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DETERMINATION OF PATIENT-SPECIFIC FUNCTIONAL AXES THROUGH TWO-LEVEL OPTIMIZATION By JEFFREY A. REINBOLT A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2003
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Copyright 2003 by Jeffrey A. Reinbolt
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This thesis is dedicated to my loving wife, Karen.
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ACKNOWLEDGMENTS I sincerely thank Dr. B. J. Fregly for his support and leadership throughout our research endeavors; moreover, I truly recognize the value of his honest, straightforward, and experience-based advice. My life has been genuinely influenced by Dr. Freglys expectations, confidence, and trust in me. I also extend gratitude to Dr. Raphael Haftka and Dr. Roger Tran-Son-Tay for their dedication, knowledge, and instruction in the classroom. For these reasons, each was selected to serve on my supervisory committee. I express thanks to both individuals for their time, contribution, and fulfillment of their committee responsibilities. I recognize Jaco for his assistance, collaboration, and suggestions. His dedication and professionalism have allowed my graduate work to be both enjoyable and rewarding. I collectively show appreciation for my family and friends. Unconditionally, they have provided me with encouragement, support, and interest in my graduate studies and research activities. My wife, Karen, has done more for me than any person could desire. On several occasions, she has taken a leap of faith with me; more importantly, she has been directly beside me. Words or actions cannot adequately express my gratefulness and adoration toward her. I honestly hope that I can provide her as much as she has given to me. I thank God for my excellent health, inquisitive mind, strong faith, valuable experiences, encouraging teachers, loving family, supportive friends, and wonderful wife. iv
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TABLE OF CONTENTS Page ACKNOWLEDGMENTS.................................................................................................iv TABLE OF CONTENTS.....................................................................................................v LIST OF TABLES...........................................................................................................viii LIST OF FIGURES...........................................................................................................xi ABSTRACT.....................................................................................................................xiii CHAPTER 1 INTRODUCTION........................................................................................................1 Arthritis: Th e Nations Lead ing Cause of Disability...................................................1 Need for Accurate Patient-Specific Models.................................................................2 Benefits of Two-Level Optimization............................................................................3 2 BACKGROUND..........................................................................................................4 Motion Capture.............................................................................................................4 Biomechanical Models.................................................................................................4 Kinematics and Dynamics............................................................................................5 Optimization.................................................................................................................5 Limitations of Previous Methods..................................................................................5 3 METHODS...................................................................................................................7 Parametric Model Structure..........................................................................................7 Hip Joint................................................................................................................8 Knee Joint..............................................................................................................8 Ankle Joint...........................................................................................................10 Two-Level Optimization Approach............................................................................11 Why Two Levels of Optimization Are Necessary..............................................11 Inner-Level Optimization....................................................................................11 Outer-Level Optimization...................................................................................12 Two-Level Optimization Evaluation..........................................................................13 Synthetic Marker Data without Noise.................................................................13 v
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Synthetic Marker Data with Noise......................................................................13 Experimental Marker Data..................................................................................14 4 RESULTS...................................................................................................................29 Synthetic Marker Data without Noise........................................................................29 Synthetic Marker Data with Noise.............................................................................29 Experimental Marker Data.........................................................................................29 5 DISCUSSION.............................................................................................................36 Assumptions, Limitations, and Future Work..............................................................36 Joint Model Selection..........................................................................................36 Design Variable Constraints................................................................................36 Objective Function Formulation..........................................................................37 Optimization Time and Parallel Computing........................................................37 Multi-Cycle and One-Half-Cycle Joint Motions.................................................38 Range of Motion and Loading Conditions..........................................................39 Optimization Using Gait Motion.........................................................................39 Comparison of Experimental Results with Literature................................................40 6 CONCLUSION...........................................................................................................43 Rationale for New Approach......................................................................................43 Synthesis of Current Work and Literature..................................................................43 GLOSSARY......................................................................................................................45 APPENDIX A NOMINAL JOINT PARAMETERS & OPTIMIZATION BOUNDS FOR SYNTHETIC MARKER DATA................................................................................52 B NOMINAL JOINT PARAMETERS & OPTIMIZATION BOUNDS FOR EXPERIMENTAL MARKER DATA.......................................................................55 C NOMINAL & OPTIMUM JOINT PARAMETERS FOR SYNTHETIC MARKER DATA WITHOUT NOISE.........................................................................................58 D NOMINAL & OPTIMUM JOINT PARAMETERS FOR SYNTHETIC MARKER DATA WITH NOISE.................................................................................................61 E NOMINAL & OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE EXPERIMENTAL MARKER DATA.......................................................................64 F NOMINAL & OPTIMUM JOINT PARAMETERS FOR FIRST ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA....................................67 vi
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G NOMINAL & OPTIMUM JOINT PARAMETERS FOR SECOND ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA....................................70 H OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE & FIRST ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA....................................73 I OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE & SECOND ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA....................................76 LIST OF REFERENCES...................................................................................................79 BIOGRAPHICAL SKETCH.............................................................................................83 vii
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LIST OF TABLES Table Page 3-1 Model degrees of freedom........................................................................................17 3-2 Hip joint parameters.................................................................................................20 3-3 Knee joint parameters...............................................................................................23 3-4 Ankle joint parameters.............................................................................................25 4-1 Two-level optimization results for synthetic marker data with random continuous numerical noise to simulate skin movement artifacts with maximum amplitude of 1 cm.............................................................................................................................31 4-2 Mean marker distance errors for nominal values and the two-level optimization results for multi-cycle experimental marker data.....................................................33 4-3 Mean marker distance errors for the two-level optimization results using first and second halves of the joint cycle motion for experimental marker data....................35 5-1 Mean marker distance errors for the inner-level objective function consisting of marker coordinate errors versus marker distance errors for multi-cycle experimental marker data...............................................................................................................41 5-2 Execution times for the inner-level objective function consisting of marker coordinate errors versus marker distance errors for multi-cycle experimental marker data...........................................................................................................................42 A-1 Nominal right hip joint parameters and optimization bounds for synthetic marker data...........................................................................................................................52 A-2 Nominal right knee joint parameters and optimization bounds for synthetic marker data...........................................................................................................................53 A-3 Nominal right ankle joint parameters and optimization bounds for synthetic marker data...........................................................................................................................54 B-1 Nominal right hip joint parameters and optimization bounds for experimental marker data...............................................................................................................55 viii
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B-2 Nominal right knee joint parameters and optimization bounds for experimental marker data...............................................................................................................56 B-3 Nominal right ankle joint parameters and optimization bounds for experimental marker data...............................................................................................................57 C-1 Nominal and optimum right hip joint parameters for synthetic marker data without noise.........................................................................................................................58 C-2 Nominal and optimum right knee joint parameters for synthetic marker data without noise............................................................................................................59 C-3 Nominal and optimum right ankle joint parameters for synthetic marker data without noise............................................................................................................60 D-1 Nominal and optimum right hip joint parameters for synthetic marker data with noise.........................................................................................................................61 D-2 Nominal and optimum right knee joint parameters for synthetic marker data with noise.........................................................................................................................62 D-3 Nominal and optimum right ankle joint parameters for synthetic marker data with noise.........................................................................................................................63 E-1 Nominal and optimum right hip joint parameters for multi-cycle experimental marker data...............................................................................................................64 E-2 Nominal and optimum right knee joint parameters for multi-cycle experimental marker data...............................................................................................................65 E-3 Nominal and optimum right ankle joint parameters for multi-cycle experimental marker data...............................................................................................................66 F-1 Nominal and optimum right hip joint parameters for first one-half-cycle experimental marker data.........................................................................................67 F-2 Nominal and optimum right knee joint parameters for first one-half-cycle experimental marker data.........................................................................................68 F-3 Nominal and optimum right ankle joint parameters for first one-half-cycle experimental marker data.........................................................................................69 G-1 Nominal and optimum right hip joint parameters for second one-half-cycle experimental marker data.........................................................................................70 G-2 Nominal and optimum right knee joint parameters for second one-half-cycle experimental marker data.........................................................................................71 ix
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G-3 Nominal and optimum right ankle joint parameters for second one-half-cycle experimental marker data.........................................................................................72 H-1 Optimum right hip joint parameters for multi-cycle and first one-half-cycle experimental marker data.........................................................................................73 H-2 Optimum right knee joint parameters for multi-cycle and first one-half-cycle experimental marker data.........................................................................................74 H-3 Optimum right ankle joint parameters for multi-cycle and first one-half-cycle experimental marker data.........................................................................................75 I-1 Optimum right hip joint parameters for multi-cycle and second one-half-cycle experimental marker data.........................................................................................76 I-2 Optimum right knee joint parameters for multi-cycle and second one-half-cycle experimental marker data.........................................................................................77 I-3 Optimum right ankle joint parameters for multi-cycle and second one-half-cycle experimental marker data.........................................................................................78 x
