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PATIENTSPECIFIC DYNAMIC MODELING TO PREDICT FUNCTIONAL OUTCOMES By JEFFREY A. REINBOLT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Jeffrey A. Reinbolt This dissertation is dedicated to my loving wife, Karen, and our wonderful son, Jacob. 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 experiencebased advice. My life has been genuinely influenced by Dr. Fregly's expectations, confidence, and trust in me. I also extend gratitude to Dr. Raphael Haftka, Dr. Carl Crane, Dr. Scott Banks, and Dr. Steven Kautz for their dedication, knowledge, and instruction both inside and outside of the classroom. For these reasons, each was selected to serve on my supervisory committee. I express thanks to all four individuals for their time, contribution, and fulfillment of their committee responsibilities. I collectively show appreciation for my family, friends, and colleagues. 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 remarkable wife. TABLE OF CONTENTS Page A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES ............ ................ ......... .......... ............ vii LIST OF FIGURES ......... ......................... ...... ........ ............ ix ABSTRACT ........ .............. ............. ...... .......... .......... xi CHAPTER 1 IN TR OD U CTION ............................................... .. ......................... .. Arthritis: The Nation's Leading Cause of Disability .................................................1 N eed for PatientSpecific Sim ulations ................................... ................... .... ... .2 B background ................................... ........................... .... ...... ........ 3 M otion C apture................................................ 3 B iom echanical M odels ................................................ ................ .........3 Kinematics and Dynamics ............................. ............ .............. .......... ..... 4 Optimization ..... ........... .... ......... ..................4 2 CREATION OF PATIENTSPECIFIC DYNAMIC MODELS FROM THREEDIMENSIONAL MOVEMENT DATA USING OPTIMIZATION............5 B ack g rou n d ...................... .. ............. .. ................................................. .5 M eth od s ........................................................................... . 6 R e su lts ....... ......... .. ............. .. ...........................................................1 0 D isc u ssio n ...................... .. ............. .. ......................................................1 1 3 EFFECT OF MODEL PARAMETER VARIATIONS ON INVERSE DYNAMICS RESULTS USING MONTE CARLO SIMULATIONS ...........30 B a ck g ro u n d ......... ............. .. ............. .. .................................................. 3 0 M e th o d s ................................................................... 3 1 R e su lts ....... ......... .. ............. .. ...........................................................3 3 D isc u ssio n ...................... .. ............. .. ......................................................3 4 4 BENEFIT OF AUTOMATIC DIFFERENTIATION FOR BIOMECHANICAL O P T IM IZ A T IO N S ............................................................................................... 3 9 B a c k g ro u n d ............................................................................................................ 3 9 M e th o d s ..............................................................................4 0 R e su lts .................................................................................................................... 4 2 D isc u ssio n .............................................................................................................. 4 3 5 APPLICATION OF PATIENTSPECIFIC DYNAMIC MODELS TO PREDICT FUNCTIONAL OUTCOMES ....................................................... 45 B a c k g ro u n d .....................................................................................................4 5 M e th o d s ..............................................................................4 6 R e su lts .................................................................................................................... 5 0 D isc u ssio n .............................................................................................................. 5 2 6 CONCLUSION..................... .........................................65 Rationale for New Approach ..............................................................................65 Synthesis of Current Work and Literature ...............................................................65 G L O S S A R Y ..............................................................................6 7 L IST O F R E FE R E N C E S .............................................................................. 74 B IO G R A PH IC A L SK E T C H ........................................................................................ 79 LIST OF TABLES Table Page 21. Descriptions of the model degrees of freedom .............. ........................................15 22. Descriptions of the hip joint parameters. ..........................................................16 23. Descriptions of the knee joint parameters. ................................... ..........18 24. Descriptions of the ankle joint parameters. ............. ............................................ 20 25. Summary of rootmeansquare (RMS) joint parameter and marker distance errors produced by the phase one optimization and anatomic landmark methods for three types of movement data. ........................................ ......................... 24 26. Comparison of joint parameters predicted by anatomical landmark methods, phase one optimization involving individual joints separately, and phase one optimization involving all joints simultaneously. ................. ............... ...........25 27. Differences between joint parameters predicted by anatomical landmark methods and phase one optimizations. ............................... ............................... 26 28. Summary of rootmeansquare (RMS) inertial parameter and pelvis residual load errors produced by the phase two optimization and anatomic landmark methods for three types of movement data. ........................................ ......................... 27 29 Comparison of inertial parameters predicted by anatomical landmark methods and phase tw o optim izations. ............................................................................. 28 210. Differences between inertial parameters predicted by anatomical landmark methods and phase two optimizations............................................. ...............29 41. Performance results for system identification problem for a 3D kinematic ankle joint model involving 252 design variables and 1800 objective function elem en ts............................................................................................... 4 4 42. Performance results for movement prediction problem for a 3D fullbody gait model involving 660 design variables and 4100 objective function elements.........44 51. Descriptions of cost function weights and phase 3 control parameters. .................54 52. Summary of fixed offsets added to normal gait for each prescribed foot path........54 53. Comparison of cost function weights and phase 3 control parameters ..................55 54. Summary of rootmeansquare (RMS) errors for tracked quantities for toeout g ait ............... .. .................................................................5 6 55. Summary of rootmeansquare (RMS) errors for tracked quantities for wide sta n c e g a it .................................................................5 7 56. Summary of rootmeansquare (RMS) errors for predicted left knee abduction torque quantities for toeout and wide stance gait. .................................................55 LIST OF FIGURES Figure Page 21. Illustration of the 3D, 14 segment, 27 DOF fullbody kinematic model linkage joined by a set of gimbal, universal, and pin joints ..............................................14 22. Illustration of the 3 DOF right hip joint center simultaneously defined in the pelvis and right femur segments and the 6 translational parameters optimized to determine the functional hip joint center location .......... ....................................16 23. Illustration of the 1 DOF right knee joint simultaneously defined in the right femur and right tibia segments and the 4 rotational and 5 translational parameters optimized to determine the knee joint location and orientation.............17 24. Illustration of the 2 DOF right ankle joint complex simultaneously defined in the right tibia, talus, and foot segments and the 5 rotational and 7 translational parameters optimized to determine the joint locations and orientations ................19 25. Illustration of the modified Cleveland Clinic marker set used during static and dynam ic m otion capture trials. ..... ...................................................................... 21 26. Phase one optimization convergence illustration series for the knee joint, where synthetic markers are blue, model markers are red, and rootmeansquare (RMS) m arker distance error is green. ............................................................................ 22 27. Phase two optimization convergence illustration series for the knee joint, where synthetic masses are blue, model masses are red, and rootmeansquare (RMS) residual pelvis forces and torques are orange and green, respectively.....................23 31. Legend of five sample statistics presented by the chosen boxplot convention........36 32. Comparison of rootmeansquare (RMS) leg joint torques and marker distance error distributions. ..................................................................... 37 33. Comparison of mean leg joint torques and marker distance error distributions. .....38 51. Comparison of left knee abduction torques for toeout gait ....................................60 52. Comparison of left knee abduction torques for wide stance gait, where original (blue) is experimental normal gait, simulation (red) is predicted toeout gait, and final (green) is experimental toeout gait. ..................................... ............... 61 53. Comparison of left knee abduction torques for simulated high tibial osteotomy (H TO ) postsurgery gait. ............................................... ............................... 62 54. Comparison of mean (solid black line) plus or minus one standard deviation (gray shaded area) for experimental left knee abduction torques. .........................63 55. Comparison of mean (solid black line) plus or minus one standard deviation (gray shaded area) for simulated left knee abduction torques..............................64 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PATIENTSPECIFIC DYNAMIC MODELING TO PREDICT FUNCTIONAL OUTCOMES By Jeffrey A. Reinbolt May 2006 Chair: Benjamin J. Fregly Major Department: Mechanical and Aerospace Engineering Movement related disorders have treatments (e.g., corrective surgeries, rehabilitation) generally characterized by variable outcomes as a result of subjective decisions based on qualitative analyses using a onesizefitsall healthcare approach. Imagine the benefit to the healthcare provider and, more importantly, the patient, if certain clinical parameters may be evaluated pretreatment in order to predict the posttreatment outcome. Using patientspecific models, movement related disorders may be treated with reliable outcomes as a result of objective decisions based on quantitative analyses using a patientspecific approach. The general objective of the current work is to predict posttreatment outcome using patientspecific models given pretreatment data. The specific objective is to develop a fourphase optimization approach to identify patientspecific model parameters and utilize the calibrated model to predict functional outcome. Phase one involves identifying joint parameters describing the positions and orientations of j points within adjacent body segments. Phase two involves identifying inertial parameters defining the mass, centers of mass, and moments of inertia for each body segment. Phase three involves identifying control parameters representing weighted components of joint torque inherent in different walking movements. Phase four involves an inverse dynamics optimization to predict function outcome. This work comprises a computational framework to create and apply patientspecific models to predict clinically significant outcomes. CHAPTER 1 INTRODUCTION Arthritis: The Nation's 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 twothirds 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 weightbearing axis 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 PatientSpecific Simulations Innovative patientspecific 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 patientspecific dynamic model may be useful for planning intended surgical parameters and predicting the outcome of HTO. The main motivation for developing the following patientspecific dynamic model and the associated multiphase optimization approach is to predict the postsurgery peak internal knee abduction moment in HTO patients. Conventional surgical planning techniques for HTO involve choosing the amount of necessary tibial angulation from standing radiographs (or xrays). Unfortunately, alignment correction estimates from static xrays do not accurately predict longterm 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 postsurgery knee abduction moment. The precision of dynamic analyses is fundamentally associated with the accuracy of kinematic and kinetic model parameters (Andriacchi and Strickland, 1985; Challis and Kerwin, 1996; Cappozzo and Pedotti, 1975; Davis, 1992; Holden and Stanhope, 1998; Holden and Stanhope, 2000; Stagni et al., 2000). Understandably, a model constructed of rigid links within a multilink chain and simple mechanical approximations of joints will not precisely match the human anatomy and function. The model should provide the best possible agreement to experimental motion data within the bounds of the dynamic model selected (Sommer and Miller, 1980). Background Motion Capture Motion capture is the use of external devices to capture the movement of a real object. One type of motioncapture 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 highspeed, highresolution 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 motioncapture 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. 