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Patient-Specific Dynamic Modeling To Predict Functional Outcomes

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Title:
Patient-Specific Dynamic Modeling To Predict Functional Outcomes
Copyright Date:
2008

Subjects

Subjects / Keywords:
Gait ( jstor )
Kidnapping ( jstor )
Kinematics ( jstor )
Kinetics ( jstor )
Knee joint ( jstor )
Knees ( jstor )
Modeling ( jstor )
Parametric models ( jstor )
Pelvis ( jstor )
Torque ( jstor )

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University of Florida
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University of Florida
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7/24/2006

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PATIENT-SPECIFIC 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 experience-based 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 Patient-Specific 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 PATIENT-SPECIFIC DYNAMIC MODELS FROM
THREE-DIMENSIONAL 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 PATIENT-SPECIFIC 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

2-1. Descriptions of the model degrees of freedom .............. ........................................15

2-2. Descriptions of the hip joint parameters. ..........................................................16

2-3. Descriptions of the knee joint parameters. ................................... ..........18

2-4. Descriptions of the ankle joint parameters. ............. ............................................ 20

2-5. Summary of root-mean-square (RMS) joint parameter and marker distance
errors produced by the phase one optimization and anatomic landmark methods
for three types of movement data. ........................................ ......................... 24

2-6. 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

2-7. Differences between joint parameters predicted by anatomical landmark
methods and phase one optimizations. ............................... ............................... 26

2-8. Summary of root-mean-square (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

2-9 Comparison of inertial parameters predicted by anatomical landmark methods
and phase tw o optim izations. ............................................................................. 28

2-10. Differences between inertial parameters predicted by anatomical landmark
methods and phase two optimizations............................................. ...............29

4-1. 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

4-2. Performance results for movement prediction problem for a 3D full-body gait
model involving 660 design variables and 4100 objective function elements.........44

5-1. Descriptions of cost function weights and phase 3 control parameters. .................54

5-2. Summary of fixed offsets added to normal gait for each prescribed foot path........54









5-3. Comparison of cost function weights and phase 3 control parameters ..................55

5-4. Summary of root-mean-square (RMS) errors for tracked quantities for toeout
g ait ............... .. .................................................................5 6

5-5. Summary of root-mean-square (RMS) errors for tracked quantities for wide
sta n c e g a it .................................................................5 7

5-6. Summary of root-mean-square (RMS) errors for predicted left knee abduction
torque quantities for toeout and wide stance gait. .................................................55















LIST OF FIGURES


Figure Page

2-1. Illustration of the 3D, 14 segment, 27 DOF full-body kinematic model linkage
joined by a set of gimbal, universal, and pin joints ..............................................14

2-2. 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

2-3. 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

2-4. 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

2-5. Illustration of the modified Cleveland Clinic marker set used during static and
dynam ic m otion capture trials. ..... ...................................................................... 21

2-6. Phase one optimization convergence illustration series for the knee joint, where
synthetic markers are blue, model markers are red, and root-mean-square (RMS)
m arker distance error is green. ............................................................................ 22

2-7. Phase two optimization convergence illustration series for the knee joint, where
synthetic masses are blue, model masses are red, and root-mean-square (RMS)
residual pelvis forces and torques are orange and green, respectively.....................23

3-1. Legend of five sample statistics presented by the chosen boxplot convention........36

3-2. Comparison of root-mean-square (RMS) leg joint torques and marker distance
error distributions. ..................................................................... 37

3-3. Comparison of mean leg joint torques and marker distance error distributions. .....38

5-1. Comparison of left knee abduction torques for toeout gait ....................................60

5-2. 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









5-3. Comparison of left knee abduction torques for simulated high tibial osteotomy
(H TO ) post-surgery gait. ............................................... ............................... 62

5-4. Comparison of mean (solid black line) plus or minus one standard deviation
(gray shaded area) for experimental left knee abduction torques. .........................63

5-5. 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

PATIENT-SPECIFIC 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 one-size-fits-all healthcare approach.

Imagine the benefit to the healthcare provider and, more importantly, the patient, if

certain clinical parameters may be evaluated pre-treatment in order to predict the

post-treatment outcome. Using patient-specific models, movement related disorders may

be treated with reliable outcomes as a result of objective decisions based on quantitative

analyses using a patient-specific approach.

The general objective of the current work is to predict post-treatment outcome

using patient-specific models given pre-treatment data. The specific objective is to

develop a four-phase optimization approach to identify patient-specific 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 patient-specific 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 two-thirds of individuals with arthritis are under 65 years old. As the

population ages, the number of people with arthritis is likely to increase significantly.

The most common forms of arthritis are osteoarthritis, rheumatoid arthritis, fibromyalgia,

and gout. Osteoarthritis of the knee joint accounts for roughly 30% ($25 billion) of the

$82 billion total arthritis costs per year in the United States.

Knee osteoarthritis symptoms of pain and dysfunction are the primary reasons for

total knee replacement (TKR). This procedure involves a resurfacing of bones

surrounding the knee joint. The end of the femur is removed and covered with a metal

implant. The end of the tibia is removed and substituted by a plastic implant. Smooth

metal and plastic articulation replaces the irregular and painful arthritic surfaces.

Approximately 100,000 Medicare patients alone endure TKR procedures each year (Heck

et al., 1998). Hospital charges for unilateral TKR are more than $30,000 and the cost of

bilateral TKR is over $50,000 (Lane et al., 1997).

An alternative to TKR is a more conservative (both economically and surgically)

corrective procedure known as high tibial osteotomy (HTO). By changing the frontal

plane alignment of the tibia with a wedge of bone, a HTO shifts the weight-bearing axis









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 Patient-Specific Simulations

Innovative patient-specific models and simulations would be valuable for

addressing problems in orthopedics and sports medicine, as well as for evaluating and

enhancing corrective surgical procedures (Arnold et al., 2000; Arnold and Delp, 2001;

Chao et al., 1993; Chao and Sim, 1995; Delp et al., 1998; Delp et al., 1996; Delp et al.,

1990; Pandy, 2001). For example, a patient-specific dynamic model may be useful for

planning intended surgical parameters and predicting the outcome of HTO.

The main motivation for developing the following patient-specific dynamic model

and the associated multi-phase optimization approach is to predict the post-surgery 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 x-rays). Unfortunately, alignment correction estimates from

static x-rays do not accurately predict long-term clinical outcome after HTO (Andriacchi,

1994; Tetsworth and Paley, 1994). Researchers have identified the peak external knee

adduction moment as an indicator of clinical outcome while investigating the gait of

HTO patients (Andriacchi, 1994; Bryan et al., 1997; Hurwitz et al., 1998; Prodromos et

al., 1985; Wang et al., 1990). Currently, no movement simulations (or other methods for

that matter) allow surgeons to choose HTO surgical parameters to achieve a chosen

post-surgery knee 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 multi-link 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 motion-capture technology is based on a passive optical technique.

Passive refers to markers, which are simply spheres covered in reflective tape, placed on

the object. Optical refers to the technology used to provide 3D data, which involves

high-speed, high-resolution video cameras. By placing passive markers on an object,

special hardware records the position of those markers in time and it generates a set of

motion data (or marker data).

Often motion capture is used to create synthetic actors by capturing the motions of

real humans. Special effects companies have used this technique to produce incredibly

realistic animations in movies such as Star Wars Episode I & II, Titanic, Batman, and

Terminator 2.

Biomechanical Models

Researchers use motion-capture technology to construct biomechanical models of

the human structure. The position of external markers may be used to estimate the









position of internal landmarks such as joint centers. The markers also enable the creation

of individual segment reference frames that define the position and orientation of each

body segment within a Newtonian laboratory reference frame. Marker data collected

from an individual are used to prescribe the motion of the biomechanical model.

Kinematics and Dynamics

Human kinematics is the study of the positions, angles, velocities, and accelerations

of body segments and joints during motion. With kinematic data and mass-distribution

data, one can study the forces and torques required to produce the recorded motion data.

Errors between the biomechanical model and the recorded motion data will inevitably

propagate to errors in the force and torque results of dynamic analyses.

Optimization

Optimization involves searching for the minimum or maximum of an objective

function by adjusting a set of design variables. For example, the objective function may

be the errors between the biomechanical model and the recorded 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 PATIENT-SPECIFIC DYNAMIC MODELS FROM
THREE-DIMENSIONAL 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

patient-specific 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 three-dimensional (3D) multi-joint 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 computationally-efficient two-phase optimization approach

for determining patient-specific 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

degree-of-freedom (DOF), parametric full-body 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 two-phase

optimization approach. For this purpose, a parametric 3D, 27 DOF, full-body gait model

whose equations of motion were derived with the symbolic manipulation software,

AutolevTM (OnLine Dynamics, Sunnyvale, CA), was used (Figure 2-1, Table 2-1).

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 2-2, Table 2-2, Figure 2-3, Table 2-3, Figure 2-4, Table 2-4).

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 2-2). From percentages of the inter-ASIS 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 2-2). 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 2-3). The talus coordinate system was created where the y-axis extends

along the line perpendicular to both the talocrural joint axis and the subtalar joint axis

(Figure 2-4). The heel and toe markers, in combination with the tibia y-axis, defined the

foot coordinate system (Figure 2-4).

Experimental kinematic and kinetic data were collected from a single subject using

a video-based 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 2-5). 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

flexion-extension followed by abduction-adduction were recorded. Similar to Leardini et

al. (1999), internal-external rotation of the hip was avoided to reduce skin and soft tissue

movement artifacts. Multiple cycles of knee flexion-extension were measured. Finally,

multiple cycles of simultaneous ankle plantarflexion-dorsiflexion and inversion-eversion

were recorded. Gait motion and ground reaction data were collected to investigate

simultaneous motion of all lower extremity joints under load-bearing 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 2-1,

Figure 2-6). For isolated joint motion trials, the design variables were 540 B-spline

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 B-spline 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)] (2-1)
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 ground-to-pelvis joint (Equation 2-2,

Figure 2-7). 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 +--- + (2-2)
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. Root-mean-square (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 2-5, Table 2-6). 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 2-7). 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 two-level 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 2-8, Table 2-9). 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 2-10). 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 two-phase optimization problem driven by motion capture

data. Optimized dynamic models can provide a more reliable foundation for future

patient-specific dynamic analyses and optimizations.

This study presented a two-phase optimization approach for tuning joint and

inertial parameters in a dynamic skeletal model to match experimental movement data

from a specific patient. For the full-body gait model used in this study, the JP

optimization satisfactorily reproduced patient-specific 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 two-phase

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 two-phase 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 patient-specific 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 patient-specific 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 two-phase 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 2-1. Illustration of the 3D, 14 segment, 27 DOF full-body kinematic model linkage
joined by a set of gimbal, universal, and pin joints.


Joint Types


Pin


Universal





Gimbal


q10









q11



q12

"*'* ,

* -


(anterior)










Table 2-1. Descriptions of the model degrees of freedom.

DOF Description

qi Pelvis anterior-posterior position.
q2 Pelvis superior-inferior position.
q3 Pelvis medial-lateral position.
q4 Pelvis anterior-posterior tilt angle.
q5 Pelvis elevation-depression angle.
q6 Pelvis internal-external rotation angle.
q7 Right hip flexion-extension angle.

q8 Right hip adduction-abduction angle.
q9 Right hip internal-external rotation angle.

qio Right knee flexion-extension angle.
qii Right ankle plantarflexion-dorsiflexion angle.

q12 Right ankle inversion-eversion angle
q13 Left hip flexion-extension angle.
q14 Left hip adduction-abduction angle.
q15 Left hip internal-external rotation angle.
q16 Left knee flexion-extension angle.
q17 Left ankle plantarflexion-dorsiflexion angle.
q18 Left ankle inversion-eversion angle
q19 Trunk anterior-posterior tilt angle.
q20 Trunk elevation-depression angle.
q21 Trunk internal-external rotation angle.

q22 Right shoulder flexion-extension angle.
q23 Right shoulder adduction-abduction angle.
q24 Right elbow flexion angle.
q25 Left shoulder flexion-extension angle.
q26 Left shoulder adduction-abduction angle.
q27 Left elbow flexion angle.










Y

Pelvis
Optimized P3 z
Hip Joint r -- -
SCenter X~
P4 P2
'P5 p;(


Zj( ; .


Y
(superior)


Z
(lateral)


X
(anterior)


Figure 2-2. 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 2-2. Descriptions of the hip joint parameters.

Hip Joint E
Parameter

pi Anterior-posterior

P2 Superior-inferior 1
p3 Medial-lateral loc<
p4 Anterior-posterior
p5 Superior-inferior 1
P6 Medial-lateral 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 2-3. 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 2-3. Descriptions of the knee joint parameters.

Knee Joint De
Parameter

pi Adduction-abduction

p2 Internal-external rota

p3 Adduction-abduction

p4 Internal-external rota

p5 Anterior-posterior lo

p6 Superior-inferior local

p7 Anterior-posterior lo

ps Superior-inferior local
p9 Medial-lateral 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 2-4. 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 2-4. Descriptions of the ankle joint parameters.

Ankle Joint Des
Des
Parameter

pi Adduction-abduction rotatic

p2 Internal-external rotation of

p3 Internal-external rotation of

p4 Internal-external rotation of
p5 Dorsi-plantar rotation of sut

p6 Anterior-posterior location

p7 Superior-inferior location ol

ps Medial-lateral location of ta

p9 Superior-inferior location ol
plo Anterior-posterior location

pll Superior-inferior location ol

p12 Medial-lateral 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 2-5. 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 2-6. Phase one optimization convergence illustration series for the knee joint, where synthetic markers are blue, model markers
are red, and root-mean-square (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 2-7. Phase two optimization convergence illustration series for the knee joint, where synthetic masses are blue, model masses
are red, and root-mean-square (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 2-5. Summary of root-mean-square (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 video-based 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 non-weight 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) 9.82e-09 3.58e-08 1.80e-08 2.39e-06
Synthetic without Noise Phase one optimization Orientation parameters () n/a 4.68e-04 6.46e-03 5.45e-03
Position parameters (mm) 4.85e-05 8.69e-04 6.35e-03 1.37e-02

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 2-6. 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 2-7. 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 2-6. 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 2-8. Summary of root-mean-square (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 video-based 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 8.04e-10 2.16e-10 5.13e-13 1.22e-10 7.11e-13
optimization

Synthetic with Noise Phase 2
16.92 4.93 5.67e-03 1.08 6.64e-03
(correct seed) optimization

Synthetic with Noise Phase 2
Synthetic with Noise Phase2 16.960.44 5.240.23 7.45e-021.81e-02 1.330.39 2.16e-022.92e-03
(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 2-9 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 -9.36e-02 -9.36e-02 -9.63e-02 -1.00e-012.29e-02 -9.33e-02
Pelvis Superior -2.37e-02 -2.37e-02 -2.33e-02 -2.49e-025.44e-03 -2.37e-02
Lateral 0.00 -2.35e-13 6.57e-13 8.06e-134.02e-13 6.07e-13
Anterior 0.00 1.54e-13 4.02e-13 7.76e-132.06e-13 4.86e-13
Thigh Superior -1.68e-01 -1.68e-01 -1.65e-01 -1.58e-011.84e-02 -1.59e-01
Lateral 0.00 -1.49e-14 1.58e-12 2.41e-128.05e-13 8.91e-13
Anterior 0.00 1.22e-14 -5.41e-13 -1.91e-136.20e-13 7.62e-13
Shank Superior -1.72e-01 -1.72e-01 -1.89e-01 -1.89e-014.48e-02 -1.80e-01
Lateral 0.00 6.59e-14 3.15e-13 5.65e-132.72e-13 4.56e-13
Anterior 9.21e-02 9.21e-02 8.73e-02 8.17e-021.90e-02 9.27e-02
Center of
Centr Foot Superior -1.60e-04 -1.60e-04 -4.19e-04 -4.12e-041.19e-04 -1.60e-04
mass (m)
Lateral 0.00 -1.81e-14 1.26e-14 9.35e-148.93e-14 1.85e-14