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LIST OF FIGURES Figure Page 3-1 The 3D, 14 segment, 27 DOF full-body kinematic model linkage joined by a set of gimbal, universal, and pin joints..............................................................................16 3-2 A 1 DOF joint axis simultaneously defined in two adjacent body segments and the geometric constraints on the optimization of each of the 9 model parameters........18 3-3 Modified Cleveland Clinic marker set used during static and dynamic motion-capture trials................................................................................................19 3-4 The 3 DOF right hip joint center simultaneously defined in the pelvis and right femur segments and the 6 translational model parameters optimized to determine the functional hip joint center location.....................................................................20 3-5 Geometric constraints on the optimization of translational and rotational model parameters for the hip, knee, and ankle joints..........................................................21 3-6 The 1 DOF right knee joint simultaneously defined in the right femur and right tibia segments and the 4 rotational and 5 translational model parameters optimized to determine the knee joint location and orientation................................................22 3-7 The 2 DOF right ankle joint complex simultaneously defined in the right tibia, talus, and foot segments and the 5 rotational and 7 translational model parameters optimized to determine the joint locations and orientations....................................24 3-8 Two-level optimization technique minimizing the 3D marker coordinate errors between the kinematic model markers and experimental marker data to determine functional joint axes for each lower-extremity joint................................................26 3-9 Inner-level optimization convergence illustration series for the knee joint, where synthetic markers are blue and model markers are red............................................27 3-10 Two-level optimization approach minimizing the 3D marker coordinate errors between the kinematic model markers and experimental marker data to determine functional joint axes.................................................................................................28 4-1 Outer-level optimization objective function fitness value convergence for synthetic marker data with random continuous numerical noise to simulate skin movement xi
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artifacts with maximum amplitude of 1 cm, where the best fitness value among all nodes is given for each iteration...............................................................................32 4-2 Outer-level optimization objective function fitness value convergence for multi-cycle experimental marker data, where the best fitness value among all nodes is given for each iteration.........................................................................................34 xii
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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science DETERMINATION OF PATIENT-SPECIFIC FUNCTIONAL AXES THROUGH TWO-LEVEL OPTIMIZATION By Jeffrey A. Reinbolt 2003 Chair: Benjamin J. Fregly Major Department: Biomedical Engineering An innovative patient-specific dynamic model would be useful for evaluating and enhancing corrective surgical procedures. This thesis presents a nested (or two-level) system identification optimization approach to determine patient-specific model parameters that best fit a three-dimensional (3D), 18 degree-of-freedom (DOF) lower-body model to an indi viduals move ment data. The whole body was modeled as a 3D, 14 segment, 27 DOF linkage joined by a set of gimbal, universal, and pin joints. For a given set of model parameters, the inner-level optimization uses a nonlinear least squares algorithm that adjusts each generalized coordinate of the lower-body model to minimize 3D marker coordinate errors between the model and motion data for each time instance. The outer-level optimization implements a parallel particle swarm algorithm that modifies each model parameter to minimize the sum of the squares of 3D marker coordinate errors computed by the inner-level optimization throughout all time instances (or the entire motion). xiii
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At the termination of each two-level optimization using synthetic marker data without noise, original marker trajectories were precisely recovered to within an arbitrarily tight tolerance (on the order of 1e-13 cm) using double precision computations. At the termination of each two-level optimization using synthetic marker data with noise representative of skin and soft tissue movement artifacts, the mean marker distance error for each joint complex was as follows: ankle = 0.51 + 0.23 cm; knee = 0.39 + 0.15 cm; and hip = 0.47 + 0.20 cm. Mean marker distance errors are approximately one-half of the 1 cm maximum amplitude specified for the noise model. At the termination of each two-level optimization using experimental marker data from one subject, the mean marker distance error for each joint complex was less than or equal to the following: ankle = 0.38 + 0.19 cm; knee = 0.55 + 0.27 cm; and hip = 0.36 + 0.20 cm. Experimental mean marker distance error results are comparable to the results of the synthetic data with noise. The two-level optimization method effectively determines patient-specific model parameters defining a 3D lower-extremity model that is well suited to a particular subject. When compared to previous values in the literature, experimental results show reasonable agreement and demonstrate the necessity for the new approach. By minimizing fitness errors between the patient-specific model and experimental motion data, the resulting kinematic model provides an accurate foundation for future dynamic analyses and optimizations. xiv
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CHAPTER 1 INTRODUCTION Arthritis: The Nations Leading Cause of Disability In 1997, the Centers for Disease Control and Prevention (CDC) reported that 43 million (or 1 in 6) Americans suffered with arthritis. A 2002 CDC study showed that 70 million (a 63% increase in 5 years; or 1 in 3) Americans have arthritis ( CDC, 2003 ). Approximately two-thirds of individuals with arthritis are under 65 years old. As the population ages, the number of people with arthritis is likely to increase significantly. The most common forms of arthritis are osteoarthritis, rheumatoid arthritis, fibromyalgia, and gout. Osteoarthritis of the knee joint accounts for roughly 30% ($25 billion) of the $82 billion total arthritis costs per year in the United States. Knee osteoarthritis symptoms of pain and dysfunction are the primary reasons for total knee replacement (TKR). This procedure involves a resurfacing of bones surrounding the knee joint. The end of the femur is removed and covered with a metal implant. The end of the tibia is removed and substituted by a plastic implant. Smooth metal and plastic articulation replaces the irregular and painful arthritic surfaces. Approximately 100,000 Medicare patients alone endure TKR procedures each year ( Heck et al., 1998 ). Hospital charges for unilateral TKR are more than $30,000 and the cost of bilateral TKR is over $50,000 ( Lane et al., 1997 ). An alternative to TKR is a more conservative (both economically and surgically) corrective procedure known as high tibial osteotomy (HTO). By changing the frontal plane alignment of the tibia with a wedge of bone, a HTO shifts the weight-bearing axis 1
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2 of the leg, and thus the mechanical stresses, from the diseased portion to the healthy section of the knee compartment. By transferring the location of mechanical stresses, the degenerative disease process may be slowed or possibly reversed. The advantages of HTO are appealing to younger and active patients who receive recommendations to avoid TKR. Need for Accurate Patient-Specific Models Innovative patient-specific models and simulations would be valuable for addressing problems in orthopedics and sports medicine, as well as for evaluating and enhancing corrective surgical procedures ( Arnold et al., 2000 ; Arnold and Delp, 2001 ; Chao et al., 1993 ; Chao and Sim, 1995 ; Delp et al., 1998 ; Delp et al., 1996 ; Delp et al., 1990 ; Pandy, 2001 ). For example, a patient-specific dynamic model may be useful for planning intended surgical parameters and predicting the outcome of HTO. The main motivation for developing a patient-specific computational model and a two-level optimization method to enhance the lower-extremity portion is to predict the post-surgery peak knee adduction moment in HTO patients. Conventional surgical planning techniques for HTO involve choosing the amount of necessary tibial angulation from standing radiographs (or x-rays). Unfortunately, alignment correction estimates from static x-rays do not accurately predict long-term clinical outcome after HTO ( Andriacchi, 1994 ; Tetsworth and Paley, 1994 ). Researchers have identified the peak external knee adduction moment as an indicator of clinical outcome while investigating the gait of HTO patients ( Andriacchi, 1994 ; Bryan et al., 1997; Hurwitz et al., 1998 ; Prodromos et al., 1985 ; Wang et al., 1990 ). Currently, no movement simulations (or other methods for that matter) allow surgeons to choose HTO surgical parameters to achieve a chosen post-surgery knee adduction moment.
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3 Movement simulations consist of models involving skeletal structure, muscle paths, musculotendon actuation, muscle excitation-contraction coupling, and a motor task goal ( Pandy, 2001 ). Development of an accurate inverse dynamic model of the skeletal structure is a significant first step toward creating a predictive patient-specific forward dynamic model to perform movement simulations. The precision of dynamic analyses is fundamentally associated with the accuracy of kinematic model parameters such as segment lengths, joint positions, and joint orientations ( Andriacchi and Strickland, 1985 ; Challis and Kerwin, 1996 ; Cappozzo et al., 1975 ; Davis, 1992 ; Holden and Stanhope, 1998 ; Holden and Stanhope, 2000 ; Stagni et al., 2000 ). Understandably, a model constructed of rigid links within a multi-link chain and simple mechanical approximations of joints will not precisely match the human anatomy and kinematics. The model should provide the best possible agreement to experimental motion data within the bounds of the joint models selected ( Sommer and Miller, 1980 ). Benefits of Two-Level Optimization This thesis presents a nested (or two-level) system identification optimization approach to determine patient-specific joint parameters that best fit a three-dimensional (3D), 18 degree-of-freedom (DOF) lower-body model to an indi viduals move ment data. The two-level technique combines the advantages of using optimization to determine both the position of model segments from marker data and the anatomical joint axes linking adjacent segments. By formulating a two-level objective function to minimize marker coordinate errors, the resulting optimum model more accurately represents experimental marker data (or a specific patient and his or her motion) when compared to a nominal model defined by joint axes prediction methods.
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CHAPTER 2 BACKGROUND Motion Capture Motion capture is the use of external devices to capture the movement of a real object. One type of motion-capture technology is based on a passive optical technique. Passive refers to markers, which are simply spheres covered in reflective tape, placed on the object. Optical refers to the technology used to provide 3D data, which involves high-speed, high-resolution video cameras. By placing passive markers on an object, special hardware records the position of those markers in time and it generates a set of motion data (or marker data). Often motion capture is used to create synthetic actors by capturing the motions of real humans. Special effects companies have used this technique to produce incredibly realistic animations in movies such as Star Wars Episode I & II, Titanic, Batman, and Terminator 2. Biomechanical Models Researchers use motion-capture technology to construct biomechanical models of the human structure. The position of external markers may be used to estimate the position of internal landmarks such as joint centers. The markers also enable the creation of individual segment reference frames that define the position and orientation of each body segment within a Newtonian laboratory reference frame. Marker data collected from an individual are used to prescribe the motion of the biomechanical model. 4
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5 Kinematics and Dynamics Human kinematics is the study of the positions, angles, velocities, and accelerations of body segments and joints during motion. With kinematic data and mass-distribution data, one can study the forces and torques required to produce the recorded motion data. Errors between the biomechanical model and the recorded motion data will inevitably propagate to errors in the force and torque results of dynamic analyses. Optimization Optimization involves searching for the minimum or maximum of an objective function by adjusting a set of design variables. For example, the objective function may be the errors between the biomechanical model and the recorded motion data. These errors are a function of the m odels genera lized coordinates and the m odels kinem atic parameters such as segment lengths, joint positions, and joint orientations. Optimization may be used to modify the design variables of the model to minimize the overall fitness errors and identify a structure that matches the experimental data very well. Limitations of Previous Methods The literature contains a number of examples that use techniques, with or without optimization, to assist in the development of subject-specific joint models within a larger computational model. Several authors have presented methodologies to predict joint locations and orientations from external landmarks without using optimization ( Bell et al., 1990 ; Inman, 1976 ; Vaughan et al., 1992 ). However, a regression model based solely upon population studies may not accurately portray an individual patient. Another study demonstrated an optimization method to determine the position and orientation of a 3 link, 6 DOF model by minimizing the distances between model-determined and experimental marker positions ( Lu and OConnor, 1999 ). A model optimally positioned
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6 without adjusting its joint parameters may not properly correspond to a certain patient. Earlier studies described optimization methods to determine a set of model parameters for a 3D, 2 DOF model by decreasing the error between the motion of the model and experimental data ( Sommer and Miller, 1980 ; Bogert et al., 1994 ). A model defined by optimal joint parameters without optimizing its segment positions may not accurately describe the motion of a patient within the bounds of the chosen joint approximations.
