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This is to certify that the
dissertation entitled
TWO-DIMENSIONAL QUASl-STATIC KNEE MODEL
FOR THE ESTIMATION OF LIGAMENT AND QUADRICEPS
FORCES AS A FUNCTION OF KNEE FLEXTION

presented by

Claudia Alejandra Angeli

has been accepted towards fulfillment
of the requirements for

 

Ph.D. degreein Kinesiology

 

Z/ZQIW Wm

Major professor

 

Date mat] IL/ 02 00,
d I

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

PLACE IN RETURN Box to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 chIRC/DatoDquBS-pns

"Jf/fi

 

FC

 

 

TWO-DIMENSIONAL QUASI-STATIC KNEE MODEL
FOR THE ESTIMATION OF LIGAMENT AND QUADRICEPS
FORCES AS A FUNCTION OF KNEE FLEXION
By

Claudia Alejandra Angeli

A DISSERTATION
Submitted to
Michigan State University
In partial fulfillment of the requirements
For the degree of
DOCTOR OF PHILOSOPHY
Department of Kinesiology

2001

ESTI

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ABSTRACT
TWO-DIMENSIONAL QUASI-STATIC KNEE MODEL FOR THE
ESTIMATION OF LIGAMENT AND QUADRICEP FORCES As A FUNCTION
OF KNEE FLEXION
By

Claudia Alejandra Angeli

The knee joint is one of the most commonly injured joints in sport activities. A
two-dimensional quasi-static knee model was developed to estimate the forces sustained
by the ligaments and generated by the quadriceps as a function of knee flexion. The tibio-
femoral and patello—femoral joints were defined mathematically and modeled as
frictionless joints. The anterior and posterior cruciate ligaments and the collateral
ligament were modeled as single fiber extensible units. The instantaneous center of
rotation was used in the estimation of the rolling and sliding characteristics of the knee
joint. External forces were used as input parameters to the model. The model was tested
using kinematic and kinetic data obtained during squatting for three different conditions:
normal, ACL-deficient and ACL-reconstructed. The results obtained with the model
showed the expected increase in anterior-posterior motion of the femur relative to the
tibia in the ACL-deficient knee. Laxity values were higher for the ACL-deficient knee

and had return to normal following the ACL-reconstmction.

Copyright by
CLAUDIA ALEIANDRA ANGELI
2001

To my family, for their constant support and encouragement.
And to Dr. Dianne Ulibarri, for inspiring me to reach for
excellence in every aspect of my life.

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ACKNOWLEDGMENTS

Special Thanks to:

Dr. Robert SoutaS-Little, for giving me the tools to be successful in the field of
biomechanics.

Patricia Soutas-Little, for the constant support and encouragement.

All my coworkers and students at the Biomechanics Evaluation Laboratory, for the great
times and constant support.

My committee members, for all the great advice.

TABLE OF CONTENTS

LIST OF TABLES

 

LIST OF FIGURES

 

KEY TO SYMBOLS

 

INTRODUCTION
I. Need for the Study
11. Purpose of the Study
111. Assumptions and Limitations
IV. Significance of the Study

 

 

 

 

 

REVIEW OF LITERATURE
1. Mathematical Models
11. Knee Laxity and Restraints
III. Ligament Mechanics
IV. Neuromuscular Activity

 

 

 

 

 

ANALYTICAL METHODS
I. Tibio-femoral Joint Definition
11. Patello-femoral Joint Definition
III. Ligament Forces
IV. Mechanical Behavior of the Tibio-femoral Joint
V. Calculation of Tibio-femoral Contact Force
VI. Calculation of Internal Moments due to Ligament Forces
VII. Calculation of External Forces
VIII. Calculation of PatellO-femoral Internal Moments _
IX. Mathematical Solution

 

 

 

 

 

 

EXPERIMENTAL METHODS
I. Subjects and Data Collection
II. Targeting protocol
III. Computational Model

 

 

 

 

RESULTS AND DISCUSSION

 

CONCLUSIONS
I. Limitations
II. Suggestions for Future Studies

 

 

 

APPENDIX A
Main Program Code

 

 

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APPENDIX B

 

Mathcad Template 1
Mathcad Template 2

APPENDIX C

 

 

 

Mathcad Template 3

APPENDIX D

 

 

Ligament Forces

 

REFERENCES

 

vii

128
129
134

137
138

144
145

147

Tablc 3: L".

——

 

LIST OF TABLES

Table l: Stiffness coefficient for the modeled ligaments. .......................... 37

Table 2: Length ratio values for the modeled ligaments at full extension. ...... 39

Table 3: Ultimate load of modeled ligaments and patella tendon. ................ 40

viii

 

 

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LIST OF FIGURES

Figure l: Graphical representation of the knee model ............................... 4
Figure 2: Four-bar linkage system .................................................... 7
Figure 3: Line of application of the contact force for (a) flat tibia]

surface and (b) concave surface ............................................ 8
Figure 4: Cruciate ligament strains as a function of flexion angle ................. 14
Figure 5: Load-deformation curve for ligaments .................................... 17

Figure 6: Involute circle with a 90 counterclockwise rotation,

representing medial view of the femoral condyle ........................ 24
Figure 7: Involute circle with identification of slope zero positions. ............... 26
Figure 8: Involute circle with identification of infinity slope positions. ............ 27

Figure 9: Representation of the femoral condyle in full extension in the
unprimed and primed coordinate systems. ................................. 28

