Leitner 1997][Krackow 1999]. For cemented TKR where the surface quality from a conventional bone saw is sufficient, the question is whether errors due to conventional cutting are large enough to warrant more sophisticated bone cutting tools such as robotic guidance of the bone saw or robotic machining. One study of cuts made in balsa wood blocks shows flexion/extension SD of 0.25° with a bias of 1° towards flexing of the saw blade up and away from the guide surface [Otani 1993]. Although several studies have been made of the bone surface quality and roughness after cutting, we could not find any published studies on overall implant orientation error in the frontal plane compared to the guide setting. 2 Methods 2.1 Ankle digitizing probe design and testing Probe Design: In making the traditional estimate of the ankle centre based on the malleoli, precision and robustness could be improved if the digitizing probe only fit onto the malleolus in a certain way, forcing the user to place the probe in a smaller range of positions. As the malleoli are roughly spherical in shape (even indistinct malleoli can often be palpated well enough to find some amount of spherical protrusion), the ankle probe (Figure la) contacts the anatomical feature with three 7.8 mm diameter balls spaced evenly on a common 12.8 mm radius about the probe centreline, with all 3 contact ball centres lying in a plane normal to the centreline. A plunger is free to move along the probe centreline and contacts the feature with a normal planar face. When the probe is pressed against a roughly spherical feature, the three contact balls force the centreline of the probe to pass through the sphere centre and the plunger face contact point lying on the sphere centreline is known (Figure lb). Unless used in a completely careless way, this design repeatably guides the plunger contact point to the extreme medial and lateral surfaces of the malleoli. Figure 1. Ankle probe in use (a) and frontal plane view of both measurement positions (b). Page 79 Study design and data analysis: To simulate conditions in a computer-assisted T K R , sets of 21 to 30 repeated measurements of the ankle centre were made by 6 different operators on 8 different embalmed cadavers for a total of 318 ankle centre measurements. To estimate intra-observer repeatability on a variety of patients, we report the standard deviation (SD) of 30 ankle centre measurements by a single operator on 7 different embalmed cadavers. To show the probe's robustness when used by different operators, three additional operators each made 21 to 30 repeated measurements on one of these specimens and we report the total range spanning the 95% confidence limits of each individual operator. To show the relation between the characteristic ankle point found with the probe and the centre of area for the articular surface, two additional operators each made 20 measures of an additional specimen, after which we dissected the talocrural joint, digitized the tibial mortise and calculated its centre of area. Measurement techniques: A l l measurements were recorded relative to a three-emitter triangular local reference frame (120 mm on a side) pinned rigidly to the proximal tibia and nominally aligned with the body planes. We used a Flashpoint 5000 localizer (Image Guided Technologies, Boulder, C O , U S A ) for all measurements, which has a typical accuracy of ~0.5 mm [Chassat 1998]. We removed indentations in the skin by massaging the tissue overlying the malleoli between measurements to prevent the probe contact balls from repeatedly falling into the same position. 2.2 Bone cutting errors Study design: For this study we define bone cutting error as the difference in orientation between the cutting guide and the implant, described as two rotations. The first is in the plane of the face of the cutting guide facing the surgeon (labeled varus/valgus for distal femoral and tibial plateau cuts, and internal/external rotation for A /P femoral cuts), and the second is about a nominally mediolateral (ML) axis defined by the intersection of the front face plane and the cutting plane of the guide (labeled flexion/extension for distal femoral and tibial plateau cuts, and anterior/posterior deviation for A / P femoral cuts). Two expert T K R surgeons, three orthopaedic residents with T K R experience and a foreign orthopaedic surgeon without T K R experience made cuts simulating typical practice on five fresh frozen human femora and eight tibia. Several cuts were made on each specimen, approximating primary and revision bone resections, for a total of 53 cuts. The specimens had no obvious abnormalities in bone quality. Cutting guides, a conventional pneumatic oscillating bone saw (CPS #1535), and saw blades supplied by Johnson & Johnson (Raynham, Mass. USA) . We used open guides for 33 cuts and slotted guides for the remainder. 