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CHAPTER 3 METHODS Parametric Model Structure A generic, parametric 3D full-body kinematic model was constructed with Autolev (Online Dynam ics, Inc., Sunnyvale, CA) as a 14 segment, 27 DOF linkage joined by a set of gimbal, universal, and pin joints ( Figure 3-1 Table 3-1 ). Comparable to Pandy's ( 2001 ) model structure, 3 translational degrees of freedom (DOFs) (q1, q2, and q3) and 3 rotational DOFs (q4, q5, and q6) express the movement of the pelvis in 3D space and the remaining 13 body segments comprise four open chains branching from the pelvis segment. The locations and orientations of the joints within corresponding body segments are described by 98 patient-specific model parameters. In other words, the patient-specific model parameters designate the geometry of the model containing the following joints types: 3 DOF hip, 1 DOF knee, 2 DOF ankle, 3 DOF back, 2 DOF shoulder, and 1 DOF elbow. Each joint is defined in two adjacent body segments and provides a mechanical approximation connecting those segments ( Figure 3-2 ). For example, the knee joint axis is simultaneously established in the femur coordinate system and the tibia coordinate system. A modified version of the Cleveland Clinic marker set ( Figure 3-3 ) and a static motion-capture trial is used to create segment coordinate systems and define static and dynamic marker locations in these coordinate systems. Institutional review board approval and proper informed consent were obtained before human involvement in the experiments. The marker data collection system was a HiRes Expert Vision System 7
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8 (Motion Analysis Corp., Santa Rosa, CA), including six HSC-180 cameras, EVa 5.11 software, and two AMTI force plates (Advanced Management Technology, Inc., Arlington, VA). Marker data were collected at 180 Hz during 3 seconds for static trials and 6 seconds for individual joint trials. The raw data were filtered using a fourth-order, zero phase-shift, low pass Butterworth Filter with a cutoff frequency set at 6 Hz. Hip Joint There are 6 translational model parameters that must be adjusted to establish a functional hip joint center for a particular patient ( Figure 3-4 Table 3-2 ). Markers placed over the left anterior superior iliac spine (ASIS), right ASIS, and superior sacrum define the pelvis segment coordinate system. From percentages of the inter-ASIS distance, a predicted (or nominal) hip joint center location within the pelvis segment is 19.3% posterior (p1), 30.4% inferior (p2), and 35.9% medial-lateral (p3) ( Bell et al., 1990 ). This nominal hip joint center is the origin of the femur coordinate system, which is subsequently defined by markers placed over the medial and lateral femoral epicondyles. An additional 3 translational model parameters (p4, p5, and p6), described in the femur coordinate system, complete the structure of the nominal hip joint center. Given the physical hip joint center is located within the pelvic region lateral to the midsagittal plane, a cube with side lengths equal to 75% of the inter-ASIS distance and its anterior-superior-medial vertex positioned at the midpoint of the inter-ASIS line provides the geometric constraints for the optimization of each model parameter ( Figure 3-5 Table A-1 Table B-1 ). Knee Joint There are 9 model parameters (5 translational and 4 rotational) that must be tailored to identify a patient-specific functional knee joint axis ( Figure 3-6 Table 3-3 ). The
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9 femoral transepicondylar axis is a good approximation of a fixed knee joint axis ( Churchill et al., 1998 ). The line (or nominal) knee joint axis, connecting the medial and lateral knee markers is defined in the femur and tibia coordinate systems ( Vaughan et al., 1992 ). Given the line passes through the midsagittal plane (x-y plane) of the femur segment, the nominal knee joint axis is positioned within the femur via 2 translational model parameters (p5 and p6) and 2 rotational model parameters (p1 and p2). The tibia coordinate system originates at the midpoint of the knee markers and is defined by additional markers located on the medial and lateral malleoli. The distal description of the nominal knee joint axis is comprised of 3 translational model parameters (p7, p8, and p9) and 2 rotational model parameters (p3 and p4) in the tibia segment. Given the anatomical knee joint DOFs are situated within the articular capsule, a cube with side lengths equal to the distance between knee markers and its center located at the midpoint of the nominal knee joint axis provides the geometric constraints for the optimization of each translational model parameter. The rotational model parameters are constrained within a circular cone defined by the 360 revolution of the nominal knee joint axis perturbed by + 30 ( Figure 3-5 Table A-2, Table B-2 ). It is not a trivial notion to eliminate a potential medial-lateral translational model parameter in the femur segment. This model parameter is considered redundant, as the knee joint axis passes through the midsagittal plane of the femur, and its inclusion may lead to possible optimization convergence problems, similar to the redundant ankle model parameter discussion of Bogert et al. ( 1994 ). By including redundant model parameters, there are an infinite number of optimum solutions within the constraints of corresponding superfluous model parameters.
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10 Ankle Joint There are 12 patient-specific model parameters (7 translational and 5 rotational) that must be customized to determine a pair of patient-specific functional ankle joint axes ( Figure 3-7 Table 3-4 ). Comparable to Bogert et al. ( 1994 ), the talocrural and subtalar joints connect the tibia, talus, and foot segments. Within the tibia segment, 3 translational model parameters (p6, p7, and p8) and 2 rotational model parameters (p1 and p2) position the nominal talocrural joint axis. The talus origin corresponds to the talocrural joint center; therefore, it is not necessary to prescribe model parameters defining the talocrural joint axis in the talus segment. The talus coordinate system is created where the y-axis extends along the line perpendicular to both the talocrural joint axis and the subtalar joint axis. The heel and toe markers, in combination with the tibia y-axis, define the foot coordinate system. There are 3 translational model parameters (p10, p11, and p12) and 2 rotational model parameters (p4 and p5) ( Inman, 1976 ) that place the nominal subtalar joint axis in the foot coordinate system. Given the anatomical ankle joint DOFs are found within the articular capsule, a cube with side lengths equal to the distance between ankle markers and its center located at the midpoint of the nominal talocrural joint axis provides the geometric constraints for the optimization of each translational model parameter. The rotational model parameters of the talocrural joint axis are restricted within a circular cone defined by the 360 revolution of the nominal talocrural joint axis varied by + 30. The rotational model parameters of the subtalar joint axis are confined within a circular cone defined by the 360 revolution of the nominal subtalar joint axis altered by + 30 ( Figure 3-5 Table A-3, Table B-3 ).
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11 Two-Level Optimization Approach Why Two Levels of Optimization Are Necessary Optimization may be used to identify a system (or determine patient-specific joint parameters) that best fit a 3D, 18 DOF lower-body m odel to an individuals m ovement data. One level of optimization is necessary to establish the models geom etry. Given a defined model, another level of optimization is required to position and orientate the models body segm ents. By formulating a two-level objective function to minimize 3D marker coordinate errors, the two-level optimization results describe a lower-body model that accurately represents experimental data. Inner-Level Optimization Given marker trajectory data, md, and a constant set of patient-specific model parameters, p the inner-level optimization ( Figure 3-8 inner boxes) minimizes the 3D marker coordinate errors, ec, between the model markers, mm, and the marker movement data, md, ( Equation 3-1 ) using a nonlinear least squares algorithm that adjusts the generalized coordinates, q of the model at each instance in time, t ( Figure 3-9 ), similar to Lu and OConnor ( 1999 ). In other words, the pose of the model is revised to match the marker movement data at each time frame of the entire motion. (q, p, t) m (t) m (q, p, t) em d c min(3-1) At the first time instance, the algorithm is seeded with exact values for the 6 generalized coordinates of the pelvis, since the marker locations directly identify the position and orientation of the pelvis coordinate system, and all remaining generalized coordinates are seeded with values equal to zero. Given the joint motion is continuous, each optimal generalized coordinate solution, including the pelvis generalized
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12 coordinates, at one time instance is used as the algorithm s seed for the next time instance. Matlab 6.1 (The MathWorks, Inc., Natick, MA), in conjunction with the Matlab Optimization Toolbox and Matlab C/C++ Compiler, was used to develop the inner-level optimization program. Outer-Level Optimization The outer-level global optimization ( Figure 3-8 outer boxes) minimizes the sum of the squares, ess, of the 3D marker coordinate errors, ec, ( Equation 3-1 ) computed by the inner-level algorithm throughout all time instances, n ( Equation 3-2 ) by modifying the patient-specific model parameters, p In other words, the geometric structure of the model is varied to best fit the marker movement data for the entire motion. n t c T c ss(q, p, t) e (q, p, t) e (q, p, n) e1 min (3-2) The outer-level optimization is adapted from the population-based Particle Swarm Optimizer (PSO) ( Kennedy and Eberhart, 1995 ). The PSO algorithm was chosen over gradient-based optimizers for its suitability to be parallelized and its ability to solve global optimization problems. It is particularly effective in the determination of joint positions and orientations of biomechanical systems ( Schutte et al., 2003 ). The work of Schutte et al. ( 2003 ) contrasted the PSO to a gradient-based optimizer (i.e., Broyden-Fletcher-Goldfarb-Shanno) that is commonly used in system identification problems involving biomechanical models. The PSO very reliably converged to the global minimum and it was insensitive to both design variable scaling and initial seeds ( Schutte et al., 2003 ). To manage computational requirements, the outer-level optimization uses a parallel version of the PSO operating on a cluster of 20 Linux-based 1.33 GHz Athlon PCs on a
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13 100 Mbps switched Fast Ethernet network. Each machine is separately seeded with a random set of initial patient-specific model parameter values. The outer-level optimization program was implemented in C on the Linux operating system with the Message Passing Interface (MPI) parallel computation libraries. Two-Level Optimization Evaluation Synthetic Marker Data without Noise To evaluate the ability of the two-level optimization approach ( Figure 3-10 ) to calibrate the generic, parametric kinematic model, synthetic movement data was generated for the ankle, knee, and hip joints based on estimated in vivo model parameters and experimental movement data. For each generated motion, the distal segment moved within the physiological range of motion and exercised each DOF for the joint. There were 50 time frames and approximately 3.5 cycles of a circumductive hip motion consisting of concurrent flexion-extension and abduction-adduction. Flexion-extension comprised 50 time frames and roughly 4 cycles of knee motion. The ankle motion involved 50 time frames and nearly 2.75 cycles of circumduction of the toe tip, where plantarflexion-dorsiflexion and inversion-eversion occurred simultaneously. The ability of the two-level optimization to recover the original model parameters used when generating the synthetic motions was assessed. Synthetic Marker Data with Noise To evaluate the ability of the two-level optimization method ( Figure 3-10 ) to calibrate the generic kinematic model to a synthetic patient, skin movement artifacts were introduced into the synthetic movement data for the ankle, knee, and hip joints. The relative movement between skin and underlying bone occurs in a continuous rather than a random fashion ( Cappozzo et al., 1993 ). Comparable to the simulated skin movement
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14 artifacts of Lu and OConnor ( 1999 ), a continuous numerical noise model of the form t A sin was used and the equation variables were randomly generated within the following bounds: amplitude (0 A 1 cm), frequency (0 25 rad/s), and phase angle (0 2 ) ( Chze et al., 1995 ). Noise was separately generated for each 3D coordinate of the marker trajectories. Again, the two-level optimization was tested for its ability to reproduce the original model parameters. Experimental Marker Data To verify the ability of the two-level optimization technique ( Figure 3-10 ) to calibrate the generic kinematic model to a particular patient, multi-cycle experimental marker trajectory data was collected from one subject. For each joint motion, the distal segment moved within the physiological range of motion and exercised each DOF for the joint. Analogous to Bogert et al. ( 1994 ), the original data were resampled non-equidistantly to eliminate weighting the data set with many data points occurring during acceleration and deceleration at the limits of the range of motion. In other words, regardless of changes in velocity during joint movements, the data was equally distributed over the entire joint range of motion. The time frames of original tracked marker data sets (right hip = 1015, right knee = 840, and right ankle = 707) were reduced to 50 time frames. The resampled data allowed a fixed amount of marker movement between frames to arrive at the number of time frames chosen, given that 50 time frames is analogous to Lu and OConnor ( 1999 ). There were nearly 2 cycles of flexion-extension and abduction-adduction involved in the hip motion. Similar to Leardini et al. ( 1999 ), internal-external rotation of the hip was avoided to reduce the effects of skin and soft tissue movement artifacts. Approximately 2 cycles of knee
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15 motion included flexion-extension. Simultaneous plantarflexion-dorsiflexion and inversion-eversion comprised roughly 2 cycles of ankle motion. Without knowledge of original model parameters, the marker coordinate errors are the only means of measuring the effectiveness of the two-level optimization. To verify the ability of the two-level optimization procedure ( Figure 3-10 ) to calibrate the generic kinematic model to a particular patient using a smaller portion of the joint motion cycle, the resampled multi-cycle experimental marker trajectory data described above was divided into the first and second halves of the individual hip, knee, and ankle joint motion cycles. The number of time frames comprising each one-half-cycle of the joint motion was as follows: ankle = 13, knee = 13, and hip = 19. Again, the two-level optimization was tested for its ability to reduce the marker coordinate errors and obtain an optimal set of model parameters.