Anterior 0.00 1.92e-13 -8.33e-13 -8.96e-132.35e-12 8.21e-13
Head & trunk Superior -1.44e-01 -1.44e-01 -1.69e-01 -1.87e-011.38e-02 -1.42e-01
Lateral 0.00 -2.56e-14 1.66e-12 2.34e-127.09e-13 4.45e-13
Anterior 0.00 4.74e-14 -3.21e-13 -2.69e-132.60e-13 6.15e-13
Upper arm Superior -1.84e-01 -1.84e-01 -1.86e-01 -1.95e-014.96e-02 -1.84e-01
Lateral 0.00 -1.79e-14 3.51e-13 4.19e-132.45e-13 7.92e-13
Anterior 0.00 5.59e-14 1.45e-13 2.32e-133.58e-13 9.22e-13
Lower arm & hand Superior -1.85e-01 -1.85e-01 -1.97e-01 -1.89e-014.36e-02 -1.85e-01
Lateral 0.00 -3.85-e14 2.86e-13 4.13e-131.59e-13 7.38e-13
Anterior 6.83e-02 6.83e-02 6.83e-02 6.60e-022.30e-02 6.83e-02
Pelvis Superior 6.22e-02 6.22e-02 6.21e-02 5.98e-021.81e-02 6.22e-02
Lateral 5.48e-02 5.48e-02 5.47e-02 5.25e-021.64e-02 5.48e-02
Anterior 1.78e-01 1.78e-01 1.80e-01 1.67e-015.39e-02 1.78e-01
Thigh Superior 3.66e-02 3.66e-02 3.65e-02 3.71e-021.35e-02 3.66e-02
Lateral 1.78e-01 1.78e-01 1.74e-01 2.06e-015.31e-02 1.78e-01
Anterior 2.98e-02 2.98e-02 2.97e-02 3.06e-021.19e-02 2.98e-02
Shank Superior 4.86e-03 4.86e-03 4.86e-03 4.01e-031.04e-03 4.86e-03
Lateral 2.84e-02 2.84e-02 2.86e-02 2.75e-026.65e-03 2.84e-02
Moment of Anterior 2.38e-03 2.38e-03 2.29e-03 2.27e-036.45e-04 2.38e-03
inertia Foot Superior 4.56e-03 4.56e-03 4.58e-03 4.90e-031.29e-03 4.56e-03
(kg*m2) Lateral 3.08e-03 3.08e-03 2.99e-03 3.02e-036.57e-04 3.08e-03
Anterior 9.51e-01 9.51e-01 9.18e-01 8.96e-013.35e-01 9.51e-01
Head & trunk Superior 2.27e-01 2.27e-01 2.27e-01 1.93e-013.62e-02 2.27e-01
Lateral 8.41e-01 8.41e-01 8.37e-01 9.99e-012.68e-01 8.41e-01
Anterior 1.53e-02 1.53e-02 1.53e-02 1.44e-025.65e-03 1.53e-02
Upper arm Superior 4.71e-03 4.71e-03 4.71e-03 4.57e-031.29e-03 4.71e-03
Lateral 1.37e-02 1.37e-02 1.37e-02 1.56e-023.29e-03 1.37e-02
Anterior 1.97e-02 1.97e-02 1.97e-02 2.06e-024.97e-03 1.97e-02
Lower arm & hand Superior 1.60e-03 1.60e-03 1.60e-03 1.41e-035.02e-04 1.60e-03
Lateral 2.05e-02 2.05e-02 2.04e-02 2.41e-023.78e-03 2.05e-02













Table 2-10.


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 2-9.


Synthetic differences Experimental
Experimental
Category Body segment Direction Random seeds differences
Correct seed differences
(10 cases)
Pelvis 4.28e-01 -7.01e-011.87 -5.66e-02
Thigh 1.02 1.451.02 -8.29e-01
Shank 2.69e-01 3.89e-013.86e-01 -3.11e-02
Mass (kg) Foot n/a 1.29e-01 1.62e-012.95e-01 3.53e-02
Head and trunk -3.38 -3.111.34 -1.75e-01
Upper arm -4.62e-02 -3.05e-014.85e-01 1.46e-03
Lower arm and hand 1.01e-01 2.15e-014.67e-01 -1.89e-03
Anterior -2.69e-03 -6.51e-032.29e-02 2.30e-04
Pelvis Superior 4.13e-04 -1.24e-035.44e-03 1.12e-05
Lateral 6.57e-13 8.06e-134.02e-13 6.07e-13
Anterior 4.02e-13 7.76e-132.06e-13 4.86e-13
Thigh Superior 3.48e-03 1.01e-021.84e-02 9.81e-03
Lateral 1.58e-12 2.41e-128.05e-13 8.91e-13
Anterior -5.41e-13 -1.91e-136.20e-13 7.62e-13
Shank Superior -1.66e-02 -1.73e-024.48e-02 -7.65e-03
Lateral 3.15e-13 5.65e-132.72e-13 4.56e-13
Anterior -4.83e-03 -1.04e-021.90e-02 6.30e-04
Center of mass
(mn) Foot Superior -5.25e-08 7.39e-061.19e-04 9.50e-13
Lateral 1.26e-14 9.35e-148.93e-14 1.85e-14
Anterior -8.33e-13 -8.96e-132.35e-12 8.21e-13
Head and trunk Superior -2.47e-02 -4.31e-021.38e-02 1.66e-03
Lateral 1.66e-12 2.34e-127.09e-13 4.45e-13
Anterior -3.21e-13 -2.69e-132.60e-13 6.15e-13
Upper arm Superior -2.40e-03 -1.15e-024.96e-02 1.10e-04
Lateral 3.51e-13 4.19e-132.45e-13 7.92e-13
Anterior 1.45e-13 2.32e-133.58e-13 9.22e-13
Lower arm and hand Superior -1.29e-02 -4.21e-034.36e-02 3.70e-04
Lateral 2.86e-13 4.13e-131.59e-13 7.38e-13
Anterior 5.53e-05 -2.24e-032.30e-02 1.95e-04
Pelvis Superior -1.11e-04 -2.41e-031.81e-02 -9.03e-04
Lateral -1.21e-04 -2.28e-031.64e-02 -1.19e-05
Anterior 2.06e-03 -1.14e-025.39e-02 2.86e-03
Thigh Superior -7.14e-05 5.67e-041.35e-02 -1.45e-04
Lateral -4.56e-03 2.73e-025.31e-02 -9.06e-03
Anterior -9.91e-05 8.51e-041.19e-02 -5.98e-04
Shank Superior 6.80e-07 -8.49e-041.04e-03 1.85e-06
Lateral 2.09e-04 -9.19e-046.65e-03 4.15e-04
Anterior -5.94e-07 -1.73e-056.45e-04 -9.07e-08
Moment of
nerti (km) Foot Superior -8.59e-08 3.16e-041.29e-03 -6.41e-07
Lateral 3.06e-06 2.87e-056.57e-04 1.36e-05
Anterior -3.30e-02 -5.56e-023.35e-01 -3.08e-02
Head and trunk Superior -6.15e-04 -3.37e-023.62e-02 -2.87e-03
Lateral -4.13e-03 1.59e-012.68e-01 -1.73e-02
Anterior -1.62e-05 -9.11e-045.65e-03 -1.37e-04
Upper arm Superior -1.79e-07 -4.14e-051.29e-03 -9.40e-07
Lateral -7.90e-07 1.97e-033.29e-03 -1.60e-06
Anterior -1.63e-05 8.91e-044.97e-03 -1.09e-04
Lower arm and hand Superior -8.95e-08 -1.95e-045.02e-04 -7.50e-07
Lateral -5.84e-05 3.59e-033.78e-03 -1.15e-05














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 flexion-extension torque has









been studied for multiple walking speeds while changing the knee center location by

10mm in the anterior-posterior 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 three-dimensional (3D) full-body 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 video-based 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 degree-of-freedom (DOF), full-body gait model (Figure 2-1,

Table 2-1) 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 full-body 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, Valero-Cuevas

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 lower-body 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 3-1).

RMS and mean hip torque errors were computed in the flexion-extension,

abduction-adduction, and internal-external rotation directions (Figure 3-2, Figure 3-3).

Mostly due to JPs, the hip flexion-extension and abduction-adduction torques showed

more variation than the internal-external 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 flexion-extension and

abduction-adduction directions (Figure 3-2, Figure 3-3). Similar to the corresponding hip

directions, the knee torques have larger spans for the abduction-adduction direction

compared to the flexion-extension 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 flexion-extension torque varied slightly more than the

abduction-adduction torque. The breadth of knee torques decreased with attenuated

percentages of parameter bounds.

RMS and mean ankle torque errors were computed in the

plantarflexion-dorsiflexion and inversion-eversion directions (Figure 3-2, Figure 3-3).

The inversion-eversion torque displayed a broader distribution compared to the

plantarflexions-dorsiflexion 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 3-2, Figure 3-3). 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

full-body 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 full-body 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 patient-specific JPs and IPs reduces the uncertainty of inverse dynamics

results.






















1 -







-1


36




I
90th percentile


.75th percentile




=t-50th percentile
(95% confidence notch)



.- 25th percentile



- 10th percentile


I
All Instantiations


Figure 3-1. 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 3-2. Comparison of root-mean-square (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 3-3. 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

three-dimensional (3D) musculoskeletal models (Pandy, 2001). When gradient-based

methods are used to solve large-scale 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 gradient-based optimizations are dramatically

improved by using an analytical Jacobian matrix (all first-order 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 freely-available

AD package, Automatic Differentiation by OverLoading in C++ (ADOL-C) (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 full-body 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 video-based 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

degrees-of-freedom. Each of the 12 generalized coordinate curves was parameterized

using 20 B-spline 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 ground-to-pelvis residual reactions) between

model and experiment. The movement prediction model possessed 27

degrees-of-freedom (21 adjusted by B-spline curves and 6 prescribed for arm motion) and

12 ground reactions. Each adjustable generalized coordinate and reaction curve was

parameterized using 20 B-spline 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.

ADOL-C was incorporated into each objective function to compute an analytical

Jacobian matrix. This package was chosen because the objective functions were

comprised of Matlab mex-files, which are dynamic link libraries of compiled C or C++

code. ADOL-C 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 built-in 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 built-in 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 built-in driver function j acobian to compute first-order derivatives
using reverse mode AD.

6. Compile the code including built-in header file adolc .h and linking with built-in
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 4-1 and Table 4-2. 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 trade-offs

between those approaches and AD require further investigation. While AD-based

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 large-scale problems, AD provides a relatively simple means for computing

analytical derivatives to improve the speed of biomechanical optimizations.










Table 4-1. 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 4-2. Performance results for movement prediction problem for a 3D full-body 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 PATIENT-SPECIFIC 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 pre-treatment in order to predict the

post-treatment outcome. For example, a patient-specific 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 post-surgery 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 lower-body 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 patient-specific 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 video-based 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 patient-specific model parameters as described in Chapter 2.

Toeout, or increased foot progression angle, gait data was gathered to calibrate the

patient-specific control parameters. Wide stance gait data was collected to evaluate the

calibrated model.

A parametric 3D, 27 degree-of-freedom (DOF), full-body gait model (Figure 2-1,

Table 2-1) 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 5-1). Specifically, the design variables were 660 B-spline 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 B-spline node was chosen to be the

corresponding value from normal gait. The inverse dynamics optimization cost function

predictt outcome) minimized the weighted (Equation 5-2, Equation 5-3, Table 5-1) 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) + (5-1)
outcome qr 1 j=1 k=l J1
6 2
S[AT (pcp
J=1 k=1


PCP= [6 W7 W8 W9 W10 Wll]


(5-2)









ATle (pcp) = w6 hp + 7AThp, + w8ATh, +
flexion adduction rotation (53)
(5-3)
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 5-2, Equation 5-3,

Table 5-1), 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 5-2) 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 6-dimensional 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 6-dimensional 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 5-2) 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 anterior-posterior 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 patient-specific 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 5-3) 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 5-1, Table 5-4).

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 flexion-extension torque at the beginning of

the cycle. Knee flexion-extension 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 internal-external rotation angle (max = 9.540 at 7% of cycle). The knee

flexion-extension 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 5-5). All kinematic and kinetic

differences are summarized in Table 5-7.

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 5-2, Table 5-4). 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 flexion-extension torque. Knee

flexion-extension 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

flexion-extension angle. The knee flexion-extension 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 5-6). All kinematic and kinetic differences are

summarized in Table 5-8.

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 5-3). 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 5-1, Table 5-4). 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 5-2, Table

5-4). In all cases, tracked quantities demonstrated good agreement (Table 5-5, Table

5-6). 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 5-3). The differences in dynamic

models created from different portions of the multi-phase optimization approach led to

variations in the computed knee abduction torques for experimental and simulated data

(Figure 5-4, Figure 5-5).

Discussion

The main motivation for developing a patient-specific inverse dynamics

optimization approach is to predict novel patient-specific gait motions (e.g., subtle gait

changes and HTO post-surgery gait). Toeout gait was successfully used to tune

patient-specific 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 flexion-extension 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 multi-phase 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 5-3). 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 patient-specific models and predictions for a population of HTO

patients pre-surgery and compare with post-surgery outcome. Examination of different

gaits (e.g., normal and pathological) requires future study as well.










Table 5-1. 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 flexion-extension torque
W7 Control parameter for tracking hip adduction-abduction torque
w8 Control parameter for tracking hip internal-external torque
W9 Control parameter for tracking knee flexion-extension torque

wlo Control parameter for tracking talocrural dorsiflexion-plantarflexion torque
wil Control parameter for tracking subtalar inversion-eversion torque


Table 5-2. 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 5-3. 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 5.79e-01 8.13 5.17

W7 8.67 2.23 6.84 3.57e-01

ws 7.31e-08 9.29e-10 7.18e-01 1.08

W9 9.14 1.62e-07 6.74 2.94e-01

wio 2.61 8.18 6.87 5.55

wil 4.24 9.63 9.18 9.17



Table 5-4. Summary of root-mean-square (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 5-5 Summary of root-mean-square (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 5-6. Summary of root-mean-square (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 5-7. Summary of root-mean-square (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

2.49e-02

1.70e-02

1.62e-01

1.67e-02

5.09e-02

3.92e-02

2.33e-01

1.73


3.22

2.55

5.39

2.68

3.21

4.71

15.09

23.35

1.55

7.46e-01

1.76e-01

2.98e-01

1.27

2.32

1.26

3.49


9.14e-01

2.20

10.94

1.03

2.05

3.89

2.45

5.36

4.05e-01

9.58e-02

1.42e-01

3.75e-01

3.24e-02

7.07e-02

2.65e-01

1.29


3.72

2.27

5.71

2.76

2.85

4.49

9.89

9.27

9.39e-01

3.43e-01

1.99e-01

2.92e-01

5.49e-01

3.98e-01

1.31

3.60


2.09

2.37

11.03

1.62

1.53

4.14

1.29

1.99

2.15e-02

2.43e-02

1.12e-01

1.50e-02

1.41e-02

1.79e-02

5.16e-01

1.52


2.79

2.39

5.35

2.52

2.85

4.87

9.97

7.09

1.55

7.50e-01

1.80e-01

2.92e-01

1.29

2.31

1.32

3.95


1.08 3.46

2.63 2.32

11.27 5.76

8.93e-01 2.69

1.78 2.58

3.49 4.36

1.58 9.33

3.53 7.86

3.11e-02 7.01e-01

2.31e-01 4.21e-01

1.10e-01 1.83e-01

2.41e-01 2.21e-01

1.60e-02 5.53e-01

2.98e-02 3.86e-01

2.51e-01 1.39

7.07e-01 4.04


6.98e-01 3.65e-01 5.43e-01 2.78e-01 7.58e-01 3.42e-01 5.64e-01 3.28e-01
















Table 5-8. Summary of root-mean-square (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)


6.64e-01

3.07

2.43

6.75e-01

2.17

6.82

1.44

2.10

2.33e-02

3.82e-02

1.93e-01

1.28e-02

1.85e-02

3.16e-02

2.76e-01

1.03

2.66


4.39

1.73

2.62

3.46

4.35

6.43

27.40

11.22

5.04e-01

5.25e-01

2.44e-01

4.27e-01

3.74e-01

4.23e-01

1.13

4.15

8.75e-01


5.50e-01

3.59

1.74

1.48

2.19

4.06

1.49

1.95

1.71e-01

1.29e-01

2.10e-01

2.49e-01

1.15e-02

2.53e-02

3.94e-01

8.09e-01

2.35


4.55

1.69

2.82

4.41

5.18

4.07

26.01

9.13

5.95e-01

3.39e-01

2.87e-01

6.24e-01

3.91e-01

3.45e-01

1.20

3.98

1.14


3.68e-01

3.21

1.91

6.28e-01

1.91

5.87

4.45

1.54

2.43e-02

6.54e-02

2.00e-01

1.46e-02

1.01e-02

1.62e-02

2.81e-01

1.23

2.56


4.29

1.73

2.58

3.48

4.33

5.53

27.52

11.22

5.25e-01

5.21e-01

2.49e-01

4.37e-01

3.71e-01

4.23e-01

1.17

4.30

8.72e-01


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

3.68e-01 8.18

1.58e-02 6.26e-01

2.53e-01 3.23e-01

9.11e-02 2.48e-01

2.79e-01 3.90e-01

1.07e-02 3.93e-01

2.39e-02 3.63e-01

6.76e-01 1.04

2.13 3.93

1.75 1.79














14|--i-----------i 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 5-1. 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 5-2. 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 5-3. Comparison of left knee abduction torques for simulated high tibial osteotomy (HTO) post-surgery 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
C-0
"o






< 3-


rI






3



--
S0
.2



-1



















0



Gait Cycle (%)


Figure 5-4. 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 5-5. 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 patient-specific inverse dynamic

model and the associated optimization approach is to predict the post-surgery 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 patient-specific dynamic

simulation.