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 massdistribution 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 data. These errors are a function of the model's generalized coordinates and the model's kinematic and kinetic parameters. 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. CHAPTER 2 CREATION OF PATIENTSPECIFIC DYNAMIC MODELS FROM THREEDIMENSIONAL MOVEMENT DATA USING OPTIMIZATION Background Forward and inverse dynamics analyses of gait can be used to study clinical problems in neural control, rehabilitation, orthopedics, and sports medicine. These analyses utilize a dynamic skeletal model that requires values for joint parameters (JPs  joint positions and orientations in the body segments) and inertial parameters (IPs  masses, mass centers, and moments of inertia of the body segments). If the specified parameter values do not match the patient's anatomy and mass distribution, then the predicted gait motions and loads may not be indicative of the clinical situation. The literature contains a variety of methods to estimate JP and IP values on a patientspecific basis. Anatomic landmark methods estimate parameter values using scaling rules developed from cadaver studies (Bell et al., 1990; Churchill et al., 1998; Inman, 1976; de Leva, 1996). In contrast, optimization methods adjust parameter values to minimize errors between model predictions and experimental measurements. Optimizations identifying JP values for threedimensional (3D) multijoint kinematic models have a high computational cost (Reinbolt et al., 2005). Optimizations identifying IP values without corresponding optimizations identifying JP values have been performed with limited success for planar models of running, jumping, and kicking motions (Vaughan et al., 1982). This study presents a computationallyefficient twophase optimization approach for determining patientspecific JP and IP values in a dynamic skeletal model given experimental movement data to match. The first phase determines JP values that best match experimental kinematic data, while the second phase determines IP values that best match experimental kinetic data. The approach is demonstrated by fitting a 3D, 27 degreeoffreedom (DOF), parametric fullbody gait model possessing 98 JPs and 84 IPs to synthetic (i.e., computer generated) and experimental movement data. Methods A sample dynamic model is needed to demonstrate the proposed twophase optimization approach. For this purpose, a parametric 3D, 27 DOF, fullbody gait model whose equations of motion were derived with the symbolic manipulation software, AutolevTM (OnLine Dynamics, Sunnyvale, CA), was used (Figure 21, Table 21). Comparable to Pandy's (2001) model structure, 3 translational and 3 rotational DOFs express the movement of the pelvis in a Newtonian reference frame and the remaining 13 segments comprised four open chains branching from the pelvis. The positions and orientations of joint axes within adjacent segment coordinate systems were defined by unique JPs. For example, the knee joint axis is simultaneously established in the femur and tibia coordinate systems. These parameters are used to designate the geometry of the following joint types: 3 DOF hip, 1 DOF knee, 2 DOF ankle, 3 DOF back, 2 DOF shoulder, and 1 DOF elbow. Each joint provides a simplified mechanical approximation to the actual anatomic articulations. Anatomic landmark methods were used to estimate nominal values for 6 hip (Bell et al., 1990), 9 knee (Churchill et al., 1998), and 12 ankle (Inman, 1976) JPs (Figure 22, Table 22, Figure 23, Table 23, Figure 24, Table 24). The segment masses, mass centers, and moments of inertia were described by unique IPs. Anatomic landmark methods were used to estimate nominal values for 7 IPs per segment (de Leva, 1996). Parameters defining the structure of the model were referenced to local coordinate systems fixed in each body segment. These coordinate systems were created from a static motion capture trial (see below). Markers placed over the left anterior superior iliac spine (ASIS), right ASIS, and superior sacrum were used to define the pelvis segment coordinate system (Figure 22). From percentages of the interASIS distance, a nominal hip joint center location was estimated within the pelvis segment (Bell et al., 1990). This nominal joint center served as the origin of the femur coordinate system, which was subsequently defined using markers placed over the medial and lateral femoral epicondyles (Figure 22). The tibia coordinate system originated at the midpoint of the knee markers and was defined by additional markers located on the medial and lateral malleoli (Figure 23). The talus coordinate system was created where the yaxis extends along the line perpendicular to both the talocrural joint axis and the subtalar joint axis (Figure 24). The heel and toe markers, in combination with the tibia yaxis, defined the foot coordinate system (Figure 24). Experimental kinematic and kinetic data were collected from a single subject using a videobased motion analysis system (Motion Analysis Corporation, Santa Rosa, CA) and two force plates (AMTI, Watertown, MA). Institutional review board approval and informed consent were obtained prior to the experiments. As described above, segment coordinate systems were created from surface marker locations measured during a static standing pose (Figure 25). Unloaded isolated joint motions were performed to exercise the primary functional axes of each lower extremity joint (hip, knee, and ankle on each side). For each joint, the subject was instructed to move the distal segment within the physiological range of motion so as to exercise all DOFs of the joint. Three trials were done for each joint with all trials performed unloaded. Multiple cycles of standing hip flexionextension followed by abductionadduction were recorded. Similar to Leardini et al. (1999), internalexternal rotation of the hip was avoided to reduce skin and soft tissue movement artifacts. Multiple cycles of knee flexionextension were measured. Finally, multiple cycles of simultaneous ankle plantarflexiondorsiflexion and inversioneversion were recorded. Gait motion and ground reaction data were collected to investigate simultaneous motion of all lower extremity joints under loadbearing physiological conditions. To evaluate the proposed optimization methodology, two types of synthetic movement data were generated from the experimental data sets. The first type was noiseless synthetic data generated by moving the model through motions representative of the isolated joint and gait experiments. The second type was synthetic data with superimposed numerical noise to simulate skin and soft tissue movement artifacts. A continuous noise model of the form A sin(ow t + (p) was used with the following uniform random parameter values: amplitude (0 < A < 1 cm), frequency (0 < co < 25 rad/s), and phase angle (0 < p < 27r) (Cheze et al., 1995). The first phase of the optimization procedure adjusted JP values and model motion to minimize errors between model and experimental marker locations (Equation 21, Figure 26). For isolated joint motion trials, the design variables were 540 Bspline nodes (q) parameterizing the generalized coordinate trajectories (20 nodes per DOF) and 6 hip, 9 knee, or 12 ankle JPs (pjp). For the gait trial, the number of JPs was reduced to 4 hip, 9 knee, and 4 ankle, due to inaccuracies in determining joint functional axes with rotations less than 250 (Cheze et al., 1998). The initial value for each Bspline node and JP was chosen to be zero to test the robustness of the optimization approach. The JP cost function (ejp) minimized the errors between model (m) and experimental (m) marker locations for each of the 3 marker coordinates over nm markers and nftime frames. The JP optimizations were performed with Matlab's nonlinear least squares algorithm (The Mathworks, Natick, MA). nf nm 3 2 e = minto[ m= (pjp,q)] (21) Pjq i=1 j=1 k=1 The second phase of the optimization procedure adjusted IP values to minimize the residual forces and torques acting on a 6 DOF groundtopelvis joint (Equation 22, Figure 27). Only the gait trial was used in this phase. The design variables for phase two were a reduced set of 20 IPs (pip 7 masses, 8 centers of mass, and 5 moments of inertia) accounting for body symmetry and limited joint ranges of motion during gait. The initial seed for each IP was the nominal value or a randomly altered value within + 50% of nominal. The IP cost function (eip) utilized a combination of pelvis residual loads (F and T) calculated over all nftime frames and differences between initial (p 'p) and current (pip) IP values. The residual pelvis forces (F) were normalized by body weight (BW) and the residual pelvis torques (T) by body weight times height (BW*H). IP differences were normalized by their respective initial values to create nondimensional errors. The IP optimizations also were performed with Matlab's nonlinear least squares algorithm. Once a IP optimization converged, the final IP values were used as the initial guess for a subsequent IP optimization, with this processing being repeated until the resulting pelvis residual loads converged. erp = mim n + + (22) PL' =1 =l1 BW ] BW H p ) ( The JP and IP optimization procedures were applied to all three data sets (i.e., synthetic data without noise, synthetic data with noise, and experimental data). For isolated joint motion trials, JPs for each joint were determined through separate optimizations. For comparison, JPs for all three joints were determined simultaneously for the gait trial. Subsequently, IPs were determined for the gait trial using the previously optimum JP values. Rootmeansquare (RMS) errors between original and recovered parameters, marker distances, and pelvis residual loads were used to quantify the procedure's performance. All optimizations were performed on a 3.4 GHz Pentium 4 PC. Results For phase one, each JP optimization using noiseless synthetic data precisely recovered the original marker trajectories and model parameters to within an arbitrarily tight convergence tolerance (Table 25, Table 26). For the other two data sets, RMS marker distance errors were at most 6.62 mm (synthetic with noise) and 4.04 mm (experimental), which are of the same order of magnitude as the amplitude of the applied continuous noise model. Differences between experimental data results and anatomic landmark methods are much larger than differences attributed to noise alone for synthetic data with noise (Table 27). Optimizations involving the isolated joint trial data sets (i.e., 1200 time frames of data) required between 108 and 380 seconds of CPU time while the gait trial data set (i.e., 208 time frames of data) required between 70 and 100 seconds of CPU time. These computation times were orders of magnitude faster than those reported using a twolevel optimization procedure (Reinbolt et al., 2005). For phase two, each IP optimization using noiseless synthetic data produced zero pelvis residual loads and recovered the original IP values to within an arbitrarily tight convergence tolerance (Table 28, Table 29). For the other two data sets, pelvis residual loads and IP errors remained small, with a random initial seed producing nearly the same pelvis residual loads but slightly higher IP errors than when the correct initial seed was used. Differences between experimental data results and anatomic landmark methods are small compared to differences attributed to noise and initial seed for synthetic data with noise (Table 210). Required CPU time ranged from 11 to 48 seconds. Discussion The accuracy of dynamic analyses of a particular patient depends on the accuracy of the associated kinematic and kinetic model parameters. Parameters are typically estimated from anatomic landmark methods. The estimated (or nominal) values may be improved by formulating a twophase optimization problem driven by motion capture data. Optimized dynamic models can provide a more reliable foundation for future patientspecific dynamic analyses and optimizations. This study presented a twophase optimization approach for tuning joint and inertial parameters in a dynamic skeletal model to match experimental movement data from a specific patient. For the fullbody gait model used in this study, the JP optimization satisfactorily reproduced patientspecific JP values while the IP optimization successfully reduced pelvis residual loads while allowing variation in the IP values away from their initial guesses. The JP and IP values found by this twophase optimization approach are only as reliable as the noisy experimental movement data used as inputs. By optimizing over all time frames simultaneously, the procedure smoothes out the effects of this noise. An optimization approach that modifies JPs and IPs simultaneously may provide even further reductions in pelvis residual loads. It cannot be claimed that models fitted with this twophase approach will reproduce the actual functional axes and inertial properties of the patient. This is clear from the results of the synthetic data with noise, where the RMS errors in the recovered parameters were not zero. At the same time, the optimized parameters for this data set corresponded to a lower cost function value in each case than did the nominal parameters from which the synthetic data were generated. Thus, it can only be claimed that the optimized model structure provides the best possible fit to the imperfect movement data. There are differences between phase one and phase two when comparing results for experimental data to synthetic data with noise. For the JP case, absolute parameter differences were higher for experimental data compared to synthetic data with noise. This suggests that noise does not hinder the process of determining JPs. However, for the IP case, absolute parameter differences were higher for synthetic data with noise compared to experimental data. In fact, noise is a limiting factor when identifying IPs. The phase one optimization determined patientspecific joint parameters similar to previous works. The optimal hip joint center