Figure 10: Representation of position vectors to locate the insertion of the
ACL on the femoral condyle. ................................................ 29

Figure 11: Graphical representation of the tibiofemoral joint with
ligamentous constraints (Full extension). .................................. 29

Figure 12: Graphical representation of the tibiofemoral joint at
60 degrees of flexion, and the relative position of

the ACL, PCL and LCL. ...................................................... 31
Figure 13: Patellar geometry. .............................................................. 33
Figure 14: Patello-femoral and tibiO-femoral joints at full extension. ............... 34
Figure 15: Patello-femoral equilibrium mechanism. ................................... 35
Figure 16: Targeting protocol for joint center calculation. ........................... 40

Figure 17: Position vectors involved in the calculation of
the instantaneous knee joint center. ......................................... 4O

ix

Figure 18: Rolling and Sliding of a circular disk (a and b respectively). ........... 43

Figure 19: Assumption of circular discs representation of the femoral

condyle geometry. ............................................................ 44
Figure 20: Representation of the ligament angle for the ACL. ....................... 47
Figure 21: Representation of the ligament angle for the PCL. ....................... 48
Figure 22: Representation of the ligament angle for the CL. ......................... 48

Figure 23: Graphical representation of the calculations of ACL internal
moments. ....................................................................... 50

Figure 24: Graphical representation of the calculations of PCL internal
moments. ....................................................................... 51

Figure 25: Graphical representation of the calculatic.-.. of CL internal

moments. ...................................................................... 51
Figure 26: Targeting configuration for the thigh and shank segments. 1 ............ 59
Figure 27: Thigh and shank segmental coordinate systems. ........................ 62
Figure 28: Rotational sequence for calculation of Euler angles. .................. 64
Figure 29: Joint coordinate system. ................................................... 65

Figure 30: Graphical Representation of theTransverse view
of the femoral condyles and depth-width relationship. ................. 67

Figure 31: Patello-femoral equilibrium mechanism. ................................. 71

Figure 32: Unit vector representation in the patello-femoral
mechanism. ................................................................... 72

Figure 33: Contact point displacement comparison. ................................. 74

Figure 34: Laxity Curve: Ligament horizontal force vs. Anterior-posterior
displacement for a normal knee. .......................................... 76

Figure 35: Laxity Curve: Ligament horizontal force vs. Anterior-posterior
displacement for an ACL deficient knee. ................................ 77

Figure 36: Laxity Curve: Ligament horizontal force vs. Anterior—posterior
displacement for an ACL reconstructed knee. ........................... 77

Figure 37: Patello-femoral Mechanism angles vs. Knee flexion.

Normal knee. ................................................................... 79
Figure 38: Patello—femoral Mechanism angles vs. Knee flexion.
ACL-deficient knee. ............................................................ 80
Figure 39: PatellO—femoral Mechanism angles vs. Knee flexion.
ACL-reconstructed knee. ....................................................... 80
Figure 40: Patellar Tendon to Quadriceps Tendon force ratio
for the normal knee. ............................................................ 83
Figure 41: Patellar Tendon to Quadriceps Tendon force ratio
for the ACL-deficient knee. .................................................. 83
Figure 42: Patellar Tendon to Quadriceps Tendon force ratio
for the ACL-reconstructed knee. ............................................. 84
Figure 43: Strain comparison for the ACL. .............................................. 86
Figure 44: Strain comparison for the PCL. ............................................... 86
Figure 45: Strain comparison for the CL. ................................................ 87
Figure 46: Quadriceps and Patellar Tendon forces for the normal knee.
(a) Horizontal Forces (b) Vertical Forces. .................................. 90
Figure 47: Quadriceps and Patellar Tendon forces for the ACL-deficient knee.
(a) Horizontal Forces (b) Vertical Forces. ................................. 92
Figure 48: Quadriceps and Patellar Tendon forces for the
ACL-reconstructed knee. (a) Horizontal Forces (b) Vertical Forces. ....94
Figure 1A: Ligament forces for the normal knee. ..................................... 145
Figure 2A: Ligament forces for the ACL-deficient knee. ............................ 146
Figure 3A: Ligament forces for the ACL-reconstructed knee. ...................... 146

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KEY TO SYMBOLS

X variable to define the involute circle

Y variable to define the involute circle

radius of the involute circle

parameter of dependency for the contraint of the involute circle
parametric equation of the involute with a prescribed rotation
parametric equation of the involute with a prescribed rotation

90 degree rotation of the involute

angle of knee flexion

x-coordinate for the insertion point of the ACL on the femur at full
extension

y—coordinate for the insertion point of the ACL on the femur at full
extension

x-coordinate for the insertion point of the PCL on the femur at full
extension

y-coordinate for the insertion point of the PCL on the femur at full
extension

x-coordinate for the insertion point of the CL on the femur at full
extension

y-coordinate for the insertion point of the CL on the femur at full
extension

x-coordinate for the insertion point of the ACL on the femur at 0 degrees
of flexion

y-coordinate for the insertion point of the ACL on the femur at 0 degrees
of flexion

x-coordinate for the insertion point of the PCL on the femur at 9 degrees
of flexion

y-coordinate for the insertion point of the PCL on the femur at O degrees
of flexion

x-coordinate for the insertion point of the CL on the femur at 9 degrees of
flexion

