DETERMINATION OF PATIENT-SPECIFIC FUNCTIONAL AXES
THROUGH TWO-LEVEL OPTIMIZATION
By
JEFFREY A. REINBOLT
A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
UNIVERSITY OF FLORIDA
2003
Copyright 2003
by
Jeffrey A. Reinbolt
This thesis is dedicated to my loving wife, Karen.
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 and Dr. Roger Tran-Son-Tay for their
dedication, knowledge, and instruction in the classroom. For these reasons, each was
selected to serve on my supervisory committee. I express thanks to both individuals for
their time, contribution, and fulfillment of their committee responsibilities.
I recognize Jaco for his assistance, collaboration, and suggestions. His dedication
and professionalism have allowed my graduate work to be both enjoyable and rewarding.
I collectively show appreciation for my family and friends. Unconditionally, they
have provided me with encouragement, support, and interest in my graduate studies and
research activities.
My wife, Karen, has done more for me than any person could desire. On several
occasions, she has taken a leap of faith with me; more importantly, she has been directly
beside me. Words or actions cannot adequately express my gratefulness and adoration
toward her. I honestly hope that I can provide her as much as she has given to me.
I thank God for my excellent health, inquisitive mind, strong faith, valuable
experiences, encouraging teachers, loving family, supportive friends, and wonderful wife.
TABLE OF CONTENTS
Page
A C K N O W L E D G M E N T S ................................................................................................. iv
TA B LE O F C O N TEN T S................................................................... ......................... v
LIST OF TABLES ........................................................ ............. viii
L IST O F F IG U R E S .......................................................... .... .. ..... .. .. .. ........... xi
A B S T R A C T ...................................................................................................... x iii
CHAPTER
1 IN TR O D U C TIO N ............................................................. .. ...... .. ............
Arthritis: The Nation's Leading Cause of Disability ..........................................1
Need for Accurate Patient-Specific Models ...................................... ............... 2
Benefits of Tw o-Level Optim ization....................................... ......................... 3
2 B A CK G R O U N D ................................................. .................... .... ........
M option C aptu re ................................................................................ 4
B iom mechanical M models .................................. ............................ .. .......... .... ....
K inem atics and D ynam ics .......................................................... ............. 5
O p tim iz atio n .......................................................... ................ 5
Lim stations of Previous M ethods.................................... ....................................... ...5
3 M E T H O D S ...................................... ........... .................... ................ 7
P aram etric M odel Structure ............................................................... .....................7
H ip J o in t ................................................................................................................8
K n e e Jo in t ...................................... ............................... ................ 8
A n k le Joint .................. ................................................. ................ 10
Two-Level Optimization Approach................ ...................................................... 11
Why Two Levels of Optimization Are Necessary ..............................................11
Inner-Level O ptim ization ................................................... ........................ 11
Outer-Level Optim ization ............................................................................12
Two-Level Optimization Evaluation .......................................................................13
Synthetic Marker Data without Noise .... .......... .......................................13
Synthetic M arker Data with N oise ............................ ................................... 13
Experimental M arker Data .................... ................ ........... 14
4 R E S U L T S .....................................................................................................2 9
Synthetic M arker Data without N oise ............................................. ............... 29
Synthetic M arker D ata w ith N oise ........................................ ........................ 29
E xperim ental M arker D ata .............................................................. .....................29
5 D ISC U S SIO N ............... ................................................................................ 36
Assumptions, Limitations, and Future W ork................................... ............... 36
Joint M odel Selection ............... ........... .. ...... ....... ................... 36
Design Variable Constraints........ ........... ... ............. .... ..... .......... 36
Objective Function Formulation.................................................................... 37
Optimization Time and Parallel Computing............................................ 37
M ulti-Cycle and One-Half-Cycle Joint M otions..............................................38
Range of Motion and Loading Conditions .................................39
Optimization Using Gait Motion...... .................................... 39
Comparison of Experimental Results with Literature ............................. .............40
6 CONCLUSION ........... ................ .... .. ... ......... ...... .............. 43
Rationale for New Approach ............... ......... ..................... 43
Synthesis of Current W ork and Literature........................................ ....... ............ 43
G L O S SA R Y ...................................................................4 5
APPENDIX
A NOMINAL JOINT PARAMETERS & OPTIMIZATION BOUNDS FOR
SYN TH ETIC M ARK ER D A TA ................................................... ............... ............52
B NOMINAL JOINT PARAMETERS & OPTIMIZATION BOUNDS FOR
EXPERIMENTAL M ARKER DATA ............................................ ............... 55
C NOMINAL & OPTIMUM JOINT PARAMETERS FOR SYNTHETIC MARKER
D A TA W ITH OU T N OISE .............................................. ..... ........................ 58
D NOMINAL & OPTIMUM JOINT PARAMETERS FOR SYNTHETIC MARKER
D A TA W ITH N O ISE ........................................................................... .............61
E NOMINAL & OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE
EXPERIMENTAL MARKER DATA ............................... .......................... 64
F NOMINAL & OPTIMUM JOINT PARAMETERS FOR FIRST
ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA...............................67
G NOMINAL & OPTIMUM JOINT PARAMETERS FOR SECOND
ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA...............................70
H OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE & FIRST
ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA...............................73
I OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE & SECOND
ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA...............................76
L IST O F R E FE R E N C E S ....................................................................... ... ................... 79
BIOGRAPHICAL SKETCH ............................................................................ 83
LIST OF TABLES
Table Page
3-1 M odel degrees of freedom ...................................................................... 17
3-2 H ip joint param eters. .......................................................... .. ............ 20
3-3 K nee joint param eters............................................ .................. ............... 23
3-4 A nkle joint param eters. ............................................................................ ....... 25
4-1 Two-level optimization results for synthetic marker data with random continuous
numerical noise to simulate skin movement artifacts with maximum amplitude of 1
c m .................................................................................... . 3 1
4-2 Mean marker distance errors for nominal values and the two-level optimization
results for multi-cycle experimental marker data ................................................33
4-3 Mean marker distance errors for the two-level optimization results using first and
second halves of the joint cycle motion for experimental marker data ................ 35
5-1 Mean marker distance errors for the inner-level objective function consisting of
marker coordinate errors versus marker distance errors for multi-cycle experimental
m arker data. ...........................................................................4 1
5-2 Execution times for the inner-level objective function consisting of marker
coordinate errors versus marker distance errors for multi-cycle experimental marker
data .................................................................................42
A-i Nominal right hip joint parameters and optimization bounds for synthetic marker
data .................................................................................52
A-2 Nominal right knee joint parameters and optimization bounds for synthetic marker
data .................................................................................53
A-3 Nominal right ankle joint parameters and optimization bounds for synthetic marker
data .................................................................................54
B-l Nominal right hip joint parameters and optimization bounds for experimental
m arker data. ...........................................................................55
B-2 Nominal right knee joint parameters and optimization bounds for experimental
m arker data. ........................................... ........................... 56
B-3 Nominal right ankle joint parameters and optimization bounds for experimental
m arker data. ........................................... ........................... 57
C-l Nominal and optimum right hip joint parameters for synthetic marker data without
n oise. ............................................................................... 5 8
C-2 Nominal and optimum right knee joint parameters for synthetic marker data
w without noise. ...................................................... ................. 59
C-3 Nominal and optimum right ankle joint parameters for synthetic marker data
w without noise. ...................................................... ................. 60
D-1 Nominal and optimum right hip joint parameters for synthetic marker data with
n o ise ............................................................................... 6 1
D-2 Nominal and optimum right knee joint parameters for synthetic marker data with
n oise. ............................................................................... 62
D-3 Nominal and optimum right ankle joint parameters for synthetic marker data with
n oise. ............................................................................... 6 3
E-1 Nominal and optimum right hip joint parameters for multi-cycle experimental
m arker data. ........................................... ........................... 64
E-2 Nominal and optimum right knee joint parameters for multi-cycle experimental
m arker data. ........................................... ........................... 65
E-3 Nominal and optimum right ankle joint parameters for multi-cycle experimental
m arker data. ........................................... ........................... 66
F-l Nominal and optimum right hip joint parameters for first one-half-cycle
experim ental m arker data. ........................................... ........................................67
F-2 Nominal and optimum right knee joint parameters for first one-half-cycle
experim ental m arker data. ........................................... ........................................68
F-3 Nominal and optimum right ankle joint parameters for first one-half-cycle
experim ental m arker data. ........................................... ........................................69
G-l Nominal and optimum right hip joint parameters for second one-half-cycle
experim ental m arker data. ........................................... ........................................70
G-2 Nominal and optimum right knee joint parameters for second one-half-cycle