location of 2.94 cm (12.01% posterior), 9.21 cm (37.63% inferior), and 9.09 cm (37.10% lateral) is comparable to 19.30%, 30.40%, and 35.90%, respectively, of the inter ASIS distance (Bell et al., 1990). The optimum femur length (42.23 cm) and tibia length (38.33 cm) are similar to 41.98 cm and 37.74 cm, respectively (de Leva, 1996). The optimum coronal plane rotation (87.36) of the talocrural joint correlates to 82.7 + 3.70 (range 740 to 940) (Inman, 1976). The optimum distance (0.58 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 (34.790) and sagittal plane rotation (31.340) of the subtalar joint corresponds to 23 11 (range 40 to 47) and 42 90 (range 20.50 to 68.50), respectively (Inman, 1976). Compared to anatomic landmark methods reported in the literature, the phase one optimization reduced RMS marker distance errors by 17% (hip), 52% (knee), 68% (ankle), and 34% (full leg). The phase two optimization determined patientspecific inertial parameters similar to previous work. The optimum masses were within an average of 1.99% (range 0.078% to 8.51%), centers of mass within range 0.047% to 5.84% (mean 1.58%), and moments of inertia within 0.99% (0.0038% to 5.09%) from the nominal values (de Leva, 1996). Compared to anatomic landmark methods reported in the literature, the phase two optimization reduced RMS pelvis residual loads by 20% (forces) and 8% (torques). Two conclusions may be drawn from these comparisons. First, the similarities suggest the results are reasonable and show the extent of agreement with past studies. Second, the differences between values indicate the extent to which tuning the parameters to the patient via optimization methods would change their values compared to anatomic landmark methods. In all cases, the twophase optimization successfully reduced cost function values for marker distance errors or pelvis residual loads below those resulting from anatomic landmark methods. S23 q 22 ^ /I q24 (,J ,f S27 v 49 915 I ^ 8 113 q q13 q6 I~ I 2 I I I I I q3  S q16 q17 y ^ ', (superior) q1 q 418 (lateral) (lateral) Figure 21. Illustration of the 3D, 14 segment, 27 DOF fullbody kinematic model linkage joined by a set of gimbal, universal, and pin joints. Joint Types Pin Universal Gimbal q10 q11 q12 "*'* , *  (anterior) Table 21. Descriptions of the model degrees of freedom. DOF Description qi Pelvis anteriorposterior position. q2 Pelvis superiorinferior position. q3 Pelvis mediallateral position. q4 Pelvis anteriorposterior tilt angle. q5 Pelvis elevationdepression angle. q6 Pelvis internalexternal rotation angle. q7 Right hip flexionextension angle. q8 Right hip adductionabduction angle. q9 Right hip internalexternal rotation angle. qio Right knee flexionextension angle. qii Right ankle plantarflexiondorsiflexion angle. q12 Right ankle inversioneversion angle q13 Left hip flexionextension angle. q14 Left hip adductionabduction angle. q15 Left hip internalexternal rotation angle. q16 Left knee flexionextension angle. q17 Left ankle plantarflexiondorsiflexion angle. q18 Left ankle inversioneversion angle q19 Trunk anteriorposterior tilt angle. q20 Trunk elevationdepression angle. q21 Trunk internalexternal rotation angle. q22 Right shoulder flexionextension angle. q23 Right shoulder adductionabduction angle. q24 Right elbow flexion angle. q25 Left shoulder flexionextension angle. q26 Left shoulder adductionabduction angle. q27 Left elbow flexion angle. Y Pelvis Optimized P3 z Hip Joint r   SCenter X~ P4 P2 'P5 p;( Zj( ; . Y (superior) Z (lateral) X (anterior) Figure 22. Illustration of the 3 DOF right hip joint center simultaneously defined in the pelvis and right femur segments and the 6 translational parameters optimized to determine the functional hip joint center location. Table 22. Descriptions of the hip joint parameters. Hip Joint E Parameter pi Anteriorposterior P2 Superiorinferior 1 p3 Mediallateral loc< p4 Anteriorposterior p5 Superiorinferior 1 P6 Mediallateral loc< descriptionn location in pelvis segment. location in pelvis segment. nation in pelvis segment. location in femur segment. location in femur segment. nation in femur segment. \ r\\ .h 17 Y Femur X \P6 " P2 pp 9 P4 Optimized' P3 Knee Joint Center p P7 Optimized Knee Joint Axis Tibia Pg Z X Y (superior) Z X (lateral) Lab (anterior) Figure 23. Illustration of the 1 DOF right knee joint simultaneously defined in the right femur and right tibia segments and the 4 rotational and 5 translational parameters optimized to determine the knee joint location and orientation. Table 23. Descriptions of the knee joint parameters. Knee Joint De Parameter pi Adductionabduction p2 Internalexternal rota p3 Adductionabduction p4 Internalexternal rota p5 Anteriorposterior lo p6 Superiorinferior local p7 Anteriorposterior lo ps Superiorinferior local p9 Mediallateral locatic scription rotation in femur segment. tion in femur segment. rotation in tibia segment. tion in tibia segment. ;ation in femur segment. ition in femur segment. ;ation in tibia segment. ition in tibia segment. in in tibia segment. Y Tibia z P6 IP7 Center Figure 24. Illustration of the 2 DOF right ankle joint complex simultaneously defined in the right tibia, talus, and foot segments and the 5 rotational and 7 translational parameters optimized to determine the joint locations and orientations. P11 .J Foot Z. Y (superior) ) Lb (lateral) Lab (anterior) c re ,~ ri, Table 24. Descriptions of the ankle joint parameters. Ankle Joint Des Des Parameter pi Adductionabduction rotatic p2 Internalexternal rotation of p3 Internalexternal rotation of p4 Internalexternal rotation of p5 Dorsiplantar rotation of sut p6 Anteriorposterior location p7 Superiorinferior location ol ps Mediallateral location of ta p9 Superiorinferior location ol plo Anteriorposterior location pll Superiorinferior location ol p12 Mediallateral location of SL cription n of talocrural in tibia segment. talocrural in tibia segment. subtalar in talus segment. subtalar in foot segment. talar in foot segment. of talocrural in tibia segment. Stalocrural in tibia segment. locrural in tibia segment. Ssubtalar in talus segment. of subtalar in foot segment. Ssubtalar in foot segment. ibtalar in foot segment. S i J' d /"4v rl r, I C.: ii, f" ;I I *.i C 1to * ^6 ^ * Figure 25. Illustration of the 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. p. y.1 I. ' I 7' .1 ' anterior * V let' SiC anterior Ci /" 1 .: Y'I anterior * y/* anterior Figure 26. Phase one optimization convergence illustration series for the knee joint, where synthetic markers are blue, model markers are red, and rootmeansquare (RMS) marker distance error is green. Given synthetic marker data without noise and a synthetic knee flexion angle = 900, A) is the initial model knee flexion = 00 and knee joint parameters causing joint dislocation, B) is the model knee flexion = 300 and improved knee joint parameters, C) is the model knee flexion = 600 and nearly correct knee joint parameters, and D) is the final model knee flexion = 900 and ideal knee joint parameters. M A i B J (d r 1' anterior Nr 0" anterior Ci N~J anterior I ?i anterior Figure 27. Phase two optimization convergence illustration series for the knee joint, where synthetic masses are blue, model masses are red, and rootmeansquare (RMS) residual pelvis forces and torques are orange and green, respectively. Given synthetic kinetic data without noise and a synthetic knee flexion motion, A) is the initial model thigh and shank inertial parameters, B) is the improved thigh and shank inertial parameters, C) is the nearly correct thigh and shank inertial parameters, and D) is the final model thigh and shank inertial parameters. A i B J ..ii b , /ri ''ii'' i  Table 25. Summary of rootmeansquare (RMS) joint parameter and marker distance errors produced by the phase one optimization and anatomic landmark methods for three types of movement data. Experimental data were from isolated joint and gait motions measured using a videobased motion analysis system with three surface markers per body segment. Synthetic marker data were generated by applying the experimental motions to a nominal kinematic model with and without superimposed numerical noise. The isolated joint motions were nonweight bearing and utilized larger joint excursions than did the gait motion. For the anatomical landmark method, only the optimization involving model motion was performed since the joint parameters were specified. The individual joint optimizations used isolated joint motion data while the full leg optimization used gait motion data. Movement data Method RMS error Hip only Knee only Ankle only Full leg Marker distances (mm) 9.82e09 3.58e08 1.80e08 2.39e06 Synthetic without Noise Phase one optimization Orientation parameters () n/a 4.68e04 6.46e03 5.45e03 Position parameters (mm) 4.85e05 8.69e04 6.35e03 1.37e02 Marker distances (mm) 6.62 5.89 5.43 5.49 Synthetic with Noise Phase one optimization Orientation parameters () n/a 0.09 3.37 3.43 Position parameters (mm) 0.55 0.59 1.21 3.65 Phase one optimization Marker distances (mm) 3.73 1.94 1.43 4.04 Experimentalndmas M r d s ( ) 47 43 44 Anatomical landmarks Marker distances (mm) 4.47 4.03 4.44 6.08 Table 26. Comparison of joint parameters predicted by anatomical landmark methods, phase one optimization involving individual joints separately, and phase one optimization involving all joints simultaneously. Body segment indicates the segment in which the associated joint parameter is fixed. For a graphical depiction of the listed joint parameters, consult Figure 2, Figure 3, and Figure 4. Single joint optimization Full leg optimization .*Body Anatomical Synthetic Synthetic w Joint Joint parameter d na l n SyntheticSynthetic Synthetic segment landmarks without Experimental without Experimental ois with noise with noise noise noise Anterior position (mm) Pelvis 53.88 53.88 53.84 32.62 53.89 53.35 29.41 Superior position (mm) Pelvis 83.26 83.26 82.76 106.53 83.25 76.40 92.15 Hip Lateral position (mm) Pelvis 78.36 78.36 79.30 90.87 78.36 79.30 90.87 Anterior position (mm) Thigh 0.00 0.00 0.49 8.88 0.01 4.35 27.08 Superior position (mm) Thigh 0.00 0.00 0.47 25.33 0.02 5.55 11.59 Lateral position (mm) Thigh 0.00 0.00 0.46 18.13 0.00 0.45 18.13 Frontal plane orientation () Thigh 0.00 0.00 0.01 4.39 0.01 0.33 7.59 Transverse plane orientation () Thigh 0.00 0.00 0.09 2.49 0.00 1.07 3.96 Frontal plane orientation (0) Shank 4.27 4.27 4.13 0.15 4.28 3.52 0.98 Transverse plane orientation () Shank 17.11 17.11 17.14 18.96 17.11 18.11 23.34 Knee Anterior position (mm) Thigh 0.00 0.00 0.74 3.01 0.00 2.32 10.97 Superior position (mm) Thigh 419.77 419.77 420.20 410.44 419.79 425.80 408.70 Anterior position (mm) Shank 0.00 0.00 0.18 0.52 0.00 7.63 7.22 Superior position (mm) Shank 0.00 0.00 0.53 2.68 0.02 4.49 10.13 Lateral position (mm) Shank 0.00 0.00 0.84 6.79 0.00 1.63 2.67 Frontal plane orientation () Transverse plane orientation () Ankle (talocral) Anterior position (mm) Superior position (mm) Lateral position (mm) Transverse plane orientation () Transverse plane orientation () Ankle Sagittal plane orientation () (subtalar) Superior position (mm) Anterior position (mm) Superior position (mm) Lateral position (mm) Shank Shank Shank Shank Shank Talus Foot Foot Talus Foot Foot Foot 12.22 0.00 0.00 377.44 0.00 35.56 23.00 42.00 10.00 93.36 54.53 6.89 12.21 0.01 0.01 377.45 0.01 35.56 23.00 42.00 10.00 93.36 54.52 6.89 16.29 3.84 0.24 379.20 2.03 40.56 23.34 42.59 9.22 93.95 53.34 6.06 11.18 23.89 4.14 395.73 7.08 27.86 34.79 31.34 5.77 87.31 39.68 3.80 12.22 0.00 0.00 377.45 0.00 35.56 23.00 42.00 10.00 93.36 54.52 6.89 12.81 8.79 2.09 379.99 2.03 40.57 23.34 42.59 9.22 93.95 53.34 6.06 2.64 18.12 3.29 393.29 7.08 27.86 34.79 31.34 5.77 87.31 39.68 3.80 Table 27. Differences between joint parameters predicted by anatomical landmark methods and phase one optimizations. Body segment indicates the segment in which the associated joint parameter is fixed. For actual joint parameter values, consult Table 26. For a graphical depiction of the listed joint parameters, consult Figure 2, Figure 3, and Figure 4. JointBody Single joint optimization Full leg optimization Joint Joint parameter Synthetic Experimental Synthetic Experimental difference difference difference difference Anterior position (mm) Pelvis 0.03 21.26 0.53 24.47 Superior position (mm) Pelvis 0.51 23.27 6.86 8.89 Hip Lateral position (mm) Pelvis 0.94 12.51 0.94 12.51 Anterior position (mm) Thigh 0.49 8.88 4.35 27.08 Superior position (mm) Thigh 0.47 25.33 5.55 11.59 Lateral position (mm) Thigh 0.46 18.13 0.46 18.13 Frontal plane orientation () Thigh 0.01 4.39 0.33 7.59 Transverse plane orientation () Thigh 0.09 2.49 1.08 3.96 Frontal plane orientation (0) Shank 0.14 4.42 0.76 5.25 Transverse plane orientation () Shank 0.03 1.85 1.01 6.23 Knee Anterior position (mm) Thigh 0.74 3.01 2.32 10.97 Superior position (mm) Thigh 0.43 9.33 6.04 11.07 Anterior position (mm) Shank 0.18 0.52 7.64 7.22 Superior position (mm) Shank 0.53 2.68 4.49 10.13 Lateral position (mm) Shank 0.84 6.79 1.63 2.67 Frontal plane orientation () Transverse plane orientation () Anterior position (mm) Superior position (mm) Lateral position (mm) Transverse plane orientation () Transverse plane orientation () Sagittal plane orientation () Superior position (mm) Anterior position (mm) Superior position (mm) Lateral position (mm) Shank Shank Shank Shank Shank Talus Foot Foot Talus Foot Foot Foot 4.08 1.04 3.85 23.89 0.24 4.14 1.76 18.29 2.03 7.08 5.01 7.70 0.34 11.79 0.59 10.66 0.78 4.23 0.59 6.05 1.19 14.85 0.82 3.09 0.59 9.58 8.79 18.12 2.09 3.29 2.54 15.85 2.03 7.08 5.01 7.70 0.34 11.79 0.59 10.66 0.78 4.23 0.59 6.05 1.19 14.85 0.82 3.09 Ankle (talocrural) Ankle (subtalar) Table 28. Summary of rootmeansquare (RMS) inertial parameter and pelvis residual load errors produced by the phase two optimization and anatomic landmark methods for three types of movement data. Experimental data were from a gait motion measured using a videobased motion analysis system with three surface markers per body segment. Synthetic marker data were generated by applying the experimental motion to a nominal kinematic model with and without superimposed numerical noise. RMS error Movement data Method Force (N) Torque (N*m) Inertia (kg*m2) Mass (kg) Center of mass (m) Phase 2 Synthetic without Noise optimize n 8.04e10 2.16e10 5.13e13 1.22e10 7.11e13 optimization Synthetic with Noise Phase 2 16.92 4.93 5.67e03 