y-coordinate for the insertion point of the CL on the femur at 0 degrees of
flexion

xii

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patellar tendon angle

angle of inclination of the patella

angle between quadriceps tendon and patello-femoral force
angle between the patellar tendon and patello-femoral force

ligament force

stiffness coefficient

change in length of the ligament

ligament length

original ligament length

ligament length of the ACL

ligament length of the PCL

ligament length of the CL

x-coordinate for the origin of the ACL on the tibia

x-coordinate for the origin of the PCL on the tibia

x-coordinate for the origin of the CL on the tibia

y-coordinate for the origin of the CL on the tibia

y-coordinate for the position of the tibial plateau

length ratio of the ligaments

subscript used to differentiate among ligaments

position vector

angular velocity

linear velocity

sliding distance of the femur relative to the tibia

rolling distance of the femur relative to the tibia

transformation matrix

normal unit vector

relative vector for the ACL along the y-axis

relative vector for the PCL along the y-axis

relative vector for the CL along the y-axis

angle between line of action of the ACL and the horizontal

angle between line of action of the PCL and the horizontal

angle between line of action of the CL and the horizontal

ligament force for jth ligament along the x-axis

ligament force for jth ligament along the y-axis

perpendicular distance for y—component of ACL force to ICR location
perpendicular distance for x-component of ACL force to ICR location
perpendicular distance for y-component of PCL force to ICR location
perpendicular distance for x-component of PCL force to ICR location
perpendicular distance for y-component of CL force to ICR location
perpendicular distance for x-component of CL force to ICR location
flexion/extension moment for the jth ligament

xiii

 

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force plate moment

moment parallel

moment perpendicular
Projection of force vector
forceplate moment about the x-axis

forceplate moment about the y-axis

resultant external force vector
external force along the x-axis

external force along the y-axis

external force along the z-axis

height from the forceplate surface to the center of the forceplate

location of the COP along the x-axis

location of the COP along the y-axis

location of the instantaneous center of rotation in the x-axis of the inertial
coordinate system

location of the instantaneous center of rotation in the y-axis of the inertial
coordinate system

location of the instantaneous center of rotation in the z-axis of the inertial
coordinate system

flexion/extension external moment
quadriceps tendon force

quadriceps tendon force along the x-axis
quadriceps tendon force along the y-axis
patellar tendon force

patellar tendon force along the x-axis

patellar tendon force along the y-axis
patello-femoral contact force

patello-femoral contact force along the x-axis
patello-femoral contact force along the y-axis
moment arm along the x-axis

moment arm along the y-axis

sum for forces along the x-axis

sum for forces along the y-axis

sum for forces about the z-axis

unit vector along the x-axis
unit vector along the y-axis

unit vector along the z-axis
position vector for anatomical target

xiv

position vector for dynamic target

x-axis in segmental coordinate system of the thigh
y-axis in segmental coordinate system of the thigh
z-axis in segmental coordinate system of the thigh
x-axis in segmental coordinate system of the shank
y-axis in segmental coordinate system of the shank
z-axis in segmental coordinate system of the shnk
posterior axis of the shank

position vector for the medial condyle

position vector for the lateral condyle

position vector for the greater trochanter

position vector for the proximal target on the shank
position vector for the distal target on the shank
position vector for the posterior target on the shank
floating axis base vector for the joint coordinate system

base vector for the joint coordinate system
base vector for the joint coordinate system

XV

INTRODUCTION

Injuries to the knee joint are among the most common injuries sustained during
sport activities. The ligaments of the knee joint are required to support high tensile loads,
and are the most often injured soft tissues in the knee joint due to the high loads sustained
during the restriction of excessive motion. Fifty percent of the knee injuries are due to
trauma to the anterior cruciate ligament (ACL) and the medial collateral ligament (MCL)
(Nicholas & Hershman, 1995).

Both cruciate ligaments are constraints for the rolling and sliding motion of the
femur relative to the tibia. Even though the knee has been classified as a hinge joint, the
articulation of the femur and the tibia allows for six degrees of freedom. The three
rotational movements and the three translational movements are coupled throughout the
full range of motion allowed by the joint. Complete loss of joint stability is observed with
a third degree sprain of the ligament in which the ligament fibers are fully torn (Macnicol,
1986). The primary objective of treatment to an injured knee should be to restore the

normal joint mechanics and stability and to prevent premature degeneration of the joint.

The soft tissues that cross the knee joint act as constraints to the kinematics of the
joint. While the ligaments are considered the primary stabilizers of the knee joint,
muscles also provide stabilization to the joint during dynamic activities. An injury to the
ACL compromises the stability of the joint by allowing excessive anterior translation of

the tibia with respect to the femur. Compromised stability due to an injury to the ACL

may be controlled by increased muscle activity.

I. Need for the Study

The knee joint has been one of the most widely researched joints of the human
body. Mathematical models have been developed to investigate various aspects of knee
joint mechanics. The inclusion of muscle, ligamentous and cartilagenous tissue increases
the number of unknown parameters in the model. This increase in unknowns is not
matched by an increase in the number of motion or equilibrium equations. The limited
number of motion equations with respect to the higher number of unknowns creates the
complex situation of indeterminacy. Constraint equations can be developed to increase
the number of equations and solve for some of the unknown variables; however,
development of these equations becomes a complex task.