experim ental m arker data. ............................................................................. 71
G-3 Nominal and optimum right ankle joint parameters for second one-half-cycle
experim ental m arker data. .............................................. ............................... 72
H-1 Optimum right hip joint parameters for multi-cycle and first one-half-cycle
experim ental m arker data. .............................................. ............................... 73
H-2 Optimum right knee joint parameters for multi-cycle and first one-half-cycle
experim ental m arker data. .............................................. ............................... 74
H-3 Optimum right ankle joint parameters for multi-cycle and first one-half-cycle
experim ental m arker data. .............................................. ............................... 75
I-1 Optimum right hip joint parameters for multi-cycle and second one-half-cycle
experim ental m arker data. .............................................. ............................... 76
I-2 Optimum right knee joint parameters for multi-cycle and second one-half-cycle
experim ental m arker data. .............................................. ............................... 77
1-3 Optimum right ankle joint parameters for multi-cycle and second one-half-cycle
experim ental m arker data. .............................................. ............................... 78
LIST OF FIGURES
Figure Page
3-1 The 3D, 14 segment, 27 DOF full-body kinematic model linkage joined by a set of
gim bal, universal, and pin joints. ..................................... ........................... ........ 16
3-2 A 1 DOF joint axis simultaneously defined in two adjacent body segments and the
geometric constraints on the optimization of each of the 9 model parameters........18
3-3 Modified Cleveland Clinic marker set used during static and dynamic
m otion-capture trials. ..................................... ...... ... ...... ....... ............19
3-4 The 3 DOF right hip joint center simultaneously defined in the pelvis and right
femur segments and the 6 translational model parameters optimized to determine
the functional hip joint center location............................................................20
3-5 Geometric constraints on the optimization of translational and rotational model
parameters for the hip, knee, and ankle joints .......................................................21
3-6 The 1 DOF right knee joint simultaneously defined in the right femur and right
tibia segments and the 4 rotational and 5 translational model parameters optimized
to determine the knee joint location and orientation. ....................... ................22
3-7 The 2 DOF right ankle joint complex simultaneously defined in the right tibia,
talus, and foot segments and the 5 rotational and 7 translational model parameters
optimized to determine the joint locations and orientations. ..................................24
3-8 Two-level optimization technique minimizing the 3D marker coordinate errors
between the kinematic model markers and experimental marker data to determine
functional joint axes for each lower-extremity joint. ....................... ................26
3-9 Inner-level optimization convergence illustration series for the knee joint, where
synthetic markers are blue and model markers are red. ........................................27
3-10 Two-level optimization approach minimizing the 3D marker coordinate errors
between the kinematic model markers and experimental marker data to determine
functional joint axes. ............................................ ............... .. .... ...... 28
4-1 Outer-level optimization objective function fitness value convergence for synthetic
marker data with random continuous numerical noise to simulate skin movement
artifacts with maximum amplitude of 1 cm, where the best fitness value among all
nodes is given for each iteration .................................................................... ..... 32
4-2 Outer-level optimization objective function fitness value convergence for
multi-cycle experimental marker data, where the best fitness value among all nodes
is given for each iteration. ............................................... .............................. 34
Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science
DETERMINATION OF PATIENT-SPECIFIC FUNCTIONAL AXES
THROUGH TWO-LEVEL OPTIMIZATION
By
Jeffrey A. Reinbolt
August 2003
Chair: Benjamin J. Fregly
Major Department: Biomedical Engineering
An innovative patient-specific dynamic model would be useful for evaluating and
enhancing corrective surgical procedures. This thesis presents a nested (or two-level)
system identification optimization approach to determine patient-specific model
parameters that best fit a three-dimensional (3D), 18 degree-of-freedom (DOF)
lower-body model to an individual's movement data.
The whole body was modeled as a 3D, 14 segment, 27 DOF linkage joined by a set
of gimbal, universal, and pin joints. For a given set of model parameters, the inner-level
optimization uses a nonlinear least squares algorithm that adjusts each generalized
coordinate of the lower-body model to minimize 3D marker coordinate errors between
the model and motion data for each time instance. The outer-level optimization
implements a parallel particle swarm algorithm that modifies each model parameter to
minimize the sum of the squares of 3D marker coordinate errors computed by the
inner-level optimization throughout all time instances (or the entire motion).
At the termination of each two-level optimization using synthetic marker data
without noise, original marker trajectories were precisely recovered to within an
arbitrarily tight tolerance (on the order of le-13 cm) using double precision
computations. At the termination of each two-level optimization using synthetic marker
data with noise representative of skin and soft tissue movement artifacts, the mean
marker distance error for each joint complex was as follows: ankle = 0.51 + 0.23 cm;
knee = 0.39 + 0.15 cm; and hip = 0.47 + 0.20 cm. Mean marker distance errors are
approximately one-half of the 1 cm maximum amplitude specified for the noise model.
At the termination of each two-level optimization using experimental marker data from
one subject, the mean marker distance error for each joint complex was less than or equal
to the following: ankle = 0.38 + 0.19 cm; knee = 0.55 + 0.27 cm; and hip = 0.36 + 0.20
cm. Experimental mean marker distance error results are comparable to the results of the
synthetic data with noise.
The two-level optimization method effectively determines patient-specific model
parameters defining a 3D lower-extremity model that is well suited to a particular subject.
When compared to previous values in the literature, experimental results show reasonable
agreement and demonstrate the necessity for the new approach. By minimizing fitness
errors between the patient-specific model and experimental motion data, the resulting
kinematic model provides an accurate foundation for future dynamic analyses and
optimizations.
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 Accurate Patient-Specific Models
Innovative patient-specific models and simulations would be valuable for
addressing problems in orthopedics and sports medicine, as well as for evaluating and
enhancing corrective surgical procedures (Arnold et al., 2000; Arnold and Delp, 2001;
Chao et al., 1993; Chao and Sim, 1995; Delp et al., 1998; Delp et al., 1996; Delp et al.,
1990; Pandy, 2001). For example, a patient-specific dynamic model may be useful for
planning intended surgical parameters and predicting the outcome of HTO.
The main motivation for developing a patient-specific computational model and a
two-level optimization method to enhance the lower-extremity portion is to predict the
post-surgery peak knee adduction moment in HTO patients. Conventional surgical
planning techniques for HTO involve choosing the amount of necessary tibial angulation
from standing radiographs (or x-rays). Unfortunately, alignment correction estimates
from static x-rays do not accurately predict long-term clinical outcome after HTO
(Andriacchi, 1994; Tetsworth and Paley, 1994). Researchers have identified the peak
external knee adduction moment as an indicator of clinical outcome while investigating
the gait of HTO patients (Andriacchi, 1994; Bryan et al., 1997; Hurwitz et al., 1998;
Prodromos et al., 1985; Wang et al., 1990). Currently, no movement simulations (or
other methods for that matter) allow surgeons to choose HTO surgical parameters to
achieve a chosen post-surgery knee adduction moment.
Movement simulations consist of models involving skeletal structure, muscle paths,
musculotendon actuation, muscle excitation-contraction coupling, and a motor task goal
(Pandy, 2001). Development of an accurate inverse dynamic model of the skeletal
structure is a significant first step toward creating a predictive patient-specific forward
dynamic model to perform movement simulations.
The precision of dynamic analyses is fundamentally associated with the accuracy of
kinematic model parameters such as segment lengths, joint positions, and joint
orientations (Andriacchi and Strickland, 1985; Challis and Kerwin, 1996; Cappozzo et
al., 1975; Davis, 1992; Holden and Stanhope, 1998; Holden and Stanhope, 2000; Stagni
et al., 2000). Understandably, a model constructed of rigid links within a multi-link chain
and simple mechanical approximations of joints will not precisely match the human
anatomy and kinematics. The model should provide the best possible agreement to
experimental motion data within the bounds of the joint models selected (Sommer and
Miller, 1980).
Benefits of Two-Level Optimization
This thesis presents a nested (or two-level) system identification optimization
approach to determine patient-specific joint parameters that best fit a three-dimensional
(3D), 18 degree-of-freedom (DOF) lower-body model to an individual's movement data.
The two-level technique combines the advantages of using optimization to determine
both the position of model segments from marker data and the anatomical joint axes
linking adjacent segments. By formulating a two-level objective function to minimize
marker coordinate errors, the resulting optimum model more accurately represents
experimental marker data (or a specific patient and his or her motion) when compared to
a nominal model defined by joint axes prediction methods.
CHAPTER 2
BACKGROUND
Motion Capture
Motion capture is the use of external devices to capture the movement of a real
object. One type of motion-capture technology is based on a passive optical technique.
Passive refers to markers, which are simply spheres covered in reflective tape, placed on
the object. Optical refers to the technology used to provide 3D data, which involves
high-speed, high-resolution video cameras. By placing passive markers on an object,
special hardware records the position of those markers in time and it generates a set of
motion data (or marker data).
Often motion capture is used to create synthetic actors by capturing the motions of
real humans. Special effects companies have used this technique to produce incredibly
realistic animations in movies such as Star Wars Episode I & II, Titanic, Batman, and
Terminator 2.
Biomechanical Models
Researchers use motion-capture technology to construct biomechanical models of
the human structure. The position of external markers may be used to estimate the
position of internal landmarks such as joint centers. The markers also enable the creation
of individual segment reference frames that define the position and orientation of each
body segment within a Newtonian laboratory reference frame. Marker data collected
from an individual are used to prescribe the motion of the biomechanical model.
Kinematics and Dynamics
Human kinematics is the study of the positions, angles, velocities, and accelerations
of body segments and joints during motion. With kinematic data and mass-distribution
data, one can study the forces and torques required to produce the recorded motion data.
Errors between the biomechanical model and the recorded motion data will inevitably
propagate to errors in the force and torque results of dynamic analyses.
Optimization
Optimization involves searching for the minimum or maximum of an objective
function by adjusting a set of design variables. For example, the objective function may
be the errors between the biomechanical model and the recorded motion data. These
errors are a function of the model's generalized coordinates and the model's kinematic
parameters such as segment lengths, joint positions, and joint orientations. Optimization
may be used to modify the design variables of the model to minimize the overall fitness
errors and identify a structure that matches the experimental data very well.