Synthesis of Current Work and Literature

The multi-phase optimization method satisfactorily determines patient-specific

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 multi-phase

optimization approach successfully reduces the fitness errors between the patient-specific

model and the experimental motion data.

The precision of dynamic analyses made for a particular patient depends on the

accuracy of the patient-specific 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

motion-capture data. By using a multi-phase 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 physics-based 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 patient-specific 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 patient-specific 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 inversion-eversion


Ankle motion


Ankle plantarflexion-dorsiflexion


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 anterior-posterior 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 medial-lateral 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 abduction-adduction



Hip flexion-extension



Hip internal-external rotation




Hip motion


Inferior

Inter-ASIS 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 weight-bearing 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
anterior-posterior axis of the pelvis.

Motion of the long axis of the thigh within the
sagittal plane as seen by an observer positioned
along the medial-lateral axis of the pelvis.

Motion of the medial-lateral axis of the thigh with
respect to the medial-lateral axis of the pelvis within
the transverse plane as seen by an observer
positioned along the longitudinal axis of the thigh.

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 abduction-adduction


Knee flexion-extension



Knee internal-external 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 anterior-posterior axis of the thigh.

Motion of the long axis of the shank within the
sagittal plane as seen by an observer positioned
along the medial-lateral axis of the thigh.

Motion of the medial-lateral axis of the shank with
respect to the medial-lateral axis of the thigh within
the transverse plane as viewed by an observer
positioned along the longitudinal axis of the shank.

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 two-dimensions, the moment of force about a
point is the product of a force and the perpendicular
distance from the line of action of the force to the
point. Typically, moments of force are calculated
about the center of rotation of a joint.

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 anterior-posterior tilt



Pelvis elevation-depression




Pelvis internal-external 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 medial-lateral axis of the laboratory.

Motion of the medial-lateral axis of the pelvis
within the coronal plane as seen by an observer
positioned along the anterior-posterior axis of the
laboratory.

Motion of the medial-lateral or anterior-posterior
axis of the pelvis within the transverse plane as seen
by an observer positioned along the longitudinal
axis of the laboratory.

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 tibial-talar 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.















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BIOGRAPHICAL SKETCH

Jeffrey A. Reinbolt was born on May 6, 1974, in Bradenton, Florida. His parents

are Charles and Joan Reinbolt. He has an older brother, Douglas, and an older sister,

Melissa. In 1992, Jeff graduated salutatorian from Southeast High School, Bradenton,

Florida. After completing his secondary education, he enrolled at the University of

Florida supported by the Florida Undergraduate Scholarship and full-time employment at

a local business. He earned a traditional 5-year engineering degree in only 4 years. In

1996, Jeff graduated with honors receiving a Bachelor of Science degree in engineering

science with a concentration in biomedical engineering. He used this foundation to assist

in the medical device development and clinical research programs of Computer Motion,

Inc., Santa Barbara, California. In this role, Jeff was Clinical Development Site Manager

for the Southeastern United States and he traveled extensively throughout the United

States, Europe, and Asia collaborating with surgeons and fellow medical researchers. In

1998, Jeff married Karen, a 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.




Full Text

PAGE 1

PATIENT-SPECIFIC DYNAMIC MODELING TO PREDICT FUNCTIONAL OUTCOMES By JEFFREY A. REINBOLT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Jeffrey A. Reinbolt

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This dissertation is dedicated to my lovi ng wife, Karen, and our wonderful son, Jacob.

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iv ACKNOWLEDGMENTS I sincerely thank Dr. B. J. Fregly for his support and leadership throughout our research endeavors; moreover, I truly rec ognize the value of his honest, straightforward, and experience-based advice. My life has been genuinely influenced by Dr. Freglys expectations, confiden ce, and trust in me. I also extend gratitude to Dr. Raphael Haftka, Dr. Carl Crane, Dr. Scott Banks, and Dr. Steven Kautz for their de dication, knowledge, and instructi on 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 en couragement, 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 adequa tely express my grat efulness 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.

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v TABLE OF CONTENTS Page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES...........................................................................................................ix ABSTRACT.......................................................................................................................xi CHAPTER 1 INTRODUCTION........................................................................................................1 Arthritis: The Nations Lead ing Cause of Disability...................................................1 Need for Patient-Specific Simulations..........................................................................2 Background...................................................................................................................3 Motion Capture......................................................................................................3 Biomechanical Models..........................................................................................3 Kinematics and Dynamics.....................................................................................4 Optimization..........................................................................................................4 2 CREATION OF PATIENT-SPECIFIC DYNAMIC MODELS FROM THREE-DIMENSIONAL MOVEMENT DATA USING OPTIMIZATION.............5 Background...................................................................................................................5 Methods........................................................................................................................6 Results........................................................................................................................ .10 Discussion...................................................................................................................11 3 EFFECT OF MODEL PARAMETER VARIATIONS ON INVERSE DYNAMICS RESULTS USING M ONTE CARLO SIMULATIONS......................30 Background.................................................................................................................30 Methods......................................................................................................................31 Results........................................................................................................................ .33 Discussion...................................................................................................................34 4 BENEFIT OF AUTOMATIC DIFFEREN TIATION FOR BIOMECHANICAL OPTIMIZATIONS.....................................................................................................39

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vi Background.................................................................................................................39 Methods......................................................................................................................40 Results........................................................................................................................ .42 Discussion...................................................................................................................43 5 APPLICATION OF PATIENT-SPECIFIC DYNAMIC MODELS TO PREDICT FUNCTIONAL OUTCOMES....................................................................................45 Background.................................................................................................................45 Methods......................................................................................................................46 Results........................................................................................................................ .50 Discussion...................................................................................................................52 6 CONCLUSION...........................................................................................................65 Rationale for New Approach......................................................................................65 Synthesis of Current Work and Literature..................................................................65 GLOSSARY......................................................................................................................67 LIST OF REFERENCES...................................................................................................74 BIOGRAPHICAL SKETCH.............................................................................................79

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vii LIST OF TABLES Table Page 2-1. Descriptions of the model degrees of freedom.........................................................15 2-2. Descriptions of the hip joint parameters..................................................................16 2-3. Descriptions of the knee joint parameters................................................................18 2-4. Descriptions of the ankle joint parameters...............................................................20 2-5. Summary of root-mean-square (RMS) joint parameter and marker distance errors produced by the phase one optimi zation and anatomic landmark methods for three types of movement data.............................................................................24 2-6. Comparison of joint parameters pred icted by anatomical landmark methods, phase one optimization involving individua l joints separately, and phase one optimization involving all joints simultaneously.....................................................25 2-7. Differences between joint parameters predicted by anatomical landmark methods and phase one optimizations......................................................................26 2-8. Summary of root-mean-square (RMS) inerti al parameter and pelvis residual load errors produced by the phase two optimi zation and anatomic landmark methods for three types of movement data.............................................................................27 2-9 Comparison of inertial parameters pr edicted by anatomical landmark methods and phase two optimizations....................................................................................28 2-10. Differences between inertial parame ters predicted by an atomical landmark methods and phase two optimizations......................................................................29 4-1. Performance results for system identi fication problem for a 3D kinematic ankle joint model involving 252 design va riables and 1800 objective function elements....................................................................................................................44 4-2. Performance results for movement prediction problem for a 3D full-body gait model involving 660 design variables and 4100 objective function elements.........44 5-1. Descriptions of cost function weig hts and phase 3 control parameters...................54 5-2. Summary of fixed offsets added to norm al gait for each prescribed foot path........54

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viii 5-3. Comparison of cost function weight s and phase 3 control parameters....................55 5-4. Summary of root-mean-square (RMS) e rrors for tracked quantities for toeout gait........................................................................................................................... .56 5-5. Summary of root-mean-square (RMS) e rrors for tracked quantities for wide stance gait.................................................................................................................57 5-6. Summary of root-mean-square (RMS) e rrors for predicted left knee abduction torque quantities for toeout and wide stance gait.....................................................55

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ix LIST OF FIGURES Figure Page 2-1. Illustration of the 3D, 14 segment, 27 DOF full-body kinematic model linkage joined by a set of gimbal, universal, and pin joints..................................................14 2-2. Illustration of the 3 DOF right hip jo int center simultaneously defined in the pelvis and right femur segments and the 6 translational parameters optimized to determine the functional hi p joint center location....................................................16 2-3. Illustration of the 1 DO F right knee joint simultaneous ly defined in the right femur and right tibia segments and the 4 rotational and 5 translational parameters optimized to determine th e knee joint location and orientation.............17 2-4. 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 th e joint locations and orientations..................19 2-5. Illustration of the modified Cleveland Clinic marker set used during static and dynamic motion capture trials..................................................................................21 2-6. Phase one optimization convergence illustration series for the knee joint, where synthetic markers are blue, model marker s are red, and root-mean-square (RMS) marker distance error is green..................................................................................22 2-7. Phase two optimization convergence illust ration series for the knee joint, where synthetic masses are blue, model masses are red, and root-mean-square (RMS) residual pelvis forces and torques are orange and green, respectively.....................23 3-1. Legend of five sample statistics pres ented by the chosen boxplot convention........36 3-2. Comparison of root-mean-square (RMS) leg joint torques and marker distance error distributions.....................................................................................................37 3-3. Comparison of mean leg joint torques and marker distance error distributions......38 5-1. Comparison of left knee abduc tion torques for toeout gait......................................60 5-2. Comparison of left knee abduction torque s for wide stance gait, where original (blue) is experimental normal gait, simu lation (red) is predicted toeout gait, and final (green) is experi mental toeout gait..................................................................61

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x 5-3. Comparison of left knee abduction tor ques for simulated high tibial osteotomy (HTO) post-surgery gait...........................................................................................62 5-4. Comparison of mean (solid black line ) plus or minus one standard deviation (gray shaded area) for experiment al left knee abduction torques............................63 5-5. Comparison of mean (solid black line ) plus or minus one standard deviation (gray shaded area) for simulated left knee abduction torques..................................64

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xi Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PATIENT-SPECIFIC DYNAMIC MODELING TO PREDICT FUNCTIONAL OUTCOMES By Jeffrey A. Reinbolt May 2006 Chair: Benjamin J. Fregly Major Department: Mechanic al and Aerospace Engineering Movement related disorders have trea tments (e.g., corrective surgeries, rehabilitation) generally characterized by va riable outcomes as a result of subjective decisions based on qualitative analyses using a one-size-fits-all healthcare approach. Imagine the benefit to the h ealthcare provider and, more importantly, the patient, if certain clinical parameters may be evalua ted pre-treatment in order to predict the post-treatment outcome. Using patient-specif ic models, movement related disorders may be treated with reliable outcomes as a resu lt of objective decisions based on quantitative analyses using a patient-specific approach. The general objective of the current work is to predic t post-treatment outcome using patient-specific models given pre-treatment data. The specific objective is to develop a four-phase optimization approach to identify patient-specific model parameters and utilize the calibrated m odel to predict functional out come. Phase one involves identifying joint parameters describing the pos itions and orientations of joints within

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xii adjacent body segments. Phase two involves identifying inertial parameters defining the mass, centers of mass, and moments of in ertia for each body segment. Phase three involves identifying control parameters repres enting 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 patient-specific models to predict clinically significant outcomes.

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1 CHAPTER 1 INTRODUCTION Arthritis: The Nations Leading Cause of Disability In 1997, the Centers for Disease Control and Prevention (CDC) reported that 43 million (or 1 in 6) Americans suffered with arthritis. A 2002 CDC study showed that 70 million (a 63% increase in 5 years; or 1 in 3) Americans have arthritis (CDC, 2003 ). Approximately two-thirds of individuals with arthritis are under 65 years old. As the population ages, the number of people with arthri tis is likely to incr ease significantly. The most common forms of arthritis are osteoa rthritis, 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 a nd dysfunction are the primary reasons for total knee replacement (TKR). This pr ocedure involves a resurfacing of bones surrounding the knee joint. The end of the fe mur 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 alon e 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 conser vative (both economically and surgically) corrective procedure known as high tibial os teotomy (HTO). By changing the frontal plane alignment of the tibia with a wedge of bone, a HTO shifts the weight-bearing axis

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2 of the leg, and thus the mechanical stresses from the diseased portion to the healthy section of the knee compartment. By transfer ring 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 patien ts who receive recommendations to avoid TKR. Need for Patient-Specific Simulations Innovative patient-specific models and simulations would be valuable for addressing problems in orthopedics and sports medicine, as well as for evaluating and enhancing corrective surgical procedures (Arnold et al. 2000 ; Arnold and Delp, 2001 ; Chao et al. 1993; Chao and Sim, 1995 ; Delp et al. 1998 ; Delp et al. 1996 ; Delp et al. 1990 ; Pandy, 2001 ). For example, a patient-specifi c dynamic model may be useful for planning intended surgical parameters a nd predicting the outcome of HTO. The main motivation for developing the following patient-specific dynamic model and the associated multi-phase optimization appr oach is to predict the post-surgery peak internal knee abduction moment in HTO pa tients. Conventional surgical planning techniques for HTO involve choosing the am ount of necessary tibial angulation from standing radiographs (or x-rays). Unfortuna tely, alignment correction estimates from static x-rays do not accurately predict long-term clinical outcome after HTO (Andriacchi, 1994 ; Tetsworth and Paley, 1994 ). Researchers have identif ied the peak external knee adduction moment as an indica tor of clinical ou tcome while investigating the gait of HTO patients (Andriacchi, 1994 ; Bryan et al. 1997 ; Hurwitz et al. 1998 ; Prodromos et al. 1985 ; Wang et al. 1990 ). Currently, no movement simulations (or other methods for that matter) allow surgeons to choose HTO surgical parameters to achieve a chosen post-surgery knee abduction moment.

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3 The precision of dynamic analyses is fundame ntally 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 multi-link chain and simple mechanical approximations of joints will not precisely match the human anatomy and fu nction. The model should provide the best possible agreement to experimental motion da ta within the bounds of the dynamic model selected (Sommer and Miller, 1980 ). Background Motion Capture Motion capture is the use of external devi ces to capture the movement of a real object. One type of motion-capture technology is based on a passive optical technique. Passive refers to markers, which are simply spheres covered in reflective tape, placed on the object. Optical refers to the technol ogy used to provide 3D data, which involves high-speed, high-resolution vide o cameras. By placing passive markers on an object, special hardware records the pos ition of those markers in ti me and it generates a set of motion data (or marker data). Often motion capture is used to create synt hetic actors by capturing the motions of real humans. Special effects companies have used this technique to produce incredibly realistic animations in movies such as Star Wars Episode I & II Titanic Batman and Terminator 2 Biomechanical Models Researchers use motion-capture technology to construct biomechanical models of the human structure. The position of extern al markers may be used to estimate the

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4 position of internal landmarks such as joint cen ters. 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 re ference 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 mass-distribution data, one can study the forces and torques requ ired 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 exampl e, the objective function may be the errors between the biomechanical model and the recorded data. These errors are a function of the models generalized coordina tes and the models kinematic and kinetic parameters. Optimization may be used to m odify the design variables of the model to minimize the overall fitness errors and identify a structure that matches the experimental data very well.

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5 CHAPTER 2 CREATION OF PATIENT-SPECIFIC DYNAMIC MODELS FROM THREE-DIMENSIONAL 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 re quires values for joint parameters (JPs joint positions and orientati ons in the body segments) and in ertial parameters (IPs masses, mass centers, and mome nts of inertia of the body segments). If the specified parameter values do not match the patients anatomy and mass di stribution, then the predicted gait motions and loads may not be indicative of the clinical situation. The literature contains a vari ety of methods to estimate JP and IP values on a patient-specific 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 three-dimensional (3D) multi-joint 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 ).