A number of mathematical models have been developed to evaluate the
interaction of the cruciate ligaments as motion constraints. However, limited work has
been performed in the evaluation of the interaction between 1i gamentous and muscle
constraints to knee motion. An anterior load on the tibia has been shown to stress the
ACL and produce an anterior displacement of the tibia relative to the femur. Contraction
of the hamstrings group would produce a load in the posterior direction, reducing the
stress on the anterior cruciate ligament and controlling the translational motion. The
interactions of the ligamentous constraints with the constraints provided by muscle
contraction are of great significance in the prediction of the rotational and translational
motions of the knee joint. The abnormal joint mechanical behavior produced by an
injured ACL can be compensated by the adaptation of the neuromuscular system to assist

in the stabilization of the knee joint (Collins & O’Connor, 1991).

 

   

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II. Purpose of the Study

The wide range of research focused on the knee is indicative of the complex
mechanics exhibited at the joint. The purpose of this study is to develop a two-
dimensional quasi-static knee model for the prediction of quadriceps muscle forces in
relation to ligament forces and angle Of knee flexion. The model will predict the internal
forces on the constraints of the joint, depending on the joint position and the external

forces applied to the system.

111. Assumptions and limitations

Due to the complexity of the knee joint motion and the limited number of
dynamic equations, several assumptions have been made to simplify the requirements of
the model. The relative motion of the femur and the tibia will be examined only in the
sagittal plane. The motion out of the sagittal plane is minimal and, therefore, an accurate
two-dimensional model can be developed. For the two-dimensional model, the medial
collateral ligament (MCL) will be combined with the lateral collateral ligament (LCL)
and only one collateral ligament (CL) will be modeled. The origin and insertion locations
of the CL will follow those of the LCL, due to the more important role of the LCL in the
restraint of both anterior and posterior translations (Daniel, Akeson & O’Connor, 1990).
Ligamentous constraints will be modeled as single nonlinear springs. From the evaluation
of previous research, it was determined that a flat tibial plateau will allow for an accurate
prediction of the translational values and the femur will be modeled as an involute circle
(Figure 1). The joint will be assumed to be frictionless and any contributions of the

meniscii in the joint mechanics will be ignored. The patella will be included in the model

 

 

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for the purpose of transferring the force from the quadriceps tendon to the patellar tendon.
The patella will be modeled as a rectangle with biarticulating surfaces, allowing the
representation of the shift in contact surfaces between the femur and the patella at high
flexion angles. Contact between the patella and the femoral condyles will be assumed to
be frictionless. The quadriceps tendon and the patellar tendon will be considered

inextensible cords.

 

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(a) Femoral Condyle

(b) Tibial Plateau
(g) (e) (c) ACL

(d) PCL

(e) CL

(0 Patella

(g) Patellar Tendon

(h) Quadriceps Force

 

 

 

Figure l: Graphical representation of the knee model.

IV. Significance of the study
The findings of this study will contribute to the field of orthopaedics as well as an
aid to injury rehabilitation. The mathematical model will allow clinicians to determine the

stresses applied to the injured ligament including the contribution of the quadriceps group

 

 

 

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muscle contractions at specified flexion angles. The ability to determine the internal
forces acting on the joint can aid in the identification of beneficial joint positions for
rehabilitation exercises, as well as maximum loads that can be supported by the soft
structures. The mathematical model also will help clinicians better understand the
mechanics of the knee joint and the interactions of primary and secondary stabilizers Of

the joint.

REVIEW OF LITERATURE

The knee joint is one of the most commonly injured joints in athletic performance.
The complexity of the kinematics associated with the knee joint has led to a large number
of research studies conducted in this area. Mathematical models have been developed to
describe the kinematics of the knee joint under simplified conditions. The mechanical
properties and mechanical behavior of the joint constraints determine the overall
kinematics of the joint. Injury to the joint constraints compromises the normal mechanics
of the knee joint. Joint function has not been assessed in terms of the joint kinematics and
interaction of soft tissue components. The objective of this literature review is to focus
primarily on the mathematical modeling done on the knee joint and secondarily on the

pertinent clinical research associated with ligamentous injuries and rehabilitation.

1. Mathematical Models

Two- and three-dimensional mathematical models of the knee joint have been
widely used to describe the joint mechanics. Due to the complicated mechanics exhibited
at the knee joint, mathematical models are simplified by describing limited aspects of the
joint motion. Differences in the results obtained with similar analytical models are due to
the wide variety of measurement techniques used to obtain the input parameters. These
differences in results lead to the reconsideration of the importance of some parameters as
predictors in the model. The literature review was focused on models that examined the
mechanics of the knee joint and included the interaction of the ligaments and muscles

surrounding the joint for stabilization purposes.

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Simplified models have been successfully used to describe the interaction between
two body segments. The four-bar kinematic linkage is one of the simplest models
developed to describe knee joint motion (Daniel et al., 1990) (Figure 2). The linkage
system changes geometry as the joint moves through the range of motion in the sagittal
plane. As the knee flexes and extends, rolling and sliding actions of the femur over the
tibia occur. This model gives an inextensible representation of the cruciate ligaments. The
validity of the model has been questioned by other authors (Hefty & Grood, 1983;
Hirokawa & Tsuruno, 1997; Lanir, 1983). Throughout the range of motion, the angles in
the four-bar model change, and at any flexion angle point of intersection of the two
ligament bars is considered the instantaneous center of rotation of the joint. Due to the
inextensible characteristics of the ligaments, the femur was shown to slide forward on the
tibia and to roll backward as the knee flexes. The opposite actions were observed during

extension of the knee joint (O’Connor, Shercliff, Biden & Goodfellow, 1989).