Limitations of Previous Methods
The literature contains a number of examples that use techniques, with or without
optimization, to assist in the development of subject-specific joint models within a larger
computational model. Several authors have presented methodologies to predict joint
locations and orientations from external landmarks without using optimization (Bell et
al., 1990; Inman, 1976; Vaughan et al., 1992). However, a regression model based solely
upon population studies may not accurately portray an individual patient. Another study
demonstrated an optimization method to determine the position and orientation of a 3
link, 6 DOF model by minimizing the distances between model-determined and
experimental marker positions (Lu and O'Connor, 1999). A model optimally positioned
6
without adjusting its joint parameters may not properly correspond to a certain patient.
Earlier studies described optimization methods to determine a set of model parameters for
a 3D, 2 DOF model by decreasing the error between the motion of the model and
experimental data (Sommer and Miller, 1980; Bogert et al., 1994). A model defined by
optimal joint parameters without optimizing its segment positions may not accurately
describe the motion of a patient within the bounds of the chosen joint approximations.
CHAPTER 3
METHODS
Parametric Model Structure
A generic, parametric 3D full-body kinematic model was constructed with
AutolevTM (Online Dynamics, Inc., Sunnyvale, CA) as a 14 segment, 27 DOF linkage
joined by a set of gimbal, universal, and pin joints (Figure 3-1, Table 3-1). Comparable
to Pandy's (2001) model structure, 3 translational degrees of freedom (DOFs) (qi, q2, and
q3) and 3 rotational DOFs (q4, q5, and q6) express the movement of the pelvis in 3D space
and the remaining 13 body segments comprise four open chains branching from the
pelvis segment. The locations and orientations of the joints within corresponding body
segments are described by 98 patient-specific model parameters. In other words, the
patient-specific model parameters designate the geometry of the model containing the
following joints types: 3 DOF hip, 1 DOF knee, 2 DOF ankle, 3 DOF back, 2 DOF
shoulder, and 1 DOF elbow. Each joint is defined in two adjacent body segments and
provides a mechanical approximation connecting those segments (Figure 3-2). For
example, the knee joint axis is simultaneously established in the femur coordinate system
and the tibia coordinate system.
A modified version of the Cleveland Clinic marker set (Figure 3-3) and a static
motion-capture trial is used to create segment coordinate systems and define static and
dynamic marker locations in these coordinate systems. Institutional review board
approval and proper informed consent were obtained before human involvement in the
experiments. The marker data collection system was a HiRes Expert Vision System
(Motion Analysis Corp., Santa Rosa, CA), including six HSC-180 cameras, EVa 5.11
software, and two AMTI force plates (Advanced Management Technology, Inc.,
Arlington, VA). Marker data were collected at 180 Hz during 3 seconds for static trials
and 6 seconds for individual joint trials. The raw data were filtered using a fourth-order,
zero phase-shift, low pass Butterworth Filter with a cutoff frequency set at 6 Hz.
Hip Joint
There are 6 translational model parameters that must be adjusted to establish a
functional hip joint center for a particular patient (Figure 3-4, Table 3-2). Markers placed
over the left anterior superior iliac spine (ASIS), right ASIS, and superior sacrum define
the pelvis segment coordinate system. From percentages of the inter-ASIS distance, a
predicted (or nominal) hip joint center location within the pelvis segment is 19.3%
posterior (pi), 30.4% inferior (p2), and 35.9% medial-lateral (p3) (Bell et al., 1990). This
nominal hip joint center is the origin of the femur coordinate system, which is
subsequently defined by markers placed over the medial and lateral femoral epicondyles.
An additional 3 translational model parameters (p4, p5, and p6), described in the femur
coordinate system, complete the structure of the nominal hip joint center.
Given the physical hip joint center is located within the pelvic region lateral to the
midsagittal plane, a cube with side lengths equal to 75% of the inter-ASIS distance and
its anterior-superior-medial vertex positioned at the midpoint of the inter-ASIS line
provides the geometric constraints for the optimization of each model parameter (Figure
3-5, Table A-i, Table B-l).
Knee Joint
There are 9 model parameters (5 translational and 4 rotational) that must be tailored
to identify a patient-specific functional knee joint axis (Figure 3-6, Table 3-3). The
femoral transepicondylar axis is a good approximation of a fixed knee joint axis
(Churchill et al., 1998). The line (or nominal) knee joint axis, connecting the medial and
lateral knee markers is defined in the femur and tibia coordinate systems (Vaughan et al.,
1992). Given the line passes through the midsagittal plane (x-y plane) of the femur
segment, the nominal knee joint axis is positioned within the femur via 2 translational
model parameters (p5 and p6) and 2 rotational model parameters (pl and p2). The tibia
coordinate system originates at the midpoint of the knee markers and is defined by
additional markers located on the medial and lateral malleoli. The distal description of
the nominal knee joint axis is comprised of 3 translational model parameters (p7, ps, and
p9) and 2 rotational model parameters (p3 and p4) in the tibia segment.
Given the anatomical knee joint DOFs are situated within the articular capsule, a
cube with side lengths equal to the distance between knee markers and its center located
at the midpoint of the nominal knee joint axis provides the geometric constraints for the
optimization of each translational model parameter. The rotational model parameters are
constrained within a circular cone defined by the 3600 revolution of the nominal knee
joint axis perturbed by + 30 (Figure 3-5, Table A-2, Table B-2).
It is not a trivial notion to eliminate a potential medial-lateral translational model
parameter in the femur segment. This model parameter is considered redundant, as the
knee joint axis passes through the midsagittal plane of the femur, and its inclusion may
lead to possible optimization convergence problems, similar to the redundant ankle model
parameter discussion of Bogert et al. (1994). By including redundant model parameters,
there are an infinite number of optimum solutions within the constraints of corresponding
superfluous model parameters.
Ankle Joint
There are 12 patient-specific model parameters (7 translational and 5 rotational)
that must be customized to determine a pair of patient-specific functional ankle joint axes
(Figure 3-7, Table 3-4). Comparable to Bogert et al. (1994), the talocrural and subtalar
joints connect the tibia, talus, and foot segments. Within the tibia segment, 3
translational model parameters (p6, P7, and ps) and 2 rotational model parameters (pl and
p2) position the nominal talocrural joint axis. The talus origin corresponds to the
talocrural joint center; therefore, it is not necessary to prescribe model parameters
defining the talocrural joint axis in the talus segment. The talus coordinate system is
created where the y-axis extends along the line perpendicular to both the talocrural joint
axis and the subtalar joint axis. The heel and toe markers, in combination with the tibia
y-axis, define the foot coordinate system. There are 3 translational model parameters
(plo, p11, and p12) and 2 rotational model parameters (p4 and ps) (Inman, 1976) that place
the nominal subtalar joint axis in the foot coordinate system.
Given the anatomical ankle joint DOFs are found within the articular capsule, a
cube with side lengths equal to the distance between ankle markers and its center located
at the midpoint of the nominal talocrural joint axis provides the geometric constraints for
the optimization of each translational model parameter. The rotational model parameters
of the talocrural joint axis are restricted within a circular cone defined by the 3600
revolution of the nominal talocrural joint axis varied by + 300. The rotational model
parameters of the subtalar joint axis are confined within a circular cone defined by the
3600 revolution of the nominal subtalar joint axis altered by + 300 (Figure 3-5, Table A-3,
Table B-3).
Two-Level Optimization Approach
Why Two Levels of Optimization Are Necessary
Optimization may be used to identify a system (or determine patient-specific joint
parameters) that best fit a 3D, 18 DOF lower-body model to an individual's movement
data. One level of optimization is necessary to establish the model's geometry. Given a
defined model, another level of optimization is required to position and orientate the
model's body segments. By formulating a two-level objective function to minimize 3D
marker coordinate errors, the two-level optimization results describe a lower-body model
that accurately represents experimental data.
Inner-Level Optimization
Given marker trajectory data, md, and a constant set of patient-specific model
parameters, p, the inner-level optimization (Figure 3-8, inner boxes) minimizes the 3D
marker coordinate errors, ec, between the model markers, mm, and the marker movement
data, md, (Equation 3-1) using a nonlinear least squares algorithm that adjusts the
generalized coordinates, q, of the model at each instance in time, t, (Figure 3-9), similar
to Lu and O'Connor (1999). In other words, the pose of the model is revised to match the
marker movement data at each time frame of the entire motion.
min e(q, p, t) = md(t) mm(q,p, t) (3-1)
At the first time instance, the algorithm is seeded with exact values for the 6
generalized coordinates of the pelvis, since the marker locations directly identify the
position and orientation of the pelvis coordinate system, and all remaining generalized
coordinates are seeded with values equal to zero. Given the joint motion is continuous,
each optimal generalized coordinate solution, including the pelvis generalized
coordinates, at one time instance is used as the algorithm's seed for the next time
instance. Matlab 6.1 (The MathWorks, Inc., Natick, MA), in conjunction with the Matlab
Optimization Toolbox and Matlab C/C++ Compiler, was used to develop the inner-level
optimization program.