1.08 6.64e03 (correct seed) optimization Synthetic with Noise Phase 2 Synthetic with Noise Phase2 16.960.44 5.240.23 7.45e021.81e02 1.330.39 2.16e022.92e03 (random seeds) optimization Phase 2 optim27.89 12.66 n/a n/a n/a optimization Experimental Anatomical An l 34.81 13.75 n/a n/a n/a landmarks Table 29 Comparison of inertial parameters predicted by anatomical landmark methods and phase two optimizations. Body segment indicates the segment in which the associated inertial parameter is fixed. Direction indicates the means in which the associated inertial parameter was applied. Anatomical Synthetic Synthetic with noise Category Body segment Direction landmarks without Correct Random seeds Experimental noise seed (10 cases) Pelvis 7.69 7.69 8.11 6.981.87 7.63 Thigh 9.74 9.74 10.76 11.191.02 8.91 Shank 2.98 2.98 3.25 3.370.39 2.95 Mass (kg) Foot n/a 1.26 1.26 1.37 1.400.29 1.29 Head & trunk 26.99 26.99 23.61 23.881.34 26.82 Upper arm 1.86 1.86 1.82 1.560.49 1.87 Lower arm & hand 1.53 1.53 1.63 1.750.47 1.53 Anterior 9.36e02 9.36e02 9.63e02 1.00e012.29e02 9.33e02 Pelvis Superior 2.37e02 2.37e02 2.33e02 2.49e025.44e03 2.37e02 Lateral 0.00 2.35e13 6.57e13 8.06e134.02e13 6.07e13 Anterior 0.00 1.54e13 4.02e13 7.76e132.06e13 4.86e13 Thigh Superior 1.68e01 1.68e01 1.65e01 1.58e011.84e02 1.59e01 Lateral 0.00 1.49e14 1.58e12 2.41e128.05e13 8.91e13 Anterior 0.00 1.22e14 5.41e13 1.91e136.20e13 7.62e13 Shank Superior 1.72e01 1.72e01 1.89e01 1.89e014.48e02 1.80e01 Lateral 0.00 6.59e14 3.15e13 5.65e132.72e13 4.56e13 Anterior 9.21e02 9.21e02 8.73e02 8.17e021.90e02 9.27e02 Center of Centr Foot Superior 1.60e04 1.60e04 4.19e04 4.12e041.19e04 1.60e04 mass (m) Lateral 0.00 1.81e14 1.26e14 9.35e148.93e14 1.85e14 Anterior 0.00 1.92e13 8.33e13 8.96e132.35e12 8.21e13 Head & trunk Superior 1.44e01 1.44e01 1.69e01 1.87e011.38e02 1.42e01 Lateral 0.00 2.56e14 1.66e12 2.34e127.09e13 4.45e13 Anterior 0.00 4.74e14 3.21e13 2.69e132.60e13 6.15e13 Upper arm Superior 1.84e01 1.84e01 1.86e01 1.95e014.96e02 1.84e01 Lateral 0.00 1.79e14 3.51e13 4.19e132.45e13 7.92e13 Anterior 0.00 5.59e14 1.45e13 2.32e133.58e13 9.22e13 Lower arm & hand Superior 1.85e01 1.85e01 1.97e01 1.89e014.36e02 1.85e01 Lateral 0.00 3.85e14 2.86e13 4.13e131.59e13 7.38e13 Anterior 6.83e02 6.83e02 6.83e02 6.60e022.30e02 6.83e02 Pelvis Superior 6.22e02 6.22e02 6.21e02 5.98e021.81e02 6.22e02 Lateral 5.48e02 5.48e02 5.47e02 5.25e021.64e02 5.48e02 Anterior 1.78e01 1.78e01 1.80e01 1.67e015.39e02 1.78e01 Thigh Superior 3.66e02 3.66e02 3.65e02 3.71e021.35e02 3.66e02 Lateral 1.78e01 1.78e01 1.74e01 2.06e015.31e02 1.78e01 Anterior 2.98e02 2.98e02 2.97e02 3.06e021.19e02 2.98e02 Shank Superior 4.86e03 4.86e03 4.86e03 4.01e031.04e03 4.86e03 Lateral 2.84e02 2.84e02 2.86e02 2.75e026.65e03 2.84e02 Moment of Anterior 2.38e03 2.38e03 2.29e03 2.27e036.45e04 2.38e03 inertia Foot Superior 4.56e03 4.56e03 4.58e03 4.90e031.29e03 4.56e03 (kg*m2) Lateral 3.08e03 3.08e03 2.99e03 3.02e036.57e04 3.08e03 Anterior 9.51e01 9.51e01 9.18e01 8.96e013.35e01 9.51e01 Head & trunk Superior 2.27e01 2.27e01 2.27e01 1.93e013.62e02 2.27e01 Lateral 8.41e01 8.41e01 8.37e01 9.99e012.68e01 8.41e01 Anterior 1.53e02 1.53e02 1.53e02 1.44e025.65e03 1.53e02 Upper arm Superior 4.71e03 4.71e03 4.71e03 4.57e031.29e03 4.71e03 Lateral 1.37e02 1.37e02 1.37e02 1.56e023.29e03 1.37e02 Anterior 1.97e02 1.97e02 1.97e02 2.06e024.97e03 1.97e02 Lower arm & hand Superior 1.60e03 1.60e03 1.60e03 1.41e035.02e04 1.60e03 Lateral 2.05e02 2.05e02 2.04e02 2.41e023.78e03 2.05e02 Table 210. Differences between inertial parameters predicted by anatomical landmark methods and phase two optimizations. Body segment indicates the segment in which the associated joint parameter is fixed. Direction indicates the means in which the associated inertial parameter was applied. For actual joint parameter values, consult Table 29. Synthetic differences Experimental Experimental Category Body segment Direction Random seeds differences Correct seed differences (10 cases) Pelvis 4.28e01 7.01e011.87 5.66e02 Thigh 1.02 1.451.02 8.29e01 Shank 2.69e01 3.89e013.86e01 3.11e02 Mass (kg) Foot n/a 1.29e01 1.62e012.95e01 3.53e02 Head and trunk 3.38 3.111.34 1.75e01 Upper arm 4.62e02 3.05e014.85e01 1.46e03 Lower arm and hand 1.01e01 2.15e014.67e01 1.89e03 Anterior 2.69e03 6.51e032.29e02 2.30e04 Pelvis Superior 4.13e04 1.24e035.44e03 1.12e05 Lateral 6.57e13 8.06e134.02e13 6.07e13 Anterior 4.02e13 7.76e132.06e13 4.86e13 Thigh Superior 3.48e03 1.01e021.84e02 9.81e03 Lateral 1.58e12 2.41e128.05e13 8.91e13 Anterior 5.41e13 1.91e136.20e13 7.62e13 Shank Superior 1.66e02 1.73e024.48e02 7.65e03 Lateral 3.15e13 5.65e132.72e13 4.56e13 Anterior 4.83e03 1.04e021.90e02 6.30e04 Center of mass (mn) Foot Superior 5.25e08 7.39e061.19e04 9.50e13 Lateral 1.26e14 9.35e148.93e14 1.85e14 Anterior 8.33e13 8.96e132.35e12 8.21e13 Head and trunk Superior 2.47e02 4.31e021.38e02 1.66e03 Lateral 1.66e12 2.34e127.09e13 4.45e13 Anterior 3.21e13 2.69e132.60e13 6.15e13 Upper arm Superior 2.40e03 1.15e024.96e02 1.10e04 Lateral 3.51e13 4.19e132.45e13 7.92e13 Anterior 1.45e13 2.32e133.58e13 9.22e13 Lower arm and hand Superior 1.29e02 4.21e034.36e02 3.70e04 Lateral 2.86e13 4.13e131.59e13 7.38e13 Anterior 5.53e05 2.24e032.30e02 1.95e04 Pelvis Superior 1.11e04 2.41e031.81e02 9.03e04 Lateral 1.21e04 2.28e031.64e02 1.19e05 Anterior 2.06e03 1.14e025.39e02 2.86e03 Thigh Superior 7.14e05 5.67e041.35e02 1.45e04 Lateral 4.56e03 2.73e025.31e02 9.06e03 Anterior 9.91e05 8.51e041.19e02 5.98e04 Shank Superior 6.80e07 8.49e041.04e03 1.85e06 Lateral 2.09e04 9.19e046.65e03 4.15e04 Anterior 5.94e07 1.73e056.45e04 9.07e08 Moment of nerti (km) Foot Superior 8.59e08 3.16e041.29e03 6.41e07 Lateral 3.06e06 2.87e056.57e04 1.36e05 Anterior 3.30e02 5.56e023.35e01 3.08e02 Head and trunk Superior 6.15e04 3.37e023.62e02 2.87e03 Lateral 4.13e03 1.59e012.68e01 1.73e02 Anterior 1.62e05 9.11e045.65e03 1.37e04 Upper arm Superior 1.79e07 4.14e051.29e03 9.40e07 Lateral 7.90e07 1.97e033.29e03 1.60e06 Anterior 1.63e05 8.91e044.97e03 1.09e04 Lower arm and hand Superior 8.95e08 1.95e045.02e04 7.50e07 Lateral 5.84e05 3.59e033.78e03 1.15e05 CHAPTER 3 EFFECT OF MODEL PARAMETER VARIATIONS ON INVERSE DYNAMICS RESULTS USING MONTE CARLO SIMULATIONS Background "One of the most valuable biomechanical variables to have for the assessment of any human movement is the time history of the moments of force at each joint" (Winter, 1980). There are countless applications involving the investigation of human movement ranging from sports medicine to pathological gait. In all cases, the moment of force, or torque, at a particular joint is a result of several factors: muscle forces, muscle moment arms, ligament forces, contact forces due to articular surface geometry, positions and orientations of axes of rotation, and inertial properties of body segments. A common simplification is muscles generate the entire joint torque by assuming ligament forces are insignificant and articular contact forces act through a chosen joint center (Challis and Kerwin, 1996). Individual muscle forces are estimated from resultant joint torques often computed by inverse dynamics analyses. In the end, inverse dynamics computations depend upon chosen model parameters, namely positions and orientations of joint axes parameters (JPs) and inertial property parameters (IPs) of body segments. The literature contains a variety of methods investigating the sensitivity of inverse dynamics torques to JPs and IPs. Two elbow joint torques were analyzed during a maximal speed dumbbell curl motion as one or more parameters were changed by the corresponding estimated uncertainty value up to 10mm for joint centers and 10% for inertial properties (Challis and Kerwin, 1996). The knee flexionextension torque has been studied for multiple walking speeds while changing the knee center location by 10mm in the anteriorposterior direction (Holden and Stanhope, 1998). Hip forces and torques have been examined for gait as planar leg inertial properties were individually varied over nine steps within 40% (Pearsall and Costigan, 1999). Knee forces and torques were evaluated for a planar harmonic oscillating motion as inertial properties varied by 5% (Andrews and Mish, 1996). Directly related to knee joint torque, knee kinematics were investigated during gait when the rotation axes varied by 100 (Croce et al., 1996). This study performs a series of Monte Carlo analyses relating to inverse dynamics using instantiations of a threedimensional (3D) fullbody gait model with isolated and simultaneous variations of JP and IP values. In addition, noise parameter (NP) variations representing kinematic noise were implemented separately. Uniform distribution of each parameter was chosen within bounds consistent with previous studies. Subsequently, kinematics were identified by optimizing the fitness of the model motion to the experimental motion. The number of instantiations was sufficient for convergence of inverse dynamics torque values. Methods Experimental kinematic and kinetic gait data were collected from a single subject using a videobased motion analysis system (Motion Analysis Corporation, Santa Rosa, CA) and two force plates (AMTI, Watertown, MA). Institutional review board approval and informed consent were obtained prior to the experiments. A parametric 3D, 27 degreeoffreedom (DOF), fullbody gait model (Figure 21, Table 21) was constructed, whose equations of motion were derived with the symbolic manipulation software, AutolevTM (OnLine Dynamics, Sunnyvale, CA). The pelvis was connected to ground via a 6 DOF joint and the remaining 13 segments comprised four open chains branching from the pelvis. The positions and orientations of joint axes within adjacent segment coordinate systems were defined by unique JPs. The segment masses, mass centers, and moments of inertia were described by unique IPs. Anatomic landmark methods were used to estimate nominal values for 84 IPs (de Leva, 1996) and 98 JPs (Bell et al., 1990, Churchill et al., 1998, Inman, 1976). Select model parameters were identified via optimization as described previously in Chapter 2 (Reinbolt and Fregly, 2005). A series of Monte Carlo analyses relating to inverse dynamics were performed using instantiations the fullbody gait model with isolated and simultaneous variations of JP and IP values within 25%, 50%, 75%, and 100% of the associated maximum. Uniform distribution of each parameter was chosen within bounds consistent with previous studies. Joint center locations were bounded by a maximum of 10mm (Challis and Kerwin, 1996, Holden and Stanhope, 1998). Joint axes orientations were bounded by a maximum of 100 (Croce et al., 1996). Inertial properties were bounded by a maximum of 10% of their original values (Andrews and Mish, 1996, Challis and Kerwin, 1996, Pearsall and Costigan, 1999). New kinematics were identified by optimizing the fitness of the model motion to the experimental motion similar to Chapter 2 (Reinbolt et al., 2005). The number of instantiations (e.g., 5000) was sufficient for convergence of inverse dynamics torque values. The mean and coefficient of variance (100*SD/mean) of all joint torques were within 2% of the final mean and coefficient of variance, respectively, for the last 10% of instantiations (Fishman, 1996, ValeroCuevas et al., 2003). A separate set of Monte Carlo inverse dynamics analyses were performed with variations of NP values within 25%, 50%, 75%, and 100% of the maximum. The NP represent the amplitude of simulated skin movement artifacts. 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 artifacts of Lu and O'Connor (1999), a continuous noise model of the form A sin(cot + (p) was used with the following uniform random parameter values: amplitude (0 < A < 1 cm), frequency (0 < co < 25 rad/s), and phase angle (0 < p < 27t) (Cheze et al., 1995). Noise was separately generated for each 3D coordinate of the marker trajectories. Similar to the lowerbody focus of Chapter 2, the distributions from each Monte Carlo analysis were compared for the left leg joint torque errors and marker distance errors (e.g., difference between model markers for the inverse dynamics simulation and true synthetic data cases). Results The distributions of mean inverse dynamics results are best summarized using a boxplot presenting five sample statistics: the minimum (or 10th percentile), the lower quartile (or 25th percentile), the median (or 50th percentile), the upper quartile (or 75th percentile) and the maximum (or 90th percentile) (Figure 31). RMS and mean hip torque errors were computed in the flexionextension, abductionadduction, and internalexternal rotation directions (Figure 32, Figure 33). Mostly due to JPs, the hip flexionextension and abductionadduction torques showed more variation than the internalexternal rotation torque. The IPs had insignificant effect on range of hip torques. The NPs had slightly larger effects compared to IPs. The distributions of torques were reduced by attenuating the model fitness based on percentages of parameter bounds. RMS and mean knee torque errors were computed in the flexionextension and abductionadduction directions (Figure 32, Figure 33). Similar to the corresponding hip directions, the knee torques have larger spans for the abductionadduction direction compared to the flexionextension direction. The NPs had slightly larger effects compared to IPs. The JPs had much more effect on knee torques compared to the IPs. For the IPs case, the flexionextension torque varied slightly more than the abductionadduction torque. The breadth of knee torques decreased with attenuated percentages of parameter bounds. RMS and mean ankle torque errors were computed in the plantarflexiondorsiflexion and inversioneversion directions (Figure 32, Figure 33). The inversioneversion torque displayed a broader distribution compared to the plantarflexionsdorsiflexion torque. The IPs had much less effect on ankle torque distributions compared to JPs. The NPs had slightly larger effects compared to IPs. The attenuated percentages of parameter bounds condensed the spread for both torques. RMS and mean marker distance errors were computed between the inverse dynamics simulation and synthetic data (Figure 32, Figure 33). The IPs showed marker distance errors equal to zero since true kinematics were used as inputs. The NPs represented marker distance errors consistent with the chosen numerical noise model. The JPs had the largest effect on marker distance error distributions. Discussion This study examined the distribution of inverse dynamics torques for a 3D fullbody gait model using a series of Monte Carlo analyses simultaneously varying JPs and IPs. It is well established that joint torque data is one of the most valuable quantities for the biomechanical investigation of human movement. As evident in the literature, investigators are concerned with the sensitivity of inverse dynamics results to uncertainties in JPs and IPs (Challis and Kerwin, 1996, Holden and Stanhope, 1998, Pearsall and Costigan, 1999, Andrews and Mish, 1996, Croce et al., 1996). Assuming parameter independence and adjusting a single parameter at a time may underestimate the resulting effect of the parameter variation. Specifically concerning IPs, adjusting one type of segment parameter may change the other two types. Similar to all Monte Carlo analyses assuming parameter independence, the resulting distributions may in fact be overestimated. A more diverse assortment of fullbody models are being simulated by not explicitly requiring particular body segment geometries. Nevertheless, implementing an optimization approach to maximize the fitness of the model to the experiment through determination of patientspecific JPs and IPs reduces the uncertainty of inverse dynamics results. 