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16 3-1. The 3D, 14 segment, 27 DOF full-body kinematic model linkage joined by a set of gimbal, universal, and pin joints. Figure
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17 3-1. Model degrees of freedom. Table DOF Description q1 Pelvis anterior-posterior position q2 Pelvis superior-inferior position q3 Pelvis medial-lateral position q4 Pelvis anterior-posterior tilt angle q5 Pelvis elevation-depression angle q6 Pelvis internal-external rotation angle q7 Right hip flexion-extension angle q8 Right hip adduction-abduction angle q9 Right hip internal-external rotation angle q10 Right knee flexion-extension angle q11 Right ankle plantarflexion-dorsiflexion angle q12 Right ankle inversion-eversion angle q13 Left hip flexion-extension angle q14 Left hip adduction-abduction angle q15 Left hip internal-external rotation angle q16 Left knee flexion-extension angle q17 Left ankle plantarflexion-dorsiflexion angle q18 Left ankle inversion-eversion angle q19 Trunk anterior-posterior tilt angle q20 Trunk elevation-depression angle q21 Trunk internal-external rotation angle q22 Right shoulder flexion-extension angle q23 Right shoulder adduction-abduction angle q24 Right elbow flexion angle q25 Left shoulder flexion-extension angle q26 Left shoulder adduction-abduction angle q27 Left elbow flexion angle
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18 3-2. A 1 DOF joint axis simultaneously defined in two adjacent body segments and the geometric constraints on the optimization of each of the 9 model parameters. Figure
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19 3-3. Modified Cleveland Clinic marker set used during static and dynamic motion-capture trials. Note: the background femur and knee markers have been omitted for clarity and the medial and lateral markers for the knee and ankle are removed following the static trial. Figure
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20 3-4. The 3 DOF right hip joint center simultaneously defined in the pelvis and right femur segments and the 6 translational model parameters optimized to determine the functional hip joint center location. Figure 3-2. Hip joint parameters. Table Hip Joint Parameter Description p1 Anterior-posterior location in pelvis segment p2 Superior-inferior location in pelvis segment p3 Medial-lateral location in pelvis segment p4 Anterior-posterior location in femur segment p5 Superior-inferior location in femur segment p6 Medial-lateral location in femur segment
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21 3-5. Geometric constraints on the optimization of translational and rotational model parameters for the hip, knee, and ankle joints. Figure
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22 3-6. The 1 DOF right knee joint simultaneously defined in the right femur and right tibia segments and the 4 rotational and 5 translational model parameters optimized to determine the knee joint location and orientation. Figure
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23 3-3. Knee joint parameters. Table Knee Joint Parameter Description p1 Adduction-abduction rotation in femur segment p2 Internal-external rotation in femur segment p3 Adduction-abduction rotation in tibia segment p4 Internal-external rotation in tibia segment p5 Anterior-posterior location in femur segment p6 Superior-inferior location in femur segment p7 Anterior-posterior location in tibia segment p8 Superior-inferior location in tibia segment p9 Medial-lateral location in tibia segment
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24 3-7. The 2 DOF right ankle joint complex simultaneously defined in the right tibia, talus, and foot segments and the 5 rotational and 7 translational model parameters optimized to determine the joint locations and orientations. Figure
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25 3-4. Ankle joint parameters. Table Ankle Joint Parameter Description p1 Adduction-abduction rotation of talocrural in tibia segment p2 Internal-external rotation of talocrural in tibia segment p3 Internal-external rotation of subtalar in talus segment p4 Internal-external rotation of subtalar in foot segment p5 Dorsi-plantar rotation of subtalar in foot segment p6 Anterior-posterior location of talocrural in tibia segment p7 Superior-inferior location of talocrural in tibia segment p8 Medial-lateral location of talocrural in tibia segment p9 Superior-inferior location of subtalar in talus segment p10 Anterior-posterior location of subtalar in foot segment p11 Superior-inferior location of subtalar in foot segment p12 Medial-lateral location of subtalar in foot segment
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26 3-8. Two-level optimization technique minimizing the 3D marker coordinate errors between the kinematic model markers and experimental marker data to determine functional joint axes for each lower-extremity joint. Figure
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27 3-9. Inner-level optim ization convergence il lustration series for the knee joint, where synthetic m a rkers are blue and m odel m a rkers are red. Given synthetic m a rker data without noise, optim ized outer-level design variables, and a synthetic knee flexion angle = 90, A) is the initial m odel knee flexion = 0, B) is the m odel knee flexion = 30, C) is the m odel knee flexion = 60, and D) is the final m o del knee flexion = 90. Figure
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28 3-10. Two-level optimization approach minimizing the 3D marker coordinate errors between the kinematic model markers and experimental marker data to determine functional joint axes. Figure
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CHAPTER 4 RESULTS Synthetic Marker Data without Noise For synthetic motions without noise, each two-level optimization precisely recovered the original marker trajectories to within an arbitrarily tight tolerance (on the order of 1e-13 cm), as illustrated in Figure 3-9 At the termination of each optimization, the optimum model parameters for the hip, knee, and ankle were recovered with mean rotational errors less than or equal to 0.045 and mean translational errors less than or equal to 0.0077 cm ( Appendix C ). Synthetic Marker Data with Noise For synthetic motions with noise, the two-level optimization of the hip, knee, and ankle resulted in mean marker distance errors equal to 0.46 cm, which is of the same order of magnitude as the selected random continuous noise model ( Table 4-1) The two-level approach determined the original model parameters with mean rotational errors less than or equal to 3.73 and mean translational errors less than or equal to 0.92 cm ( Appendix D ). The outer-level fitness history converged rapidly ( Figure 4-1 ) and the hip, knee, and ankle optimizations terminated with a mean wall clock time of 41.02 hours. Experimental Marker Data For multi-cycle experimental motions, the mean marker distance error of the optimal hip, knee, and ankle solutions was 0.41 cm, which is a 0.43 cm improvement over the mean nominal error of 0.84 cm ( Table 4-2 ). For each joint complex, the optimum model parameters improved upon the nominal parameter data (or values found 29
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30 in the literature) by mean rotational values less than or equal to 6.18 and mean translational values less than or equal to 1.05 cm ( Appendix E ). When compared to the synthetic data with noise, the outer-level fitness history of the multi-cycle experimental data optimization converged at approximately the same rate and resulted in an improved final solution for both the ankle and the hip ( Figure 4-2 ). On the contrary, the higher objective function values for the knee are evidence of the inability of the fixed pin joint to represent the screw-home motion ( Blankevoort et al., 1988 ) of the multi-cycle experimental knee data. The multi-cycle hip, knee, and ankle optimizations terminated with a mean wall clock time of 35.94 hours. For one-half-cycle experimental motions, the mean marker distance error of the optimal hip, knee, and ankle solutions was 0.30 cm for the first half and 0.30 cm for the second half ( Table 4-3 ). The fitness of both the ankle and the hip were comparable to the multi-cycle joint motion results. However, the knee fitness values were improved due to the reduced influence (i.e., 1 time frame of data as opposed to 9) of the screw-home motion of the knee. For each joint complex, the optimum model parameters improved upon the nominal parameter data (or values found in the literature) by mean rotational values less than or equal to 11.08 and mean translational values less than or equal to 2.78 cm ( Appendix F Appendix G ). In addition, the optimum model parameters for one-half-cycle motion differed from those for the multi-cycle motion by mean rotational values less than or equal to 15.77 and mean translational values less than or equal to 2.95 cm ( Appendix H Appendix I ). The one-half-cycle hip, knee, and ankle optimizations terminated with a mean wall clock time of 11.77 hours.
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31 4-1. Two-level optimization results for synthetic marker data with random continuous numerical noise to simulate skin movement artifacts with maximum amplitude of 1 cm. Table Synthetic Data with Noise Hip Knee Ankle Mean marker distance error (cm) 0.474603 + 0.2022480.392331 + 0.1459290.514485 + 0.233956 Mean rotational parameter error () n/a 2.158878 + 1.2887033.732191 + 3.394553 Mean translational parameter error (cm) 0.161318 + 0.0394490.321930 + 0.1279970.923724 + 0.471443
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32 4-1. Outer-level optimization objective function fitness value convergence for synthetic marker data with random continuous numerical noise to simulate skin movement artifacts with maximum amplitude of 1 cm, where the best fitness value among all nodes is given for each iteration. Figure
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33 4-2. Mean marker distance errors for nominal values and the two-level optimization results for multi-cycle experimental marker data. Table Experimental Data Hip Knee Ankle Nominal mean marker distance error (cm) 0.499889 + 0.1779471.139884 + 0.6185670.885437 + 0.478530 Optimum mean marker distance error (cm) 0.342262 + 0.1670790.547787 + 0.2697260.356279 + 0.126559 Mean marker distance error attenuation (cm) 0.157627 + 0.1662360.592097 + 0.4436800.529158 + 0.438157
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34 4-2. Outer-level optimization objective function fitness value convergence for multi-cycle experimental marker data, where the best fitness value among all nodes is given for each iteration. Figure
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35 4-3. Mean marker distance errors for the two-level optimization results using first and second halves of the joint cycle motion for experimental marker data. Table Experimental Data Hip Knee Ankle First half: mean marker distance error (cm) 0.335644 + 0.1633700.189551 + 0.0729960.384786 + 0.193149 Second half: mean marker distance error (cm) 0.361179 + 0.2007740.202413 + 0.1010630.338886 + 0.128596
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CHAPTER 5 DISCUSSION Assumptions, Limitations, and Future Work Joint Model Selection If the current model cannot adequately reproduce future experimental motions, the chosen joint models may be modified. For example, the flexion-extension of the knee is not truly represented by a fixed pin joint ( Churchill et al., 1998 ). When comparing the fitness of the optimum knee joint model to multi-cycle experimental marker data, the agreement was quite good for all knee flexion angles with the exception of those approaching full extension. By eliminating knee flexion angles less than 20, which comprised 18% of the flexion-extension data, the mean marker distance error was reduced to 0.48 + 0.23 cm (11.89% decrease) using the optimum model parameters from the full data set. A pin joint knee may be sufficiently accurate for many modeling applications. A 2 DOF knee model ( Hollister et al., 1993 ) may account for the screw-home motion of the knee joint occurring between 0 and 20 ( Blankevoort et al., 1988 ). If greater fidelity to actual bone motion is necessary, a 6 DOF knee joint may be implemented with kinematics determined from fluoroscopy ( Rahman et al., 2003 ). Design Variable Constraints Certain joint parameters must be constrained to zero with the purpose of preventing the unnecessary optimization of redundant parameters. Case in point, the medial-lateral translational model parameter placing the knee joint center in the femur segment must be constrained to zero. On the other hand, this model parameter may be used as a design 36
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37 variable, granted the medial-lateral translational model parameter placing the knee joint center in the tibia segment is constrained to zero. If both medial-lateral translational model parameters are used as redundant design variables, the outer-level optimization has an infinite number of solutions within the constraints of both parameters. Through the elimination (i.e., constraining to zero) of redundant model parameters, the outer-level optimization encounters less convergence problems in globally minimizing the objective function. Objective Function Formulation The inner-level optimization objective function should be comprised of marker coordinate errors rather than marker distance errors. A substantial amount of information (i.e., of the number of errors) describing the fitness value is lost with computation of marker distance errors. In other words, a marker distance error provides only the radius of a sphere surrounding an experimental marker and it does not afford the location of a model marker on the surface of the sphere. However, a set of three marker coordinate errors describes both the magnitude and direction of an error vector between an experimental marker and a model marker. By using marker coordinate errors, the inner-level optimization has improved convergence ( Table 5-1 ) and shorter execution time ( Table 5-2 ). Optimization Time and Parallel Computing To reduce the computation time, it is necessary to use an outer-level optimization algorithm in a parallel environment on a network cluster of processors. The PSO algorithm was chosen over gradient-based optimizers for its suitability to be parallelized and its ability to solve global optimization problems. The large computation time is a result of the random set of initial values used to seed each node of the parallel algorithm.