Outer-Level Optimization
The outer-level global optimization (Figure 3-8, outer boxes) minimizes the sum of
the squares, ess, of the 3D marker coordinate errors, ec, (Equation 3-1) computed by the
inner-level algorithm throughout all time instances, n, (Equation 3-2) by modifying the
patient-specific model parameters, p. In other words, the geometric structure of the
model is varied to best fit the marker movement data for the entire motion.
min e(q,p, n) = [ec(q,p, tj [e(q, p, t)] (3-2)
t=1
The outer-level optimization is adapted from the population-based Particle Swarm
Optimizer (PSO) (Kennedy and Eberhart, 1995). The PSO algorithm was chosen over
gradient-based optimizers for its suitability to be parallelized and its ability to solve
global optimization problems. It is particularly effective in the determination of joint
positions and orientations of biomechanical systems (Schutte et al., 2003). The work of
Schutte et al. (2003) contrasted the PSO to a gradient-based optimizer (i.e.,
Broyden-Fletcher-Goldfarb-Shanno) that is commonly used in system identification
problems involving biomechanical models. The PSO very reliably converged to the
global minimum and it was insensitive to both design variable scaling and initial seeds
(Schutte et al., 2003).
To manage computational requirements, the outer-level optimization uses a parallel
version of the PSO operating on a cluster of 20 Linux-based 1.33 GHz Athlon PC's on a
100 Mbps switched Fast Ethernet network. Each machine is separately seeded with a
random set of initial patient-specific model parameter values. The outer-level
optimization program was implemented in C on the Linux operating system with the
Message Passing Interface (MPI) parallel computation libraries.
Two-Level Optimization Evaluation
Synthetic Marker Data without Noise
To evaluate the ability of the two-level optimization approach (Figure 3-10) to
calibrate the generic, parametric kinematic model, synthetic movement data was
generated for the ankle, knee, and hip joints based on estimated in vivo model parameters
and experimental movement data. For each generated motion, the distal segment moved
within the physiological range of motion and exercised each DOF for the joint. There
were 50 time frames and approximately 3.5 cycles of a circumductive hip motion
consisting of concurrent flexion-extension and abduction-adduction. Flexion-extension
comprised 50 time frames and roughly 4 cycles of knee motion. The ankle motion
involved 50 time frames and nearly 2.75 cycles of circumduction of the toe tip, where
plantarflexion-dorsiflexion and inversion-eversion occurred simultaneously. The ability
of the two-level optimization to recover the original model parameters used when
generating the synthetic motions was assessed.
Synthetic Marker Data with Noise
To evaluate the ability of the two-level optimization method (Figure 3-10) to
calibrate the generic kinematic model to a synthetic patient, skin movement artifacts were
introduced into the synthetic movement data for the ankle, knee, and hip joints. The
relative movement between skin and underlying bone occurs in a continuous rather than a
random fashion (Cappozzo et al., 1993). Comparable to the simulated skin movement
artifacts of Lu and O'Connor (1999), a continuous numerical noise model of the form
A sin(co t + p) was used and the equation variables were randomly generated within the
following bounds: amplitude (0
angle (0 < (p < 22) (Cheze et al., 1995). Noise was separately generated for each 3D
coordinate of the marker trajectories. Again, the two-level optimization was tested for its
ability to reproduce the original model parameters.
Experimental Marker Data
To verify the ability of the two-level optimization technique (Figure 3-10) to
calibrate the generic kinematic model to a particular patient, multi-cycle experimental
marker trajectory data was collected from one subject. For each joint motion, the distal
segment moved within the physiological range of motion and exercised each DOF for the
joint. Analogous to Bogert et al. (1994), the original data were resampled
non-equidistantly to eliminate weighting the data set with many data points occurring
during acceleration and deceleration at the limits of the range of motion. In other words,
regardless of changes in velocity during joint movements, the data was equally
distributed over the entire joint range of motion. The time frames of original tracked
marker data sets (right hip = 1015, right knee = 840, and right ankle = 707) were reduced
to 50 time frames. The resampled data allowed a fixed amount of marker movement
between frames to arrive at the number of time frames chosen, given that 50 time frames
is analogous to Lu and O'Connor (1999). There were nearly 2 cycles of
flexion-extension and abduction-adduction involved in the hip motion. Similar to
Leardini et al. (1999), internal-external rotation of the hip was avoided to reduce the
effects of skin and soft tissue movement artifacts. Approximately 2 cycles of knee
motion included flexion-extension. Simultaneous plantarflexion-dorsiflexion and
inversion-eversion comprised roughly 2 cycles of ankle motion. Without knowledge of
original model parameters, the marker coordinate errors are the only means of measuring
the effectiveness of the two-level optimization.
To verify the ability of the two-level optimization procedure (Figure 3-10) to
calibrate the generic kinematic model to a particular patient using a smaller portion of the
joint motion cycle, the resampled multi-cycle experimental marker trajectory data
described above was divided into the first and second halves of the individual hip, knee,
and ankle joint motion cycles. The number of time frames comprising each
one-half-cycle of the joint motion was as follows: ankle = 13, knee = 13, and hip = 19.
Again, the two-level optimization was tested for its ability to reduce the marker
coordinate errors and obtain an optimal set of model parameters.
q 23
q24
q'
II S"f
q 7
Si
q10
qil
q12
q27
q1"4
L .4
-,k
q2
q3
L---------.
q16
i17
% (superior)
Sq18
q, .. .
Figure 3-1. 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
Z
(lateral)
(anterior)
Table 3-1. 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
Figure 3-2. A 1 DOF joint axis simultaneously defined in two adjacent body segments
and the geometric constraints on the optimization of each of the 9 model
parameters.
OK
1C -
r
I,,
/
0,
It
/[
/
a i
/'
,'
A -
'O
.
b '''
Figure 3-3. Modified Cleveland Clinic marker set used during static and dynamic
motion-capture trials. Note: the background femur and knee markers have
been omitted for clarity and the medial and lateral markers for the knee and
ankle are removed following the static trial.
*19
sire
It
'c1o
1
*i
.h
Pelvis
Optimized
Hip Joint
SCenter \
F--X
\ P5 P1
Femur x
\I'
"J ;
Sw
It.;
J"
Y
(superior)
Z a
(lateral)
Sx
(anterior)
Figure 3-4. The 3 DOF right hip joint center simultaneously defined in the pelvis and
right femur segments and the 6 translational model parameters optimized to
determine the functional hip joint center location.
Table 3-2. Hip joint parameters.
Hip Joint
Parameter
pi
P2
P3
P4
P5
P6
Description
Anterior-posterior location in pelvis segment
Superior-inferior location in pelvis segment
Medial-lateral location in pelvis segment
Anterior-posterior location in femur segment
Superior-inferior location in femur segment
Medial-lateral location in femur segment
ct.mt
LJ --
Figure 3-5. Geometric constraints on the optimization of translational and rotational
model parameters for the hip, knee, and ankle joints.
ZP5
\P
p2
p4
Optimized P3
Knee Joint
Center Y p8
f Y P7
Optimized
Knee Joint
Axis Tibia -' P
Z X
Y I
(superior)
z x i
(lateral) Lab (anterior)
Figure 3-6. The 1 DOF right knee joint simultaneously defined in the right femur and
right tibia segments and the 4 rotational and 5 translational model parameters
optimized to determine the knee joint location and orientation.
Table 3-3. Knee joint paramete
Knee Joint
Parameter
pi
P2
P3
P4
P5
P6
P7
P8
P9
:rs.
Description
Adduction-abduction rotation in femur segment
Internal-external rotation in femur segment
Adduction-abduction rotation in tibia segment
Internal-external rotation in tibia segment
Anterior-posterior location in femur segment
Superior-inferior location in femur segment
Anterior-posterior location in tibia segment
Superior-inferior location in tibia segment
Medial-lateral location in tibia segment
Tibia
Z- -
P6
r Pg
P7
Optimized
Talocrural Joint
-' Center
Talus
Optimized Z
Talocrural Joint Optimized
Axis Ps Subtalar Joint
P3 Center
P4
Optimized
Subtalar Joint Y
Axis (superior)
Y /L / P11
SFoot P12 -- P10
Z Z X
--" (lateral) Lab (anterior)
Figure 3-7. The 2 DOF right ankle joint complex simultaneously defined in the right tibia,
talus, and foot segments and the 5 rotational and 7 translational model
parameters optimized to determine the joint locations and orientations.
Table 3-4. Ankle joint pa
Ankle Joint
Parameter
pi
P2
P3
P4
P5
P6
P7
P8
P9
Pio
Pll
P12
rameters.
Description
Adduction-abduction rotation of talocrural in tibia segment
Internal-external rotation of talocrural in tibia segment
Internal-external rotation of subtalar in talus segment
Internal-external rotation of subtalar in foot segment
Dorsi-plantar rotation of subtalar in foot segment
Anterior-posterior location of talocrural in tibia segment
Superior-inferior location of talocrural in tibia segment
Medial-lateral location of talocrural in tibia segment
Superior-inferior location of subtalar in talus segment
Anterior-posterior location of subtalar in foot segment
Superior-inferior location of subtalar in foot segment
Medial-lateral location of subtalar in foot segment
Joint Axes
Experiments Hip
Outer Optimization
Parallel Particle Swarm
Knee
Outer Optimization
SParallel Particle Swarm
izalion
squares
I II
nation
uares
2
Figure 3-8. Two-level optimization technique minimizing the 3D marker coordinate
errors between the kinematic model markers and experimental marker data to
determine functional joint axes for each lower-extremity joint.