1  1 36 I 90th percentile .75th percentile =t50th percentile (95% confidence notch) . 25th percentile  10th percentile I All Instantiations Figure 31. Legend of five sample statistics presented by the chosen boxplot convention. _1. 12 8. 12 I 8 m 8 4 12 .r <1/'12  0 12 I 4 i 12 12 8a U))' <0^ HJ ^ ^ 25 50 75 100 IP Bounds (%) Figure 32. Comparison of rootmeansquare (RMS) leg joint torques and marker distance error distributions. First column contains distributions from varying only joint parameters (JP). Second column contains distributions from varying only inertial parameters (IP). Third column contains distributions from varying both JPs and IPs. Fourth column contains distributions from vary only the noise amplitude parameter (NP). l~rli I__=~ I__~ I___~ I____ I I~ I0 3 3 3oi  i g 3 3i o 31 3 IL 0 Z o 3 0, 3 E 24 12 if ^ ^ I ':' 25 50 75 100 i n :.u,n: IP Bounds (%) I~ 25 50 75 100 JP and IP Bounds (%) Figure 33. Comparison of mean leg joint torques and marker distance error distributions. First column contains distributions from varying only joint parameters (JP). Second column contains distributions from varying only inertial parameters (IP). Third column contains distributions from varying both JPs and IPs. Fourth column contains distributions from vary only the noise amplitude parameter (NP). 25 50 75 100 NP Bounds (%) CHAPTER 4 BENEFIT OF AUTOMATIC DIFFERENTIATION FOR BIOMECHANICAL OPTIMIZATIONS Background Optimization algorithms are often used to solve biomechanical system identification or movement prediction problems employing complicated threedimensional (3D) musculoskeletal models (Pandy, 2001). When gradientbased methods are used to solve largescale problems involving hundreds of design variables, the computational cost of performing repeated simulations to calculate finite difference gradients can be extremely high. In addition, as 3D movement model complexity increases, there is a considerable increase in the computational expense of repeated simulations. Frequently, in spite of advances in processor performance, optimizations remain limited by computation time. Both speed and robustness of gradientbased optimizations are dramatically improved by using an analytical Jacobian matrix (all firstorder derivatives of dependent objective function variables with respect to independent design variables) rather than relying on finite difference approximations. The objective function may involve thousands or perhaps millions of lines of computer code, so the task of computing analytical derivatives by hand or even symbolically may prove impractical. For more than eight years, the Network Enabled Optimization Server (NEOS) at Argonne National Laboratory has been using Automatic Differentiation (AD), also called Algorithmic Differentiation, to compute Jacobians of remotely supplied user code (Griewank, 2000). AD is a technique for computing derivatives of arbitrarily complex computer programs by mechanical application of the chain rule of differential calculus. AD exploits the fact every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations. By applying the chain rule repeatedly to these operations, derivatives can be computed automatically and accurately to working precision. This study evaluates the benefit of using AD methods to calculate analytical Jacobians for biomechanical optimization problems. For this purpose, a freelyavailable AD package, Automatic Differentiation by OverLoading in C++ (ADOLC) (Griewank et al., 1996), was applied to two biomechanical optimization problems. The first is a system identification problem for a 3D kinematic ankle joint model involving 252 design variables and 1800 objective function elements. The second is a movement prediction problem for a 3D fullbody gait model involving 660 design variables and 4100 objective function elements. Both problems are solved using a nonlinear least squares optimization algorithm. Methods Experimental kinematic and kinetic data were collected from a single subject using a videobased motion analysis system (Motion Analysis Corporation, Santa Rosa, CA) and two force plates (AMTI, Watertown, MA). Institutional review board approval and informed consent were obtained prior to the experiments. The ankle joint problem was first solved without AD using the Matlab (The Mathworks, Inc., Natick, MA) nonlinear least squares optimizer as in Chapter 2 (Reinbolt and Fregly, 2005). The optimization approach simultaneously adjusted joint parameter values and model motion to minimize errors between model and experimental marker locations. The ankle joint model possessed 12 joint parameters and 12 degreesoffreedom. Each of the 12 generalized coordinate curves was parameterized using 20 Bspline nodal points (240 total). Altogether, there were 252 design variables. The problem contained 18 error quantities for each of the 100 time frames of data. The Jacobian matrix consisted of 1800 rows and 252 columns estimated by finite difference approximations. The movement prediction problem was first solved without AD using the same Matlab nonlinear least squares optimizer (Fregly et al., 2005). The optimization approach simultaneously adjusted model motion and ground reactions to minimize knee adduction torque and 5 categories of tracking errors (foot path, center of pressure, trunk orientation, joint torque, and fictitious groundtopelvis residual reactions) between model and experiment. The movement prediction model possessed 27 degreesoffreedom (21 adjusted by Bspline curves and 6 prescribed for arm motion) and 12 ground reactions. Each adjustable generalized coordinate and reaction curve was parameterized using 20 Bspline nodal points (660 total design variables). The problem contained 41 error quantities for each of the 100 time frames of data. The Jacobian matrix consisted of 4100 rows and 660 columns estimated by finite difference approximations. ADOLC was incorporated into each objective function to compute an analytical Jacobian matrix. This package was chosen because the objective functions were comprised of Matlab mexfiles, which are dynamic link libraries of compiled C or C++ code. ADOLC was implemented into the C++ source code by the following steps: 1. Mark the beginning and end of active section (portion computing dependent variables from independent variables) using builtin functions trace on and trace off, respectively. 2. Select a set of active variables (those considered differentiable at some point in the program execution) and change type from double to builtin type double. 3. Define a set of independent variables using the output stream operator (<<=). 4. Define a set of dependent variables using the input stream operator (>>=). 5. Call the builtin driver function j acobian to compute firstorder derivatives using reverse mode AD. 6. Compile the code including builtin header file adolc .h and linking with builtin library file adolc. lib. All optimizations with and without AD were performed on a 1.73 GHz Pentium M laptop with 2.00 GB of RAM. The computation time performance was compared. Results For each problem, performance comparisons of optimizing with and without AD are summarized in Table 41 and Table 42. The use of AD increased the computation time per objective function evaluation. However, the number of function evaluations necessary per optimization iteration was far less with AD. For the system identification problem, the computation time required per optimization iteration with AD was approximately 20.5% (or reduced by a factor of 4.88) of the time required without AD. For the movement prediction problem, the computation time required per optimization iteration with AD was approximately 51.1% (or reduced by a factor of 1.96) of the time required without AD. Although computation times varied with and without AD, the optimization results remained virtually identical. Discussion The main motivation for investigating the use of AD for biomechanical optimizations was to improve computational speed of obtaining solutions. Speed improvement for the movement prediction optimization in particular was not as significant as anticipated. Further investigation is necessary to determine the effect of AD characteristics such as forward mode vs. reverse mode and source code transformation vs. operator overloading on computational speed. Whichever AD method is used, having analytical derivatives eliminates inaccurate search directions and sensitivity to design variable scaling which can plague optimizations that use finite difference gradients. If central (more accurate) instead of forward differencing was used in the movement prediction optimization without AD, the performance improvement would have been a factor of four instead of two. Special dynamics formulations can also be utilized to compute analytical derivatives concurrently while evaluating the equations of motion, and the tradeoffs between those approaches and AD require further investigation. While ADbased analytical derivatives may be less efficient computationally than those derived using special dynamics formulations, AD provides effortless updating of the derivative calculations should the biomechanical model used in the optimization be changed. For largescale problems, AD provides a relatively simple means for computing analytical derivatives to improve the speed of biomechanical optimizations. Table 41. Performance results for system identification problem for a 3D kinematic ankle joint model involving 252 design variables and 1800 objective function elements. Performance Criteria Time per Function Evaluation (s) Number of Function Evaluations per Optimization Iteration Time per Optimization Iteration (s) Without AD 0.0189 252 4.77 Table 42. Performance results for movement prediction problem for a 3D fullbody gait model involving 660 design variables and 4100 objective function elements. Performance Criteria Without AD With AD Time per Function Evaluation (s) 0.217 73.1 Number of Function Evaluations per Optimization Iteration 660 1 Time per Optimization Iteration (s) 143 73.1 With AD 0.978 1 0.978 CHAPTER 5 APPLICATION OF PATIENTSPECIFIC DYNAMIC MODELS TO PREDICT FUNCTIONAL OUTCOMES Background Imagine the benefit to the healthcare provider and, more importantly, the patient, if certain clinical parameters may be evaluated pretreatment in order to predict the posttreatment outcome. For example, a patientspecific dynamic model may be useful for planning intended surgical parameters and predicting the outcome of HTO. Researchers have identified the peak external knee adduction torque 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 postsurgery knee adduction torque. There are a few gait studies evaluating effects of kinematic changes on the knee adduction torque. One study examined the effects of increased foot progression angle, or toeout gait, on minimizing knee adduction torque during walking, stair ascent, and stair descent (Manal and Guo, 2005). Another study investigated the minimization of knee adduction torque by changing lowerbody kinematics without changing foot path during gait (Fregly et al., 2005). No computational framework is available to perform a virtual HTO surgery and predict the resulting knee adduction torque. This study presents the development and application of an inverse dynamics optimization approach to predict novel patientspecific gait motions, such as gait following HTO surgery and the resulting internal knee abduction torque. To further tailor the kinematic and kinetic model developed in Chapter 2, response surface (RS) methods are used to determine weights for a cost function specific to the patient and movement task, comprising phase three of the optimization procedure. Other biomechanical studies (Chang et al., 1999, Hong et al., 2001) have used RS optimization methods, but this study is the first to perform inverse dynamic optimizations of gait. To increase the speed and robustness of optimizations, automatic differentiation (AD) is used to compute analytic derivatives of the inverse dynamics optimization objective function. Methods Experimental kinematic and kinetic gait data were collected from a single subject using a videobased motion analysis system (Motion Analysis Corporation, Santa Rosa, CA) and two force plates (AMTI, Watertown, MA). Institutional review board approval and informed consent were obtained prior to the experiments. Normal gait data was recorded to calibrate the patientspecific model parameters as described in Chapter 2. Toeout, or increased foot progression angle, gait data was gathered to calibrate the patientspecific control parameters. Wide stance gait data was collected to evaluate the calibrated model. A parametric 3D, 27 degreeoffreedom (DOF), fullbody gait model (Figure 21, Table 21) was constructed, whose equations of motion were derived with the symbolic manipulation software, AutolevTM (OnLine Dynamics, Sunnyvale, CA). The pelvis was connected to ground via a 6 DOF joint and the remaining 13 segments comprised four open chains branching from the pelvis. Anatomic landmark methods were used to estimate nominal