8 cuts were made with a 0.8 mm thick, 75 mm long blade (Synvasive 'Stablecut' #11-0470) and the remainder were made with 1 mm thick, 90 mm long blades (Johnson & Johnson #26-6050) renewed at the surgeon's request. To simulate orientation of the components after placement on the cut bone, we made dummy components from flat aluminum plate by copying the profile of the tibial trays and the distal footprints of the femoral components in a Johnson & Johnson PFC implant series.. For each test cut, the surgeon pinned the guide to the specimen and we measured the front face and cutting surfaces of the guide to provide the reference frame for reporting errors in varus/valgus (internal/external rotation for A / P cuts) and flexion/extension (anterior/posterior deviation for A / P cuts). The surgeon then made the cut and did any trimming and checking as would normally be done in T K R . We then re-measured the guide, placed a dummy implant on the cut bone surface to simulate placement of an implant with proper bone coverage, and measured the plane o f the dummy implant. Measurement techniques and data analysis: A l l measurements were made using the Flashpoint 5000 localizer (described above). To determine the variability associated with placing a dummy implant on a bone cut and measuring its orientation, we made 30 repeated placements and measurements on a single bone cut. The resulting SD was <0.10° for varus/valgus errors and <0.17° for flexion/extension errors (95% confidence). For test cuts, we computed the mean and SD of all cutting errors in each direction. We are continuing to enroll surgeons in this study, so until we have a sufficient number to represent the general surgical population and until more cuts are made with slotted guides, we wil l not try to extract the effects of guide style, or type of cut. Page 80 3 Results 3.1 Ankle digitizing probe results The average SD of ankle centre location for any one set of repeated measurements is 0.75 mm in the mediolateral ( M L ) direction and 1.12 mm in the anteroposterior (AP) direction (indicating that the new probe will give a result within ±1.9 mm (ML) and +2.9 mm (AP) 95% of the time, Figure 2). This corresponds to an angular variability of 0.15° SD in varus/valgus and 0.2° SD in flexion/extension when registering a short (375 mm) tibial mechanical axis. The maximum range of a single operator on a single specimen is 6.5 mm (ML) and 10 mm (AP), giving a maximum tibial axis range of 1.0° varus/valgus and 1.5° flexion/extension. A n k l e c e n t r e p r e c i s i o n i n M L a n d A P d i r e c t i o n s , a l l t r i a l s , n = 2 1 t o 3 0 r e p e a t e d m e a s u r e s i n e a c h t r i a l 3.50 E I 3.00 2.50 n -5 <!.uu > i= S c ° 3 1-50 1 S L O O s 0.50 0.00 HAnkle centre SD, ML • Ankle centre SD, AP Fft fc 1 2D 5D 8D 13D 18D 20D 21A 21B 21C Test (Specimen # & Operator letter) 21D 27E 27F Figure 2. Range of variation in estimated ankle centre locations in the mediolateral (ML) and anteroposterior (AP) directions based on multiple specimens and operators. A truly robust method would have no significant differences between the operator means. A l l means are within 0.8 mm in both A P and M L directions (Figure 3) and the range of the 95% confidence limits of the means is 1.6 mm (ML) and 1.3 mm (AP). Mean ankle centre locations in ML direction. (21 - 22 trials by each operator, all on a single specimen) I u o c 5 Specimen S £ U CJ u in Q < Mean ankle centre locations in AP direction. (21 - 22 trials by each operator, all on a single specimen) •80 00 81 00 82 00 4— •83 00 -84 00 85 00 kn\A Specimen Figure 3. Interoperator differences in estimated ankle centre locations in the ML and AP directions on a single specimen (ankle centre locations are relative to a local reference frame pinned rigidly to the proximal tibia and nominally aligned with the body planes). On the one specimen available for dissection, 2 operators both obtained mean results within 2 to 3 mm (ML) and 2 to 5 mm (AP) of the talocrural articular surface centroid (Figure 4). Page 81 40 301 20! 10 0 -10 -20 -30 -40! Probe means 2 to 3 mm lateral of centroid T Probe means 2 to 5 mm posterior of centroid -40 -20 0 20 40 lateral ML axis (mm) medial Transverse plane through ankle: • = digitized surface perimeter + = centroid of tibial mortise O = operator E Figure 4. Comparing mean ankle centre locations, obtained by two operators, with centroid of tibial mortise, obtained by direct digitization, all on one specimen. 