Ankle Outer Optimization
-' Parallel Particle Swarm
Processor # 20 Inner Optimizalion
SNonlinear Least Squares
SProcessor # Opiz Inner Optimization L
SNonlinear Least Squares
Processor f 2 Inner Optimization /
T Nonlinear Least Squares
Processor # 1 Inner Optimization
Nonlinear Least Squares
V
Error Time j
Error Time J f /
a Error Time n/
Error Time #2
Frame #1
a
111
71<
.,9)
U
E
C +
(V
og
+ +1
b LU
II :E
Ca
II II
$ (D
c S
(D ir
a0m
*U
c)
.* ** 8
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Sf
E
o
m(
* .
2
0I I
@7 ;
0I
"r6
/
JoiJedns
JOaedns
A -~
jouadns
-A
--a
S ;-
jouadns
o C
-e
a -
a E0
0Sct
OO
mo
*- j a)
aa
Co
O co
C > II
oa a
-oc
^^ 0
0o0
Ct0
C o
So\.
0i b
.
Initialize outer-level
parallel particle
swarm optimization
f
Minimize outer-level
objective function (i.e.,
3D marker coordinate
errors for all time
frames of inner-level
optimization)
Adjust outer-level
design variables (i.e.,
model parameters)
False
Initialize inner-level
non-linear least
squares optimization
False
Minimize inner-level
objective function (i.e.,
3D marker coordinate
errors for current time
frame i)
Terminate inner-level
non-linear least
squares optimization
True
Terminate outer-level
parallel particle
swarm optimization
Figure 3-10. Two-level optimization approach minimizing the 3D marker coordinate
errors between the kinematic model markers and experimental marker data
to determine functional joint axes.
CHAPTER 4
RESULTS
Synthetic Marker Data without Noise
For synthetic motions without noise, each two-level optimization precisely
recovered the original marker trajectories to within an arbitrarily tight tolerance (on the
order of le-13 cm), as illustrated in Figure 3-9. At the termination of each optimization,
the optimum model parameters for the hip, knee, and ankle were recovered with mean
rotational errors less than or equal to 0.0450 and mean translational errors less than or
equal to 0.0077 cm (Appendix C).
Synthetic Marker Data with Noise
For synthetic motions with noise, the two-level optimization of the hip, knee, and
ankle resulted in mean marker distance errors equal to 0.46 cm, which is of the same
order of magnitude as the selected random continuous noise model (Table 4-1). The
two-level approach determined the original model parameters with mean rotational errors
less than or equal to 3.730 and mean translational errors less than or equal to 0.92 cm
(Appendix D). The outer-level fitness history converged rapidly (Figure 4-1) and the hip,
knee, and ankle optimizations terminated with a mean wall clock time of 41.02 hours.
Experimental Marker Data
For multi-cycle experimental motions, the mean marker distance error of the
optimal hip, knee, and ankle solutions was 0.41 cm, which is a 0.43 cm improvement
over the mean nominal error of 0.84 cm (Table 4-2). For each joint complex, the
optimum model parameters improved upon the nominal parameter data (or values found
in the literature) by mean rotational values less than or equal to 6.180 and mean
translational values less than or equal to 1.05 cm (Appendix E). When compared to the
synthetic data with noise, the outer-level fitness history of the multi-cycle experimental
data optimization converged at approximately the same rate and resulted in an improved
final solution for both the ankle and the hip (Figure 4-2). On the contrary, the higher
objective function values for the knee are evidence of the inability of the fixed pin joint to
represent the screw-home motion (Blankevoort et al., 1988) of the multi-cycle
experimental knee data. The multi-cycle hip, knee, and ankle optimizations terminated
with a mean wall clock time of 35.94 hours.
For one-half-cycle experimental motions, the mean marker distance error of the
optimal hip, knee, and ankle solutions was 0.30 cm for the first half and 0.30 cm for the
second half (Table 4-3). The fitness of both the ankle and the hip were comparable to the
multi-cycle joint motion results. However, the knee fitness values were improved due to
the reduced influence (i.e., 1 time frame of data as opposed to 9) of the screw-home
motion of the knee. For each joint complex, the optimum model parameters improved
upon the nominal parameter data (or values found in the literature) by mean rotational
values less than or equal to 11.080 and mean translational values less than or equal to
2.78 cm (Appendix F, Appendix G). In addition, the optimum model parameters for
one-half-cycle motion differed from those for the multi-cycle motion by mean rotational
values less than or equal to 15.770 and mean translational values less than or equal to
2.95 cm (Appendix H, Appendix I). The one-half-cycle hip, knee, and ankle
optimizations terminated with a mean wall clock time of 11.77 hours.
Table 4-1. Two-level optimization results for synthetic marker data with random
continuous numerical noise to simulate skin movement artifacts with
maximum amplitude of 1 cm.
Synthetic Data Hip Knee Ankle
with Noise
Mean marker
Mean marker 0.474603 + 0.202248 0.392331 + 0.145929 0.514485 + 0.233956
distance error (cm)
Mean rotational
n/a 2.158878 + 1.288703 3.732191 + 3.394553
parameter error () -
Mean translational
0.Mean translational 161318 + 0.039449 0.321930 + 0.127997 0.923724 + 0.471443
parameter error (cm)
500
-Hip
Knee
400 -Ankle
"E
u 300
S200
U.
100
0
0 5000 10000 15000 20000 25000
Function Evaluations
Figure 4-1. Outer-level optimization objective function fitness value convergence for
synthetic marker data with random continuous numerical noise to simulate
skin movement artifacts with maximum amplitude of 1 cm, where the best
fitness value among all nodes is given for each iteration.
Table 4-2. Mean marker distance errors for nominal values and the two-level
optimization results for multi-cycle experimental marker data.
Experimental Data Hip Knee Ankle
Nominal mean
marker distance 0.499889 + 0.177947 1.139884 + 0.618567 0.885437 + 0.478530
error (cm)
Optimum mean
marker distance 0.342262 + 0.167079 0.547787 + 0.269726 0.356279 + 0.126559
error (cm)
Mean marker
distance error 0.157627 + 0.166236 0.592097 + 0.443680 0.529158 + 0.438157
attenuation (cm)
600
-Hip
500 -Knee
-Ankle
400
N
* 300
200
100
0
0 5000 10000 15000 20000 25000
Function Evaluations
Figure 4-2. Outer-level optimization objective function fitness value convergence for
multi-cycle experimental marker data, where the best fitness value among all
nodes is given for each iteration.
35
Table 4-3. Mean marker distance errors for the two-level optimization results using first
and second halves of the joint cycle motion for experimental marker data.
Experimental Data Hip Knee Ankle
First half: mean
marker distance 0.335644 + 0.163370 0.189551 + 0.072996 0.384786 + 0.193149
error (cm)
Second half: mean
marker distance 0.361179 + 0.200774 0.202413 + 0.101063 0.338886 + 0.128596
error (cm)
CHAPTER 5
DISCUSSION
Assumptions, Limitations, and Future Work
Joint Model Selection
If the current model cannot adequately reproduce future experimental motions, the
chosen joint models may be modified. For example, the flexion-extension of the knee is
not truly represented by a fixed pin joint (Churchill et al., 1998). When comparing the
fitness of the optimum knee joint model to multi-cycle experimental marker data, the
agreement was quite good for all knee flexion angles with the exception of those
approaching full extension. By eliminating knee flexion angles less than 200, which
comprised 18% of the flexion-extension data, the mean marker distance error was
reduced to 0.48 + 0.23 cm (11.89% decrease) using the optimum model parameters from
the full data set. A pin joint knee may be sufficiently accurate for many modeling
applications. A 2 DOF knee model (Hollister et al., 1993) may account for the
screw-home motion of the knee joint occurring between 00 and 200 (Blankevoort et al.,
1988). If greater fidelity to actual bone motion is necessary, a 6 DOF knee joint may be
implemented with kinematics determined from fluoroscopy (Rahman et al., 2003).
Design Variable Constraints
Certain joint parameters must be constrained to zero with the purpose of preventing
the unnecessary optimization of redundant parameters. Case in point, the medial-lateral
translational model parameter placing the knee joint center in the femur segment must be
constrained to zero. On the other hand, this model parameter may be used as a design
variable, granted the medial-lateral translational model parameter placing the knee joint
center in the tibia segment is constrained to zero. If both medial-lateral translational
model parameters are used as redundant design variables, the outer-level optimization has
an infinite number of solutions within the constraints of both parameters. Through the
elimination (i.e., constraining to zero) of redundant model parameters, the outer-level
optimization encounters less convergence problems in globally minimizing the objective
function.
Objective Function Formulation
The inner-level optimization objective function should be comprised of marker
coordinate errors rather than marker distance errors. A substantial amount of information
(i.e., % of the number of errors) describing the fitness value is lost with computation of
marker distance errors. In other words, a marker distance error provides only the radius
of a sphere surrounding an experimental marker and it does not afford the location of a
model marker on the surface of the sphere. However, a set of three marker coordinate
errors describes both the magnitude and direction of an error vector between an
experimental marker and a model marker. By using marker coordinate errors, the
inner-level optimization has improved convergence (Table 5-1) and shorter execution
time (Table 5-2).
Optimization Time and Parallel Computing
To reduce the computation time, it is necessary to use an outer-level optimization
algorithm in a parallel environment on a network cluster of processors. The PSO
algorithm was chosen over gradient-based optimizers for its suitability to be parallelized
and its ability to solve global optimization problems. The large computation time is a
result of the random set of initial values used to seed each node of the parallel algorithm.