values for 84 inertial parameters (de Leva, 1996) and 98 joint parameters (Bell et al., 1990, Churchill et al., 1998, Inman, 1976). Select model parameters were identified via optimization as described previously in Chapter 2 (Reinbolt and Fregly, 2005). The inverse dynamics optimization procedure adjusted model motion and ground reactions to minimize errors between model and experimental, or prescribed, quantities to be tracked (Equation 51). Specifically, the design variables were 660 Bspline nodes (q, r) parameterizing the generalized coordinate (q's) trajectories and ground reaction (r's) data (20 nodes per curve). The initial value for each Bspline node was chosen to be the corresponding value from normal gait. The inverse dynamics optimization cost function predictt outcome) minimized the weighted (Equation 52, Equation 53, Table 51) tracking errors summed over the number of time frames (nf) for the following quantities: 6 DOFs for each prescribed foot path (qfoot), 2 transverse plane translations for pelvis (qpeivs,), 3 DOFs for trunk orientation (qtrunk), 2 transverse plane translations for center of pressure for each foot (CoPfoot), 3 residual pelvis forces (Fpe ivs), 3 residual pelvis torques (Tpeivis), and leg joint torques (Tleg). The inverse dynamics optimizations were performed with Matlab's nonlinear least squares algorithm (The Mathworks, Natick, MA) using analytical Jacobians computed by AD. 2 6 2 3 w1 (Aoot ) + w2 (Ae,. )+ W3 (Aqu + =l1 k=l =l1 =l1 nf 2 2 3 predict =mm w4 (ACoot)yk +W5 p(A elvis +ATpelvs) + (51) outcome qr 1 j=1 k=l J1 6 2 S[AT (pcp J=1 k=1 PCP= [6 W7 W8 W9 W10 Wll] (52) ATle (pcp) = w6 hp + 7AThp, + w8ATh, + flexion adduction rotation (53) (53) w9ATknee + WloATtalocrural + wTubtalar flexlon dorsiflexlon inversion Before predicting outcomes with the inverse dynamics optimization, the phase 3 optimization calibrated the 6 control parameters, or w6 w11 (Equation 52, Equation 53, Table 51), to the patient's toe out gait motion. Each prescribed foot path was the only tracked quantity which was not experimentally measured beforehand. For 3 DOFs, fixed offsets (Table 52) were defined to approximate the predicted toeout foot path without using the exact foot path. All other tracked quantities involved the experimentally measured normal gait motion. A computationally efficient optimization was developed using RS methods to determine how well the simulated knee abduction torque matched the experimental knee abduction torque for toeout gait. Response surfaces were developed to approximate the knee abduction torque predicted by the model for various control parameter weights. A separate quadratic RS was created for each time frame of torque data (125 response surfaces) predicted by the model where the 6 control parameter weights were treated as design variables. Although separate response surface were constructed, the correlation coefficients between them was not computed. Each quadratic RS required solution of 28 unknown polynomial coefficients. A Hammersley Quasirandom Sequence was used to generate 64 sampled data points within the bounds of 0 to 10 for the 6dimensional space. The RS coefficients were then determined via a linear least squares fit in Matlab (The Mathworks, Natick, MA). Using the response surfaces as surrogates for repeated inverse dynamic optimizations, an optimization problem was formulated to minimize errors between model predicted knee abduction torque and experimentally realized abduction torque for toeout gait. A Hammersley Quasirandom Sequence was used to generate 1000 initial seed points within the bounds of 0 to 10 for the 6dimensional space. The 6 control parameters resulting in the lowest cost function value were used. The kinematic and kinetic changes for the left leg were compared. The calibrated model was evaluated by predicting wide stance gait. Similar to toeout gait, fixed offsets (Table 52) were defined for 3 DOFs to approximate the predicted wide stance foot path without using the exact foot path. Again, all other tracked quantities involved the experimentally measured normal gait motion. The kinematic and kinetic changes for the left leg were compared. The calibrated model was applied to predict the outcome of HTO surgery. All tracked quantities involved the experimentally measured normal gait motion, including the prescribed foot path (e.g., all foot path offsets equaled zero). The virtual HTO surgery was performed as a lateral opening wedge osteotomy. The anteriorposterior axis of rotation was located 10 cm inferior and 5 cm lateral to the midpoint of the transepicondylar axis. The sensitivity of the knee abduction torque to the extent, or wedge angle, of the virtual HTO surgery was illustrated in for 3, 5, and 7. To demonstrate the effects of prior calibrating optimizations which determined joint parameters (phase one from Chapter 2) and inertial parameters (phase two from Chapter 2), the entire process of creation and application of patientspecific dynamic models was repeated for three additional cases: 1) phase three only (or without phase one or two), 2) phases one and three, and 3) phases two and three. The intermediate calibration and final evaluation effects on knee abduction torque results were compared. Results Using the response surfaces as surrogates for repeated inverse dynamic optimizations using the model calibrated with phases one, two, and three, the optimization determined control parameters (Table 53) that minimized errors between model predicted knee abduction torque and experimentally obtained torque for toeout gait with 0.08 % body weight times height (BW*H) and 0.18 %BW*H errors for 1st and 2nd peaks, respectively (RMS for entire cycle = 0.14 %BW*H) (Figure 51, Table 54). Changes for hip torques were matched well (RMS = 0.48 %BW*H) with the exception a minor deviation (2.82 %BW*H) in the hip flexionextension torque at the beginning of the cycle. Knee flexionextension torque changes were matched with an RMS difference of 0.22 %BW*H. Both ankle torques remained close to the initial normal gait values (RMS = 0.02 %BW*H) without matching the small changes for toeout gait (RMS = 0.48 %BW*H). Overall changes for hip joint angles were matched (RMS = 4.110) with the exception of internalexternal rotation angle (max = 9.540 at 7% of cycle). The knee flexionextension angle remained more flexed than necessary (RMS = 2.690, max = 3.880 at 16% of cycle). Both ankle joint angles generally matched the changes well (RMS = 3.580), but deviations were seen near 60% of cycle with a maximum of 9.670. Other tracked quantities demonstrated good agreement (Table 55). All kinematic and kinetic differences are summarized in Table 57. Using the control parameters tuned with toeout gait for the model calibrated with phases one, two, and three, the optimization predicted knee abduction torque for wide stance with RMS of 0.23 %BW*H (0.03 %BW*H and 0.11 %BW*H errors for 1st and 2nd peaks, respectively) (Figure 52, Table 54). Changes for hip torques were matched well (RMS = 0.46 %BW*H) with the exception a minor deviations (1.54 %BW*H at 4% of cycle and 3.27 %BW*H at 100% of cycle) in the hip flexionextension torque. Knee flexionextension torque changes were matched with an RMS difference of 0.39 %BW*H. Both ankle torques remained close to the initial normal gait values (RMS = 0.008 %BW*H) without matching the small changes for wide stance gait (RMS = 0.36 %BW*H). Overall changes for hip joint angles were matched (RMS = 3.130) with the maximum difference (7.81 %BW*H) occurring at 86% of cycle for the hip flexionextension angle. The knee flexionextension angle remained more extended than necessary (RMS = 3.060, max = 3.310 at 76% of cycle, or 2nd peak). Both ankle joint angles generally matched the changes satisfactorily (RMS = 5.180), but deviations were demonstrated near the end of cycle for both talocrural (max = 11.51 at 90% of cycle) and subtalar angles (max = 11.420 at 100% of cycle). Other tracked quantities demonstrated good agreement (Table 56). All kinematic and kinetic differences are summarized in Table 58. There were noticeable effects when using dynamic models created without phase one or phase two of the optimization procedure. The resulting control parameters from phase three varied widely (Table 53). The predicted knee abduction torque for toeout gait showed considerable differences using models without phase one included, while the model prediction including phases one and three was comparable to the model with phases one, two, and three (Figure 51, Table 54). For wide stance gait, the knee abduction torque was not predicted well when using models without phase one, but the prediction including phases one and three showed some improvement (Figure 52, Table 54). In all cases, tracked quantities demonstrated good agreement (Table 55, Table 56). Concerning the virtual HTO, the sensitivities to wedge angle showed the expected incremental reductions in knee abduction torque for only those models including phases one and three or phases one, two, and three (Figure 53). The differences in dynamic models created from different portions of the multiphase optimization approach led to variations in the computed knee abduction torques for experimental and simulated data (Figure 54, Figure 55). Discussion The main motivation for developing a patientspecific inverse dynamics optimization approach is to predict novel patientspecific gait motions (e.g., subtle gait changes and HTO postsurgery gait). Toeout gait was successfully used to tune patientspecific control parameters. Wide stance gait predictions match the experiment rather well. The response surface approach greatly increased the computation efficiency of determining the control parameters. The current approach has several limitations. A pin joint knee model was chosen to represent the flexionextension motion that dominates other knee motions during gait. A rigid foot model without ground contact model was selected to increase the computational speed and exploit the ability to prescribe a desired foot path for the inverse dynamics optimization. A few kinematic and kinetic quantities were not predicted as well as others. Selection of tracking term weights is subjective and was performed in a manner to constrain the foot paths alone and track the other quantities. The control parameters identified during phase 3 depend on the weights chosen for the tracking terms. Although referred to as control parameters (joint torque = control), the cost function weights do not have an obvious physical explanation tied to neural control strategies. It cannot be claimed that predictions made with this multiphase approach will reproduce the actual functional outcome for every patient and every data set. This is clear from the calibration results for toeout gait and the evaluation results for wide stance gait, where the RMS errors in the knee abduction torque predictions were not zero. At the same time, the predicted curves matched the 1st and 2nd peaks well. Thus, it can only be claimed that the optimized model structure provides the best possible prediction given the imperfect movement data inputs. For the model calibrated with phase one and two, the optimization predictions for virtual HTO surgeries showed sensitivities in agreement with previous clinical studies (Bryan et al., 1997, Wada et al., 1998) (Figure 53). For the 30 case, the knee abduction torque was reduced by 0.78 %BW*H and 0.71 %BW*H for the 1st and 2nd peaks, respectively. Bryan et al. (1997) showed an approximately 1.1 %BW*H (slightly higher than optimization prediction) decrease for 30 of change. For the 50 case, the knee abduction torque was reduced by 0.95 %BW*H and 1.26 %BW*H for the 1st and 2nd peaks, respectively. Bryan et al. (1997) showed an approximately 1.2 %BW*H (very similar to optimization prediction) decrease for 50 of change. For the 70 case, the knee abduction torque was reduced by 1.23 %BW*H and 1.74 %BW*H for the 1st and 2nd peaks, respectively. Bryan et al. (1997) showed an approximately 1.5 %BW*H (slightly lower than optimization prediction) decrease for 70 of change. Cost function weights calibrated to a particular gait motion accurately predict several kinematic and kinetic quantities for a different gait motion. Further investigation is necessary to determine the efficacy of the predictions for HTO. It would be advantageous to create patientspecific models and predictions for a population of HTO patients presurgery and compare with postsurgery outcome. Examination of different gaits (e.g., normal and pathological) requires future study as well. Table 51. Descriptions of cost function weights and phase 3 control parameters. Weight Description wi Scale factor for tracking foot paths W2 Scale factor for tracking pelvis transverse plane translations W3 Scale factor for tracking trunk orientations W4 Scale factor for tracking center of pressure translations W5 Scale factor for tracking residual pelvis forces and torques W6 Control parameter for tracking hip flexionextension torque W7 Control parameter for tracking hip adductionabduction torque w8 Control parameter for tracking hip internalexternal torque W9 Control parameter for tracking knee flexionextension torque wlo Control parameter for tracking talocrural dorsiflexionplantarflexion torque wil Control parameter for tracking subtalar inversioneversion torque Table 52. Summary of fixed offsets added to normal gait for each prescribed foot path. Offset Side Toeout Gait Wide Stance Gait Right 41.78 5.26 Anterior Translation (mm) Left 21.23 3.50 Right 28.45 62.57 Lateral Translation (mm) Left 34.86 42.18 Right 15.79 0.75 Transverse Plane Rotation () Right Left 13.12 0.29 Table 53. Comparison of cost function weights and phase 3 control parameters. Phases 1, 2, Weight Phase 3 only Phases 1 and 3 Phases 2 and 3 ' and 3 wi 10.00 10.00 10.00 10.00 W2 1.00 1.00 1.00 1.00 W3 1.00 1.00 1.00 1.00 w4 1.00 1.00 1.00 1.00 W5 1.00 1.00 1.00 1.00 W6 8.71 5.79e01 8.13 5.17 W7 8.67 2.23 6.84 3.57e01 ws 7.31e08 9.29e10 7.18e01 1.08 W9 9.14 1.62e07 6.74 2.94e01 wio 2.61 8.18 6.87 5.55 wil 4.24 9.63 9.18 9.17 Table 54. Summary of rootmeansquare (RMS) errors for predicted left knee abduction torque quantities for toeout and wide stance gait. RMS error (% BW*H) Gait trial Phases 1, 2, Phase 3 only Phases 1 and 3 Phases 2 and 3 P and 3 Toeout 0.66 0.13 0.69 0.14 Wide stance 0.68 0.37 0.68 0.21 Table 55 