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6 This study presents a computationally-e fficient two-phase optimization approach for determining patient-specific JP and IP values in a dynamic skeletal model given experimental movement data to match. The fi rst phase determines JP values that best match experimental kinematic data, while the se cond phase determines IP values that best match experimental kinetic data. The approach is demonstrated by fitting a 3D, 27 degree-of-freedom (DOF), parametric fu ll-body gait model possessing 98 JPs and 84 IPs to synthetic (i.e., computer generate d) and experimental movement data. Methods A sample dynamic model is needed to demonstrate the proposed two-phase optimization approach. For this purpose, a parametric 3D, 27 DOF, full-body gait model whose equations of motion were derived w ith the symbolic manipulation software, Autolev (OnLine Dynamics, Sunnyvale, CA), was used (Figure 2-1 Table 2-1 ). Comparable to Pandy's (2001) model structur e, 3 translational and 3 rotational DOFs express the movement of the pelvis in a Ne wtonian reference frame and the remaining 13 segments comprised four open chains bran ching 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 prov ides a simplified mechanical approximation to the actual anatomic articulations. Anatom ic 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 2-2 Table 2-2 Figure 2-3 Table 2-3 Figure 2-4 Table 2-4 ). The segment masses, mass centers, and moments of inertia were described by unique IPs.

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7 Anatomic landmark methods were used to estimate nominal values for 7 IPs per segment (de Leva, 1996 ). Parameters defining the structure of the m odel were referenced to local coordinate systems fixed in each body segment. These coordinate systems were created from a static motion capture trial (see below). Mark ers placed over the left anterior superior iliac spine (ASIS), right ASIS, and superior sacrum were used to define the pelvis segment coordinate system (Figure 2-2 ). From percentages of the inter-ASIS distance, a nominal hip joint center location was es timated within the pelvis segment (Bell et al. 1990 ). This nominal joint center served as th e origin of the femur coordinate system, which was subsequently defined using marker s placed over the medial and lateral femoral epicondyles (Figure 2-2 ). 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 2-3 ). The talus coordinate system was created where the y-axis extends along the line perpendicular to bot h the talocrural joint axis and the subtalar joint axis (Figure 2-4 ). The heel and toe markers, in combin ation with the tibia y-axis, defined the foot coordinate system (Figure 2-4 ). Experimental kinematic and ki netic data were collected from a single subject using a video-based motion analysis system (Mo tion Analysis Corporation, Santa Rosa, CA) and two force plates (AMTI, Watertown, MA). Institutional revi ew board approval and informed consent were obtained prior to th e experiments. As described above, segment coordinate systems were created from surface marker locations measured during a static standing pose (Figure 2-5 ). Unloaded isolated joint motions were performed to exercise the primary functional axes of each lower ex tremity joint (hip, knee, and ankle on each

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8 side). For each joint, the subject was instru cted 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 tr ials performed unloaded. Mult iple cycles of standing hip flexion-extension followed by abduction-a dduction were recorded. Similar to Leardini et al. (1999 ), internal-external rotation of the hip wa s avoided to reduce skin and soft tissue movement artifacts. Multiple cycles of knee flexion-extension were measured. Finally, multiple cycles of simultaneous ankle plantarflexion-dorsiflexion and inversion-eversion were recorded. Gait motion and ground reac tion data were collected to investigate simultaneous motion of all lower extremity joints under load-bearing physiological conditions. To evaluate the proposed optimizati on methodology, two types of synthetic movement data were generated from the expe rimental data sets. The first type was noiseless synthetic data gene rated by moving the model th rough 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 t A sin was used with the following uniform random parameter values: amplitude (0 A 1 cm), frequency (0 25 rad/s), and phase angle (0 2 ) (Chze et al. 1995 ). The first phase of the optimization proce dure adjusted JP values and model motion to minimize errors between model and experimental marker locations (Equation 2-1 Figure 2-6 ). For isolated joint motion trials, the design variables were 540 B-spline nodes ( q ) parameterizing the generalized coordi nate trajectories (20 nodes per DOF) and 6 hip, 9 knee, or 12 ankle JPs ( pJP). For the gait trial, the nu mber of JPs was reduced to 4

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9 hip, 9 knee, and 4 ankle, due to inaccuracies in determining joint functional axes with rotations less than 25 (Chze et al. 1998 ). The initial value for each B-spline node and JP was chosen to be zero to test the robustn ess 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 nf time frames. The JP optimizations were perfor med with Matlabs nonlinear least squares algorithm (The Mathworks, Natick, MA). 2 11 3 1 ) ( min nf i nm jk JP ijk ijk JPm' mJPq p eq p The second phase of the optimization proce dure adjusted IP values to minimize the residual forces and torques acting on a 6 DOF ground-to-pelvis joint (Equation 2-2 Figure 2-7 ). 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 lim ited 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 combinati on of pelvis residual loads ( F and T ) calculated over all nf time frames and differe nces between initial ( p IP) 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 respectiv e initial values to create nondimensional errors. The IP optimizations also were pe rformed with Matlabs nonlinear least squares algorithm. Once a IP optimization converged, the final IP values were used as the initial (2-1)

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10 guess for a subsequent IP optimization, with this processing being repeated until the resulting pelvis residual loads converged. 2 1 3 1 2 2 ) ( ) ( min IP IP IP nf ij IP ij IP ij IPH BW T BW FIPp' p' p p p ep 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 de termined for the gait tria l using the previously optimum JP values. Root-mean-square (RMS ) errors between orig inal and recovered parameters, marker distances, and pelvis re sidual loads were used to quantify the procedures performance. All optimizations were performed on a 3.4 GHz Pentium 4 PC. Results For phase one, each JP optimization us ing noiseless synthetic data precisely recovered the original marker trajectories a nd model parameters to within an arbitrarily tight convergence tolerance (Table 2-5 Table 2-6 ). 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 differenc es attributed to noise alone for synthetic data with noise (Table 2-7 ). Optimizations involving the isol ated 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 (2-2)

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11 CPU time. These computation times were orders of magnitude faster than those reported using a two-level optimization procedure (Reinbolt et al. 2005 ). For phase two, each IP optimization using noiseless synthetic data produced zero pelvis residual loads and recove red the original IP values to within an arbitrarily tight convergence tolerance (Table 2-8 Table 2-9 ). For the other two data sets, pelvis residual loads and IP errors remained small, with a random initial seed produc ing nearly the same pelvis residual loads but slight ly 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 2-10 ). 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 m odel parameters. Parameters are typically estimated from anatomic landmark methods. The estimated (or nominal) values may be improved by formulating a two-phase optimization problem driven by motion capture data. Optimized dynamic models can provide a more reliable foundation for future patient-specific dynamic analyses and optimizations. This study presented a two-phase optimi zation approach for tuning joint and inertial parameters in a dynamic skeletal m odel to match experimental movement data from a specific patient. For the fullbody gait model used in this study, the JP optimization satisfactorily reproduced patient-specific JP values while the IP optimization successfully reduced pelvis residu al loads while allowing variation in the IP values away from their initial guesses. Th e JP and IP values found by this two-phase optimization approach are only as reliable as the noisy experimental movement data used

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12 as inputs. By optimizing over all time fr ames simultaneously, the procedure smoothes out the effects of this noise. An optimi zation approach that modifies JPs and IPs simultaneously may provide even further reductions in pelvis residual loads. It cannot be claimed that models fitted w ith this two-phase approach will reproduce the actual functional axes and inertial propert ies 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 f unction value in each case than did the nominal parameters from which the synthetic data were generate d. Thus, it can only be claimed that the optimized model structure provides the best pos sible fit to the impe rfect movement data. There are differences between phase one a nd phase two when comparing results for experimental data to synthetic data with noise. For the JP case, absolute parameter differences were higher for expe rimental data compared to synthetic data with noise. This suggests that noise does not hinder the pr ocess of determining JPs. However, for the IP case, absolute parameter differences we re higher for synthetic data with noise compared to experimental data. In fact, noi se is a limiting factor when identifying IPs. The phase one optimization determined pati ent-specific joint parameters similar to previous works. The optimal hip joint cente r location of 2.94 cm (12.01% posterior), 9.21 cm (37.63% inferior), and 9.09 cm (37.10% la teral) is comparable to 19.30%, 30.40%, and 35.90%, respectively, of the interASIS 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 pl ane rotation (8 7.36) of the talocrural joint correlates to 82.7 3.7 (range 74 to 94) (Inman, 1976 ). The optimum

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13 distance (0.58 cm) between the ta locrural joint and the subtal ar joint is analogous to 1.24 0.29 cm (Bogert et al. 1994 ). The optimum transverse plane rotation (34.79) and sagittal plane rotation (31.34) of the subtalar joint correspon ds to 23 11 (range 4 to 47) and 42 9 (range 20.5 to 68.5), respectively (Inman, 1976 ). Compared to anatomic landmark methods reported in the lit erature, 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 patient-specific inertial parameters similar to previous work. The optimum masses we re 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 re ported in the litera ture, 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 reasonabl e and show the extent of agre ement 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 two-pha se optimization successfully reduced cost function values for marker distance errors or pelvis residual loads below those resulting from anatomic landmark methods.

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14 2-1. Illustration of the 3D, 14 segment, 27 DOF full-body kinematic model linkage joined by a set of gimbal, universal, and pin joints. Figure

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15 2-1. Descriptions of the model degrees of freedom. DOF Description q1 Pelvis anterior-posterior position. q2 Pelvis superior-inferior position. q3 Pelvis medial-lateral position. q4 Pelvis anterior-posterior tilt angle. q5 Pelvis elevation-depression angle. q6 Pelvis internal-external rotation angle. q7 Right hip flexion-extension angle. q8 Right hip adduction-abduction angle. q9 Right hip internal-ext ernal rotation angle. q10 Right knee flexion-extension angle. q11 Right ankle plantarflexi on-dorsiflexion angle. q12 Right ankle invers ion-eversion angle q13 Left hip flexion-extension angle. q14 Left hip adduction-abduction angle. q15 Left hip internal-external rotation angle. q16 Left knee flexion-extension angle. q17 Left ankle plantarflexi on-dorsiflexion angle. q18 Left ankle inversion-eversion angle q19 Trunk anterior-posterior tilt angle. q20 Trunk elevation-depression angle. q21 Trunk internal-external rotation angle. q22 Right shoulder flexion-extension angle. q23 Right shoulder adduction-abduction angle. q24 Right elbow flexion angle. q25 Left shoulder flexion-extension angle. q26 Left shoulder adduction-abduction angle. q27 Left elbow flexion angle. Table

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16 2-2. Illustration of the 3 DOF right hip jo int center simultaneously defined in the pelvis and right femur segments and th e 6 translational parameters optimized to determine the functional hip joint center location. 2-2. Descriptions of th e hip joint parameters. Hip Joint Parameter Description p1 Anterior-posterior loca tion in pelvis segment. p2 Superior-inferior locatio n in pelvis segment. p3 Medial-lateral locati on in pelvis segment. p4 Anterior-posterior loca tion in femur segment. p5 Superior-inferior loca tion in femur segment. p6 Medial-lateral location in femur segment. Figure Table

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17 2-3. Illustration of the 1 DO F right knee joint simultaneous ly 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. Figure

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18 2-3. Descriptions of the knee joint parameters. Knee Joint Parameter Description p1 Adduction-abduction rotation in femur segment. p2 Internal-external rotation in femur segment. p3 Adduction-abduction rotati on in tibia segment. p4 Internal-external rotation in tibia segment. p5 Anterior-posterior loca tion in femur segment. p6 Superior-inferior loca tion in femur segment. p7 Anterior-posterior loca tion in tibia segment. p8 Superior-inferior location in tibia segment. p9 Medial-lateral location in tibia segment. Table

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19 2-4. Illustration of the 2 DO F 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. Figure

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20 2-4. Descriptions of the ankle joint parameters. Ankle Joint Parameter Description p1 Adduction-abduction rotation of ta locrural in tibia segment. p2 Internal-external rotation of talocrural in tibia segment. p3 Internal-external rotation of subtalar in talus segment. p4 Internal-external rotation of subtalar in foot segment. p5 Dorsi-plantar rotation of s ubtalar in foot segment. p6 Anterior-posterior location of talocrural in tibia segment. p7 Superior-inferior location of talocrural in tibia segment. p8 Medial-lateral location of talocrural in tibia segment. p9 Superior-inferior lo cation of subtalar in talus segment. p10 Anterior-posterior location of subtalar in foot segment. p11 Superior-inferior lo cation of subtalar in foot segment. p12 Medial-lateral location of subtalar in foot segment. Table

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21 2-5. Illustration of the modified Cleveland Clinic marker set used during static and dynamic motion capture trials. No te: the background femur and knee markers have been omitted for clarity and the medial and lateral markers for the knee and ankle are removed following the static trial. Figure

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22 2-6. Phase one optimization convergence illustration series for the knee joint, where synthetic markers are blue, model markers are red, and root-mean-square (RMS) marker distance error is green. Given synthe tic marker data without noise and a synthetic knee flexion angle = 90, A) is the initial model knee flexion = 0 a nd knee joint parameters causing joint dislocation, B) is the model knee flexion = 30 and improved kn ee joint parameters, C) is th e model knee flexion = 60 and nearly correct knee joint parameters, and D) is the fina l model knee flexion = 90 and ideal knee joint parameters. Figure

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23 2-7. Phase two optimization convergence illustration series for th e knee joint, where synthetic masses are blue, model masses are red, and root-mean-square (RMS) residual pelvis forces and torques are orange an d green, respectively. Given synthetic kinetic data without noise and a synthetic knee flexion motion, A) is th e initial model thi gh 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 mode l thigh and shank inertial parameters. Figure

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24 2-5. Summary of root-mean-square (RMS) jo int parameter and marker distance errors produced by the phase one optimization and anatomic landmark methods for three t ypes of movement data. Experimental da ta were from isolated joint and gait motions measured using a video-based motion analysis system with three surface markers per body segment. Synthetic marker data were generated by applyi ng the experimental motions to a nomi nal kinematic model with and without superimposed numerical noise. The isolat ed joint motions were non-weight bearing and utilized larger joint excursions than did the gait motion. For the an atomical landmark method, only the optimization involving model motion was performed since the joint parameters were speci ed. 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) 9.82e-09 3.58e-08 1.80e-08 2.39e-06 Orientation parameters () n/a 4.68e-04 6.46e-03 5.45e-03 Synthetic without Noise Phase one optimization Position parameters (mm) 4.85e-05 8.69e-04 6.35e-03 1.37e-02 Marker distances (mm) 6.62 5.89 5.43 5.49 Orientation parameters () n/a 0.09 3.37 3.43 Synthetic with Noise Phase one optimization 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 Experimental Anatomical landmarks Marker distances (mm) 4.47 4.03 4.44 6.08 Table

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252-6. Comparison of joint parameters predicted by anatomical la ndmark methods, phase one optim ization involving individual joints separately, and phase one optimiza tion involving all joints simultaneously. Body segment indicates the segment in which the associated joint parameter is xed. For a graphical depiction of th e listed joint parameters, consult Figure 2 Figure 3 and Figure 4 Single joint optimization Full leg optimization Joint Joint parameter Body segment Anatomical landmarks Synthetic without noise Synthetic with noise Experimental Synthetic without noise Synthetic with noise Experimental 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 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 Hip 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 () 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 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 Knee Lateral position (mm) Shank 0.00 0.00 -0.84 -6.79 0.00 -1.63 2.67 Frontal plane orientation () Shank -12.22 -12.21 -16.29 -11.18 -12.22 -12.81 -2.64 Transverse plane orientation () Shank 0.00 0.01 3.84 -23.89 0.00 8.79 -18.12 Anterior position (mm) Shank 0. 00 0.01 0.24 4.14 0.00 -2.09 -3.29 Superior position (mm) Shank -377.44 -377. 45 -379.20 -395.73 -377 .45 -379.99 -393.29 Ankle (talocrural) Lateral position (mm) Shank 0.00 -0.01 2.03 7.08 0.00 2.03 7.08 Transverse plane orientation () Talus -35.56 -35.56 -40.56 -27.86 -35.56 -40.57 -27.86 Transverse plane orientation () Foot -23.00 -23.00 -23.34 -34.79 -23.00 -23.34 -34.79 Sagittal plane orientation () Foot 42.00 42.00 42.59 31.34 42.00 42.59 31.34 Superior position (mm) Talus -10.00 -10.00 -9.22 -5.77 10.00 -9.22 -5.77 Anterior position (mm) Foot 93.36 93.36 93.95 87.31 93.36 93.95 87.31 Superior position (mm) Foot 54.53 54.52 53.34 39.68 54.52 53.34 39.68 Ankle (subtalar) Lateral position (mm) Foot -6.89 -6 .89 -6.06 -3.80 -6 .89 -6.06 -3.80 Table