3.2 Bone cutting results Both the guide movement and implant errors follow a normal distribution (checked using normal scores plots). Based on the actual variances, the test has 95% power to detect an implant alignment error of 0.4° in varus/valgus and 0.7° in flexion/extension. The mean cutting error in the frontal plane is about 0.3° degrees in the varus/valgus direction for both surgeon types (Figure 5). Precision in varus/valgus is within a SD of 0.54° for the experts and 1.25° for the less experienced surgeons (95% confidence); guide movement accounts for about 30% of the expert surgeons' variance and 34% of the less experienced surgeons' variance. Page 82 Mean 0.29° (95% CI 0.11 to 0.46) SD 0.37° (95% CI 0.28 to 0.54) Range 1.30° Mean 1.03° (95% CI 0.49 to 1.58) SD 1.16°(95%CI 0.89 to 1.70) Range 4.65° • Mean 0.31° (95% CI -0.05 to 0.66) SD 0.83° (95% CI 0.68 to 1.25) Range 3.71° Mean 0.74° (95% CI 0.05 to 1.42) SD 1.58° (95% CI 1.22 to 2.24) Range 7.02° Figure 5. Varus/valgus (left) and flexion/extension (right) implant error results for 43 bone cuts: 20 made by 2 expert surgeons (top) and 23 by 2 less experienced surgeons (bottom).3 Figure 6. Internal/external rotation (left) and anterior/posterior deviation (right) implant error results for 10 bone cuts.' In flexion/extension, there is a bias towards flexing of the saw blade up and away from the guide surface of over 1 ° (P = 0.0006) for the expert surgeons. The greatest flexion/extension errors in the current study occurred on large specimens where the saw blade did not reach the end of the cut before the saw body hit the guide, requiring the cut to be finished off 'freehand' at the far cortex. Precision in flexion/extension for both surgeon groups is within a SD of 2.24°, where guide movement accounts for about 37% of this variance. The measurement variance in either direction accounts for only about 1% of the observed variance. The results of ten A/P femoral cuts executed by an expert surgeon (six cuts) and a resident (four cuts) on four femora are shown in Figure 6. Error in internal/external rotation (about the femoral mechanical axis) averages to a SD of 0.50° (0.34° for the expert and 0.67° for the resident). In the sagittal plane, angular deviation of the resected bone plane relative to the cutting guide surface resulted in a SD of 0.63° (0.70° for the expert and 0.50° for the resident). 4 Discussion 4.1 Ankle digitizing probe We found excellent repeatability in using digitization to locate a characteristic point within the ankle joint complex. Our ankle probe only introduces a frontal plane SD of 0.15°, which compares favourably to Leitner's digitization result of 0.29°. A future repeatability study that will compare our probe to an alternative method such as using a point probe to digitize the malleoli is also required to validate the robustness of this Positive mean rotations are counter-clockwise about axes pointing out of the page. Mean 0.03° (95% CI -0.33 to 0.39) SD 0.50° (95% CI 0.35 to 0.92) Range 1.73° Mean -0.02° (95% CI -0.47 to 0.43) SD 0.63° (95% CI 0.43 to 1.15) Page 83 new design. We plan to do this in vivo because the skin dimpling encountered in cadaveric studies artificially decreases the repeatability of locating the same points with the point probe. We are currently working on developing an ankle centre location protocol that uses the talocrural and subtalar joint axes to define a biomechanically meaningful centre. More specimens need to be measured and dissected to show that a reliable relationship between the digitized point and a biomechanically relevant ankle centre point can be established. 4.2 Bone cutting In contrast to the excellent repeatability found at the ankle, bone cutting introduces a significantly higher contribution to the overall alignment variance. While the expert surgeons have limited their varus/valgus errors to under 1° (0.37° SD), errors as high as 2° (0.83° SD) were found with the less experienced surgeons. This corresponds to a 13mm ML error in ankle registration, which is much higher than errors experienced with the robust ankle probe. While only ten A/P femoral cuts were made, the internal/external rotation errors follow similar trends to those found in varus/valgus. The expert surgeon's SD in internal/external rotation error is almost equal to that in varus/valgus (0.34° and 0.37° respectively) and