Summary of rootmeansquare (RMS) errors for tracked quantities for toeout gait. RMS error Quantity Description Phase 3 Phases 1 Phases 2 Phases 1, 2, only and 3 and 3 and 3 6 DOFs for right (qfoot)rzght prescribed foot path 6 DOFs for left (qfoot)left prescribed foot path 2 transverse plane qpelvls translations for pelvis 3 DOFs for trunk qtrunk orientation 2 transverse plane (CoPfoot)rzght translations for right center of pressure 2 transverse plane (CoPfoot)left translations for left center of pressure 3 residual pelvis fpelvls forces 3 residual pelvis Tvls torques 2.55 mm 1.580 0.94 mm 1.270 3.64 mm 2.030 2.02 mm 1.380 1.53 mm 0.960 0.57 mm 0.860 2.35 mm 1.560 1.36 mm 0.970 7.65 mm 5.14 mm 9.91 mm 6.61 mm 0.320 0.360 0.520 0.380 9.68 mm 9.78 mm 5.68 mm 7.30 mm 2.01 mm 4.17 mm 1.68 mm 2.73 mm 4.23 N 4.27 N 3.82 N 4.11N 3.93 Nm 3.37 Nm 2.99 Nm 2.41 Nm Table 56. Summary of rootmeansquare (RMS) errors for tracked quantities for wide stance gait. RMS error Quantity Description Phase 3 Phases 1 Phases 2 Phases 1, 2, only and 3 and 3 and 3 6 DOFs for right (qfoot)rght prescribed foot path 6 DOFs for left (qfoot)left prescribed foot path 2 transverse plane qwelvs translations for pelvis 3 DOFs for trunk qtrunk orientation 2 transverse plane (CoPfoot)rzght translations for right center of pressure 2 transverse plane (CoPfoot)left translations for left center of pressure 3 residual pelvis rfpelvls forces 3 residual pelvis Tvls torques 1.07 mm 0.990 0.99 mm 1.04 1.19 mm 1.360 2.07 mm 1.290 0.85 mm 1.090 0.60 mm 1.160 0.45 mm 0.370 0.32 mm 0.240 6.47 mm 6.42 mm 6.22 mm 7.33 mm 0.780 0.830 0.840 0.750 4.45 mm 5.73 mm 4.74 mm 4.40 mm 1.80 mm 1.74 mm 3.33 mm 0.79 mm 3.56N 3.85 N 3.65 N 3.77 N 3.17Nm 2.73 Nm 3.50Nm 2.48 Nm Table 57. Summary of rootmeansquare (RMS) differences between optimization results and original normal gait or final toeout gait kinematic and kinetic quantities. RMS difference from optimization Quanitity Phase 3 only Phases 1 and 3 Phases 2 and 3 Phases 1, 2, and 3 Original Final Original Final Original Final Original Final Hip flexion angle (o) Hip abduction angle (o) Hip rotation angle (o) Knee flexion angle (o) Ankle flexion angle (o) Ankle inversion angle (o) Anterior center of pressure (mm) Lateral center of pressure (mm) Hip flexion torque (% BW*H) Hip abduction torque (% BW*H) Hip rotation torque (% BW*H) Knee flexion torque (% BW*H) Ankle flexion torque (% BW*H) Ankle inversion torque (% BW*H) Anterior ground reaction force (% BW) Superior ground reaction force (% BW) Lateral ground reaction force (% BW) 1.29 1.73 10.96 1.61 1.85 4.43 11.35 22.41 2.49e02 1.70e02 1.62e01 1.67e02 5.09e02 3.92e02 2.33e01 1.73 3.22 2.55 5.39 2.68 3.21 4.71 15.09 23.35 1.55 7.46e01 1.76e01 2.98e01 1.27 2.32 1.26 3.49 9.14e01 2.20 10.94 1.03 2.05 3.89 2.45 5.36 4.05e01 9.58e02 1.42e01 3.75e01 3.24e02 7.07e02 2.65e01 1.29 3.72 2.27 5.71 2.76 2.85 4.49 9.89 9.27 9.39e01 3.43e01 1.99e01 2.92e01 5.49e01 3.98e01 1.31 3.60 2.09 2.37 11.03 1.62 1.53 4.14 1.29 1.99 2.15e02 2.43e02 1.12e01 1.50e02 1.41e02 1.79e02 5.16e01 1.52 2.79 2.39 5.35 2.52 2.85 4.87 9.97 7.09 1.55 7.50e01 1.80e01 2.92e01 1.29 2.31 1.32 3.95 1.08 3.46 2.63 2.32 11.27 5.76 8.93e01 2.69 1.78 2.58 3.49 4.36 1.58 9.33 3.53 7.86 3.11e02 7.01e01 2.31e01 4.21e01 1.10e01 1.83e01 2.41e01 2.21e01 1.60e02 5.53e01 2.98e02 3.86e01 2.51e01 1.39 7.07e01 4.04 6.98e01 3.65e01 5.43e01 2.78e01 7.58e01 3.42e01 5.64e01 3.28e01 Table 58. Summary of rootmeansquare (RMS) differences between optimization results and original normal gait or final wide stance gait kinematic and kinetic quantities. RMS difference from optimization Quanitity Phase 3 only Phases 1 and 3 Phases 2 and 3 Phases 1, 2, and 3 Original Final Original Final Original Final Original Final Hip flexion angle (o) Hip abduction angle (o) Hip rotation angle (o) Knee flexion angle (o) Ankle flexion angle (o) Ankle inversion angle (o) Anterior center of pressure (mm) Lateral center of pressure (mm) Hip flexion torque (% BW*H) Hip abduction torque (% BW*H) Hip rotation torque (% BW*H) Knee flexion torque (% BW*H) Ankle flexion torque (% BW*H) Ankle inversion torque (% BW*H) Anterior ground reaction force (% BW) Superior ground reaction force (% BW) Lateral ground reaction force (% BW) 6.64e01 3.07 2.43 6.75e01 2.17 6.82 1.44 2.10 2.33e02 3.82e02 1.93e01 1.28e02 1.85e02 3.16e02 2.76e01 1.03 2.66 4.39 1.73 2.62 3.46 4.35 6.43 27.40 11.22 5.04e01 5.25e01 2.44e01 4.27e01 3.74e01 4.23e01 1.13 4.15 8.75e01 5.50e01 3.59 1.74 1.48 2.19 4.06 1.49 1.95 1.71e01 1.29e01 2.10e01 2.49e01 1.15e02 2.53e02 3.94e01 8.09e01 2.35 4.55 1.69 2.82 4.41 5.18 4.07 26.01 9.13 5.95e01 3.39e01 2.87e01 6.24e01 3.91e01 3.45e01 1.20 3.98 1.14 3.68e01 3.21 1.91 6.28e01 1.91 5.87 4.45 1.54 2.43e02 6.54e02 2.00e01 1.46e02 1.01e02 1.62e02 2.81e01 1.23 2.56 4.29 1.73 2.58 3.48 4.33 5.53 27.52 11.22 5.25e01 5.21e01 2.49e01 4.37e01 3.71e01 4.23e01 1.17 4.30 8.72e01 1.66 4.17 3.21 1.98 2.49 3.18 1.12 2.79 2.08 5.17 5.00 4.72 1.05 26.31 3.68e01 8.18 1.58e02 6.26e01 2.53e01 3.23e01 9.11e02 2.48e01 2.79e01 3.90e01 1.07e02 3.93e01 2.39e02 3.63e01 6.76e01 1.04 2.13 3.93 1.75 1.79 14ii i   s A Original B Optimization Q< Final Sc D < 1 C D "o < 1 0 20 40 60 80 100 0 20 40 60 80 100 Gait Cycle (%) Gait Cycle (%) Figure 51. Comparison of left knee abduction torques for toeout gait. Original (blue) is experimental normal gait, simulation (red) is predicted toeout gait, and final (green) is experimental toeout gait. Plot A) contains curves for model using phase 3 only. Plot B) contains curves for model using phases 1 and 3. Plot C) contains curves for model using phases 2 and 3. Plot D) contains curves for model using phases 1, 2, and 3. 14   i i i s A Original B o Optimization Q< Final 2 0I I I I C D <3 2 0 20 40 60 80 100 0 20 40 60 80 100 Gait Cycle (%) Gait Cycle (%) Figure 52. Comparison of left knee abduction torques for wide stance gait, where original (blue) is experimental normal gait, simulation (red) is predicted toeout gait, and final (green) is experimental toeout gait. Plot A) contains curves for model using phase 3 only. Plot B) contains curves for model using phases 1 and 3. Plot C) contains curves for model using phases 2 and 3. Plot D) contains curves for model using phases 1, 2, and 3. CB 0 A o B 30 2  a "0 CD a, Z5 Sc D a r 0 20 40 60 80 100 0 20 40 60 80 100 Gait Cycle (%) Gait Cycle (%) Figure 53. Comparison of left knee abduction torques for simulated high tibial osteotomy (HTO) postsurgery gait. Plot A) contains curves for model using phase 3 only. Plot B) contains curves for model using phases 1 and 3. Plot C) contains curves for model using phases 2 and 3. Plot D) contains curves for model using phases 1, 2, and 3. 63 0 A 3o o < 3 2 0 1 84 C0 "o < 3 rI 3  S0 .2 1 0 Gait Cycle (%) Figure 54. Comparison of mean (solid black line) plus or minus one standard deviation (gray shaded area) for experimental left knee abduction torques. Plot A) contains distribution for normal gait. Plot B) contains distribution for toeout gait. Plot C) contains distribution for wide stance gait. <3 I 0' o 64 4 A 1 4 o <3 < 3 cO CD 1 B o <3 a3 < 1 0 <0 20 40 60 80 100 Gait Cycle (%) Figure 55. Comparison of mean (solid black line) plus or minus one standard deviation (gray shaded area) for simulated left knee abduction torques. Plot A) contains distribution for toeout gait. Plot B) contains distribution for wide stance gait. CHAPTER 6 CONCLUSION Rationale for New Approach The main motivation for developing a 27 DOF patientspecific inverse dynamic model and the associated optimization approach is to predict the postsurgery peak external 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 and kinetic model (Andriacchi and Strickland, 1985; Challis and Kerwin, 1996; Cappozzo and Pedotti, 1975; Davis, 1992; Holden and Stanhope, 1998; Holden and Stanhope, 2000; Stagni et al., 2000). Development of an accurate model tailored to a specific patient forms the foundation for creating a predictive patientspecific dynamic simulation. Synthesis of Current Work and Literature The multiphase optimization method satisfactorily determines patientspecific model parameters defining a 3D model that is well suited to a particular patient. Two conclusions may be drawn from comparing and contrasting the 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 optimization is necessary 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 multiphase optimization approach successfully reduces the fitness errors between the patientspecific model and the experimental motion data. The precision of dynamic analyses made for a particular patient depends on the accuracy of the patientspecific kinematic and kinetic 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 motioncapture data. By using a multiphase 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 dynamic analyses and optimizations. Simulations based on inverse dynamics optimization subject to reality constraints predict physicsbased motion. The results accurately represent a specific patient within the bounds of the chosen dynamic model, experimental task, and data quality. Given the agreement of the current results with previous studies, there is significant benefit in the construction and optimization a patientspecific functional dynamic model to assist in the planning and outcome prediction of surgical procedures. This work comprises a unique computational framework to create and apply objective patientspecific models (different from current intuitive models based on clinical experience or regression models based on population studies) to predict clinically significant outcomes. GLOSSARY Abduction Acceleration Active markers Adduction Ankle inversioneversion Ankle motion Ankle plantarflexiondorsiflexion Anterior Coccyx Constraint functions Coronal plane Degree of freedom (DOF) Movement away from the midline of the body in the coronal plane. The time rate of change of velocity. Joint and segment markers used during motion capture that emit a signal. Movement towards the midline of the body in the coronal plane. Motion of the long axis of the foot within the coronal plane as seen by an observer positioned along the anteriorposterior axis of the shank. The ankle angles reflect the motion of the foot segment relative to the shank segment. Motion of the plantar aspect of the foot within the sagittal plane as seen by an observer positioned along the mediallateral axis of the shank. The front or before, also referred to as ventral. The tailbone located at the distal end of the sacrum. Specific limits that must be satisfied by the optimal design. The plane that divides the body or body segment into anterior and posterior parts. 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 Distal Dorsiflexion Epicondyle Variables that change to optimize the design. Away from the point of attachment or origin. Movement of the foot towards the anterior part of the tibia in the sagittal plane. Process that develops proximal to an articulation and provides additional surface area for muscle attachment. Version Extension External (lateral) rotation External moment Femur Flexion Force Force plate Forward dynamics A turning outward. Movement that rotates the bones comprising a joint away from each other in the sagittal plane. 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. The load applied to the human body due to the ground reaction forces, gravity and external forces. The longest and heaviest bone in the body. It is located between the hip joint and the knee joint. Movement that rotates the bones comprising a joint towards each other in the sagittal plane. A push or a pull and is produced when one object acts on another. 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. 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. A manner of walking or moving on foot. Generalized coordinates High tibial osteotomy (HTO) Hip abductionadduction Hip flexionextension Hip internalexternal rotation Hip motion Inferior InterASIS distance Internal (medial) rotation Internal joint moments 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. 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 weightbearing axis from the diseased medial compartment to the healthy lateral compartment of the knee. Motion of a long axis of the thigh within the coronal plane as seen by an observer positioned along the anteriorposterior axis of the pelvis. Motion of the long axis of the thigh within the sagittal plane as seen by an observer positioned along the mediallateral axis of the pelvis. Motion of the mediallateral axis of the thigh with respect to the mediallateral axis of the pelvis within the transverse plane as seen by an observer positioned along the longitudinal axis of the thigh. The hip angles reflect the motion of the thigh segment relative to the pelvis. Below or at a lower level (towards the feet). The length of measure between the left anterior superior iliac spine (ASIS) and the right ASIS. 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. The net result of all the internal forces acting about the joint which include moments due to muscles, ligaments, joint friction and structural constraints. Gait Inverse dynamics The joint moment is usually calculated around a joint center. 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 Kinematics Kinetics Knee abductionadduction Knee flexionextension Knee internalexternal rotation Knee motion Lateral A turning inward. 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. 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. Motion of the long axis of the shank within the coronal plane as seen by an observer positioned along the anteriorposterior axis of the thigh. Motion of the long axis of the shank within the sagittal plane as seen by an observer positioned along the mediallateral axis of the thigh. Motion of the mediallateral axis of the shank with respect to the mediallateral axis of the thigh within the transverse plane as viewed by an observer positioned along the longitudinal axis of the shank. The knee angles reflect the motion of the shank segment relative to the thigh segment. Away from the body's longitudinal axis, or away from the midsagittal plane. Malleolus Markers Medial Midsagittal plane Model parameters Moment of force Motion capture Objective functions Parametric Passive markers Pelvis Broadened distal portion of the tibia and fibula providing lateral stability to the ankle. 