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262-7. Differences between joint parameters predicted by anat omical landmark methods and phase one optimizations. Body segment indicates the segment in whic h the associated joint parameter is xed. For actual joint parameter values, consult Table 2-6 For a graphical depiction of th e listed joint parameters, consult Figure 2 Figure 3 and Figure 4 Single joint optimization Full leg optimization Joint Joint parameter Body segment Synthetic difference Experimental difference Synthetic difference Experimental 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 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 Hip 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 () Shank -0.14 -4.42 -0.76 -5.25 Transverse plane orientation () Shank -0.03 -1.85 -1.01 -6.23 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 Knee Lateral position (mm) Sha nk -0.84 -6.79 -1.63 2.67 Frontal plane orientation () Shank -4.08 1.04 -0.59 9.58 Transverse plane orientation () Shank 3.85 -23.89 8.79 -18.12 Anterior position (mm) Sh ank 0.24 4.14 -2.09 -3.29 Superior position (mm) Shank -1.76 -18.29 -2.54 -15.85 Ankle (talocrural) Lateral position (mm) Shank 2.03 7.08 2.03 7.08 Transverse plane orientation () Talus -5.01 7.70 -5.01 7.70 Transverse plane orientation () Foot -0.34 -11.79 -0.34 -11.79 Sagittal plane orientation () Foot 0.59 -10.66 0.59 -10.66 Superior position (mm) Talus 0.78 4.23 0.78 4.23 Anterior position (mm) Foot 0.59 -6.05 0.59 -6.05 Superior position (mm) Foot -1.19 -14.85 -1.19 -14.85 Ankle (subtalar) Lateral position (mm) Foot 0.82 3.09 0.82 3.09 Table

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27 2-8. Summary of root-mean-square (RMS) inertial parameter a nd pelvis residual load erro rs produced by the phase two optimization and anatomic landmark methods for three types of movement data. Experiment al data were from a gait motion measured using a video-based moti on analysis system with three surface markers per body segment. Synthetic marker data were generated by applyi ng the experimental motion to a nomina l 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) Synthetic without Noise Phase 2 optimization 8.04e-10 2.16e-10 5.13e-13 1.22e-10 7.11e-13 Synthetic with Noise (correct seed) Phase 2 optimization 16.92 4.93 5.67e-03 1.08 6.64e-03 Synthetic with Noise (random seeds) Phase 2 optimization 16.96.44 5.24.23 7.45e-02.81e-02 1.33.39 2.16e-02.92e-03 Phase 2 optimization 27.89 12.66 n/a n/a n/a Experimental Anatomical landmarks 34.81 13.75 n/a n/a n/a Table

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28 2-9 Comparison of inertial parameters pr edicted by anatomical landmark methods and phase two optimizations. Body segm ent indicates the segment in which the associated inertial parameter is xed. Direction indicates the means in which the associated inertial parameter was applied. Synthetic with noise Category Body segment Direction Anatomical landmarks Synthetic without noise Correct seed Random seeds (10 cases) Experimental Pelvis 0 7.69 0 7.69 0 8.11 0 6.98.87 0 7.63 Thigh 0 9.74 0 9.74 10.76 11.19.02 0 8.91 Shank 0 2.98 0 2.98 0 3.25 0 3.37.39 0 2.95 Foot 0 1.26 0 1.26 0 1.37 0 1.40.29 0 1.29 Head & trunk 26.99 26.99 23.61 23.88.34 26.82 Upper arm 0 1.86 0 1.86 0 1.82 0 1.56.49 0 1.87 Mass (kg) Lower arm & hand n/a 0 1.53 0 1.53 0 1.63 0 1.75.47 0 1.53 Anterior -9.36e-02 -9.36e-02-9.63e -02 -1.00e-01.29e-02 -9.33e-02 Superior -2.37e-02 -2.37e-02-2.33e -02 -2.49e-02.44e-03 -2.37e-02 Pelvis Lateral 0.00 -2.35e-13 6.57e-13 8.06e-13.02e-13 6.07e-13 Anterior 0.00 1.54e-13 4.02e-13 7.76e-13.06e-13 4.86e-13 Superior -1.68e-01 -1.68e-01-1.65e -01 -1.58e-01.84e-02 -1.59e-01 Thigh Lateral 0.00 -1.49e-14 1.58e-12 2.41e-12.05e-13 8.91e-13 Anterior 0.00 1.22e-14-5.41e-13 -1.91e-13.20e-13 7.62e-13 Superior -1.72e-01 -1.72e-01-1.89e -01 -1.89e-01.48e-02 -1.80e-01 Shank Lateral 0.00 6.59e-14 3.15e-13 5.65e-13.72e-13 4.56e-13 Anterior 9.21e-02 9.21e-02 8.73e-02 8.17e-02.90e-02 9.27e-02 Superior -1.60e-04 -1.60e-04-4.19e -04 -4.12e-04.19e-04 -1.60e-04 Foot Lateral 0.00 -1.81e-14 1.26e-14 9.35e-14.93e-14 1.85e-14 Anterior 0.00 1.92e-13-8.33e-13 -8.96e-13.35e-12 8.21e-13 Superior -1.44e-01 -1.44e-01-1.69e -01 -1.87e-01.38e-02 -1.42e-01 Head & trunk Lateral 0.00 -2.56e-14 1.66e-12 2.34e-12.09e-13 4.45e-13 Anterior 0.00 4.74e-14-3.21e-13 -2.69e-13.60e-13 6.15e-13 Superior -1.84e-01 -1.84e-01-1.86e -01 -1.95e-01.96e-02 -1.84e-01 Upper arm Lateral 0.00 -1.79e-14 3.51e-13 4.19e-13.45e-13 7.92e-13 Anterior 0.00 5.59e-14 1.45e-13 2.32e-13.58e-13 9.22e-13 Superior -1.85e-01 -1.85e-01-1.97e -01 -1.89e-01.36e-02 -1.85e-01 Center of mass (m) Lower arm & hand Lateral 0.00 -3.85-e14 2.86e-13 4.13e-13.59e-13 7.38e-13 Anterior 6.83e-02 6.83e-02 6.83e-02 6.60e-02.30e-02 6.83e-02 Superior 6.22e-02 6.22e-02 6.21e-02 5.98e-02.81e-02 6.22e-02 Pelvis Lateral 5.48e-02 5.48e-02 5.47e-02 5.25e-02.64e-02 5.48e-02 Anterior 1.78e-01 1.78e-01 1.80e-01 1.67e-01.39e-02 1.78e-01 Superior 3.66e-02 3.66e-02 3.65e-02 3.71e-02.35e-02 3.66e-02 Thigh Lateral 1.78e-01 1.78e-01 1.74e-01 2.06e-01.31e-02 1.78e-01 Anterior 2.98e-02 2.98e-02 2.97e-02 3.06e-02.19e-02 2.98e-02 Superior 4.86e-03 4.86e-03 4.86e-03 4.01e-03.04e-03 4.86e-03 Shank Lateral 2.84e-02 2.84e-02 2.86e-02 2.75e-02.65e-03 2.84e-02 Anterior 2.38e-03 2.38e-03 2.29e-03 2.27e-03.45e-04 2.38e-03 Superior 4.56e-03 4.56e-03 4.58e-03 4.90e-03.29e-03 4.56e-03 Foot Lateral 3.08e-03 3.08e-03 2.99e-03 3.02e-03.57e-04 3.08e-03 Anterior 9.51e-01 9.51e-01 9.18e-01 8.96e-01.35e-01 9.51e-01 Superior 2.27e-01 2.27e-01 2.27e-01 1.93e-01.62e-02 2.27e-01 Head & trunk Lateral 8.41e-01 8.41e-01 8.37e-01 9.99e-01.68e-01 8.41e-01 Anterior 1.53e-02 1.53e-02 1.53e-02 1.44e-02.65e-03 1.53e-02 Superior 4.71e-03 4.71e-03 4.71e-03 4.57e-03.29e-03 4.71e-03 Upper arm Lateral 1.37e-02 1.37e-02 1.37e-02 1.56e-02.29e-03 1.37e-02 Anterior 1.97e-02 1.97e-02 1.97e-02 2.06e-02.97e-03 1.97e-02 Superior 1.60e-03 1.60e-03 1.60e-03 1.41e-03.02e-04 1.60e-03 Moment of inertia (kg*m2) Lower arm & hand Lateral 2.05e-02 2.05e-02 2.04e-02 2.41e-02.78e-03 2.05e-02 Table

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29 2-10. Differences between inertial parame ters predicted by an atomical landmark methods and phase two optimizations. Body segment indicates the segment in which the associated joint parameter is xed. Direction indicates the means in which the associated inertial parameter was applied. For actual joint parameter values, consult Table 2-9 Synthetic differences Category Body segment Direction Correct seed Random seeds (10 cases) Experimental differences Pelvis 4.28e-01 -7.01e-01.87 -5.66e-02 Thigh 1.02 1.45.02 -8.29e-01 Shank 2.69e-01 3.89e-01.86e-01 -3.11e-02 Foot 1.29e-01 1.62e-01.95e-01 3.53e-02 Head and trunk -3.38 -3.11.34 -1.75e-01 Upper arm -4.62e-02 -3.05e-01.85e-01 1.46e-03 Mass (kg) Lower arm and hand n/a 1.01e-01 2.15e-01.67e-01 -1.89e-03 Anterior -2.69e-03 -6.51e-03.29e-02 2.30e-04 Superior 4.13e-04 -1.24e-03.44e-03 1.12e-05 Pelvis Lateral 6.57e-13 8.06e-13.02e-13 6.07e-13 Anterior 4.02e-13 7.76e-13.06e-13 4.86e-13 Superior 3.48e-03 1.01e-02.84e-02 9.81e-03 Thigh Lateral 1.58e-12 2.41e-12.05e-13 8.91e-13 Anterior -5.41e-13 -1.91e-13.20e-13 7.62e-13 Superior -1.66e-02 -1. 73e-02.48e-02 -7.65e-03 Shank Lateral 3.15e-13 5.65e-13.72e-13 4.56e-13 Anterior -4.83e-03 -1.04e-02.90e-02 6.30e-04 Superior -5.25e-08 7.39e-06.19e-04 9.50e-13 Foot Lateral 1.26e-14 9.35e-14.93e-14 1.85e-14 Anterior -8.33e-13 -8.96e-13.35e-12 8.21e-13 Superior -2.47e-02 -4.31e-02.38e-02 1.66e-03 Head and trunk Lateral 1.66e-12 2.34e-12.09e-13 4.45e-13 Anterior -3.21e-13 -2.69e-13.60e-13 6.15e-13 Superior -2.40e-03 -1.15e-02.96e-02 1.10e-04 Upper arm Lateral 3.51e-13 4.19e-13.45e-13 7.92e-13 Anterior 1.45e-13 2.32e-13.58e-13 9.22e-13 Superior -1.29e-02 -4.21e-03.36e-02 3.70e-04 Center of mass (m) Lower arm and hand Lateral 2.86e-13 4.13e-13.59e-13 7.38e-13 Anterior 5.53e-05 -2.24e-03.30e-02 1.95e-04 Superior -1.11e-04 -2. 41e-03.81e-02 -9.03e-04 Pelvis Lateral -1.21e-04 -2.28e-03.64e-02 -1.19e-05 Anterior 2.06e-03 -1.14e-02.39e-02 2.86e-03 Superior -7.14e-05 5.67e-04.35e-02 -1.45e-04 Thigh Lateral -4.56e-03 2.73e-02.31e-02 -9.06e-03 Anterior -9.91e-05 8.51e-04.19e-02 -5.98e-04 Superior 6.80e-07 -8.49e-04.04e-03 1.85e-06 Shank Lateral 2.09e-04 -9.19e-04.65e-03 4.15e-04 Anterior -5.94e-07 -1.73e-05.45e-04 -9.07e-08 Superior -8.59e-08 3.16e-04.29e-03 -6.41e-07 Foot Lateral 3.06e-06 2.87e-05.57e-04 1.36e-05 Anterior -3.30e-02 -5.56e-02.35e-01 -3.08e-02 Superior -6.15e-04 -3. 37e-02.62e-02 -2.87e-03 Head and trunk Lateral -4.13e-03 1.59e-01.68e-01 -1.73e-02 Anterior -1.62e-05 -9.11e-04.65e-03 -1.37e-04 Superior -1.79e-07 -4. 14e-05.29e-03 -9.40e-07 Upper arm Lateral -7.90e-07 1.97e-03.29e-03 -1.60e-06 Anterior -1.63e-05 8.91e-04.97e-03 -1.09e-04 Superior -8.95e-08 -1. 95e-04.02e-04 -7.50e-07 Moment of inertia (kg*m2) Lower arm and hand Lateral -5.84e-05 3.59e-03.78e-03 -1.15e-05 Table

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30 CHAPTER 3 EFFECT OF MODEL PARAMETER VARIATIONS ON INVERSE DYNAMICS RESULTS USING M ONTE 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 appli cations involving the inves tigation of human movement ranging from sports medicine to pathological gait. In all cas es, 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 iner tial 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 vari ety of methods investigating the sensitivity of inverse dynamics torques to JPs and IPs. Two el bow 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 flexion-extension torque has

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31 been studied for multiple walking speeds while changing the knee center location by mm in the anterior-posterior 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 % (Pearsall and Costigan, 1999 ). Knee forces and torques were evaluated for a planar harmoni c oscillating motion as inertial properties varied by % (Andrews and Mish, 1996 ). Directly related to knee joint torque, knee kinematics were investigated during gait when the rotation axes varied by (Croce et al. 1996 ). This study performs a series of Monte Carl o analyses relating to inverse dynamics using instantiations of a three-dimensional (3D) full-body gait model with isolated and simultaneous variations of JP and IP values. In addition, noise para meter (NP) variations representing kinematic noise were implemente d 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 instanti ations was sufficient for convergence of inverse dynamics torque values. Methods Experimental kinematic and kinetic gait da ta were collected fr om a single subject using a video-based 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 degree-of-freedom (DOF), full-body gait model (Figure 2-1 Table 2-1 ) was constructed, whose equations of mo tion were derived with the symbolic manipulation software, Autolev (OnLine Dyna mics, Sunnyvale, CA). The pelvis was

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32 connected to ground via a 6 DOF joint and th e remaining 13 segments comprised four open chains branching from the pelvis. The positions and orientations of joint axes within adjacent segment coordinate system s were defined by unique JPs. The segment masses, mass centers, and mome nts of inertia were described by unique IPs. Anatomic landmark methods were used to esti mate 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 de scribed previously in Chapter 2 (Reinbolt and Fregly, 2005 ). A series of Monte Carlo an alyses relating to invers e dynamics were performed using instantiations the full-body gait model w ith 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 locatio ns were bounded by a maximum of mm (Challis and Kerwin, 1996 Holden and Stanhope, 1998 ). Joint axes orientations were bounded by a maximum of (Croce et al. 1996 ). Inertial properties were bounded by a maximum of % 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 torque s were within 2% of the final mean and coefficient of variance, respectively, for the la st 10% of instantiations (Fishman, 1996 Valero-Cuevas et al. 2003 ).

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33 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 m ovement 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 simulate d skin movement artifacts of Lu and OConnor (1999 ), a continuous noise model of the form t A sin was used with the following uniform random parameter values: amplitude (0 A 1 cm), frequency (0 25 rad/s), and phase angle (0 2 ) (Chze et al. 1995 ). Noise was separately generated for each 3D coordinate of the marker trajectories. Similar to the lower-body 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 marker s for the inverse dynamics simulation and true synthetic data cases). Results The distributions of mean inverse dynami cs results are best summarized using a boxplot presenting five sample st atistics: 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 3-1 ). RMS and mean hip torque errors were computed in the flexion-extension, abduction-adduction, and internal-e xternal rotation directions (Figure 3-2 Figure 3-3 ). Mostly due to JPs, the hip flexion-exte nsion and abduction-adduction torques showed more variation than the internal-external rota tion torque. The IPs ha d insignificant effect on range of hip torques. The NPs had slight ly larger effects compared to IPs. The

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34 distributions of torques were reduced by attenuating the mo del fitness based on percentages of parameter bounds. RMS and mean knee torque errors were computed in the flexion-extension and abduction-adduction directions (Figure 3-2 Figure 3-3 ). Similar to the corresponding hip directions, the knee torques have larger sp ans for the abduction-adduction direction compared to the flexion-extension directi on. The NPs had slightly larger effects compared to IPs. The JPs had much more e ffect on knee torques compared to the IPs. For the IPs case, the flexion-extension to rque varied slightly more than the abduction-adduction torque. The breadth of knee torques decrease d with attenuated percentages of parameter bounds. RMS and mean ankle torque er rors were computed in the plantarflexion-dorsiflexion and i nversion-eversion directions (Figure 3-2 Figure 3-3 ). The inversion-eversion torque displayed a broader distributio n compared to the plantarflexions-dorsiflexion torque. The IPs had much less effect on ankle torque distributions compared to JPs. The NPs had s lightly larger effects compared to IPs. The attenuated percentages of parameter bounds c ondensed the spread for both torques. RMS and mean marker distance errors were computed between the inverse dynamics simulation and synthetic data (Figure 3-2 Figure 3-3 ). 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 ma rker distance error distributions. Discussion This study examined the distribution of inverse dynamics torques for a 3D full-body gait model using a series of Monte Carlo analyses simultaneously varying JPs

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35 and IPs. It is well established that joint tor que data is one of the most valuable quantities for the biomechanical investig ation of human movement. As evident in the literature, investigators are concerned with the sens itivity 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 adju sting 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 ot her two types. Similar to all Monte Carlo analyses assuming parameter independence, th e resulting distributions may in fact be overestimated. A more diverse assortment of full-body models are being simulated by not explicitly requiring partic ular body segment geometries. Nevertheless, implementing an optimization approach to maximize the fitn ess of the model to the experiment through determination of patient-speci fic JPs and IPs reduces the un certainty of inverse dynamics results.