Femur

V B

A

 

C Tibia D

Figure 2: Four-bar linkage system

The load supported by the cruciate ligaments has been demonstrated to be linked
to the external forces applied to the femur and tibia, as well as to the geometry of the

tibial plateau. Chan and Seedhom (1995) investigated the effects of the tibial plateau's

geometry on I?
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geometry on the prediction of forces on both cruciate ligaments. The ACL and posterior
cruciate ligament (PCL) loads were examined under a pure external anterior-posterior (A-
P) force applied to the tibia and also under a pure external axial compressive force
combined with the A-P force using the four-bar kinematic linkage model. The analysis
was performed for concave, convex and flat tibial plateau surfaces. The results of the
different geometric configurations demonstrated that, for a concave surface, the loadings
on the ACL and PCL were decreased when compared to those for the flat surface. This
reduction in the forces sustained by the cruciate ligaments was due to the line of
application of the tibio-femoral contact force (Figure 3). The addition of an axial
compressive force would decrease the loading on the ligaments. The results were opposite
for the convex surface, as the predicted forces for the cruciate ligaments were greater for
the same A-P force. The line of application of the contact force is influenced by the
degree of flexion at the knee joint. Therefore, knee flexion plays a role in the loading of
the cruciate ligaments. The authors concluded that due to the concavity of the medial
compartment and the convexity of the lateral knee compartment, a model with a flat tibial

plateau would closely predict the actual forces on the cruciate ligaments.

PC

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and (b) concave surface

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In a later study, Chan and Seedhorm (1999) Obtained experimental and theoretical
data on the effects of tibia geometry on ligament forces. The results supported their first
study, concluding that a concave tibia provides protection by lowering the tension
imposed on the ligaments during the application of an anterior-posterior force. The
tension values obtained in the experimental results were lower than those Obtained
through theoretical methods. The discrepancy in the results was attributed to the
A limitations and simplifications of the theoretical model.

In a similar study, Imran and O’Connor (1997) examined the effects of tibial
surface geometry and ligament orientations in the estimation of the forces produced at the
anterior and posterior cruciate ligaments. In this study, the four-bar kinematic linkage
model was analyzed under anterior-posterior loading and a loading condition simulating
isometric quadriceps contraction. Ligament loading was found to be affected more by the
changes in tibial surface geometry and the tilt of the tibial plateau in the isometric
quadriceps exercises when than in the A-P loading condition. The curved tibial surfaces
produced an increase in the ligament forces when the horizontal component of both the
contact force and the ligament force acted in opposite directions, and decreased the
ligament force when the two horizontal component forces acted in the same direction.
The flat surface maintained a constant direction for the contact force and therefore, did
not affect the ligament forces produced throughout the range of motion.

Knowledge of the origin and insertion sites for the ligaments is necessary, as fiber
orientation and length are common input parameters to theoretical models. Fuss (1989)
obtained the origin and insertion locations for the cruciate ligaments from cadaver

specimens. The cruciate ligaments are multi-bundle fibers, which undergo tension under

posit-or
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different stress conditions. The representation of these ligaments by a single bar segment
is supported by the guiding bundle theory. The guiding bundles are the fibers that
maintain the same distance between the origin and insertion sites. Other fibers within the
same ligament will undergo tension under different conditions due to the constantly
changing distance between the origin and insertion sites (Fuss, 1989). The functional
position of the knee was described by Fuss as the position where the majority of the fibers
in the ligament are under tension. The functional position for the ACL is full extension of
the knee joint, while the PCL’s functional position is full flexion of the knee joint.

The four-bar linkage system has been shown to be successful in the prediction of
ligament forces and in the description of the interaction between the femur and tibia.
Limitations of this model include the representation of the cruciate ligaments as
inextensible bars as well as the omission of the collateral ligaments and their influence in
joint motion and stability. Zavatsky and O’Connor (1992) investigated the recruitment
pattern of ligament fibers based on the previously described four-bar linkage model
during passive flexion. A neutral fiber was defined by joining a point on the tibial
attachment with a similar point on the femoral attachment that remained at a constant
distance through the range of motion. This representation allowed for a multiple fiber
attachment site to be identified at each bone. Fibers then were mapped between the
attachment sites. The translations and rotations of the bones relative to each other during
flexion of the knee, resulted in shape changes of the ligaments within the joint, which
would have an effect on the prediction of ligament loads.

The identification of multi-bundle ligaments adds accuracy to the model by

separating ligament bundles with different mechanical characteristics. In an attempt to