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38 By seeding one of the nodes with a relatively optimal set of initial values, the computation time may be significantly decreased. By doubling the number of parallel processors, the computation time declines nearly 50%. Decreasing the number of time frames of marker data additionally reduces the computation time. For example, the mean optimization time using experimental data for 50 time frames equals 35.94 hours, 19 time frames equals 12.82 hours, and 13 time frames equals 11.24 hours. Further study is necessary to establish the minimum number of marker data time frames required to effectively determine joint axes parameters. Multi-Cycle and One-Half-Cycle Joint Motions The two-level optimization results vary depending on whether marker data time frames consist of multi-cycle or one-half-cycle joint motions. In other words, the determination of patient-specific model parameters is significantly influenced by the marker trajectories contained within the chosen set of data. Given a set of marker data, the two-level optimization establishes invariable model parameters that best fit the mathematical model to the measured experimental motion. Understandably, a model constructed from one marker data set may not adequately represent a considerably different marker data set. To perform accurate dynamic analyses, joint motions used to generate the model should be consistent with those motions that will be used in the analyses. The small differences between sets of two-level optimization results for the hip and knee joint motions indicate the reliability of the model parameter values. Much larger differences occurred between sets of model parameters determined for the ankle joint. Two major factors contributing to these differences are the rotational ankle model parameters p1 and p3. On one hand, the model parameters may truly vary throughout the
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39 ankle motion and may not be represented by constant values. On the other hand, the objective function may be insensitive to changes in these model parameters indicating a design space that does not permit the reasonable determination of certain design variables. Future study is necessary to investigate the sensitivity of 3D marker coordinate errors to particular model parameters. Range of Motion and Loading Conditions To provide the largest range of motion, all experimental data was collected with each joint unloaded and freely exercising all DOFs; however, the same two-level optimization may be performed on loaded data as well. The patient-specific model parameters may change under loaded conditions ( Bogert et al., 1994 ). Moreover, loaded conditions limit the range of motion for several DOFs. Several authors ( Bell et al., 1990 ; Bogert et al., 1994 ) report inaccuracies in determining functional axes from limited motion, but a subsequent study ( Piazza et al., 2001 ) found the hip joint may be determined from motions as small as 15. Piazza et al. ( 2001 ) suggest future studies are necessary to explore the use of normal gait motions, rather than special joint motions, to determine functional axes. Optimization Using Gait Motion The two-level optimization approach and synthetic data evaluation method may be used to investigate the use of gait motion to determine functional joint axes. Each set of joint parameters may be established separately or collectively (i.e., entire single leg or both legs at once). Additional investigation is necessary to assess the differences in joint parameters obtained through individual optimizations and simultaneous whole leg optimizations. Furthermore, the joint parameters determined from gait motions may be
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40 compared to those parameters obtained from special joint motions with larger amounts of movement. Authors ( Bogert et al., 1994 ; Chze et al., 1995 ; Lu and OConnor, 1999 ) have set precedence for performing numerical (or synthetic data) simulations to evaluate a new technique. Although it is not a necessary task, there is additional benefit in supporting the numerical findings with data from one human subject. With the additional data, the joint parameters computed from unloaded joint motions may be measured against those parameters attained from unloaded (i.e., swing phase) and loaded (i.e., stance phase) gait motions. To expand upon the evaluation of the new technique and show general applicability, future work is necessary to study more than one human subject. Comparison of Experimental Results with Literature The two-level optimization determined patient-specific joint axes locations and orientations similar to previous works. The optimum hip joint center location of 7.52 cm (27.89% posterior), 9.27 cm (34.38% inferior), and 8.86 cm (32.85% lateral) are respectively comparable to 19.3%, 30.4%, and 35.9% ( Bell et al., 1990 ). The optimum femur length (40.46 cm) and tibia length (40.88 cm) are similar to 42.22 cm and 43.40 cm, respectively ( de Leva, 1996 ). The optimum coronal plane rotation (73.36) of the talocrural joint correlates to 82.7 + 3.7 (range 74 to 94) ( Inman, 1976 ). The optimum distance (2.14 cm) between the talocrural joint and the subtalar joint is analogous to 1.24 + 0.29 cm ( Bogert et al., 1994 ). The optimum transverse plane rotation (13.19) and sagittal plane rotation (45.26) of the subtalar joint corresponds to 23 + 11 (range 4 to 47) and 42 + 9 (range 20.5 to 68.5), respectively ( Inman, 1976 ).
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41 5-1. Mean marker distance errors for the inner-level objective function consisting of marker coordinate errors versus marker distance errors for multi-cycle experimental marker data. Table Experimental Data Hip Knee Ankle Marker distance objective function: mean marker distance error (cm) 0.863941 + 0.3287941.043909 + 0.4651860.674187 + 0.278451 Marker coordinate objective function: mean marker distance error (cm) 0.342262 + 0.1670790.547787 + 0.2697260.356279 + 0.126559
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42 5-2. Execution times for the inner-level objective function consisting of marker coordinate errors versus marker distance errors for multi-cycle experimental marker data. Table Experimental Data Hip Knee Ankle Marker distance objective function: execution time (s) 464.377 406.205 308.293 Marker coordinate objective function: execution time (s) 120.414 106.003 98.992
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CHAPTER 6 CONCLUSION Rationale for New Approach The main motivation for developing a 27 DOF patient-specific computational model and a two-level optimization method to enhance the lower-extremity portion is to predict the post-surgery peak knee adduction moment in HTO patients, which has been identified as an indicator of clinical outcome ( Andriacchi, 1994 ; Bryan et al., 1997 ; Hurwitz et al., 1998 ; Prodromos et al., 1985 ; Wang et al., 1990 ). The accuracy of prospective dynamic analyses made for a unique patient is determined in part by the fitness of the underlying kinematic model ( Andriacchi and Strickland, 1985 ; Challis and Kerwin, 1996 ; Cappozzo et al., 1975; Davis, 1992 ; Holden and Stanhope, 1998 ; Holden and Stanhope, 2000 ; Stagni et al., 2000 ). Development of an accurate kinematic model tailored to a specific patient forms the groundwork toward creating a predictive patient-specific dynamic simulation. Synthesis of Current Work and Literature The two-level optimization method satisfactorily determines patient-specific model parameters defining a 3D lower-extremity model that is well suited to a particular patient. Two conclusions may be drawn from comparing and contrasting the two-level optimization results to previous values found in the literature. The similarities between numbers suggest the results are reasonable and show the extent of agreement with past studies. The differences between values indicate the two-level optimization is necessary 43
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44 and demonstrate the degree of inaccuracy inherent when the new approach is not implemented. Through the enhancement of model parameter values found in the literature, the two-level optimization approach successfully reduces the fitness errors between the patient-specific model and the experimental motion data. More specifically, to quantify the improvement of the current results compared to previous values found in the literature, the mean marker distance errors were reduced by 31.53% (hip), 51.94% (knee), and 59.76% (ankle). The precision of dynamic analyses made for a particular patient depends on the accuracy of the patient-specific kinematic parameters chosen for the dynamic model. Without expensive medical images, model parameters are only estimated from external landmarks that have been identified in previous studies. The estimated (or nominal) values may be improved by formulating an optimization problem using motion-capture data. By using a two-level optimization technique, researchers may build more accurate biomechanical models of the individual human structure. As a result, the optimal models will provide reliable foundations for future dynamic analyses and optimizations.
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GLOSSARY Abduction Movement away from the midline of the body in the coronal plane. Acceleration The time rate of change of velocity. Active markers Joint and segment markers used during motion capture that emit a signal. Adduction Movement towards the midline of the body in the coronal plane. Ankle inversion-eversion Motion of the long axis of the foot within the coronal plane as seen by an observer positioned along the anterior-posterior axis of the shank. Ankle motion The ankle angles reflect the motion of the foot segment relative to the shank segment. Ankle plantarflexion-dorsiflexion Motion of the plantar aspect of the foot within the sagittal plane as seen by an observer positioned along the medial-lateral axis of the shank. Anterior The front or before, also referred to as ventral. Circumduction Movement of the distal tip of a segment described by a circle. Coccyx The tailbone located at the distal end of the sacrum. Constraint functions Specific limits that must be satisfied by the optimal design. Coronal plane The plane that divides the body or body segment into anterior and posterior parts. Couple A set of force vectors whose resultant is equal to zero. Two force vectors with equal magnitudes and opposite directions is an example of a simple couple. 45
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46 Degree of freedom (DOF) A single coordinate of relative motion between two bodies. Such a coordinate responds without constraint or imposed motion to externally applied forces or torques. For translational motion, a DOF is a linear coordinate along a single direction. For rotational motion, a DOF is an angular coordinate about a single, fixed axis. Design variables Variables that change to optimize the design. Distal Away from the point of attachment or origin. Dorsiflexion Movement of the foot towards the anterior part of the tibia in the sagittal plane. Epicondyle Process that develops proximal to an articulation and provides additional surface area for muscle attachment. Eversion A turning outward. Extension Movement that rotates the bones comprising a joint away from each other in the sagittal plane. External (lateral) rotation Movement that rotates the distal segment laterally in relation to the proximal segment in the transverse plane, or places the anterior surface of a segment away from the longitudinal axis of the body. External moment The load applied to the human body due to the ground reaction forces, gravity and external forces. Femur The longest and heaviest bone in the body. It is located between the hip joint and the knee joint. Flexion Movement that rotates the bones comprising a joint towards each other in the sagittal plane. Fluoroscopy Examination of body structures using an X-ray machine that combines an X-ray source and a fluorescent screen to enable real-time observation. Force A push or a pull and is produced when one object acts on another.