By seeding one of the nodes with a relatively optimal set of initial values, the
computation time may be significantly decreased. By doubling the number of parallel
processors, the computation time declines nearly 50%. Decreasing the number of time
frames of marker data additionally reduces the computation time. For example, the mean
optimization time using experimental data for 50 time frames equals 35.94 hours, 19 time
frames equals 12.82 hours, and 13 time frames equals 11.24 hours. Further study is
necessary to establish the minimum number of marker data time frames required to
effectively determine joint axes parameters.
Multi-Cycle and One-Half-Cycle Joint Motions
The two-level optimization results vary depending on whether marker data time
frames consist of multi-cycle or one-half-cycle joint motions. In other words, the
determination of patient-specific model parameters is significantly influenced by the
marker trajectories contained within the chosen set of data. Given a set of marker data,
the two-level optimization establishes invariable model parameters that best fit the
mathematical model to the measured experimental motion. Understandably, a model
constructed from one marker data set may not adequately represent a considerably
different marker data set. To perform accurate dynamic analyses, joint motions used to
generate the model should be consistent with those motions that will be used in the
analyses.
The small differences between sets of two-level optimization results for the hip and
knee joint motions indicate the reliability of the model parameter values. Much larger
differences occurred between sets of model parameters determined for the ankle joint.
Two major factors contributing to these differences are the rotational ankle model
parameters pi and p3. On one hand, the model parameters may truly vary throughout the
ankle motion and may not be represented by constant values. On the other hand, the
objective function may be insensitive to changes in these model parameters indicating a
design space that does not permit the reasonable determination of certain design
variables. Future study is necessary to investigate the sensitivity of 3D marker coordinate
errors to particular model parameters.
Range of Motion and Loading Conditions
To provide the largest range of motion, all experimental data was collected with
each joint unloaded and freely exercising all DOFs; however, the same two-level
optimization may be performed on loaded data as well. The patient-specific model
parameters may change under loaded conditions (Bogert et al., 1994). Moreover, loaded
conditions limit the range of motion for several DOFs. Several authors (Bell et al., 1990;
Bogert et al., 1994) report inaccuracies in determining functional axes from limited
motion, but a subsequent study (Piazza et al., 2001) found the hip joint may be
determined from motions as small as 150. Piazza et al. (2001) suggest future studies are
necessary to explore the use of normal gait motions, rather than special joint motions, to
determine functional axes.
Optimization Using Gait Motion
The two-level optimization approach and synthetic data evaluation method may be
used to investigate the use of gait motion to determine functional joint axes. Each set of
joint parameters may be established separately or collectively (i.e., entire single leg or
both legs at once). Additional investigation is necessary to assess the differences in joint
parameters obtained through individual optimizations and simultaneous whole leg
optimizations. Furthermore, the joint parameters determined from gait motions may be
compared to those parameters obtained from special joint motions with larger amounts of
movement.
Authors (Bogert et al., 1994; Cheze et al., 1995; Lu and O'Connor, 1999) have set
precedence for performing numerical (or synthetic data) simulations to evaluate a new
technique. Although it is not a necessary task, there is additional benefit in supporting
the numerical findings with data from one human subject. With the additional data, the
joint parameters computed from unloaded joint motions may be measured against those
parameters attained from unloaded (i.e., swing phase) and loaded (i.e., stance phase) gait
motions. To expand upon the evaluation of the new technique and show general
applicability, future work is necessary to study more than one human subject.
Comparison of Experimental Results with Literature
The two-level optimization determined patient-specific joint axes locations and
orientations similar to previous works. The optimum hip joint center location of 7.52 cm
(27.89% posterior), 9.27 cm (34.38% inferior), and 8.86 cm (32.85% lateral) are
respectively comparable to 19.3%, 30.4%, and 35.9% (Bell et al., 1990). The optimum
femur length (40.46 cm) and tibia length (40.88 cm) are similar to 42.22 cm and 43.40
cm, respectively (de Leva, 1996). The optimum coronal plane rotation (73.360) of the
talocrural joint correlates to 82.7 + 3.7 (range 740 to 940) (Inman, 1976). The optimum
distance (2.14 cm) between the talocrural joint and the subtalar joint is analogous to 1.24
+ 0.29 cm (Bogert et al., 1994). The optimum transverse plane rotation (13.190) and
sagittal plane rotation (45.260) of the subtalar joint corresponds to 23 + 11 (range 40 to
47) and 42 + 90 (range 20.50 to 68.50), respectively (Inman, 1976).
Table 5-1. Mean marker distance errors for the inner-level objective function consisting
of marker coordinate errors versus marker distance errors for multi-cycle
experimental marker data.
Experimental Data
Hip
Knee
Ankle
Marker distance
objective function
0oece fn n .863941 + 0.328794 1.043909 + 0.465186 0.674187 + 0.278451
mean marker
distance error (cm)
Marker coordinate
objective function
0oece fncn .342262 + 0.167079 0.547787 + 0.269726 0.356279 + 0.126559
mean marker
distance error (cm)
42
Table 5-2. Execution times for the inner-level objective function consisting of marker
coordinate errors versus marker distance errors for multi-cycle experimental
marker data.
Experimental Data Hip Knee Ankle
Marker distance
objective function: 464.377 406.205 308.293
execution time (s)
Marker coordinate
objective function: 120.414 106.003 98.992
execution time (s)
CHAPTER 6
CONCLUSION
Rationale for New Approach
The main motivation for developing a 27 DOF patient-specific computational
model and a two-level optimization method to enhance the lower-extremity portion is to
predict the post-surgery peak knee adduction moment in HTO patients, which has been
identified as an indicator of clinical outcome (Andriacchi, 1994; Bryan et al., 1997;
Hurwitz et al., 1998; Prodromos et al., 1985; Wang et al., 1990). The accuracy of
prospective dynamic analyses made for a unique patient is determined in part by the
fitness of the underlying kinematic model (Andriacchi and Strickland, 1985; Challis and
Kerwin, 1996; Cappozzo et al., 1975; Davis, 1992; Holden and Stanhope, 1998; Holden
and Stanhope, 2000; Stagni et al., 2000). Development of an accurate kinematic model
tailored to a specific patient forms the groundwork toward creating a predictive
patient-specific dynamic simulation.
Synthesis of Current Work and Literature
The two-level optimization method satisfactorily determines patient-specific model
parameters defining a 3D lower-extremity model that is well suited to a particular patient.
Two conclusions may be drawn from comparing and contrasting the two-level
optimization results to previous values found in the literature. The similarities between
numbers suggest the results are reasonable and show the extent of agreement with past
studies. The differences between values indicate the two-level optimization is necessary
and demonstrate the degree of inaccuracy inherent when the new approach is not
implemented.
Through the enhancement of model parameter values found in the literature, the
two-level optimization approach successfully reduces the fitness errors between the
patient-specific model and the experimental motion data. More specifically, to quantify
the improvement of the current results compared to previous values found in the
literature, the mean marker distance errors were reduced by 31.53% (hip), 51.94% (knee),
and 59.76% (ankle).
The precision of dynamic analyses made for a particular patient depends on the
accuracy of the patient-specific kinematic parameters chosen for the dynamic model.
Without expensive medical images, model parameters are only estimated from external
landmarks that have been identified in previous studies. The estimated (or nominal)
values may be improved by formulating an optimization problem using motion-capture
data. By using a two-level optimization technique, researchers may build more accurate
biomechanical models of the individual human structure. As a result, the optimal models
will provide reliable foundations for future dynamic analyses and optimizations.
Abduction
Acceleration
Active markers
Adduction
Ankle inversion-eversion
Ankle motion
Ankle plantarflexion-dorsiflexion
Anterior
Circumduction
Coccyx
Constraint functions
Coronal plane
Couple
GLOSSARY
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.
Movement of the distal tip of a segment described
by a circle.
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 set of force vectors whose resultant is equal to
zero. Two force vectors with equal magnitudes and
opposite directions is an example of a simple
couple.
Degree of freedom (DOF)
Design variables
Distal
Dorsiflexion
Epicondyle
Version
Extension
External (lateral) rotation
External moment
Femur
Flexion
Fluoroscopy
Force
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.
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.
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 ajoint
towards each other in the sagittal plane.
Examination of body structures using an X-ray
machine that combines an X-ray source and a
fluorescent screen to enable real-time observation.
A push or a pull and is produced when one object
acts on another.
Force plate
Forward dynamics
Gait
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
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).
Inter-ASIS distance
Internal (medial) rotation
Internal joint moments
Inverse dynamics
Inversion
Kinematics
Kinetics
Knee abduction-adduction
Knee flexion-extension
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.
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.
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.
Knee internal-external rotation
Knee motion
Lateral
Malleolus
Markers
Medial
Midsagittal plane
Model parameters
Moment of force
Motion capture
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.
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.
Non-equidistant
Objective functions
Parametric
Passive markers
Pelvis
Pelvis anterior-posterior tilt
Pelvis elevation-depression
Pelvis internal-external rotation
Pelvis motion
Plantarflexion
Posterior
Proximal
Range of motion
The opposite of equal amounts of distance between
two or more points, or not equally distanced.
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.
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.
Sacrum
Sagittal plane
Skin movement artifacts
Stance phase
Subtalar joint
Superior
Synthetic markers
Swing phase
Talocrural joint
Talus
Tibia
Transepicondylar
Transverse plane
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.
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.
Velocity
APPENDIX A
NOMINAL JOINT PARAMETERS & OPTIMIZATION BOUNDS
FOR SYNTHETIC MARKER DATA
Table A-1. Nominal right hip joint parameters and optimization bounds for synthetic
marker data.