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36 3-1. Legend of five sample statistics pr esented by the chosen boxplot convention. Figure

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37 3-2. Comparison of root-mean-square (RMS) leg joint torques and marker distance error distributions. First column cont ains 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 colu mn contains distributions from vary only the noise amplitude parameter (NP). Figure

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38 3-3. Comparison of mean leg joint torques and marker distance error distributions. First column contains distributions fro m varying only joint parameters (JP). Second column contains di stributions from varying only inertial parameters (IP). Third column contains distributi ons from varying both JPs and IPs. Fourth column contains distributions from vary only the noise amplitude parameter (NP). Figure

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39 CHAPTER 4 BENEFIT OF AUTOMATIC DIFFERENTIATION FOR BIOMECHANICAL OPTIMIZATIONS Background Optimization algorithms are often used to solve biomechanical system identification or movement predic tion problems employing complicated three-dimensional (3D) musculoskeletal models (Pandy, 2001 ). When gradient-based methods are used to solve large-scale probl ems involving hundreds of design variables, the computational cost of performing repeated simulations to calculate finite difference gradients can be extremely high. In addi tion, as 3D movement model complexity increases, there is a considerable increase in the computational expense of repeated simulations. Frequently, in spite of advan ces in processor performance, optimizations remain limited by computation time. Both speed and robustness of gradient-b ased optimizations are dramatically improved by using an analytical Jacobian matrix (all first-order deri vatives of dependent objective function variables with respect to independent design vari ables) rather than relying on finite difference approximati ons. 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 sym bolically may prove impractical. For more than eight years, the Networ k Enabled Optimization Server (NEOS) at Argonne National Laboratory has been using Automatic Diffe rentiation (AD), also called Algorithmic Differentiation, to compute Jacobians of remotely supplied user code (Griewank, 2000 ).

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40 AD is a technique for computing derivatives of arbitrarily complex computer programs by mechanical applicatio n of the chain rule of differentia l calculus. AD exploits the fact every computer program, no matter how compli cated, executes a sequence of elementary arithmetic operations. By applying the ch ain rule repeatedly to these operations, derivatives can be computed automatically and accurately to working precision. This study evaluates the benefit of usi ng AD methods to calculate analytical Jacobians for biomechanical optimization probl ems. For this purpose, a freely-available AD package, Automatic Di erentiation by OverLoading in C++ (ADOL-C) (Griewank et al. 1996 ), was applied to two biomechanical opt imization problems. The first is a system identification problem for a 3D ki nematic ankle joint model involving 252 design variables and 1800 objective function elements The second is a movement prediction problem for a 3D full-body gait model invo lving 660 design variables and 4100 objective function elements. Both problems are solved using a nonlinear least squares optimization algorithm. Methods Experimental kinematic and ki netic data were collected from a single subject using a video-based motion analysis system (Mo tion Analysis Corporation, Santa Rosa, CA) and two force plates (AMTI, Watertown, MA). Institutional revi ew board approval and informed consent were obtained prior to the experiments. The ankle joint problem was first solv ed without AD using the Matlab (The Mathworks, Inc., Natick, MA) nonlinear leas t 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

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41 locations. The ankle joint model possessed 12 joint parameters and 12 degrees-of-freedom. Each of the 12 generali zed coordinate curves was parameterized using 20 B-spline nodal points (240 total). Alto gether, there were 252 design variables. The problem contained 18 error quantities for e ach of the 100 time frames of data. The Jacobian matrix consisted of 1800 rows a nd 252 columns estimated by finite difference approximations. The movement prediction problem was fi rst 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 categorie s of tracking errors (foot pa th, center of pressure, trunk orientation, joint torque, and fictitious ground-to-pelvis residual reactions) between model and experiment. The movement prediction model possessed 27 degrees-of-freedom (21 adjusted by B-spline cu rves and 6 prescribed for arm motion) and 12 ground reactions. Each adjustable genera lized coordinate and reaction curve was parameterized using 20 B-spline nodal points (660 total design variab les). 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. ADOL-C was incorporated in to each objective function to compute an analytical Jacobian matrix. This package was chosen because the objective functions were comprised of Matlab mex-files, which are dyna mic link libraries of compiled C or C++ code. ADOL-C was implemented into the C ++ source code by the following steps:

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42 1. Mark the beginning and end of active section (portion computing dependent variables from independent vari ables) using built-in functions trace_on and trace_off respectively. 2. Select a set of active variable s (those considered differen tiable at some point in the program execution) and change t ype from double to built-in type adouble 3. Define a set of independent variable s using the output stream operator ( <<= ). 4. Define a set of dependent variables using the input stream operator ( >>= ). 5. Call the built-in driver function jacobian to compute first-order derivatives using reverse mode AD. 6. Compile the code includi ng built-in header file adolc.h and linking with built-in 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 computati on time performance was compared. Results For each problem, performance comparisons of optimizing with and without AD are summarized in Table 4-1 and Table 4-2 The use of AD increased the computation time per objective function eval uation. However, the numbe r of function evaluations necessary per optimization itera tion 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.

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43 Discussion The main motivation for investigatin g the use of AD for biomechanical optimizations was to improve computationa l speed of obtaining solutions. Speed improvement for the movement prediction optimization in particular was not as significant as anticipated. Fu rther investigation is necessary to determine the effect of AD characteristics such as forward m ode vs. reverse mode and source code transformation vs. operator overloading on co mputational speed. Whichever AD method is used, having analytical derivatives elim inates inaccurate search directions and sensitivity to design variable scaling which can plague optimizations that use finite difference gradients. If central (more accura te) 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 trade-offs between those approaches and AD require further investigation. While AD-based analytical derivatives may be less efficien t computationally than those derived using special dynamics formulations, AD provide s effortless updating of the derivative calculations should the biomechanical model used in the optimization be changed. For large-scale problems, AD provides a re latively simple means for computing analytical derivatives to improve the sp eed of biomechanical optimizations.

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44 4-1. Performance results for system identification problem for a 3D kinematic ankle joint model involving 252 design variables and 1800 objective function elements. Performance Criteria Without AD With AD Time per Function Evaluation (s) 0.0189 0.978 Number of Function Evaluations per Optimization Iteration 252 1 Time per Optimization Iteration (s) 4.77 0.978 4-2. Performance results for movement prediction problem for a 3D full-body 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 Table Table

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45 CHAPTER 5 APPLICATION OF PATIENT-SPECIFIC DYNAMIC MODELS TO PREDICT FUNCTIONAL OUTCOMES Background Imagine the benefit to the h ealthcare provider and, more importantly, the patient, if certain clinical parameters may be evalua ted pre-treatment in order to predict the post-treatment outcome. For example, a pa tient-specific dynamic model may be useful for planning intended surgical parameters and predicting the outcome of HTO. Researchers have identified th e 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 chos en post-surgery knee adduction torque. There are a few gait studies evaluating effects of kinematic changes on the knee adduction torque. One study examined the effect s of increased foot progression angle, or toeout gait, on minimizing knee adduction torq ue during walking, stai r ascent, and stair descent (Manal and Guo, 2005 ). Another study investigated the minimization of knee adduction torque by changing lower-body kinema tics without changing foot path during gait (Fregly et al. 2005 ). No computational framework is available to perform a virtual HTO surgery and predict the resu lting knee adduction torque. This study presents the development and application of an inverse dynamics optimization approach to predict novel pa tient-specific gait motions, such as gait

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46 following HTO surgery and the resulting intern al knee abduction torque. To further tailor the kinematic and kinetic mode l developed in Chapter 2, response surface (RS) methods are used to determine weights for a cost f unction specific to the patient and movement task, comprising phase three of the optimizati on 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 op timizations of gait. To increase the speed and robustness of optimizati ons, automatic differentiation (AD) is used to compute analytic derivatives of the inverse dy namics optimization objective function. Methods Experimental kinematic and kinetic gait da ta were collected fr om a single subject using a video-based 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 patient-specific m odel parameters as described in Chapter 2. Toeout, or increased foot progression angle, gait data was gathered to calibrate the patient-specific control parameters. Wide st ance gait data was collec ted to evaluate the calibrated model. A parametric 3D, 27 degree-of-freedom (DOF), full-body gait model (Figure 2-1 Table 2-1 ) was constructed, whose equations of mo tion were derived with the symbolic manipulation software, Autolev (OnLine Dyna mics, Sunnyvale, CA). The pelvis was connected to ground via a 6 DOF joint and th e remaining 13 segments comprised four open chains branching from the pelvis. An atomic 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

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47 parameters were identified via optimizati on as described previously in Chapter 2 (Reinbolt and Fregly, 2005 ). The inverse dynamics optimization pro cedure adjusted model motion and ground reactions to minimize errors be tween model and experimental, or prescribed, quantities to be tracked (Equation 5-1 ). Specifically, the design vari ables were 660 B-spline 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 B-spline node was chosen to be the corresponding value from normal gait. The inverse dynamics optimization cost function ( epredict outcome) minimized the weighted (Equation 5-2 Equation 5-3 Table 5-1 ) 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 tr anslations for pelvis ( qpelvis), 3 DOFs for trunk orientation ( qtrunk), 2 transverse plane translat ions for center of pressure for each foot ( CoPfoot), 3 residual pelvis forces ( Fpelvis), 3 residual pelvis torques ( Tpelvis), and leg joint torques ( Tleg). The inverse dynamics optimizations were performed with Matlabs nonlinear least squares algor ithm (The Mathworks, Natick, MA) using analytical Jacobians computed by AD. nf i jk ijk leg j ij pelvis pelvis jk ijk foot j ij trunk j ij pelvis jk ijk foot outcome predictT T F w CoP w q w q w q w e1 6 1 2 1 2 3 1 2 2 5 2 1 2 1 2 4 3 1 2 3 2 1 2 2 2 1 6 1 2 1) ( ) ( ) ( ) ( ) ( ) ( minCP r q,p 11 10 9 8 7 6w w w w w w CPp (5-1) (5-2)

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48 inversion subtalar on dorsiflexi talocrural flexion knee rotation hip adduction hip flexion hip legT w T w T w T w T w T w T 11 10 9 8 7 6 ) (CPp Before predicting outcomes with the i nverse dynamics optimization, the phase 3 optimization calibrated the 6 control parameters, or w6 w11 (Equation 5-2 Equation 5-3 Table 5-1 ), to the patients 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 5-2 ) were defined to approximate the pr edicted toeout foot path without using the exact foot path. All other tracked quantities involved the experimentally measured normal gait motion. A computationa lly efficient optimization was developed using RS methods to determ ine how well the simulated kn ee abduction torque matched the experimental knee abducti on 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 quadr atic RS was created for each time frame of torque data (125 response surf aces) predicted by the model where the 6 control parameter weights were treated as desi gn variables. Although sepa rate response surface were constructed, the correlation coefficients betw een them was not computed. Each quadratic RS required solution of 28 unknown pol ynomial 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 f it in Matlab (The Mathworks, Natick, MA). Using the response surfaces as surrogates for repeated inve rse dynamic optimizations, an optimization problem was formulated to minimize errors between model predicted knee abduction torque and experimentally realized abducti on torque for toeout gait. A Hammersley (5-3)

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49 Quasirandom Sequence was used to generate 1000 initial seed point s 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 pred icting wide stance gait. Similar to toeout gait, fixed offsets (Table 5-2 ) were defined for 3 DOFs to approximate the predicted wide stance foot path without us ing the exact foot pat h. 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 pred ict the outcome of HTO surgery. All tracked quantities i nvolved the experimentally meas ured normal gait motion, including the prescribed foot path (e .g., all foot path offsets equa led zero). The virtual HTO surgery was performed as a late ral opening wedge osteotomy. The anterior-posterior axis of rotation was located 10 cm inferior a nd 5 cm lateral to the midpoint of the transepicondylar axis. The sensitivity of the knee abducti on torque to the extent, or wedge angle, of the virtual HTO surgery wa s illustrated in for 3, 5, and 7. To demonstrate the effects of prior ca librating optimizations which determined joint parameters (phase one from Chapter 2) and inertial parameters (phase two from Chapter 2), the entire proce ss of creation and applicati on of patient-specific 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 kn ee abduction torque results were compared.

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50 Results Using the response surfaces as surro gates for repeated inverse dynamic optimizations using the model calibrated with phases one, two, and three, the optimization determined control parameters (Table 5-3 ) that minimized errors between model predicted knee abduction to rque and experimentally obta ined 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 5-1 Table 5-4 ). Changes for hip torques were matched we ll (RMS = 0.48 %BW*H) with the exception a minor deviation (2.82 %BW*H) in the hip flex ion-extension torque at the beginning of the cycle. Knee flexion-extension torque ch anges 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 sma ll changes for toeout gait (RMS = 0.48 %BW*H). Overall changes for hip joint a ngles were matched (RMS = 4.11) with the exception of internal-external rotation angle (max = 9.54 at 7% of cycle). The knee flexion-extension angle remained more flex ed than necessary (RMS = 2.69, max = 3.88 at 16% of cycle). Both a nkle joint angles generally ma tched the changes well (RMS = 3.58), but deviations were seen near 60% of cycle with a maximum of 9.67. Other tracked quantities demons trated good agreement (Table 5-5 ). All kinematic and kinetic differences are summarized in Table 5-7 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 5-2 Table 5-4 ). Changes for hip torques were matched well (RMS = 0.46 %BW*H) with the exception a minor deviations (1.54 %BW*H at 4%

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51 of cycle and 3.27 %BW*H at 100% of cycle) in the hip flexion-exte nsion torque. Knee flexion-extension torque changes were matched with an RMS difference of 0.39 %BW*H. Both ankle torques remained clos e to the initial normal gait values (RMS = 0.008 %BW*H) without matching the small ch anges for wide stance gait (RMS = 0.36 %BW*H). Overall changes for hip joint a ngles were matched (RMS = 3.13) with the maximum difference (7.81 %BW*H) occu rring at 86% of cycle for the hip flexion-extension angle. The knee flexion-ex tension angle remained more extended than necessary (RMS = 3.06, max = 3.31 at 76% of cycle, or 2nd peak). Both ankle joint angles generally matched the changes satisfact orily (RMS = 5.18), but deviations were demonstrated near the end of cycle for both talocrural (max = 11.51 at 90% of cycle) and subtalar angles (max = 11.42 at 100% of cycle). Other tracked quantities demonstrated good agreement (Table 5-6 ). All kinematic and kinetic differences are summarized in Table 5-8 There were noticeable effects when usi ng dynamic models crea ted without phase one or phase two of the optimization procedur e. The resulting cont rol parameters from phase three varied widely (Table 5-3 ). The predicted knee ab duction torque for toeout gait showed considerable differences using models without phase one included, while the model prediction including phases one and th ree was comparable to the model with phases one, two, and three (Figure 5-1 Table 5-4 ). 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 th ree showed some improvement (Figure 5-2 Table 5-4 ). In all cases, tracked quanti ties demonstrated good agreement (Table 5-5 Table 5-6 ). Concerning the virtual HTO, the sensitiv ities to wedge angle showed the expected

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52 incremental reductions in knee abduction to rque for only those models including phases one and three or phases one, two, and three (Figure 5-3 ). The differences in dynamic models created from different portions of the multi-phase optimization approach led to variations in the computed knee abduction to rques for experimental and simulated data (Figure 5-4 Figure 5-5 ). Discussion The main motivation for developing a patient-specific inverse dynamics optimization approach is to predict novel pa tient-specific gait motions (e.g., subtle gait changes and HTO post-surgery gait). Toe out gait was successfully used to tune patient-specific control parameters. Wide stance gait predictions match the experiment rather well. The response surface approach gr eatly increased the computation efficiency of determining the control parameters. The current approach has seve ral limitations. A pin join t knee model was chosen to represent the flexion-extensi on 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 ki netic quantities were not predicted as well as others. Selection of tracking term weights is subjec tive and was performed in a manner to constrain the foot paths alone a nd track the other quan tities. The control parameters identified during phase 3 depend on th e 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 expl anation tied to neural control strategies. It cannot be claimed that predictions ma de with this multi-phase approach will reproduce the actual func tional outcome for every patient a nd every data set. This is

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53 clear from the calibration results for toeout gait and the evaluation results for wide stance gait, where the RMS errors in the knee abducti on 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 structur e provides the best pos sible 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 5-3 ). For the 3 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 3 of change. For the 5 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 5 of change. For the 7 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 7 of change. Cost function weights calibra ted 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 patient-specific m odels and predictions for a population of HTO patients pre-surgery and compare with post-su rgery outcome. Examination of different gaits (e.g., normal and pathological ) requires future study as well.