10

trientatir
tines, an
lateal cc
and orig
research
each of I

.trrnate

EDIE. an

examine the accuracy of the prediction of ligament forces, Mommersteeg, Huiskes,
Blackvoort, Kooloos, Kauer and Maathuis (1996b) develOped a three-dimensional knee
model with multi-bundle ligaments. The bundles were defined according to the
orientations of the ligament fibers in cadaver specimens. The authors defined seven, six,
three, and three bundles for the anterior cruciate, posterior cruciate, medial collateral and
lateral collateral ligaments, respectively. Optimization was used to determine the stiffness
and original length of each ligament bundle. The optimization technique allowed the
researchers to determined the number of bundles that best fit the constraint behavior of
each of the four ligaments (Mommersteeg et al., 1996a). The ligament forces were
estimated as a function of the relative position of the two rigid bodies, the femur and
tibia, and the stiffness of the ligament fibers. The results from the experimental data were
the same as those predicted from the mathematical model. In this study, the knee joint
ligaments were modeled with non-uniform mechanical characteristics and different
bundle orientations. Optimization techniques were used to determine the recruitment of
the different ligament bundles during the range of motion of the joint Mommersteeg et
al., 1996b). A later study demonstrated the use of inverse dynamics to predict
simultaneous forces sustained by the knee ligaments (Mommersteeg et al., 1997). The
method of inverse dynamics allowed the prediction of the load carried by several fiber
bundles of the main ligaments in the knee joint. Even though accurate results were
obtained from this method, the limitations associated with the input parameters and
assumptions would outweigh the advantages of using this method over more simplistic

and similarly accurate methods.

11

 

 

(IV)

[9

Fri

11.]

The loads supported by the ligaments have been shown to be dependent on the
geometric characteristics of the femur and tibia. The forces exerted on the ligaments also
are dependent on the external loads generated by muscle contraction. A sagittal model of
the knee joint was developed by Shelbume and Pandy (1997) to examine the effects Of
muscular contractions and bone geometry on the forces sustained by the ligaments of the
joint. Eleven elastic bundles were used to model the ligaments of the joint. Hill-type
contractile elements were used to model eleven muscles crossing the knee joint. The lines
of pull of the muscles were represented as straight lines except for the gastrocnemious,
senritendinosis and semimembranosis that wrap around the femoral condyles. The authors
were able to determine the range of flexion over which ligaments were loaded relative to

the amount of muscle contraction.

II. Knee Laxity and Restraints

A large number of studies have been performed on cadaver specimens to obtain a
better understanding of the function of each ligament in the overall stabilization of the
knee joint throughout the range of motion. In-situ evaluations of the restraining
characteristics of ligaments are performed by sequentially cutting the ligaments and
evaluating the joint under simulated loading conditions. A greater displacement of the
adjacent bones following the removal of a ligament would imply a stabilization role of
the ligament. Butler, Noyes & Grood (1980) proposed a method of measuring the force
required to sustain a predetermined displacement following the cut of ligaments. The test
simulated the anterior drawer clinical test, considering the cruciate ligaments as primary

restraints and the collateral ligaments as secondary restraints. The authors found an

12

 

exte:

m3tg

Emit:

average restraining force of 333 N when the knee was positioned at 30° of flexion and 5

mm of anterior displacement of the tibia over the femur was allowed. During anterior
drawer tests, the manual force applied is approximately 45 to 50 N. Approximately 85 %
of the anterior displacement of the tibia over the femur is controlled by the anterior
cruciate ligament. Following an injury to the anterior cruciate ligament, the anterior laxity
is increased and the interaction between the secondary restraints is modified to provide
stability to the joint.

In a similar study, Piziali, Seering, Nagel and Schurrnan (1980) tested the forces
exerted on the knee ligaments during medial-latera .g. placements. The results of this
study indicated that the anterior cruciate ligament was the primary restraint for medial
tibial displacement, while the lateral tibial displacement was primarily constrained by the
collateral ligaments. The posterior cruciate ligament is also a restraint for lateral motion,
becoming tense as a result of smaller displacements as compared to the ACL. Piziali et al.
showed the significant coupling effects present during the application of medial-lateral
external forces. Such coupling of ligament restraints is not apparent in the anterior-
posterior displacement of the tibia over the femur, which supports the accuracy of models

limited to motion in the sagittal plane.

1]]. Ligament mechanics

Quantification of the load-deformation curves of the ligaments of the knee is
important in the understanding of the mechanical behavior of the joint. Most information
on the mechanical behavior of ligaments has been obtained from cadaver specimens.

Using strain gauges to evaluate the deformation of the ligaments, the ACL was found to

13

 

be most lax at 35° of flexion (Figure 4). When the coupled motion of flexion and rotation
was evaluated, at a flexion angle Of 30° internal rotation tightened the ACL; external
rotation produced the opposite result. The most lax position for the PCL was also found
to be 35° of flexion (Figure 4). The tibial collateral ligament is most lax during complete
flexion. With the knee positioned at 30° of flexion, the coupled motion of external
rotation and/or abduction increased the strain on the ligament (Kennedy, Hawkins &

Willis, 1977).

 

 

 

 

 

 

 

 

 

 

10"
— ACL
'— PCL
C _
E , L
8 <75
33
l l l I
l I T fi
0 20 40 60 80
Flexion angle
-5 J5

 

Figure 4: Cruciate ligament strains as a function of flexion angle.

The measurement of the cruciate ligament forces at different flexion angles has
been of great interest to researchers (Sakane, Fox, Woo, Livesay, Li & Fu, 1997; Takai,
Woo, Livesay, Adams & Fu, 1993). Studies of force distribution in the ACL take into
account the ligament bundles and possible differences in mechanical properties. The

anterior-medial fibers of the ACL show similar force distribution along the full range of

14

 

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motion when tested under two different values of applied anterior force. However, the
posterior-medial fibers are highly affected by the flexion angle, undergoing their highest

stresses at 15° of flexion (Sakane et al., 1997). These results agree with results presented

by Fuss (1989), where the anterior-medial fibers of the ACL were identified as the
guiding fibers. When three ligament bundles were examined (T akai et al., 1993), the
length of the anterior-medial fibers increased with knee flexion, while a decrease in
length was seen in the posterior-medial fibers. A constant length was maintained by the
intermediate fibers. In this study, Takai and coworkers (1993) used a six degrees of
freedom linkage system to calculate the forces applied to the AC ligament bundles. At
full extension, the load was distributed equally between the anterior and posterior ACL

fibers. At knee flexion angles larger than 45°, up to 95% of the load was carried by the

anterior fibers of the cruciate ligament.