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47 Force plate A transducer that is set in the floor to measure about some specified point, the force and torque applied by the foot to the ground. These devices provide measures of the three components of the resultant ground reaction force vector and the three components of the resultant torque vector. Forward dynamics Analysis to determine the motion of a mechanical system, given the topology of how bodies are connected, the applied forces and torques, the mass properties, and the initial condition of all degrees of freedom. Gait A manner of walking or moving on foot. Generalized coordinates A set of coordinates (or parameters) that uniquely describes the geometric position and orientation of a body or system of bodies. Any set of coordinates that are used to describe the motion of a physical system. High tibial osteotomy (HTO) Surgical procedure that involves adding or removing a wedge of bone to or from the tibia and changing the frontal plane limb alignment. The realignment shifts the weight-bearing axis from the diseased medial compartment to the healthy lateral compartment of the knee. Hip abduction-adduction Motion of a long axis of the thigh within the coronal plane as seen by an observer positioned along the anterior-posterior axis of the pelvis. Hip flexion-extension Motion of the long axis of the thigh within the sagittal plane as seen by an observer positioned along the medial-lateral axis of the pelvis. Hip internal-external rotation Motion of the medial-lateral axis of the thigh with respect to the medial-lateral axis of the pelvis within the transverse plane as seen by an observer positioned along the longitudinal axis of the thigh. Hip motion The hip angles reflect the motion of the thigh segment relative to the pelvis. Inferior Below or at a lower level (towards the feet).
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48 Inter-ASIS distance The length of measure between the left anterior superior iliac spine (ASIS) and the right ASIS. Internal (medial) rotation Movement that rotates the distal segment medially in relation to the proximal segment in the transverse plane, or places the anterior surface of a segment towards the longitudinal axis of the body. Internal joint moments The net result of all the internal forces acting about the joint which include moments due to muscles, ligaments, joint friction and structural constraints. The joint moment is usually calculated around a joint center. Inverse dynamics Analysis to determine the forces and torques necessary to produce the motion of a mechanical system, given the topology of how bodies are connected, the kinematics, the mass properties, and the initial condition of all degrees of freedom. Inversion A turning inward. Kinematics Those parameters that are used in the description of movement without consideration for the cause of movement abnormalities. These typically include parameters such as linear and angular displacements, velocities and accelerations. Kinetics General term given to the forces that cause movement. Both internal (muscle activity, ligaments or friction in muscles and joints) and external (ground or external loads) forces are included. The moment of force produced by muscles crossing a joint, the mechanical power flowing to and from those same muscles, and the energy changes of the body that result from this power flow are the most common kinetic parameters used. Knee abduction-adduction Motion of the long axis of the shank within the coronal plane as seen by an observer positioned along the anterior-posterior axis of the thigh. Knee flexion-extension Motion of the long axis of the shank within the sagittal plane as seen by an observer positioned along the medial-lateral axis of the thigh.
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49 Knee internal-external rotation Motion of the medial-lateral axis of the shank with respect to the medial-lateral axis of the thigh within the transverse plane as viewed by an observer positioned along the longitudinal axis of the shank. Knee motion The knee angles reflect the motion of the shank segment relative to the thigh segment. Lateral Away from th e bodys longitudinal axis, or away from the midsagittal plane. Malleolus Broadened distal portion of the tibia and fibula providing lateral stability to the ankle. Markers Active or passive objects (balls, hemispheres or disks) aligned with respect to specific bony landmarks used to help determine segment and joint position in motion capture. Medial Toward the bodys longitudinal axis, or toward the midsagittal plane. Midsagittal plane The plane that passes through the midline and divides the body or body segment into the right and left halves. Model parameters A set of coordinates that uniquely describes the model segments lengths, joint locations, and joint orientations, also referred to as joint parameters. Any set of coordinates that are used to describe the geometry of a model system. Moment of force The moment of force is calculated about a point and is the cross product of a position vector from the point to the line of action for the force and the force. In two-dimensions, the moment of force about a point is the product of a force and the perpendicular distance from the line of action of the force to the point. Typically, moments of force are calculated about the center of rotation of a joint. Motion capture Interpretation of computerized data that documents an individual's motion.
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50 Non-equidistant The opposite of equal amounts of distance between two or more points, or not equally distanced. Objective functions Figures of merit to be minimized or maximized. Parametric Of or relating to or in terms of parameters, or factors that define a system. Passive markers Joint and segment markers used during motion capture that reflect visible or infrared light. Pelvis Consists of the two hip bones, the sacrum, and the coccyx. It is located between the proximal spine and the hip joints. Pelvis anterior-posterior tilt Motion of the long axis of the pelvis within the sagittal plane as seen by an observer positioned along the medial-lateral axis of the laboratory. Pelvis elevation-depression Motion of the medial-lateral axis of the pelvis within the coronal plane as seen by an observer positioned along the anterior-posterior axis of the laboratory. Pelvis internal-external rotation Motion of the medial-lateral or anterior-posterior axis of the pelvis within the transverse plane as seen by an observer positioned along the longitudinal axis of the laboratory. Pelvis motion The position of the pelvis as defined by a marker set (for example, plane formed by the markers on the right and left anterior superior iliac spine (ASIS) and a marker between the 5th lumbar vertebrae and the sacrum) relative to a laboratory coordinate system. Plantarflexion Movement of the foot away from the anterior part of the tibia in the sagittal plane. Posterior The back or behind, also referred to as dorsal. Proximal Toward the point of attachment or origin. Range of motion Indicates joint motion excursion from the maximum angle to the minimum angle.
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51 Sacrum Consists of the fused components of five sacral vertebrae located between the 5th lumbar vertebra and the coccyx. It attaches the axial skeleton to the pelvic girdle of the appendicular skeleton via paired articulations. Sagittal plane The plane that divides the body or body segment into the right and left parts. Skin movement artifacts The relative movement between skin and underlying bone. Stance phase The period of time when the foot is in contact with the ground. Subtalar joint Located between the distal talus and proximal calcaneous, also known as the talocalcaneal joint. Superior Above or at a higher level (towards the head). Synthetic markers Computational representations of passive markers located on the kinematic model. Swing phase The period of time when the foot is not in contact with the ground. Talocrural joint Located between the distal tibia and proximal talus, also known as the tibial-talar joint. Talus The largest bone of the ankle transmitting weight from the tibia to the rest of the foot. Tibia The large medial bone of the lower leg, also known as the shinbone. It is located between the knee joint and the talocrural joint. Transepicondylar The line between the medial and lateral epicondyles. Transverse plane The plane at right angles to the coronal and sagittal planes that divides the body into superior and inferior parts. Velocity The time rate of change of displacement.
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APPENDIX A NOMINAL JOINT PARAMETERS & OPTIMIZATION BOUNDS FOR SYNTHETIC MARKER DATA A-1. Nominal right hip joint parameters and optimization bounds for synthetic marker data. Table Right Hip Joint Parameter Nominal Lower Bound Upper Bound p1 (cm) -6.022205 -20.530245 0 p2 (cm) -9.307044 -20.530245 0 p3 (cm) 8.759571 0 20.530245 p4 (cm) 0 -14.508040 6.022205 p5 (cm) 0 -11.223200 9.307044 p6 (cm) 0 -8.759571 11.770674 52
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53 A-2. Nominal right knee joint parameters and optimization bounds for synthetic marker data. Table Right Knee Joint Parameter Nominal Lower Bound Upper Bound p1 () 0 -30 30 p2 () 0 -30 30 p3 () -5.079507 -35.079507 24.920493 p4 () 16.301928 -13.698072 46.301928 p5 (cm) 0 -7.836299 7.836299 p6 (cm) -37.600828 -45.437127 -29.764528 p7 (cm) 0 -7.836299 7.836299 p8 (cm) 0 -7.836299 7.836299 p9 (cm) 0 -7.836299 7.836299
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54 A-3. Nominal right ankle joint parameters and optimization bounds for synthetic marker data. Table Right Ankle Joint Parameter Nominal Lower Bound Upper Bound p1 () 18.366935 -11.633065 48.366935 p2 () 0 -30 30 p3 () 40.230969 10.230969 70.230969 p4 () 23 -7 53 p5 () 42 12 72 p6 (cm) 0 -6.270881 6.270881 p7 (cm) -39.973202 -46.244082 -33.702321 p8 (cm) 0 -6.270881 6.270881 p9 (cm) -1 -6.270881 0 p10 (cm) 8.995334 2.724454 15.266215 p11 (cm) 4.147543 -2.123338 10.418424 p12 (cm) 0.617217 -5.653664 6.888097
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APPENDIX B NOMINAL JOINT PARAMETERS & OPTIMIZATION BOUNDS FOR EXPERIMENTAL MARKER DATA B-1. Nominal right hip joint parameters and optimization bounds for experimental marker data. Table Right Hip Joint Parameter Nominal Lower Bound Upper Bound p1 (cm) -5.931423 -20.220759 0 p2 (cm) -9.166744 -20.220759 0 p3 (cm) 8.627524 0 20.220759 p4 (cm) 0 -14.289337 5.931423 p5 (cm) 0 -11.054015 9.166744 p6 (cm) 0 -8.627524 11.593235 55