is half that of the less experienced surgeon. However, we do not see the sagittal plane bias of the saw blade flexing up and away from the guide surface as with flexion/extension errors. This flexion/extension sagittal plane bias is similar to that found by Otani [Otani 1993]. It may be possible that the amount of bone resected in an A/P femoral cut is not enough to cause the blade to flex away from the guide surface and there is no need for any freehand trimming. Although we have noted marked differences in repeatability in bone cutting between the two expert surgeons and the four surgeons with less TKR experience, we have not yet studied enough surgeons to make a statistically valid argument that this is the case. As mentioned above, we are continuing to enroll surgeons of various experience levels in this study and hope soon to be in a position to test the effect of experience on repeatability in bone cutting. These results are also based on the use of both slotted and open guides; here too we expect to be able to test for differences in performance when using different guide types once we have enrolled more surgeons into the study. Finally, although the surgeons who have participated have commented that the cutting process on the benchtop is essentially identical to that in the operating room, we do not yet have any OR data to support this impression. As our system moves closer to clinical implementation, we will be in a position to test this hypothesis. 4.3 Cumulative repeatability There are numerous contributors to variability in our computer-assisted TKR procedure; these include variance and bias in estimating the hip, knee and ankle centres, setting the correct cutting guide pose, cutting the bone, and cementing the implant into place. To date, we have only measured values for the hip and ankle and for the bone cuts. Assuming that the knee measurements will contribute roughly the same variability as those at the hip and ankle, and assuming that the angular precision of locating the cutting guides will be similar to that of reading the planar probe, we estimate the overall repeatability of our proposed procedure by summing the variances of the various measurements. When we do this using the data for all six surgeons, we find that our predicted repeatability, before cementing, is roughly 1° (SD). The cutting errors, particularly for the less experienced surgeons, contribute the greatest fraction to the overall variance (-90%), so further development of this procedure should focus on the cutting issue and the precision of the cementing process, which is currently unknown. Page 84 • Hip (SD0.05") • Knee (SD 0.085°) • Ankle (SD0.15°) • Guide Placement (SD 0.24°) • Cuts + Implants (SD 0.92°) Figure 7. Contributions to overall varus/valgus variance in computer assisted TKR procedure. 5 Conclusions We conclude that even inexperienced surgeons can likely achieve a SD within 1.5° for overall varus/valgus alignment in computer-assisted TKR using our proposed procedure consisting of a non-invasive hip centre identification technique, a robust ankle digitizing technique and a conventional cutting technique based on a computer-guided cutting guide. Expert surgeons exhibit significantly better cutting precision, so we expect their overall precision to be better (<1.2°). In flexion/extension, alignment precision will be worse due to greater cutting errors in this direction. Acknowledgements We thank Drs. Thomas Oxland of the Division of Orthopaedic Engineering Research, Vancouver Hospital, and Vlad Stanescu, UBC Dept. of Anatomy, for their assistance. References 1. Chassat F, Lavallee S: Experimental Protocol of Accuracy Evaluation of 6-D Localizers for Computer- Integrated Surgery: Application to Four Optical Localizers. In: Wells, W; Colchester, A; Delp, S. (eds): Medical Imaging and Computer Assisted Intervention (MICCAI'98). Lecture Notes in Computer Science Vol. 1496, Springer-Verlag 1998, pp. 421-430. 2. Jeffery RS, Morris RW, Denham RA: Coronal Alignment After Total Knee Replacement. J. of Bone and Joint Surgery (British Ed.) 73-B, 1991, pp. 709-714. 3. Inkpen, KB: Precision and Accuracy in Computer-Assisted Total Knee Replacement. Master's thesis, Dept. of Mechanical Engineering, University of British Columbia, Vancouver BC Canada. 1999a. 4. Inkpen KB, Hodgson AJ: Accuracy and Repeatability of Joint Centre Location in Computer-Assisted Knee Surgery. In: Taylor C, Colchester A (eds): Medical Imaging and Computer Assisted Intervention (MICCAI'99). Lecture Notes in Computer Science Vol. 1679, Springer-Verlag 1999b, pp. 1072-1079. 