The prediction of ligament force and ligament lengthening due to stress is critical
in the overall description of joint mechanics and equilibrium. The nonlinearity associated
with the ligamentous structures increases the complexity of the mathematical model.
Several models have predicted changes in length by evaluating the relative motion
between the insertion and origin sites during the joint’s range of motion (Hefty & Grood,
1983). These models do not take into account the structural characteristics of the
ligaments and the non-linearity associated with wrapping of the ligaments around bony
structures. In the case of the cruciate ligaments, as the knee flexes and internally rotates
the ligaments wrap around each other. The length of the ligaments can no longer be
determined by the difference between the origin and insertion points. Considering

ligament wrapping would affect the prediction of ligament length, ligament tension and

15

 

.'\I
I
ll

C0:

the direction of the force vector. The authors did not validate the importance of the three-
dimensional model using experimental data; however, they stated that the wrapping effect
would cause considerable changes to be observed in the predicted parameters (Hefty &
Grood, 1983).

Hirokawa and Tsuruno (1997), considering the shear and twisting applied to the
ligament bundles, also emphasized the restrictions associated with measuring ligament
length by the relative position of the origin and insertion sites when analyzing the
deformation of the ACL. Consideration of the wrapping of ligaments is of critical
importance in the prediction of the MCL length (Blackevoort & Huiskes, 1991). The
abduction/adduction rotations are affected by the incorrectly modeled ligament in the
description of joint motion without consideration of the MCL wrapping around the bony
edge of the tibia. Blackevoort and coworkers (1991) showed that including the
interaction of the ligament and the bone allowed for greater stabilization when a val gus
moment was applied.

Woo, Johnson and Smith (1993) presented a review of mathematical models
developed to describe the mechanical behavior of tendons and ligaments. The
nonlinearity associated with the behavior of ligaments under tension increased the
complexity of the model. A linear load-deformation relationship is seen at large loads due
to an increase in stiffness (Figure 5). The recruitment of fibrils under the application of
load also affects the linear characteristics of the load-deformation curve. At maximum
load, under the recnritment of all the fibrils, the ligament shows linear characteristics.
Ligaments and tendons have been modeled as elastic as well as viscoelastic elements. The

mathematical representation of the ligament is dependent on the complexity Of the overall

16

knee model as well as the parameters to be predicted. A model of an incompressive
hyper-elastic ligament was used by Hirokawa and coworkers (1997) in an attempt to
analyze the deformation of the anterior cruciate ligament. The results of the experimental
data indicated that the central portion of the ligament underwent the least amount of strain
due to the limited shear, bending and twisting applied to the central fibers. The
application of an anterior force on the tibia produced increased strain on the anterior
fibers of the ligament. The results of this study also showed that the strain along the
length of the ligament is nonuniform, indicating increased strain values near the insertion

points.

Load

 

 

Deformation

Figure 5: Load-deformation curve for ligaments

ligaments and tendons are viscoelastic tissues. Repeated cyclic loadings induce a
stable response, which resembles that of elastic material (Lanir, 1983). A recent study
predicted the load-deformation behavior of ligaments when treated as quasi-linear
viscoelastic materials, taking into consideration strain rate (Pioletti, Rakatomanana,

Benvenuti & Leyvraz, 1998). Elastic and viscous potentials were expressed as functions

17

of stress and strain. The nonlinear characteristics of collagen fibers arise from the
straightening of the crimped fibers during the initial three percent of the strain. Relaxation
tests conducted at different levels of strain showed the stress to be a function of the strain
squared (Haut & Little, 1972). The tissue’s make up is assumed to be of fibrils at different
states of crimp. As load is applied, the tissue straightens and becomes able to bear load.
The load carried by the tissue is a function of the fiber’s constitutive law (Hurschler,
Loitz-Ramage & Vanderby, 1997).

Ligament failure can be modeled mathematically using selected stress-strain
criteria. Hurschler et a1. (1997) identified that failure occurs as a large number of fibers
fail within the tissue as a result of tissue stretch. This increase in fiber failure results in a
decrease in tissue stiffness. Mathematical models usually assume the same elastic
modulus and strain limit for all fibers comprising the tissue. This assumption constrains

the fibers to fail in the same sequence as they are recruited (Liao & Belkoff, 1999).

IV. Neuromuscular Activity

A complete analysis of knee joint mechanics and function includes the
contribution of the muscles crossing the joint. Muscle contraction will produce joint
motion or stabilization by compensating for internal and external forces. Muscles may be
categorized as secondary stabilizing structures that assist the ligaments in the prevention
of excessive motion when necessary. Knee joint function following an injury is evaluated
by full range of motion, joint stability and muscle strength. The fact that muscle strength
is considered part of the clinical evaluation should encourage the analysis of muscle

forces in mathematical knee models.