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56 B-2. Nominal right knee joint parameters and optimization bounds for experimental marker data. Table Right Knee Joint Parameter Nominal Lower Bound Upper Bound p1 () 0 -30 30 p2 () 0 -30 30 p3 () -4.070601 -34.070601 25.929399 p4 () 1.541414 -28.458586 31.541414 p5 (cm) 0 -7.356876 7.356876 p6 (cm) -39.211319 -46.568195 -31.854442 p7 (cm) 0 -7.356876 7.356876 p8 (cm) 0 -7.356876 7.356876 p9 (cm) 0 -7.356876 7.356876
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57 B-3. Nominal right ankle joint parameters and optimization bounds for experimental marker data. Table Right Ankle Joint Parameter Nominal Lower Bound Upper Bound p1 () 8.814964 -21.185036 38.814964 p2 () 0 -30 30 p3 () 26.890791 -3.109209 56.890791 p4 () 23 -7 53 p5 () 42 12 72 p6 (cm) 0 -5.662309 5.662309 p7 (cm) -41.131554 -46.793862 -35.469245 p8 (cm) 0 -5.662309 5.662309 p9 (cm) -1 -5.662309 0 p10 (cm) 9.113839 3.451530 14.776147 p11 (cm) 3.900829 -1.761479 9.563138 p12 (cm) 1.116905 -4.545403 6.779214
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APPENDIX C NOMINAL & OPTIMUM JOINT PARAMETERS FOR SYNTHETIC MARKER DATA WITHOUT NOISE C-1. Nominal and optimum right hip joint parameters for synthetic marker data without noise. Table Right Hip Joint Parameter Nominal Optimized Error p1 (cm) -6.022205 -6.022205 0.000000 p2 (cm) -9.307044 -9.307041 0.000003 p3 (cm) 8.759571 8.759578 0.000007 p4 (cm) 0 0.000004 0.000004 p5 (cm) 0 0.000015 0.000015 p6 (cm) 0 -0.000008 0.000008 58
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59 C-2. Nominal and optimum right knee joint parameters for synthetic marker data without noise. Table Right Knee Joint Parameter Nominal Optimized Error p1 () 0 -0.040222 0.040222 p2 () 0 -0.051509 0.051509 p3 () -5.079507 -5.050744 0.028763 p4 () 16.301928 16.242914 0.059015 p5 (cm) 0 -0.009360 0.009360 p6 (cm) -37.600828 -37.589068 0.011760 p7 (cm) 0 -0.014814 0.014814 p8 (cm) 0 -0.002142 0.002142 p9 (cm) 0 -0.000189 0.000189
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60 C-3. Nominal and optimum right ankle joint parameters for synthetic marker data without noise. Table Right Ankle Joint Parameter Nominal Optimized Error p1 () 18.366935 18.364964 0.001971 p2 () 0 -0.011809 0.011809 p3 () 40.230969 40.259663 0.028694 p4 () 23 23.027088 0.027088 p5 () 42 42.002080 0.002080 p6 (cm) 0 0.000270 0.000270 p7 (cm) -39.973202 -39.972852 0.000350 p8 (cm) 0 -0.000287 0.000287 p9 (cm) -1 -1.000741 0.000741 p10 (cm) 8.995334 8.995874 0.000540 p11 (cm) 4.147543 4.147353 0.000190 p12 (cm) 0.617217 0.616947 0.000270
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APPENDIX D NOMINAL & OPTIMUM JOINT PARAMETERS FOR SYNTHETIC MARKER DATA WITH NOISE D-1. Nominal and optimum right hip joint parameters for synthetic marker data with noise. Table Right Hip Joint Parameter Nominal Optimized Error p1 (cm) -6.022205 -5.854080 0.168125 p2 (cm) -9.307044 -9.434820 0.127776 p3 (cm) 8.759571 8.967520 0.207949 p4 (cm) 0 0.092480 0.092480 p5 (cm) 0 -0.180530 0.180530 p6 (cm) 0 0.191050 0.191050 61
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62 D-2. Nominal and optimum right knee joint parameters for synthetic marker data with noise. Table Right Knee Joint Parameter Nominal Optimized Error p1 () 0 -3.295650 3.295650 p2 () 0 -1.277120 1.277120 p3 () -5.079507 -5.604100 0.524593 p4 () 16.301928 12.763780 3.538148 p5 (cm) 0 0.375600 0.375600 p6 (cm) -37.600828 -37.996910 0.396082 p7 (cm) 0 0.489510 0.489510 p8 (cm) 0 0.144040 0.144040 p9 (cm) 0 -0.204420 0.204420
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63 D-3. Nominal and optimum right ankle joint parameters for synthetic marker data with noise. Table Right Ankle Joint Parameter Nominal Optimized Error p1 () 18.366935 15.130096 3.236838 p2 () 0 8.007498 8.007498 p3 () 40.230969 32.975096 7.255873 p4 () 23 23.122015 0.122015 p5 () 42 42.038733 0.038733 p6 (cm) 0 -0.398360 0.398360 p7 (cm) -39.973202 -39.614220 0.358982 p8 (cm) 0 -0.755127 0.755127 p9 (cm) -1 -2.816943 1.816943 p10 (cm) 8.995334 10.210540 1.215206 p11 (cm) 4.147543 3.033673 1.113870 p12 (cm) 0.617217 -0.190367 0.807584
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APPENDIX E NOMINAL & OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE EXPERIMENTAL MARKER DATA E-1. Nominal and optimum right hip joint parameters for multi-cycle experimental marker data. Table Right Hip Joint Parameter Nominal Optimized Improvement p1 (cm) -5.931423 -7.518819 1.587396 p2 (cm) -9.166744 -9.268741 0.101997 p3 (cm) 8.627524 8.857706 0.230182 p4 (cm) 0 -2.123433 2.123433 p5 (cm) 0 0.814089 0.814089 p6 (cm) 0 1.438188 1.438188 64
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65 E-2. Nominal and optimum right knee joint parameters for multi-cycle experimental marker data. Table Right Knee Joint Parameter Nominal Optimized Improvement p1 () 0 -0.586205 0.586205 p2 () 0 14.854951 14.854951 p3 () -4.070601 -2.724374 1.346227 p4 () 1.541414 2.404475 0.863061 p5 (cm) 0 -1.422101 1.422101 p6 (cm) -39.211319 -39.611720 0.400401 p7 (cm) 0 -0.250043 0.250043 p8 (cm) 0 -0.457104 0.457104 p9 (cm) 0 1.471656 1.471656
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66 E-3. Nominal and optimum right ankle joint parameters for multi-cycle experimental marker data. Table Right Ankle Joint Parameter Nominal Optimized Improvement p1 () 8.814964 16.640499 7.825535 p2 () 0 9.543288 9.543288 p3 () 26.890791 27.359342 0.468551 p4 () 23 13.197304 9.802696 p5 () 42 45.259512 3.259512 p6 (cm) 0 1.650689 1.650689 p7 (cm) -41.131554 -41.185800 0.054246 p8 (cm) 0 -1.510034 1.510034 p9 (cm) -1 -2.141939 1.141939 p10 (cm) 9.113839 11.244080 2.130241 p11 (cm) 3.900829 3.851262 0.049567 p12 (cm) 1.116905 0.283095 0.833810
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APPENDIX F NOMINAL & OPTIMUM JOINT PARAMETERS FOR FIRST ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA F-1. Nominal and optimum right hip joint parameters for first one-half-cycle experimental marker data. Table Right Hip Joint Parameter Nominal Optimized Improvement p1 (cm) -5.931423 -7.377948 1.446525 p2 (cm) -9.166744 -9.257734 0.090990 p3 (cm) 8.627524 8.124560 0.502964 p4 (cm) 0 -2.050133 2.050133 p5 (cm) 0 0.813034 0.813034 p6 (cm) 0 0.656323 0.656323 67
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68 F-2. Nominal and optimum right knee joint parameters for first one-half-cycle experimental marker data. Table Right Knee Joint Parameter Nominal Optimized Improvement p1 () 0 7.621903 7.621903 p2 () 0 12.823259 12.823259 p3 () -4.070601 -0.642569 3.428032 p4 () 1.541414 11.252668 9.711254 p5 (cm) 0 -1.217316 1.217316 p6 (cm) -39.211319 -38.611100 0.600219 p7 (cm) 0 -1.252732 1.252732 p8 (cm) 0 -0.003903 0.003903 p9 (cm) 0 1.480035 1.480035
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69 F-3. Nominal and optimum right ankle joint parameters for first one-half-cycle experimental marker data. Table Right Ankle Joint Parameter Nominal Optimized Improvement p1 () 8.814964 -15.959751 24.774715 p2 () 0 -4.522393 4.522393 p3 () 26.890791 18.986137 7.904654 p4 () 23 28.588479 5.588479 p5 () 42 36.840527 5.159473 p6 (cm) 0 3.624386 3.624386 p7 (cm) -41.131554 -43.537980 2.406426 p8 (cm) 0 -3.370814 3.370814 p9 (cm) -1 -2.246233 1.246233 p10 (cm) 9.113839 12.155750 3.041911 p11 (cm) 3.900829 0.488739 3.412090 p12 (cm) 1.116905 -1.207070 2.323975
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APPENDIX G NOMINAL & OPTIMUM JOINT PARAMETERS FOR SECOND ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA G-1. Nominal and optimum right hip joint parameters for second one-half-cycle experimental marker data. Table Right Hip Joint Parameter Nominal Optimized Improvement p1 (cm) -5.931423 -7.884120 1.952697 p2 (cm) -9.166744 -10.160573 0.993829 p3 (cm) 8.627524 9.216565 0.589041 p4 (cm) 0 -2.935484 2.935484 p5 (cm) 0 0.313918 0.313918 p6 (cm) 0 1.936742 1.936742 70
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71 G-2. Nominal and optimum right knee joint parameters for second one-half-cycle experimental marker data. Table Right Knee Joint Parameter Nominal Optimized Improvement p1 () 0 7.216444 7.216444 p2 () 0 12.986174 12.986174 p3 () -4.070601 -0.228411 3.842190 p4 () 1.541414 10.970612 9.429198 p5 (cm) 0 -1.300621 1.300621 p6 (cm) -39.211319 -38.785646 0.425673 p7 (cm) 0 -1.190227 1.190227 p8 (cm) 0 -0.130610 0.130610 p9 (cm) 0 1.293016 1.293016
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72 G-3. Nominal and optimum right ankle joint parameters for second one-half-cycle experimental marker data. Table Right Ankle Joint Parameter Nominal Optimized Improvement p1 () 8.814964 31.399921 22.584957 p2 () 0 1.211118 1.21112 p3 () 26.890791 51.518589 24.627798 p4 () 23 26.945919 3.945919 p5 () 42 45.021534 3.021534 p6 (cm) 0 -3.971358 3.971358 p7 (cm) -41.131554 -36.976040 4.155514 p8 (cm) 0 -0.154441 0.154441 p9 (cm) -1 -3.345873 2.345873 p10 (cm) 9.113839 7.552444 1.561395 p11 (cm) 3.900829 7.561219 3.660390 p12 (cm) 1.116905 1.108033 0.008872
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APPENDIX H OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE & FIRST ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA H-1. Optimum right hip joint parameters for multi-cycle and first one-half-cycle experimental marker data. Table Right Hip Joint Parameter Multi-Cycle Optimized First-Half-Cycle Optimized Difference p1 (cm) -7.518819 -7.377948 0.140871 p2 (cm) -9.268741 -9.257734 0.011007 p3 (cm) 8.857706 8.124560 0.733146 p4 (cm) -2.123433 -2.050133 0.073300 p5 (cm) 0.814089 0.813034 0.001055 p6 (cm) 1.438188 0.656323 0.781865 73
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74 H-2. Optimum right knee joint parameters for multi-cycle and first one-half-cycle experimental marker data. Table Right Knee Joint Parameter Multi-Cycle Optimized First-Half-Cycle Optimized Difference p1 () -0.586205 7.621903 8.208108 p2 () 14.854951 12.823259 2.031692 p3 () -2.724374 -0.642569 2.081805 p4 () 2.404475 11.252668 8.848193 p5 (cm) -1.422101 -1.217316 0.204785 p6 (cm) -39.611720 -38.611100 1.000620 p7 (cm) -0.250043 -1.252732 1.002689 p8 (cm) -0.457104 -0.003903 0.453201 p9 (cm) 1.471656 1.480035 0.008379
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75 H-3. Optimum right ankle joint parameters for multi-cycle and first one-half-cycle experimental marker data. Table Right Ankle Joint Parameter Multi-Cycle Optimized First-Half-Cycle Optimized Difference p1 () 16.640499 -15.959751 32.600250 p2 () 9.543288 -4.522393 14.065681 p3 () 27.359342 18.986137 8.373205 p4 () 13.197304 28.588479 15.391175 p5 () 45.259512 36.840527 8.418985 p6 (cm) 1.650689 3.624386 1.973697 p7 (cm) -41.185800 -43.537980 2.352180 p8 (cm) -1.510034 -3.370814 1.860780 p9 (cm) -2.141939 -2.246233 0.104294 p10 (cm) 11.244080 12.155750 0.911670 p11 (cm) 3.851262 0.488739 3.362523 p12 (cm) 0.283095 -1.207070 1.490165