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. Toward the body's longitudinal axis, or toward the midsagittal plane. The plane that passes through the midline and divides the body or body segment into the right and left halves. 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. 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 twodimensions, 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. Interpretation of computerized data that documents an individual's motion. Figures of merit to be minimized or maximized. Of or relating to or in terms of parameters, or factors that define a system. Joint and segment markers used during motion capture that reflect visible or infrared light. Consists of the two hip bones, the sacrum, and the coccyx. It is located between the proximal spine and the hip joints. Pelvis anteriorposterior tilt Pelvis elevationdepression Pelvis internalexternal rotation Pelvis motion Plantarflexion Posterior Proximal Range of motion Sacrum Sagittal plane Skin movement artifacts Stance phase Motion of the long axis of the pelvis within the sagittal plane as seen by an observer positioned along the mediallateral axis of the laboratory. Motion of the mediallateral axis of the pelvis within the coronal plane as seen by an observer positioned along the anteriorposterior axis of the laboratory. Motion of the mediallateral or anteriorposterior axis of the pelvis within the transverse plane as seen by an observer positioned along the longitudinal axis of the laboratory. 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. Movement of the foot away from the anterior part of the tibia in the sagittal plane. The back or behind, also referred to as dorsal. Toward the point of attachment or origin. Indicates joint motion excursion from the maximum angle to the minimum angle. 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. The plane that divides the body or body segment into the right and left parts. The relative movement between skin and underlying bone. The period of time when the foot is in contact with the ground. Subtalar joint Superior Synthetic markers Swing phase Talocrural joint Talus Tibia Transepicondylar Transverse plane Velocity Located between the distal talus and proximal calcaneous, also known as the talocalcaneal joint. Above or at a higher level (towards the head). Computational representations of passive markers located on the kinematic model. The period of time when the foot is not in contact with the ground. Located between the distal tibia and proximal talus, also known as the tibialtalar joint. The largest bone of the ankle transmitting weight from the tibia to the rest of the foot. The large medial bone of the lower leg, also known as the shinbone. It is located between the knee joint and the talocrural joint. The line between the medial and lateral epicondyles. The plane at right angles to the coronal and sagittal planes that divides the body into superior and inferior parts. The time rate of change of displacement. LIST OF REFERENCES Andrews, J.G., and Mish, S.P., 1996. "Methods for Investigating the Sensitivity of Joint Resultants to Body Segment Parameter Variations." Journal ofBiomechanics, Volume 29, Number 5, Pages 651654. Andriacchi, T.P., 1994. "Dynamics of Knee Malalignment." Orthopedic Clinics of North America, Volume 25, Number 3, Pages 395403. Andriacchi, T.P., and Strickland, A.B., 1985. "Gait Analysis as a Tool to Assess Joint Kinetics." In: Berme, N., Engin, A.E., Correia da Silva, K.M. (Editors), Biomechanics of Normal and Pathological Human Articulating Joints. Martinus Nijhoff Publishers, Dordrecht, The Netherlands, Pages 83102. Arnold, A.S, Asakawa, D.J, and Delp, S.L., 2000. "Do the Hamstrings and Adductors Contribute to Excessive Internal Rotation of the Hip in Persons with Cerebral Palsy?" Gait & Posture, Volume 11, Number 3, Pages 181190. Arnold, A.S., and Delp, S.L., 2001. "Rotational Moment Arms of the Hamstrings and Adductors Vary with Femoral Geometry and Limb Position: Implications for the Treatment of InternallyRotated Gait." Journal ofBiomechanics, Volume 34, Number 4, Pages 437447. Bell, A.L., Pedersen, D.R., and Brand, R.A., 1990. "A Comparison of the Accuracy of Several Hip Center Location Prediction Methods." Journal ofBiomechanics, Volume 23, Number 6, Pages 617621. Bogert, A.J. van den, Smith, G.D., and Nigg, B.M., 1994. "In Vivo Determination of the Anatomical Axes of the Ankle Joint Complex: An Optimization Approach." Journal ofBiomechanics, Volume 27, Number 12, Pages 14771488. Bryan, J.M., Hurwitz, D.E., Bach, B.R., Bittar, T., and Andriacchi, T.P., 1997. "A Predictive Model of Outcome in High Tibial Osteotomy." In Proceedings of the 43rd Annual Meeting of the Orthopaedic Research Society, San Francisco, California, February 913, Volume 22, Paper 718. Cappozzo, A., Catani, F., and Leardini, A., 1993. "Skin Movement Artifacts in Human Movement Photogrammetry." In Proceedings of the XIVth Congress of the International Society ofBiomechanics, Paris, France, July 48, Pages 238239. Cappozzo, A., Leo, T., and Pedotti, A., 1975. "A General Computing Method for the Analysis of Human Locomotion." Journal ofBiomechanics, Volume 8, Number 5, Pages 307320. Center for Disease Control (CDC), 2003. Targeting Arthritis: The Nation's Leading Cause ofDisability. 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. "Quantification of the Uncertainties in Resultant Joint Moments Computed in a Dynamic Activity." Journal of Sports Sciences, Volume 14, Number 3, Pages 219231. Chang, P.B., Williams, B.J., Santer, T.J., Notz, W.I., and Bartel, D.L., 1999. "Robust Optimization of Total Joint Replacements Incorporating Environmental Variables." Journal ofBiomechanical Engineering, Volume 121, Number 3, Pages 304310. Chao, E.Y., and Sim, F.H., 1995. "ComputerAided PreOperative Planning in Knee Osteotomy." Iowa Orthopedic Journal, Volume 15, Pages 418. Chao, E.Y.S., Lynch, J.D., and Vanderploeg, M.J., 1993. "Simulation and Animation of Musculoskeletal Joint System. Journal ofBiomechanical Engineering, Volume 115, Number 4, Pages 562568. Cheze, L., Fregly, B.J., and Dimnet, J., 1995. "A Solidification Procedure to Facilitate Kinematic Analyses Based on Video System Data." Journal ofBiomechanics, Volume 28, Number 7, Pages 879884. Cheze, L., Fregly, B.J., and Dimnet, J., 1998. "Determination of Joint Functional Axes from Noisy Marker Data Using the Finite Helical Axis." Human Movement Science, Volume 17, Number 1, Pages 115. Churchill, D.L., Incavo, S.J., Johnson, C.C., and Beynnon, B.D., 1998. "The Transepicondylar Axis Approximates the Optimal Flexion Axis of the Knee." Clinical 01 thIlqpdi 1 % and Related Research, Volume 356, Number 1, Pages 111118. Croce, U.D., Leardini, A., Lorenzo, C., Cappozzo, A., 2005. "Human Movement Analysis Using Stereophotogrammetry Part 4: Assessment of Anatomical Landmark Misplacement and Its Effects on Joint Kinematics." Gait and Posture, Volume 21, Number 2, Pages 226237. Davis, B.L., 1992. "Uncertainty in Calculating Joint Moments During Gait." In Proceedings of the 8th Meeting ofEuropean Society ofBiomechanics, Rome, Italy, June 2124, Page 276. de Leva, P., 1996. "Adjustments to ZatsiorskySeluyanov's Segment Inertia Parameters." Journal ofBiomechanics, Volume 29, Number 9, Pages 12231230. Delp, S.L., Arnold, A.S., and Piazza, S.J., 1998. "GraphicsBased Modeling and Analysis of Gait Abnormalities." BioMedical Materials and Engineering, Volume 8, Number 3/4, Pages 227240. Delp, S.L., Arnold, A.S., Speers, R.A., and Moore, C.A., 1996. "Hamstrings and Psoas Lengths During Normal and Crouch Gait: Implications for MuscleTendon Surgery." Journal of Orthopaedic Research, Volume 14, Number 1, Pages 144151. Delp, S.L., Loan, J.P., Hoy, M.G., Zajac, F.E., Topp E.L., and Rosen, J.M., 1990. "An Interactive GraphicsBased Model of the Lower Extremity to Study Orthopaedic Surgical Procedures." IEEE Transactions on Biomedical Engineering, Volume 37, Number 8, Pages 757767. Fantozzi, S., Stagni, R., Cappello, A., and Leardini, A., 2005. "Effect of Different Inertial Parameter Sets on Joint Moment Calculation During Stair Ascending and Descending." Medical Engineering & Physics, Volume 27, Pages 537541 Fishman, G.S., 1996. Monte Carlo: Concepts, Algti hi/in, and Applications. SpringerVerlag New York, Inc., New York, NY. Fregly, B.J., Rooney, K.L., and Reinbolt, J. A., 2005. "Predicted Gait Modifications to Reduce the Peak Knee Adduction Torque." In Proceedings of the XXth Congress of the International Society ofBiomechanics and 29th Annual Meeting of the American Society ofBiomechanics, Cleveland, Ohio, July 31August 5, Page 283. Griewank, A., 2000. "Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation." Society for Industrial and Applied Mathematics, Philadelphia, PA. Griewank, A., Juedes, D., and Utke, J., 1996. "ADOLC: a Package for the Automatic Differentiation of Algorithms Written in C/C++." Association for Computing Machinery Transactions on Mathematical Software, Volume 22, Number 2, Pages 131167. 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 661689. Holden, J.P., and Stanhope, S.J., 1998. "The Effect of Variation in Knee Center Location Estimates on Net Knee Joint Moments." Gait & Posture, Volume 7, Number 1, Pages 16. Holden, J.P., and Stanhope, S.J., 2000. "The Effect of Uncertainty in Hip Center Location Estimates on Hip Joint Moments During Walking at Different Speeds." Gait & Posture, Volume 11, Number 2, Pages 120121. Hong, J.H., Mun, M.S., and Song, S.H., 2001. "An Optimum Design Methodology Development Using a Statistical Technique for Vehicle Occupant Safety." In Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, Volume 215, Number 7, Pages 795801. Hurwitz, D.E., Sumner, D.R., Andriacchi, T.P., and Sugar, D.A., 1998. "Dynamic Knee Loads During Gait Predict Proximal Tibial Bone Distribution." Journal of Biomechanics, Volume 31, Number 5, Pages 423430. Inman, V.T., 1976. The Joints of the Ankle. Williams and Wilkins Company, Baltimore, Maryland. Lane, G.J., Hozack, W.J., Shah, S., Rothman, R.H., Booth, R.E. Jr., Eng, K., Smith, P., 1997. "Simultaneous Bilateral Versus Unilateral Total Knee Arthroplasty. Outcomes Analysis." Clinical 0i ilheq,,di \% and Related Research, Volume 345, Number 1, Pages 106112. Leardini, A., Cappozzo, A., Catani, F., ToksvigLarsen, S., Petitto, A., Sforza, V., Cassanelli, G., and Giannini, S., 1999. "Validation of a Functional Method for the Estimation of Hip Joint Centre Location." Journal ofBiomechanics, Volume 32, Number 1, Pages 99103. Lu, T.W., and O'Connor, J.J., 1999. "Bone Position Estimation from Skin Marker Coordinates Using Global Optimisation with Joint Constraints." Journal of Biomechanics, Volume 32, Number 2, Pages 129134. Manal, K., and Guo, M., 2005. "Foot Progression Angle and the Knee Adduction Moment in Individuals with Medial Knee Osteoarthritis." In Proceedings of the XXth Congress of the International Society ofBiomechanics and 29th Annual Meeting of the American Society ofBiomechanics, Cleveland, Ohio, July 31August 5, Page 899. Pandy, M.G., 2001. "Computer Modeling and Simulation of Human Movement." Annual Reviews in Biomedical Engineering, Volume 3, Number 1, Pages 245273. Pearsall D.J., and Costigan P.A., 1999. "The Effect of Segment Parameter Error on Gait Analysis Results." Gait and Posture, Volume 9, Number 3, Pages 173183. Prodromos, C.C., Andriacchi, T.P., and Galante, J.O., 1985. "A Relationship Between Gait and Clinical Changes Following High Tibial Osteotomy." Journal of Bone Joint Surgery (American), Volume 67, Number 8, Pages 11881194. Reinbolt, J.A., and Fregly, B.J., 2005. "Creation of PatientSpecific Dynamic Models from ThreeDimensional Movement Data Using Optimization." In Proceedings of the 10th International Symposium on Computer Simulation in Biomechanics, Cleveland, OH, July 2830. Reinbolt, J.A., Schutte, J.F., Fregly, B.J., Haftka, R.T., George, A.D., and Mitchell, K.H., 2005. "Determination of PatientSpecific MultiJoint Kinematic Models Through TwoLevel Optimization." Journal ofBiomechanics, Volume 38, Number 3, Pages 621626. Sommer III, H.J., and Miller, N.R., 1980. "A Technique for Kinematic Modeling of Anatomical Joints." Journal ofBiomechanical Engineering, Volume 102, Number 4, Pages 311317. 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 14791487. Tetsworth, K., and Paley, D., 1994. "Accuracy of Correction of Complex LowerExtremity Deformities by the Ilizarov Method." Clinical 0i Ithit'e li' \ and Related Research, Volume 301, Number 1, Pages 102110. ValeroCuevas, F.J., Johanson, M.E., and Towles, J.D., 2003. "Towards a Realistic Biomechanical Model of the Thumb: the Choice of Kinematic Description May Be More Critical Than the Solution Method or the Variability/Uncertainty of Musculoskeletal Parameters." Jouirnal ofBiomechanics, Volume 36, Number 7, Pages 10191030. Vaughan, C.L., Andrews, J.G., and Hay, J.G., 1982. "Selection of Body Segment Parameters by Optimization Methods." Journal ofBiomechanical Engineering, Volume 104, Number 1, Pages 3844. Wada, M., Imura, S., Nagatani, K., Baba, H., Shimada, S., and Sasaki, S., 1998. "Relationship Between Gait and Clinical Results After High Tibial Osteotomy." Clinical 0i Ith1/qie'd li \ and Related Research, Volume 354, Number 1, Pages 180188. 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 Proximal Tibial Osteotomy." Journal of Bone and Joint Surgery (American), Volume 72, Number 6, Pages 905913. Winter, D.A., 1980. "Overall Principle of Lower Limb Support During Stance Phase of Gait." Journal ofBiomechanics, Volume 13, Number 11, Pages 923927. 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 fulltime employment at a local business. He earned a traditional 5year 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 fellow 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 with a graduate research assistantship in the Computational Biomechanics Laboratory. In 2003, Jeff graduated with a perfect 4.0 GPA receiving a Master of Science degree in biomedical engineering with a concentration in biomechanics. At that time, he decided to continue his graduate education and research activities through the pursuit of a Doctor of Philosophy in mechanical engineering. In 2005, Jeff and Karen were blessed with the 80 birth of their first child, Jacob. Jeff would like to further his creative involvement in problem solving and the design of solutions to overcome healthcare challenges in the future. 