Right Hip Joint
Parameter
pi (cm)
p2 (cm)
p3 (cm)
p4 (cm)
p5 (cm)
p6 (cm)
Nominal
-6.022205
-9.307044
8.759571
0
0
0
Lower Bound
-20.530245
-20.530245
0
-14.508040
-11.223200
-8.759571
Upper Bound
0
0
20.530245
6.022205
9.307044
11.770674
Table A-2. Nominal right knee joint parameters and optimization bounds for synthetic
marker data.
Right Knee Joint
Parameter
pi ()
P2 ()
P3 ()
P4()
P5 (cm)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Nominal
0
0
-5.079507
16.301928
0
-37.600828
0
0
0
Lower Bound
-30
-30
-35.079507
-13.698072
-7.836299
-45.437127
-7.836299
-7.836299
-7.836299
Upper Bound
30
30
24.920493
46.301928
7.836299
-29.764528
7.836299
7.836299
7.836299
Table A-3. Nominal right ankle joint parameters and optimization bounds for synthetic
marker data.
Right Ankle Joint
Parameter
pi ()
P2 (o)
P3 (o)
P4()
P5 (0)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Pio (cm)
Pll (cm)
P12 (cm)
Nominal
18.366935
0
40.230969
23
42
0
-39.973202
0
-1
8.995334
4.147543
0.617217
Lower Bound
-11.633065
-30
10.230969
-7
12
-6.270881
-46.244082
-6.270881
-6.270881
2.724454
-2.123338
-5.653664
Upper Bound
48.366935
30
70.230969
53
72
6.270881
-33.702321
6.270881
0
15.266215
10.418424
6.888097
APPENDIX B
NOMINAL JOINT PARAMETERS & OPTIMIZATION BOUNDS
FOR EXPERIMENTAL MARKER DATA
Table B-1. Nominal right hip joint parameters and optimization bounds for experimental
marker data.
Right Hip Joint
Parameter
pi (cm)
p2 (cm)
p3 (cm)
p4 (cm)
p5 (cm)
p6 (cm)
Nominal
-5.931423
-9.166744
8.627524
0
0
0
Lower Bound
-20.220759
-20.220759
0
-14.289337
-11.054015
-8.627524
Upper Bound
0
0
20.220759
5.931423
9.166744
11.593235
Table B-2. Nominal right knee joint parameters and optimization bounds for
experimental marker data.
Right Knee Joint
Parameter
pi ()
P2 ()
P3 ()
P4()
P5 (cm)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Nominal
0
0
-4.070601
1.541414
0
-39.211319
0
0
0
Lower Bound
-30
-30
-34.070601
-28.458586
-7.356876
-46.568195
-7.356876
-7.356876
-7.356876
Upper Bound
30
30
25.929399
31.541414
7.356876
-31.854442
7.356876
7.356876
7.356876
Table B-3. Nominal right ankle joint parameters and optimization bounds for
experimental marker data.
Right Ankle Joint
Parameter
pi ()
P2 (o)
P3 (o)
P4()
P5 (0)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Pio (cm)
Pll (cm)
P12 (cm)
Nominal
8.814964
0
26.890791
23
42
0
-41.131554
0
-1
9.113839
3.900829
1.116905
Lower Bound
-21.185036
-30
-3.109209
-7
12
-5.662309
-46.793862
-5.662309
-5.662309
3.451530
-1.761479
-4.545403
Upper Bound
38.814964
30
56.890791
53
72
5.662309
-35.469245
5.662309
0
14.776147
9.563138
6.779214
APPENDIX C
NOMINAL & OPTIMUM JOINT PARAMETERS FOR SYNTHETIC MARKER
DATA WITHOUT NOISE
Table C-1. Nominal and optimum right hip joint parameters for synthetic marker data
without noise.
Right Hip Joint .
SNominal Optimized Error
Parameter
pl (cm) -6.022205 -6.022205 0.000000
p2 (cm) -9.307044 -9.307041 0.000003
p3 (cm) 8.759571 8.759578 0.000007
p4 (cm) 0 0.000004 0.000004
p5 (cm) 0 0.000015 0.000015
p6 (cm) 0 -0.000008 0.000008
Table C-2. Nominal and optimum right knee joint parameters for synthetic marker data
without noise.
Right Knee Joint
Parameter
pi ()
P2 ()
P3 ()
P4()
P5 (cm)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Nominal
0
0
-5.079507
16.301928
0
-37.600828
0
0
0
Optimized
-0.040222
-0.051509
-5.050744
16.242914
-0.009360
-37.589068
-0.014814
-0.002142
-0.000189
Error
0.040222
0.051509
0.028763
0.059015
0.009360
0.011760
0.014814
0.002142
0.000189
Table C-3. Nominal and optimum right ankle joint parameters for synthetic marker data
without noise.
Right Ankle Joint
Parameter
pi(0)
P2 (o)
P3 (o)
P4()
P5 (0)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Pio (cm)
Pll (cm)
P12 (cm)
Nominal
18.366935
0
40.230969
23
42
0
-39.973202
0
-1
8.995334
4.147543
0.617217
Optimized
18.364964
-0.011809
40.259663
23.027088
42.002080
0.000270
-39.972852
-0.000287
-1.000741
8.995874
4.147353
0.616947
Error
0.001971
0.011809
0.028694
0.027088
0.002080
0.000270
0.000350
0.000287
0.000741
0.000540
0.000190
0.000270
APPENDIX D
NOMINAL & OPTIMUM JOINT PARAMETERS FOR SYNTHETIC MARKER
DATA WITH NOISE
Table D-1. Nominal and optimum right hip joint parameters for synthetic marker data
with noise.
Right Hip Joint .
SNominal Optimized Error
Parameter
pi (cm) -6.022205 -5.854080 0.168125
p2 (cm) -9.307044 -9.434820 0.127776
p3 (cm) 8.759571 8.967520 0.207949
p4 (cm) 0 0.092480 0.092480
p5 (cm) 0 -0.180530 0.180530
p6 (cm) 0 0.191050 0.191050
Table D-2. Nominal and optimum right knee joint parameters for synthetic marker data
with noise.
Right Knee Joint
Parameter
pi ()
P2 ()
P3 ()
P4()
P5 (cm)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Nominal
0
0
-5.079507
16.301928
0
-37.600828
0
0
0
Optimized
-3.295650
-1.277120
-5.604100
12.763780
0.375600
-37.996910
0.489510
0.144040
-0.204420
Error
3.295650
1.277120
0.524593
3.538148
0.375600
0.396082
0.489510
0.144040
0.204420
Table D-3. Nominal and optimum right ankle joint parameters for synthetic marker data
with noise.
Right Ankle Joint
Parameter
pi(0)
P2 (o)
P3 (o)
P4()
P5 (0)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Pio (cm)
Pll (cm)
P12 (cm)
Nominal
18.366935
0
40.230969
23
42
0
-39.973202
0
-1
8.995334
4.147543
0.617217
Optimized
15.130096
8.007498
32.975096
23.122015
42.038733
-0.398360
-39.614220
-0.755127
-2.816943
10.210540
3.033673
-0.190367
Error
3.236838
8.007498
7.255873
0.122015
0.038733
0.398360
0.358982
0.755127
1.816943
1.215206
1.113870
0.807584
APPENDIX E
NOMINAL & OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE
EXPERIMENTAL MARKER DATA
Table E-1. Nominal and optimum right hip joint parameters for multi-cycle experimental
marker data.
Right Hip Joint
Parameter
pi (cm)
p2 (cm)
p3 (cm)
p4 (cm)
p5 (cm)
p6 (cm)
Nominal
-5.931423
-9.166744
8.627524
0
0
0
Optimized
-7.518819
-9.268741
8.857706
-2.123433
0.814089
1.438188
Improvement
1.587396
0.101997
0.230182
2.123433
0.814089
1.438188
Table E-2. Nominal and optimum right knee joint parameters for multi-cycle
experimental marker data.
Right Knee Joint
Parameter
pi ()
P2 ()
P3 ()
P4()
P5 (cm)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Nominal
0
0
-4.070601
1.541414
0
-39.211319
0
0
0
Optimized
-0.586205
14.854951
-2.724374
2.404475
-1.422101
-39.611720
-0.250043
-0.457104
1.471656
Improvement
0.586205
14.854951
1.346227
0.863061
1.422101
0.400401
0.250043
0.457104
1.471656
Table E-3. Nominal and optimum right ankle joint parameters for multi-cycle
experimental marker data.
Right Ankle Joint
Parameter
pi(0)
P2 (o)
P3 (o)
P4()
P5 (0)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Pio (cm)
Pll (cm)
P12 (cm)
Nominal
8.814964
0
26.890791
23
42
0
-41.131554
0
-1
9.113839
3.900829
1.116905
Optimized
16.640499
9.543288
27.359342
13.197304
45.259512
1.650689
-41.185800
-1.510034
-2.141939
11.244080
3.851262
0.283095
Improvement
7.825535
9.543288
0.468551
9.802696
3.259512
1.650689
0.054246
1.510034
1.141939
2.130241
0.049567
0.833810
APPENDIX F
NOMINAL & OPTIMUM JOINT PARAMETERS FOR FIRST ONE-HALF-CYCLE
EXPERIMENTAL MARKER DATA
Table F-1. Nominal and optimum right hip joint parameters for first one-half-cycle
experimental marker data.