5. Krackow, K.A., Bayers-Thering, M. , Phillips, M.J. (1999). A New Technique for Determining Proper Mechanical Axis Alignment During Total Knee Arthroplasty: Progress Toward Computer-Assisted TKA. Orthopedics., 22(7), 698-702. Page 85 6. Leardini, A., O'Connor, J.J., Catani, F., & Giannini, S. (1999). Kinematics of the human ankle complex in passive flexion; a single degree of freedom system. J.Biomech., 32(1), 111-118. 7. Leitner F, Picard F, Minfelde R, Schulz H-J, Cinquin P, Saragaglia D: Computer Assisted Knee Surgical Total Replacement. In: Troccaz J, Grimson E, Mosges R (eds): CVRMed-MRCAS '97. Lecture Notes in Computer Science Vol. 1205, Springer-Verlag 1997, pp. 627-638. 8. Otani T, Whiteside LA, White SE: Cutting Errors in Preparation of Femoral Components in Total Knee Arthroplasty. Journal of Arthroplasty, Vol 8, No 5, Oct. 1993 pp. 503-510. 9. Bogert, A.J. van den, Smith GD, Nigg BM. In Vivo Determination of the Anatomical Axes of the Ankle Joint Complex: an Optimization Approach, Journal of Biomechanics, Vol. 27(12), 1477 - 1488, 1994. Page 86 Repeatability and Accuracy of Bone Cutting and Ankle Digitization in Computer-Assisted Total Knee Replacements Kevin B. Inkpen', Amony). Hodgson', Christopher Plaskos', Cameron A. Shulc1 and Robert W. MeGtaw2 Departments of 'Mechanical lingineering and ^Orthopaedics University of British Columbia, Vancouver, BC, Canada Eraail.vahodgson@mech.ubc.ca Motivation In conventional total knee replacement surgery, a significant fraction of implants have varus/valgus alignment errors large enough to reduce the lifespan of the implant, so we are developing a more accurate computer-assisted procedure aimed at reducing the standard deviation (SD) of the implant's alignment to under 1°. In this study we measured the contributions to overall alignment error of two steps in our proposed procedure: ankle digitization and manual bone cutting. ' IP Study Design Bone Cutting Bone Culling Krror: Difference in orientation between the cutting guide. A, and the prepared bone bed, \ SB, described and labeled as I wo fixed frame rotations I - v . flexion/extension (F/E) error: tibial & distal femoral. anterior/posterior (A/I*) deviation error: A/P cuts. ^ varus/valgus (V/V) error: tibial & distal femoral cuts, 1 internal/external rotation error: .VP cuts. 2 expert T K R surgeons • 4 less experienced surgeons • Oscillating bone saw • Dummy implants • .8 & I mm thick blades • Open & slotted guides * 5 femora &. 8 tibia • Total of 53 bone curs« Results Experts 0.29° ± 0.37°: 11.30'] 1.03° ±1.16° [1.65°j Varus/Valgus [Kangc°| Less Exp, 0.31-10.83 [3.71°] Flexion/Extension , 0.74° ±1.58' • A [7.021 Internal/External Rotation A titerior/Posterwr Deviation Surgeons 0.03'± 0.50s " M -0.02'";± 0.63° 1173*1 12.021 Guide Movement (SD) 1 2 O tl« r> 0.6 0.* 0.2 0 VurusA'ulgu. , | | |B Less Exp. Flexiort/Exlension Discussion • w s r x f K S D Expert SD < '.ess Exp. SD ' /-XK bias of ~l = -•> saw blade flexing up from cutting guide • No bias for A/P femoral cuts. • Guide movement significanl, especially for less experienced surgeons. ' Standard deviation of guide movement -50% dial of overall cutting errors. Ankle Digitization • Self Aligning Probe Design Plunger R e 5 u | | m g Three emitter frame fixed contact , to plunger and centreline points., •"-"T"-- Probe in ^ 1 lateral ' , position * v j | Three contact balls \ Fourth li-rrjitter fixed relative to contact nulls Spherical Approximations Probe in medial position Results Intraoperalor Precision Medial lateral 0.75 mm or 0.11° Anterior Posterior 1.12 mm or 0.17° Based on 375 mm Tibial Mechanical i Axis length intraoperalor Precision (I operator 7 specimens) SO fa Me.-tisl I.areml Dire. i \ . i f I I Tnteroperator Precision All ankle centre means were within 0.8 mm in both M l - and AP directions 'Interoperator precision (3 operators 1 specimen) Comparison to Anatomy (2 operators I specimen) Prcbe mom 2 to 3mmlii!«nl to. ca iii oid • Diililijedpcrimctti of tibial inoitiw: Comparison to Anatomy 2 operators obtained means within 2-3 mm lateral to and 2-5 mm posterior to the centroid of tibial mortise articular surface Conclusions Cumulative Repeatability - ±1 Quantified contributions to total variability: • Hip, knee, and ankle registration • Setting guide pose and making cuts ("miinn errors (for all surgeons) contribute - *)S% of this 1" variance! • Hp {SD0 05T BGuce Placement f Sis (SD 0 071 <*^*3 OKnee (SD0.0&51 j 095% DArWe [SO 0 15") OCuts * Implants ^ — ~ (SO 0 92 ^ Page 87

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