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APPENDIX I OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE & SECOND ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA I-1. Optimum right hip joint parameters for multi-cycle and second one-half-cycle experimental marker data. Table Right Hip Joint Parameter Multi-Cycle Optimized Second-Half-Cycle Optimized Difference p1 (cm) -7.518819 -7.884120 0.365301 p2 (cm) -9.268741 -10.160573 0.891832 p3 (cm) 8.857706 9.216565 0.358859 p4 (cm) -2.123433 -2.935484 0.812051 p5 (cm) 0.814089 0.313918 0.500171 p6 (cm) 1.438188 1.936742 0.498554 76
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77 I-2. Optimum right knee joint parameters for multi-cycle and second one-half-cycle experimental marker data. Table Right Knee Joint Parameter Multi-Cycle Optimized Second-Half-Cycle Optimized Difference p1 () -0.586205 7.216444 7.802649 p2 () 14.854951 12.986174 1.868777 p3 () -2.724374 -0.228411 2.495963 p4 () 2.404475 10.970612 8.566137 p5 (cm) -1.422101 -1.300621 0.121480 p6 (cm) -39.611720 -38.785646 0.826074 p7 (cm) -0.250043 -1.190227 0.940184 p8 (cm) -0.457104 -0.130610 0.326494 p9 (cm) 1.471656 1.293016 0.178640
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78 I-3. Optimum right ankle joint parameters for multi-cycle and second one-half-cycle experimental marker data. Table Right Ankle Joint Parameter Multi-Cycle Optimized Second-Half-Cycle Optimized Difference p1 () 16.640499 31.399921 14.759422 p2 () 9.543288 1.211118 8.332170 p3 () 27.359342 51.518589 24.159247 p4 () 13.197304 26.945919 13.748615 p5 () 45.259512 45.021534 0.237978 p6 (cm) 1.650689 -3.971358 5.622047 p7 (cm) -41.185800 -36.976040 4.209760 p8 (cm) -1.510034 -0.154441 1.355593 p9 (cm) -2.141939 -3.345873 1.203934 p10 (cm) 11.244080 7.552444 3.691636 p11 (cm) 3.851262 7.561219 3.709957 p12 (cm) 0.283095 1.108033 0.824938
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80 Cappozzo, A., Leo, T., and Pedotti, A., 1975. A General Com puting Method for the Analysis of Human Locomo tion. Journal of Biomechanics Volume 8, Number 5, Pages 307-320. CDC, 2003. Targeting Ar thritis: The Nations Leading Cause of Disability Centers for Disease Control and Prevention, National Center for Chronic Disease Prevention and Health Promotion, Atlanta, Georgia. Accessed: http://www.cdc.gov/nccdphp/ aag/pdf/aag_arthritis2003.pdf February, 2003. Challis, J.H. and Kerwin, D.G., 1996. Quantif ication of the Uncertainties in Resultant Joint Moments Computed in a Dynami c Activity. Journal of Sports Sciences Volume 14, Number 3, Pages 219-231. Chao, E.Y. and Sim, F.H., 1995. Com puter-Aided Pre-Operative Planning in Knee Osteotomy. Iowa Orthopedic Journal Volume 15, Pages 4-18. Chao, E.Y.S., Lynch, J.D., and Vanderploeg, M.J., 1993. Sim ulation and Animation of Musculoskeletal Joint System. Journal of Biomechanical Engineering Volume 115, Number 4, Pages 562-568. Churchill, D.L., Incavo, S.J., Johnson, C.C., and Beynnon, B.D., 1998. The Transepicondylar Axis Approximates the Optimal Flex ion Axis of the Knee. Clinical Orthopaedics and Related Research Volume 356, Number 1, Pages 111-118. Chze, L., Fregly, B.J., and Dimnet, J., 1995. A Solidification Procedure to Facilitate Kinematic Analyses Based on Video System Da ta. Journal of Biomechanics Volume 28, Number 7, Pages 879-884. Davis, B.L., 1992. Uncertainty in Calcul ating Joint Mome nts During Gait. In Proceedings of the 8th Meeting of European Society of Biomechanics Rome, Italy, June 21-24, Page 276. de Leva, P., 1996. Adjustm ents to Zatsiorsky-Se luyanovs Segm ent Inertia Parameters. Journal of Biomechanics Volume 29, Number 9, Pages 1223-1230. Delp, S.L., Arnold, A.S., and Piazza, S.J., 1998. Graphics-Based Modeling and Analysis of Gait Abnorma lities. Bio-Medical Materials and Engineering Volume 8, Number 3/4, Pages 227-240. Delp, S.L., Arnold, A.S., Speers, R.A., and Moore, C.A., 1996. Ha mstrings and Psoas Lengths During Normal and Crouch Gait: Implications for Muscle-Tendon Surgery. Journal of Orthopaedic Research Volume 14, Number 1, Pages 144-151.
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81 Delp, S.L., Loan, J.P., Hoy, M.G., Zajac, F.E., Topp E.L., and Rosen, J.M., 1990. An Interactive Graphics-Based Model of the Lower Extremity to Study Orthopaedic Surgical Procedures. IEEE Transactions on Biomedical Engineering Volume 37, Number 8, Pages 757-767. Heck, D.A., Melfi, C.A., Mamlin, L.A., Katz, B.P., Arthur, D.S., Dittus, R.S., and Freund, D.A., 1998. "Revision Rates Following Knee Replacement in the United States." Medical Care Volume 36, Number 5, Pages 661-689. Holden, J.P. and Stanhope, S.J., 1998. The Ef fect of Variation in Knee Center Location Estimates on Net Knee Joint Mome nts. Gait & Posture Volume 7, Number 1, Pages 1-6. Holden, J.P. and Stanhope, S.J., 2000. The Ef fect of Uncertainty in Hip Center Location Estimates on Hip Joint Moments During Wa lking at Different Speeds. Gait & Posture Volume 11, Number 2, Pages 120-121. Hollister, A.M., Jatana, S., Singh, A.K., Sullivan, W.W. and Lupichuk, A.G., 1993. The Axes of Rota tion of the Knee. Clinical Orthopaedics and Related Research Volume 290, Number 1, Pages 259-268. Hurwitz, D.E., Sumner, D.R., Andriacchi, T. P., and Sugar, D.A., 1998. Dyna mic Knee Loads During Gait Predict Proxima l Tibial Bone Distribution. Journal of Biomechanics Volume 31, Number 5, Pages 423-430. Inma n, V.T., 1976. The Joints of the Ankle. W illiams and Wilkins Company, Baltimore, Maryland. Kennedy, J. and Eberhart, R.C., 1995. P article Swarm Optimization. In Proceedings of the 1995 IEEE International Conference on Neural Networks Perth, Australia, November 27 December 1, Volume 4, Pages 1942-1948. Lane, G.J., Hozack, W.J., Shah, S., Rothman, R.H., Booth, R.E. Jr., Eng, K., Smith, P., 1997. Simultaneous Bilate ral Versus Unilateral Total Knee Arthroplasty. Outcom es Analysis. Clinical Orthopaedics and Related Research Volume 345, Number 1, Pages 106-112. Leardini, A., Cappozzo, A., Catani, F., Toksvig-Larsen, S., Petitto, A., Sforza, V., Cassanelli, G., and Giannini, S., 1999. V alidation of a Functional Method for the Estim ation of Hip Joint Centre Location. Journal of Biomechanics Volume 32, Number 1, Pages 99-103. Lu, T.-W and OConnor, J.J., 1999. Bone Position Estimation from Skin Marker Coordinates Using Global Optimisat ion with Joint Constraints. Journal of Biomechanics Volume 32, Number 2, Pages 129-134. Pandy, M.G., 2001. Computer Modeling and Simulation of Human Moveme nt. Annual Reviews in Biomedical Engineering Volume 3, Number 1, Pages 245-273.
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82 Piazza, S.J., Okita, N., and Cavanagh, P.R., 2001. Accuracy of the Functional Meth od of Hip Joint Center Location: Effects of Limited Motion and Varied Implem entation. Journal of Biomechanics Volume 34, Number 7, Pages 967-973. Prodromos, C.C., Andriacchi, T.P., and Gala nte, J.O., 1985. A Relation ship Between Gait and Clinical Changes Following High Tibial Osteotom y. Journal of Bone Joint Surgery (American) Volume 67, Number 8, Pages 1188-1194. Rahman, H., Fregly, B.J., and Banks, S.A., 2003. Accurate Measurem ent of Three-Dimensional Natural Knee Kinematics Using Single-Pl ane Fluoroscopy. In Proceedings of the 2003 Summer Bionengineering Conference The American Society of Mechanical Engineers, Key Biscayne, Florida, June 25-29. Schutte, J.F., Koh, B., Reinbolt, J.A., Haftka, R.T., George, A.D., and Fregly, B.J., 2003. Scale-Independent Biom echanical Optim ization. In Proceedings of the 2003 Summer Bioengineering Conference The American Society of Mechanical Engineers, Key Biscayne, Florida, June 25-29. Sommer III, H.J. and Miller, N.R., 1980. A Technique for Kinem atic Modeling of Anatom ical Joints. Journal of Biomechanical Engineering Volume 102, Number 4, Pages 311-317. Stagni, R., Leardini, A., Benedetti, M.G., Cappozzo, A., and Cappello, A., 2000. Effects of Hip Joint Centre Mislocation on Gait Analysis Results. Journal of Biomechanics Volume 33, Number 11, Pages 1479-1487. Tetsworth, K. and Paley, D., 1994. A ccuracy of Correction of Complex Lower-Extremity Deformities by the Ilizarov Me thod. Clinical Orthopaedics and Related Research Volume 301, Number 1, Pages 102-110. Vaughan, C.L., Davis, B.L., and OConnor, J.C., 1992. Dynamics of Human Gait Human Kinetics Publishers, Champaign, Illinois, Page 26. Wang, J.-W., Kuo, K.N., Andriacchi, T.P., and Galante, J.O., 1990. The Influence of Walking Mechanics and Time on the Results of Proxima l Tibial Osteotomy. Journal of Bone and Joint Surgery (American), Volume 72, Number 6, Pages 905-913.
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BIOGRAPHICAL SKETCH Jeffrey A. Reinbolt was born on May 6, 1974 in Bradenton, Florida. His parents are Charles and Joan Reinbolt. He has an older brother, Douglas, and an older sister, Melissa. In 1992, Jeff graduated salutatorian from Southeast High School, Bradenton, Florida. After completing his secondary education, he enrolled at the University of Florida supported by the Florida Undergraduate Scholarship and full-time employment at a local business. He earned a traditional 5-year engineering degree in only 4 years. In 1996, Jeff graduated with honors receiving a Bachelor of Science degree in engineering science with a concentration in biomedical engineering. He used this foundation to assist in the medical device development and clinical research programs of Computer Motion, Inc., Santa Barbara, California. In this role, Jeff was Clinical Development Site Manager for the Southeastern United States and he traveled extensively throughout the United States, Europe, and Asia collaborating with surgeons and fellow medical researchers. In 1998, Jeff married Karen, a student he met during his undergraduate studies. After more than 4 years in the medical device industry, he decided to continue his academic career at the University of Florida. In 2001, Jeff began his graduate studies in Biomedical Engineering and he was appointed a graduate research assistantship in the Computational Biomechanics Laboratory. He plans to continue his graduate education and research activities through the pursuit of a Doctor of Philosophy in mechanical engineering. Jeff would like to further his creative involvement in problem solving and the design of solutions to overcome healthcare challenges. 83