Right Hip Joint
Parameter
pi (cm)
p2 (cm)
p3 (cm)
p4 (cm)
p5 (cm)
p6 (cm)
Nominal
-5.931423
-9.166744
8.627524
0
0
0
Optimized
-7.377948
-9.257734
8.124560
-2.050133
0.813034
0.656323
Improvement
1.446525
0.090990
0.502964
2.050133
0.813034
0.656323
Table F-2. Nominal and optimum right knee joint parameters for first one-half-cycle
experimental marker data.
Right Knee Joint
Parameter
pi ()
P2 ()
P3 ()
P4()
P5 (cm)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Nominal
0
0
-4.070601
1.541414
0
-39.211319
0
0
0
Optimized
7.621903
12.823259
-0.642569
11.252668
-1.217316
-38.611100
-1.252732
-0.003903
1.480035
Improvement
7.621903
12.823259
3.428032
9.711254
1.217316
0.600219
1.252732
0.003903
1.480035
Table F-3. Nominal and optimum right ankle joint parameters for first one-half-cycle
experimental marker data.
Right Ankle Joint
Parameter
pi(0)
P2 (o)
P3 (o)
P4()
P5 (0)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Pio (cm)
Pll (cm)
P12 (cm)
Nominal
8.814964
0
26.890791
23
42
0
-41.131554
0
-1
9.113839
3.900829
1.116905
Optimized
-15.959751
-4.522393
18.986137
28.588479
36.840527
3.624386
-43.537980
-3.370814
-2.246233
12.155750
0.488739
-1.207070
Improvement
24.774715
4.522393
7.904654
5.588479
5.159473
3.624386
2.406426
3.370814
1.246233
3.041911
3.412090
2.323975
APPENDIX G
NOMINAL & OPTIMUM JOINT PARAMETERS FOR SECOND ONE-HALF-CYCLE
EXPERIMENTAL MARKER DATA
Table G-1. Nominal and optimum right hip joint parameters for second one-half-cycle
experimental marker data.
Right Hip Joint
Parameter
pi (cm)
p2 (cm)
p3 (cm)
p4 (cm)
p5 (cm)
p6 (cm)
Nominal
-5.931423
-9.166744
8.627524
0
0
0
Optimized
-7.884120
-10.160573
9.216565
-2.935484
0.313918
1.936742
Improvement
1.952697
0.993829
0.589041
2.935484
0.313918
1.936742
Table G-2. Nominal and optimum right knee joint parameters for second one-half-cycle
experimental marker data.
Right Knee Joint
Parameter
pi ()
P2 ()
P3 ()
P4()
P5 (cm)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Nominal
0
0
-4.070601
1.541414
0
-39.211319
0
0
0
Optimized
7.216444
12.986174
-0.228411
10.970612
-1.300621
-38.785646
-1.190227
-0.130610
1.293016
Improvement
7.216444
12.986174
3.842190
9.429198
1.300621
0.425673
1.190227
0.130610
1.293016
Table G-3. Nominal and optimum right ankle joint parameters for second one-half-cycle
experimental marker data.
Right Ankle Joint
Parameter
pi(0)
P2 (o)
P3 (o)
P4()
P5 (0)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Pio (cm)
Pl (cm)
P12 (cm)
Nominal
8.814964
0
26.890791
23
42
0
-41.131554
0
-1
9.113839
3.900829
1.116905
Optimized
31.399921
1.211118
51.518589
26.945919
45.021534
-3.971358
-36.976040
-0.154441
-3.345873
7.552444
7.561219
1.108033
Improvement
22.584957
1.21112
24.627798
3.945919
3.021534
3.971358
4.155514
0.154441
2.345873
1.561395
3.660390
0.008872
APPENDIX H
OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE & FIRST
ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA
Table H-1. Optimum right hip joint parameters for multi-cycle and first one-half-cycle
experimental marker data.
Right Hip Joint
Parameter
pi (cm)
p2 (cm)
p3 (cm)
p4 (cm)
p5 (cm)
p6 (cm)
Multi-Cycle
Optimized
-7.518819
-9.268741
8.857706
-2.123433
0.814089
1.438188
First-Half-Cycle
Optimized
-7.377948
-9.257734
8.124560
-2.050133
0.813034
0.656323
Difference
0.140871
0.011007
0.733146
0.073300
0.001055
0.781865
Table H-2. Optimum right knee joint parameters for multi-cycle and first one-half-cycle
experimental marker data.
Right Knee Joint
Parameter
pi ()
P2 ()
P3 ()
P4()
P5 (cm)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Multi-Cycle
Optimized
-0.586205
14.854951
-2.724374
2.404475
-1.422101
-39.611720
-0.250043
-0.457104
1.471656
First-Half-Cycle
Optimized
7.621903
12.823259
-0.642569
11.252668
-1.217316
-38.611100
-1.252732
-0.003903
1.480035
Difference
8.208108
2.031692
2.081805
8.848193
0.204785
1.000620
1.002689
0.453201
0.008379
Table H-3. Optimum right ankle joint parameters for multi-cycle and first one-half-cycle
experimental marker data.
Right Ankle Joint
Parameter
pi ()
P2 (o)
P3 (o)
P4()
P5 (0)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Pio (cm)
Pl (cm)
P12 (cm)
Multi-Cycle
Optimized
16.640499
9.543288
27.359342
13.197304
45.259512
1.650689
-41.185800
-1.510034
-2.141939
11.244080
3.851262
0.283095
First-Half-Cycle
Optimized
-15.959751
-4.522393
18.986137
28.588479
36.840527
3.624386
-43.537980
-3.370814
-2.246233
12.155750
0.488739
-1.207070
Difference
32.600250
14.065681
8.373205
15.391175
8.418985
1.973697
2.352180
1.860780
0.104294
0.911670
3.362523
1.490165
APPENDIX I
OPTIMUM JOINT PARAMETERS FOR MULTI-CYCLE & SECOND
ONE-HALF-CYCLE EXPERIMENTAL MARKER DATA
Table I-1. Optimum right hip joint parameters for multi-cycle and second one-half-cycle
experimental marker data.
Right Hip Joint
Parameter
pi (cm)
p2 (cm)
p3 (cm)
p4 (cm)
p5 (cm)
p6 (cm)
Multi-Cycle
Optimized
-7.518819
-9.268741
8.857706
-2.123433
0.814089
1.438188
Second-Half-Cycle
Optimized
-7.884120
-10.160573
9.216565
-2.935484
0.313918
1.936742
Difference
0.365301
0.891832
0.358859
0.812051
0.500171
0.498554
Table 1-2. Optimum right knee joint parameters for multi-cycle and second
one-half-cycle experimental marker data.
Right Knee Joint
Parameter
pi ()
P2 ()
P3 ()
P4()
P5 (cm)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Multi-Cycle
Optimized
-0.586205
14.854951
-2.724374
2.404475
-1.422101
-39.611720
-0.250043
-0.457104
1.471656
Second-Half-Cycle
Optimized
7.216444
12.986174
-0.228411
10.970612
-1.300621
-38.785646
-1.190227
-0.130610
1.293016
Difference
7.802649
1.868777
2.495963
8.566137
0.121480
0.826074
0.940184
0.326494
0.178640
Table 1-3. Optimum right ankle joint parameters for multi-cycle and second
one-half-cycle experimental marker data.
Right Ankle Joint
Parameter
pi ()
P2 (o)
P3 (o)
P4()
P5 (0)
P6 (cm)
P7 (cm)
P8 (cm)
P9 (cm)
Pio (cm)
Pl (cm)
P12 (cm)
Multi-Cycle
Optimized
16.640499
9.543288
27.359342
13.197304
45.259512
1.650689
-41.185800
-1.510034
-2.141939
11.244080
3.851262
0.283095
Second-Half-Cycle
Optimized
31.399921
1.211118
51.518589
26.945919
45.021534
-3.971358
-36.976040
-0.154441
-3.345873
7.552444
7.561219
1.108033
Difference
14.759422
8.332170
24.159247
13.748615
0.237978
5.622047
4.209760
1.355593
1.203934
3.691636
3.709957
0.824938
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BIOGRAPHICAL SKETCH
Jeffrey A. Reinbolt was born on May 6, 1974 in Bradenton, Florida. His parents
are Charles and Joan Reinbolt. He has an older brother, Douglas, and an older sister,
Melissa. In 1992, Jeff graduated salutatorian from Southeast High School, Bradenton,
Florida. After completing his secondary education, he enrolled at the University of
Florida supported by the Florida Undergraduate Scholarship and full-time employment at
a local business. He earned a traditional 5-year engineering degree in only 4 years. In
1996, Jeff graduated with honors receiving a Bachelor of Science degree in engineering
science with a concentration in biomedical engineering. He used this foundation to assist
in the medical device development and clinical research programs of Computer Motion,
Inc., Santa Barbara, California. In this role, Jeff was Clinical Development Site Manager
for the Southeastern United States and he traveled extensively throughout the United
States, Europe, and Asia collaborating with surgeons and fellow medical researchers. In
1998, Jeff married Karen, a student he met during his undergraduate studies. After more
than 4 years in the medical device industry, he decided to continue his academic career at
the University of Florida. In 2001, Jeff began his graduate studies in Biomedical
Engineering and he was appointed a graduate research assistantship in the Computational
Biomechanics Laboratory. He plans to continue his graduate education and research
activities through the pursuit of a Doctor of Philosophy in mechanical engineering. Jeff
would like to further his creative involvement in problem solving and the design of
solutions to overcome healthcare challenges.