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54 5-1. Descriptions of cost function we ights and phase 3 control parameters. Weight Description w1 Scale factor for tracking foot paths w2 Scale factor for tracking pelvis transverse plane translations w3 Scale factor for track ing trunk orientations w4 Scale factor for tracking cente r of pressure translations w5 Scale factor for tracking resi dual pelvis forces and torques w6 Control parameter for tracking hip flexion-extension torque w7 Control parameter for tracking hip adduction-abduction torque w8 Control parameter for tracking hip internal-external torque w9 Control parameter for tracking knee flexion-extension torque w10 Control parameter for tracking talocrur al dorsiflexion-plantarflexion torque w11 Control parameter for tracking s ubtalar inversion-eversion torque 5-2. Summary of fixed offsets added to no rmal 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 (o) Left 13.12 -0.29 Table Table

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55 5-3. Comparison of cost function weig hts and phase 3 control parameters. Weight Phase 3 only Phases 1 and 3 Phases 2 and 3 Phases 1, 2, and 3 w1 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 5.79e-01 8.13 5.17 w7 8.67 2.23 6.84 3.57e-01 w8 7.31e-08 9.29e-10 7.18e-01 1.08 w9 9.14 1.62e-07 6.74 2.94e-01 w10 2.61 8.18 6.87 5.55 w11 4.24 9.63 9.18 9.17 5-4. Summary of root-mean-square (RMS) e rrors for predicted left knee abduction torque quantities for toeo ut and wide stance gait. RMS error (% BW*H) Gait trial Phase 3 only Phases 1 and 3Phases 2 and 3 Phases 1, 2, and 3 Toeout 0.66 0.13 0.69 0.14 Wide stance 0.68 0.37 0.68 0.21 Table Table

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56 5-5 Summary of root-mean-square (RMS) e rrors for tracked quantities for toeout gait. RMS error Quantity Description Phase 3 only Phases 1 and 3 Phases 2 and 3 Phases 1, 2, and 3 ( qfoot)right 6 DOFs for right prescribed foot path 2.55 mm 1.58 3.64 mm 2.03 1.53 mm 0.96 2.35 mm 1.56 ( qfoot)left 6 DOFs for left prescribed foot path 0.94 mm 1.27 2.02 mm 1.38 0.57 mm 0.86 1.36 mm 0.97 qpelvis 2 transverse plane translations for pelvis 7.65 mm 5.14 mm 9.91 mm 6.61 mm qtrunk 3 DOFs for trunk orientation 0.32 0.36 0.52 0.38 ( CoPfoot)right 2 transverse plane translations for right center of pressure 9.68 mm 9.78 mm 5.68 mm 7.30 mm ( CoPfoot)left 2 transverse plane translations for left center of pressure 2.01 mm 4.17 mm 1.68 mm 2.73 mm Fpelvis 3 residual pelvis forces 4.23 N 4.27 N 3.82 N 4.11 N Tpelvis 3 residual pelvis torques 3.93 Nm 3.37 Nm 2.99 Nm 2.41 Nm Table

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57 5-6. Summary of root-mean-square (RMS) e rrors for tracked quantities for wide stance gait. RMS error Quantity Description Phase 3 only Phases 1 and 3 Phases 2 and 3 Phases 1, 2, and 3 ( qfoot)right 6 DOFs for right prescribed foot path 1.07 mm 0.99 1.19 mm 1.36 0.85 mm 1.09 0.45 mm 0.37 ( qfoot)left 6 DOFs for left prescribed foot path 0.99 mm 1.04 2.07 mm 1.29 0.60 mm 1.16 0.32 mm 0.24 qpelvis 2 transverse plane translations for pelvis 6.47 mm 6.42 mm 6.22 mm 7.33 mm qtrunk 3 DOFs for trunk orientation 0.78 0.83 0.84 0.75 ( CoPfoot)right 2 transverse plane translations for right center of pressure 4.45 mm 5.73 mm 4.74 mm 4.40 mm ( CoPfoot)left 2 transverse plane translations for left center of pressure 1.80 mm 1.74 mm 3.33 mm 0.79 mm Fpelvis 3 residual pelvis forces 3.56 N 3.85 N 3.65 N 3.77 N Tpelvis 3 residual pelvis torques 3.17 Nm 2.73 Nm 3.50 Nm 2.48 Nm Table

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58 5-7. Summary of root-mean-square (RMS) differences between optimi zation results and original normal gait or final toeout gait kinematic and kinetic quantities. RMS difference from optimization Phase 3 only Phases 1 and 3 Phases 2 and 3 Phases 1, 2, and 3 Quanitity Original Final Original Final Original Final Original Final Hip flexion angle () 1.29 3.22 9.14e-01 3.72 2. 09 2.79 1.08 3.46 Hip abduction angle () 1.73 2. 55 2.20 2.27 2. 37 2.39 2.63 2.32 Hip rotation angle () 10.96 5. 39 10.94 5.71 11. 03 5.35 11.27 5.76 Knee flexion angle () 1.61 2.68 1.03 2.76 1.62 2.52 8.93e-01 2.69 Ankle flexion angle () 1.85 3. 21 2.05 2.85 1. 53 2.85 1.78 2.58 Ankle inversion angle () 4.43 4. 71 3.89 4.49 4. 14 4.87 3.49 4.36 Anterior center of pressure (mm) 11.35 15.09 2.45 9.89 1.29 9.97 1.58 9.33 Lateral center of pressure (mm) 22.41 23.35 5.36 9.27 1.99 7.09 3.53 7.86 Hip flexion torque (% BW*H) 2.49e-02 1.55 4.05e-01 9.39e-01 2.15e02 1.55 3.11e-02 7.01e-01 Hip abduction torque (% BW*H) 1.70e-02 7.46e01 9.58e-02 3.43e-01 2.43e02 7.50e-01 2.31e-01 4.21e-01 Hip rotation torque (% BW*H) 1.62e-01 1.76e-01 1.42e-01 1.99e-01 1.12e-01 1.80e-01 1.10e-01 1.83e-01 Knee flexion torque (% BW*H) 1.67e-02 2.98e-01 3.75e-01 2.92e-01 1.50e-02 2.92e-01 2.41e-01 2.21e-01 Ankle flexion torque (% BW*H) 5.09e-02 1. 27 3.24e-02 5.49e-01 1.41e02 1.29 1.60e-02 5.53e-01 Ankle inversion torque (% BW*H) 3.92e-02 2. 32 7.07e-02 3.98e-01 1.79e02 2.31 2.98e-02 3.86e-01 Anterior ground reaction force (% BW) 2. 33e-01 1.26 2.65e-01 1.31 5.16e-01 1.32 2.51e-01 1.39 Superior ground reaction force (% BW ) 1.73 3.49 1.29 3.60 1.52 3.95 7.07e-01 4.04 Lateral ground reaction force (% BW) 6.98e-01 3.65e-01 5.43e-01 2.78e-01 7. 58e-01 3.42e-01 5.64e-01 3.28e-01 Table

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59 5-8. Summary of root-mean-square (RMS) differences between opt imization results and original normal gait or final wide stance gait kinematic and kinetic quantities. RMS difference from optimization Phase 3 only Phases 1 and 3 Phases 2 and 3 Phases 1, 2, and 3 Quanitity Original Final Original Final Original Final Original Final Hip flexion angle () 6.64e-01 4.39 5.50e-01 4.55 3.68e01 4.29 1.66 4.17 Hip abduction angle () 3.07 1. 73 3.59 1.69 3. 21 1.73 3.21 1.98 Hip rotation angle () 2.43 2. 62 1.74 2.82 1. 91 2.58 2.49 3.18 Knee flexion angle () 6.75e-01 3. 46 1.48 4.41 6.28e01 3.48 1.12 2.79 Ankle flexion angle () 2.17 4. 35 2.19 5.18 1. 91 4.33 2.08 5.17 Ankle inversion angle () 6.82 6. 43 4.06 4.07 5. 87 5.53 5.00 4.72 Anterior center of pressure (mm) 1.44 27.40 1.49 26.01 4.45 27.52 1.05 26.31 Lateral center of pressure (mm) 2. 10 11.22 1.95 9.13 1. 54 11.22 3.68e-01 8.18 Hip flexion torque (% BW*H) 2.33e-02 5.04e-01 1.71e-01 5.95e-01 2.43e-02 5.25e-01 1.58e-02 6.26e-01 Hip abduction torque (% BW*H) 3.82e-02 5.25e01 1.29e-01 3.39e-01 6.54e02 5.21e-01 2.53e-01 3.23e-01 Hip rotation torque (% BW*H) 1.93e-01 2.44e-01 2.10e-01 2.87e-01 2.00e-01 2.49e-01 9.11e-02 2.48e-01 Knee flexion torque (% BW*H) 1.28e-02 4.27e-01 2.49e-01 6.24e-01 1.46e-02 4.37e-01 2.79e-01 3.90e-01 Ankle flexion torque (% BW*H) 1.85e-02 3.74e01 1.15e-02 3.91e-01 1.01e02 3.71e-01 1.07e-02 3.93e-01 Ankle inversion torque (% BW*H) 3.16e-02 4.23e01 2.53e-02 3.45e-01 1.62e02 4.23e-01 2.39e-02 3.63e-01 Anterior ground reaction force (% BW) 2. 76e-01 1.13 3.94e-01 1.20 2.81e-01 1.17 6.76e-01 1.04 Superior ground reaction force (% BW ) 1.03 4.15 8.09e-01 3.98 1.23 4.30 2.13 3.93 Lateral ground reaction force (% BW) 2.66 8.75e-01 2.35 1.14 2.56 8.72e-01 1.75 1.79 Table

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60 5-1. Comparison of left knee abduction torques for toeout gait. Original (blue) is e xperimental normal gait, simulation (red) is predicted toeout gait, and final (green) is experimental toeout gait. Plot A) cont ains curves for model using phase 3 only. Plot B) contains curves for model using phases 1 and 3. Plot C) cont ains curves for model using phases 2 and 3. Plot D) contains curves for model using phases 1, 2, and 3. Figure

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61 5-2. Comparison of left knee abduction torque s for wide stance gait, where original (blue) is experimental normal gait, simulation (red) is predicted toeout gait, a nd final (green) is experimental toeout ga it. Plot A) contai ns curves for model using phase 3 only. Plot B) c ontains curves for model using phases 1 and 3. Plot C) contains curves for model using phases 2 and 3. Plot D) contains cu rves for model using phases 1, 2, and 3. Figure

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62 5-3. Comparison of left knee abduction torque s for simulated high tibial osteotomy (HTO ) post-surgery 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. Figure

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63 5-4. Comparison of mean (solid black line ) plus or minus one standard deviation (gray shaded area) for experimental le ft knee abduction torques. Plot A) contains distribution for nor mal gait. Plot B) contai ns distribution for toeout gait. Plot C) contains dist ribution for wide stance gait. Figure

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64 5-5. Comparison of mean (solid black line ) plus or minus one standard deviation (gray shaded area) for simulated left kn ee abduction torques. Plot A) contains distribution for toeout gait. Plot B) contains distribu tion for wide stance gait. Figure

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65 CHAPTER 6 CONCLUSION Rationale for New Approach The main motivation for developing a 27 DOF patient-specific inverse dynamic model and the associated optimization appro ach is to predict the post-surgery peak external knee adduction moment in HTO patie nts, 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 determin ed 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 cr eating a predictive patient-specific dynamic simulation. Synthesis of Current Work and Literature The multi-phase optimization method satisfactorily determines patient-specific model parameters defining a 3D model that is well suited to a part icular patient. Two conclusions may be drawn from comparing an d 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 optim ization is necessary and demonstrate the degree of inaccuracy inherent when the new approach is not implemented. Through the

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66 enhancement of model parameter values found in the literature, the multi-phase optimization approach successfully reduces the fitness errors between the patient-specific model and the experimental motion data. The precision of dynamic analyses made fo r a particular patient depends on the accuracy of the patient-specific kinematic a nd kinetic parameters chosen for the dynamic model. Without expensive medical images, model parameters are only estimated from external landmarks that have been identifie d in previous studies. The estimated (or nominal) values may be improved by form ulating an optimization problem using motion-capture data. By using a multi-phase 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 optim ization subject to reality constraints predict physics-based motion. The results accura tely represent a spec ific patient within the bounds of the chosen dynamic model, experi mental task, and data quality. Given the agreement of the current results with previous studies, there is significant benefit in the construction and optimization a patient-specific functional dynamic mode l to assist in the planning and outcome prediction of surgical procedures. This work comprises a unique computational framework to create and apply objective patient-specific models (different from current intuitive models based on clin ical experience or regr ession models based on population studies) to predict clin ically significant outcomes.

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67 GLOSSARY Abduction Movement away from the midline of the body in the coronal plane. Acceleration The time rate of change of velocity. Active markers Joint and segmen t markers used during motion capture that emit a signal. Adduction Movement towards the midline of the body in the coronal plane. Ankle inversion-eversion Motion of th e long axis of the foot within the coronal plane as seen by an observer positioned along the anterior-posterior axis of the shank. Ankle motion The ankle angles reflect the motion of the foot segment relative to the shank segment. Ankle plantarflexion-dorsiflexi on Motion of the plantar asp ect of the foot within the sagittal plane as seen by an observer positioned along the medial-lateral axis of the shank. Anterior The front or before, also referred to as ventral. Coccyx The tailbone located at the distal end of the sacrum. Constraint functions Specific limits th at must be satisfied by the optimal design. Coronal plane The plane that divides the body or body segment into anterior and posterior parts. Degree of freedom (DOF) A single coordi nate of relative motion between two bodies. Such a coordinate responds without constraint or imposed moti on to externally applied forces or torques. For translational motion, a DOF is a linear coordinate al ong a single direction. For rotational motion, a DOF is an angular coordinate about a single, fixed axis.

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68 Design variables Variables that change to optimize the design. Distal Away from the point of attachment or origin. Dorsiflexion Movement of the foot towards the anterior part of the tibia in the sagittal plane. Epicondyle Process that develops proximal to an articulation and provides additional su rface area for muscle attachment. Eversion A turning outward. Extension Movement that rotates the bones comprising a joint away from each other in the sagittal plane. External (lateral) rotation Movement that rotates the distal segment laterally in relation to the proximal segment in the transverse plane, or places the anterior surface of a segment away from the longitudinal axis of the body. External moment The load applied to the human body due to the ground reaction forces, gravity and external forces. Femur The longest and heaviest bone in the body. It is located between the hip joint and the knee joint. Flexion Movement that rotates the bones comprising a joint towards each other in the sagittal plane. Force A push or a pull and is produced when one object acts on another. Force plate A transducer that is set in the floor to measure about some specified point, the force and torque applied by the foot to the ground. These devices provide measures of the three co mponents of the resultant ground reaction force vector and the three components of the resultant torque vector. Forward dynamics Analysis to dete rmine 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.

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69 Gait A manner of walk ing or moving on foot. Generalized coordinates A set of coordinates (or pa rameters) that uniquely describes the geometric pos ition and orientation of a body or system of bodies. Any set of coordinates that are used to describe the motion of a physical system. High tibial osteotomy (HTO) Surgical procedure that involves adding or removing a wedge of bone to or from the tibia and changing the frontal plane limb alignment. The realignment shifts the weight-bearing axis from the diseased medial compartment to the healthy lateral compartment of the knee. Hip abduction-adduction Motion of a long ax is of the thigh within the coronal plane as seen by an observer positioned along the anterior-posterior axis of the pelvis. Hip flexion-extension Motion of the long axis of the thigh within the sagittal plane as seen by an observer positioned along the medial-lateral axis of the pelvis. Hip internal-external rotation Motion of the medial-lateral axis of the thigh with respect to the medial-lateral axis of the pelvis within the transverse plane as seen by an observer positioned along the longitudinal axis of the thigh. Hip motion The hip angles refl ect the motion of the thigh segment relative to the pelvis. Inferior Below or at a lower level (towards the feet). Inter-ASIS distance The length of m easure between the left anterior superior iliac spine (ASIS) and the right ASIS. Internal (medial) rotation Movement that rotates the distal segment medially in relation to the proximal segment in the transverse plane, or places the anterior surface of a segment towards the longitudinal axis of the body. Internal joint moments The net result of all the internal forces acting about the joint which include moments due to muscles, ligaments, joint friction and structural constraints.

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70 The joint moment is usually calculated around a joint center. Inverse dynamics Analysis to de termine the forces and torques necessary to produce the motion of a mechanical system, given the topology of how bodies are connected, the kinematics, the mass properties, and the initial condition of all degrees of freedom. Inversion A turning inward. Kinematics Those parameters that are used in the description of movement without consider ation for the cause of movement abnormalities. These typically include parameters such as linear and angular displacements, velocities and accelerations. Kinetics General term given to the forces that cause movement. Both internal (muscle activity, ligaments or friction in muscles and joints) and external (ground or external loads) forces are included. The moment of force produced by muscles crossing a joint, the mechanical power flowing to and from those same muscles, and the energy changes of the body that result from this power flow are the most common kinetic parameters used. Knee abduction-adduction Motion of the l ong axis of the shank within the coronal plane as seen by an observer positioned along the anterior-posterior axis of the thigh. Knee flexion-extension Motion of the long axis of the shank within the sagittal plane as seen by an observer positioned along the medial-lateral axis of the thigh. Knee internal-external rotation Motion of the medial-lateral axis of the shank with respect to the medial-lateral axis of the thigh within the transverse plane as viewed by an observer positioned along the longitudinal axis of the shank. Knee motion The knee angles reflect the motion of the shank segment relative to the thigh segment. Lateral Away from the bodys longitudinal axis, or away from the midsagittal plane.

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71 Malleolus Broadened distal por tion of the tibia and fibula providing lateral stability to the ankle. Markers Active or passive objects (balls, hemispheres or disks) aligned with respect to specific bony landmarks used to help determine segment and joint position in motion capture. Medial Toward the bodys longit udinal axis, or toward the midsagittal plane. Midsagittal plane The plane that passes through the midline and divides the body or body segm ent into the right and left halves. Model parameters A set of coordi nates that uniquely describes the model segments lengths, joint locations, and joint orientations, also referred to as joint parameters. Any set of coordinates that are used to describe the geometry of a model system. Moment of force The moment of fo rce is calculated about a point and is the cross product of a position vector from the point to the line of action for the force and the force. In two-dimensions, the mo ment of force about a point is the product of a fo rce and the perpendicular distance from the line of action of the force to the point. Typically, moments of force are calculated about the center of rotation of a joint. Motion capture Interpretation of co mputerized data that documents an individual's motion. Objective functions Figures of mer it to be minimized or maximized. Parametric Of or relating to or in terms of parameters, or factors that define a system. Passive markers Joint and segm ent markers used during motion capture that reflect visible or infrared light. Pelvis Consists of the two hip bones, the sacrum, and the coccyx. It is located between the proximal spine and the hip joints.

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72 Pelvis anterior-posterior tilt Motion of the long axis of the pelvis within the sagittal plane as seen by an observer positioned along the medial-lateral axis of the laboratory. Pelvis elevation-depression Motion of th e medial-lateral axis of the pelvis within the coronal plane as seen by an observer positioned along the anterior-posterior axis of the laboratory. Pelvis internal-external rotation Motion of the medial-lateral or anterior-posterior axis of the pelvis within th e transverse plane as seen by an observer positioned along the longitudinal axis of the laboratory. Pelvis motion The position of the pelvis as defined by a marker set (for example, plane formed by the markers on the right and left anterior su perior iliac spine (ASIS) and a marker between the 5th lumbar vertebrae and the sacrum) relative to a laboratory coordinate system. Plantarflexion Movement of the foot away from the anterior part of the tibia in the sagittal plane. Posterior The back or behind, also referred to as dorsal. Proximal Toward the point of attachment or origin. Range of motion Indicates joint mo tion excursion from the maximum angle to the minimum angle. Sacrum Consists of the fused components of five sacral vertebrae located between the 5th lumbar vertebra and the coccyx. It attaches the axial skeleton to the pelvic girdle of the appendicular skeleton via paired articulations. Sagittal plane The plane that divides the body or body segment into the right and left parts. Skin movement artifacts The rela tive movement between skin and underlying bone. Stance phase The period of time when the foot is in contact with the ground.

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73 Subtalar joint Located between the distal talus and proximal calcaneous, also known as th e talocalcaneal joint. Superior Above or at a highe r level (towards the head). Synthetic markers Computational re presentations of passive markers located on the kinematic model. Swing phase The period of time when the foot is not in contact with the ground. Talocrural joint Located between th e distal tibia and proximal talus, also known as the tibial-talar joint. Talus The largest bone of the ankle transmitting weight from the tibia to the rest of the foot. Tibia The large medial bone of the lower leg, also known as the shinbone. It is located between the knee joint and the talocrural joint. Transepicondylar The line betw een the medial and lateral epicondyles. Transverse plane The plane at right angles to the coronal and sagittal planes that divides the body into superior and inferior parts. Velocity The time rate of change of displacement.

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74 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 of Biomechanics Volume 29, Number 5, Pages 651-654. Andriacchi, T.P., 1994. Dynamics of Knee Malalignment. Orthopedic Clinics of North America Volume 25, Number 3, Pages 395-403. Andriacchi, T.P., and Strickland, A.B., 1985. G ait 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 83-102. 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 181-190. Arnold, A.S., and Delp, S.L., 2001. Rotationa l Moment Arms of the Hamstrings and Adductors Vary with Femoral Geometry and Limb Position: Implications for the Treatment of Internally-Rotated Gait. Journal of Biomechanics Volume 34, Number 4, Pages 437-447. Bell, A.L., Pedersen, D.R., and Brand, R.A., 1990. A Comparison of the Accuracy of Several Hip Center Locati on Prediction Methods. Journal of Biomechanics Volume 23, Number 6, Pages 617-621. Bogert, A.J. van den, Smith, G.D., and Nigg, B.M., 1994. In Vivo Determination of the Anatomical Axes of the Ankle Joint Co mplex: An Optimization Approach. Journal of Biomechanics Volume 27, Number 12, Pages 1477-1488. Bryan, J.M., Hurwitz, D.E., Bach, B.R., Bitta r, T., and Andriacchi, T.P., 1997. A Predictive Model of Outcome in High Tibial Osteotomy. In Proceedings of the 43rd Annual Meeting of the Or thopaedic Research Society San Francisco, California, February 9-13, 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 Socie ty of Biomechanics Paris, France, July 4-8, Pages 238-239.

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75 Cappozzo, A., Leo, T., and Pedotti, A., 1975. A General Computing Method for the Analysis of Human Locomotion. Journal of Biomechanics Volume 8, Number 5, Pages 307-320. Center for Disease Control (CDC), 2003. Targeting Arthritis: The Nations Leading Cause of Disability Centers for Disease Control and Prevention, National Center for Chronic Disease Prevention and Hea lth 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 219-231. 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 of Biomechanical Engineering Volume 121, Number 3, Pages 304-310. Chao, E.Y., and Sim, F.H., 1995. Compute r-Aided Pre-Operative Planning in Knee Osteotomy. Iowa Orthopedic Journal Volume 15, Pages 4-18. Chao, E.Y.S., Lynch, J.D., and Vanderploeg, M.J., 1993. Simulation and Animation of Musculoskeletal Joint System. Journal of Biomechanical Engineering Volume 115, Number 4, Pages 562-568. Chze, L., Fregly, B.J., and Dimnet, J., 1995. A Solidification Procedure to Facilitate Kinematic Analyses Based on Video System Data. Journal of Biomechanics Volume 28, Number 7, Pages 879-884. Chze, 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 1-15. 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 Orthopaedics and Related Research Volume 356, Number 1, Pages 111-118. Croce, U.D., Leardini, A., Lorenzo, C., Cappozzo, A., 2005. Human Movement Analysis Using Stereophotogrammetry Pa rt 4: Assessment of Anatomical Landmark Misplacement and Its Effects on Joint Kinematics. Gait and Posture Volume 21, Number 2, Pages 226-237. Davis, B.L., 1992. Uncertainty in Calcul ating Joint Moments During Gait. In Proceedings of the 8th Meeting of European Society of Biomechanics Rome, Italy, June 21-24, Page 276.

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76 de Leva, P., 1996. Adjustments to Zatsiorsky-Seluyanovs Segment Inertia Parameters. Journal of Biomechanics Volume 29, Number 9, Pages 1223-1230. Delp, S.L., Arnold, A.S., and Piazza, S.J., 1998. Graphics-Based Modeling and Analysis of Gait Abnormalities. Bio-Medical Materials and Engineering Volume 8, Number 3/4, Pages 227-240. Delp, S.L., Arnold, A.S., Speers, R.A., and Moore, C.A., 1996. Hamstrings and Psoas Lengths During Normal and Crouch Gait: Implications for Muscle-Tendon Surgery. Journal of Orthopaedic Research Volume 14, Number 1, Pages 144-151. Delp, S.L., Loan, J.P., Hoy, M.G., Zajac, F.E., Topp E.L., and Rosen, J.M., 1990. An Interactive Graphics-Based Model of th e Lower Extremity to Study Orthopaedic Surgical Procedures. IEEE Transactions on Biomedical Engineering Volume 37, Number 8, Pages 757-767. Fantozzi, S., Stagni, R., Ca ppello, 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 537-541 Fishman, G.S., 1996. Monte Carlo: Con cepts, Algorithms, and Applications. Springer-Verlag 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 of Biom echanics and 29th Annual Meeting of the American Society of Biomechanics Cleveland, Ohio, July 31-August 5, Page 283. Griewank, A., 2000. Evaluating Derivatives: Pr inciples and Techniques of Algorithmic Differentiation. Society for Industrial and Applied Mathematics, Philadelphia, PA. Griewank, A., Juedes, D., and Utke, J., 1996. ADOL-C: 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 131-167. Heck, D.A., Melfi, C.A., Mamlin, L.A., Ka tz, B.P., Arthur, D.S., Dittus, R.S., and Freund, D.A., 1998. "Revision Rates Following Knee Replacement in the United States." Medical Care Volume 36, Number 5, Pages 661-689. Holden, J.P., and Stanhope, S.J., 1998. The Ef fect of Variation in Knee Center Location Estimates on Net Knee Joint Moments. Gait & Posture Volume 7, Number 1, Pages 1-6.

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77 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 120-121. 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 Mec hanical Engineers, Part D: Journal of Automobile Engineering Volume 215, Number 7, Pages 795-801. Hurwitz, D.E., Sumner, D.R., Andriacchi, T.P., and Sugar, D.A., 1998. Dynamic Knee Loads During Gait Predict Proxima l Tibial Bone Distribution. Journal of Biomechanics Volume 31, Number 5, Pages 423-430. 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 Bilate ral Versus Unilateral To tal Knee Arthroplasty. Outcomes Analysis. Clinical Orthopaedics and Related Research Volume 345, Number 1, Pages 106-112. Leardini, A., Cappozzo, A., Catani, F., Toks vig-Larsen, S., Petitto, A., Sforza, V., Cassanelli, G., and Giannini, S., 1999. V alidation of a Functional Method for the Estimation of Hip Joint Centre Location. Journal of Biomechanics Volume 32, Number 1, Pages 99-103. Lu, T.-W., and OConnor, J.J., 1999. Bone Position Estimation from Skin Marker Coordinates Using Global Optimisat ion with Joint Constraints. Journal of Biomechanics Volume 32, Number 2, Pages 129-134. Manal, K., and Guo, M., 2005. Foot Progression Angle and the Knee Adduction Moment in Individuals with Medi al Knee Osteoarthritis. In Proceedings of the XXth Congress of the International So ciety of Biomechanics and 29th Annual Meeting of the American Society of Biomechanics Cleveland, Ohio, July 31-August 5, Page 899. Pandy, M.G., 2001. Computer Modeling and Simulation of Human Movement. Annual Reviews in Biomedical Engineering Volume 3, Number 1, Pages 245-273. 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 173-183. Prodromos, C.C., Andriacchi, T.P., and Gala nte, J.O., 1985. A Relationship Between Gait and Clinical Changes Followi ng High Tibial Osteotomy. Journal of Bone Joint Surgery (American) Volume 67, Number 8, Pages 1188-1194.

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78 Reinbolt, J.A., and Fregly, B.J., 2005. Cr eation of Patient-Specific Dynamic Models from Three-Dimensional Movement Data Using Optimization. In Proceedings of the 10th International Symposium on Computer Simulation in Biomechanics Cleveland, OH, July 28-30. Reinbolt, J.A., Schutte, J.F., Fregly, B.J., Ha ftka, R.T., George, A.D., and Mitchell, K.H., 2005. Determination of Patient-Specific Multi-Joint Kinematic Models Through Two-Level Optimization. Journal of Biomechanics Volume 38, Number 3, Pages 621-626. Sommer III, H.J., and Miller, N.R., 1980. A Technique for Kinematic Modeling of Anatomical Joints. Journal of Biomechanical Engineering Volume 102, Number 4, Pages 311-317. Stagni, R., Leardini, A., Benedetti, M.G., Cappozzo, A., and Cappello, A., 2000. Effects of Hip Joint Centre Misloc ation on Gait Analysis Results. Journal of Biomechanics Volume 33, Number 11, Pages 1479-1487. Tetsworth, K., and Paley, D., 1994. A ccuracy of Correction of Complex Lower-Extremity Deformities by the Ilizarov Method. Clinical Orthopaedics and Related Research Volume 301, Number 1, Pages 102-110. Valero-Cuevas, 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 Met hod or the Variability/Uncertainty of Musculoskeletal Parameters. Jouirnal of Biomechanics Volume 36, Number 7, Pages 1019-1030. Vaughan, C.L., Andrews, J.G., and Hay, J.G., 1982. Selection of Body Segment Parameters by Optimization Methods. Journal of Biomechanical Engineering Volume 104, Number 1, Pages 38-44. 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 Orthopaedics and Related Research Volume 354, Number 1, Pages 180-188. Wang, J.-W., Kuo, K.N., Andriacchi, T.P., and Galante, J.O., 1990. The Influence of Walking Mechanics and Time on the Resu lts of Proximal Tibial Osteotomy. Journal of Bone and Joint Surgery (American), Volume 72, Number 6, Pages 905-913. Winter, D.A., 1980. Overall Principle of Lower Limb Support During Stance Phase of Gait. Journal of Biomechanics Volume 13, Number 11, Pages 923-927.

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79 BIOGRAPHICAL SKETCH Jeffrey A. Reinbolt was born on May 6, 1974, in Bradenton, Florida. His parents are Charles and Joan Reinbolt. He has an ol der brother, Douglas, and an older sister, Melissa. In 1992, Jeff graduated salutatorian from Southeast High School, Bradenton, Florida. After completing his secondary educ ation, he enrolled at the University of Florida supported by the Florida Undergraduat e Scholarship and full-time employment at a local business. He earned a traditional 5-ye ar engineering degree in only 4 years. In 1996, Jeff graduated with honors receiving a Bach elor of Science degree in engineering science with a concentration in biomedical engineering. He us ed this foundation to assist in the medical device development and clini cal research programs of Computer Motion, Inc., Santa Barbara, California. In this ro le, Jeff was Clinical Development Site Manager for the Southeastern United States and he traveled extensively throughout the United States, Europe, and Asia collabo rating with surgeons and fell ow medical researchers. In 1998, Jeff married Karen, a fellow student he met during his undergraduate studies. After more than 4 years in the medical devi ce industry, he decided to continue his academic career at the Univer sity of Florida. In 2001, Je ff began his graduate studies with a graduate research assistantship in th e 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 throug h the pursuit of a Doctor of Philosophy in mechanical engineering. In 2005, Jeff and Karen were blessed with the

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80 birth of their first child, Jacob. Jeff would like to further his cr eative involvement in problem solving and the design of solutions to overcome healthcare challenges in the future.