18

Muscle forces as part of mathematical models add to the complexity of the
computational methods. The number of muscles crossing a joint, which add to the
number of unknown parameters, is a major limiting factor to simplifications in models. A
large number of unknown muscle forces results in an indeterrninant system. Optimization
techniques have been used to predict accurate model solutions in the case of redundant
systems. Electromyography also can aid in the determination of muscle activity and the
reduction of the number of unknowns to allow solution of the dynamic equations. Collins
and O’Connor (1991) developed a mathematical model to predict the muscle-1i gament
interaction during walking. A two-dimensional four r linkage system was used to
determine the forces transmitted by three muscles, the two cruciate ligaments and the
contact force. The constraints of the model reduced the redundancy of the twenty possible
solutions. The researchers demonstrated single muscular activity at specific periods of the
gait cycle thus raising questions concerning the assumption of multiple muscle
contractions for joint stabilization. The results from the experimental testing would be
assumed to be different in the case of injured subjects. Simultaneous muscle contractions
are expected to assist in the stabilization of the knee joint. In a later study, Lu and
O’Connor (1996) used the four-bar kinematic linkage model to calculate the moment
arms of five muscles crossing the knee joint. Moment arms were calculated from a twO-
dimensional anatomically based model for the quadriceps, biceps femoris,
semitendinosis, gastrocnenrious and semimembranosis muscles. The ligaments were
modeled as bands of fibers. The results of the model calculations showed a general

agreement with experimental data reported by Collins and O'Connor (1991). The two-

19

dimensionality of the model did not pose any limitations to the moment arms and muscle
line-of-action predictions.

Translations at the knee joint are mostly constrained by the cruciate ligaments.
Hsieh and Draganich (1998) investigated the effects of quadriceps contraction on the
rotational and translational movements at the knee joint. The researchers found a linear
relationship between anterior translation, internal rotation and abduction of the tibia with
respect to the femur, during an applied quadriceps load. The fiber length of the ACL, PCL
and MCL also increased linearly with an increase in quadriceps load. The changes in joint
kinematics due to quadriceps contraction were relatively small when compared to the
changes experienced in ligament length. The effects of ligament loading due to muscle
contraction would be of great importance in the rehabilitation process.

Aune, Nordsletten, Skjeldal, Madsen and Ekeland (1995) investigated the changes
in mechanical properties of the ACL due to simultaneous contraction of the hamstrings
and gastrocnemius during loading of the ligament. The ACL was loaded in tension while
the ischiatic nerve was stimulated to obtain a tetanic contraction of the hamstrings and
gastrocnemious. The results of this study showed that there was a 70% increase in the
load needed to rupture the ligament and a 154% increase in the energy stored.
Additionally, the protection mechanism provided by the muscles was dependent on the
rate and magnitude of contraction. Contraction patterns of the hamstrings group have
been studied using EMG analysis during isometric exercises (Solomonow, Baratta, Zhou,
Shoji, Bose, Beck & D’ambrosia, 1987). These researchers focused on the load regulation
characteristics of the hamstrings group when the ACL has a mechanical disadvantage and

is unable to stabilize the joint. The importance of the hamstrings activity in reducing the

20

loads in the ACL and assisting in the anterior tibial translation constraint also was shown
in a later study performed on cadaver specimens (Li, Rudy, Sakane, Kanamori, Ma &
Woo, 1999). Mechanical advantage of the line of pull of the hamstrings, as well as the
ACL, was shown to contribute to the ability to provide stability to the joint.

The rehabilitation process following an injury to the ACL is associated with
hamstrings and quadriceps strengthening. Knowledge of the magnitude of strain placed
on the joint ligaments as a result of muscle contraction could prevent excessive loading of
the ligament during the early rehabilitation period. Modeling of ligament bundles and
musculotendinous units assist in the determination of the effects of muscle contraction on
the ligaments’ mechanical properties (Shelbume & Pandy, 1997). Model calculations
showed that due to the geometry of the knee joint alone, the ACL is loaded for the first

10° of flexion. Contraction of the flexor and extensor groups will assist in the reduction

of ligament stress throughout the remainder of the flexion range.

Strain in the newly reconstructed ligaments should be minimized during
rehabilitation exercises. Loading patterns in the ACL have been shown to be dependent
on both knee flexion angles and load placement (Zavatsky, Beard & O’Connor, 1994).
The authors calculated the “critical position” for load placement, which resulted in no
ligament forces needed for equilibrium. Zavatsky et al. found that as the flexion angle
increased the critical position moved distally along the shank segment. The maximum

angle where a critical position could be found within the lower limb, was found to be 90°

of flexion.
Simultaneous measurement of forces sustained by the constraints of the knee joint

can aid in the full understanding of knee joint mechanics. In a study measuring the

21

contact forces and quadriceps tendon and patellar ligament forces, the authors found that
the force sustained by the quadriceps tendon increased with increased angle of knee
flexion (Singerman, Berilla, Archdeacon & Peyser, 1999). Similarly, the anterior-
posterior patellar contact force increased with increased flexion. The tibiofemoral contact
force acted posterior on the joint from full extension to approximately 52 degrees of
flexion. The tibiofemoral contact force acted anteriorly at higher flexion angles.
Singerrnan et a1, (1999) cut the cruciate ligaments to evaluated the effects on the
measured forces. Absence of the ACL resulted in a slight decrease of the quadriceps
tendon force and an increase in the shear forces applied at the joint. The quadriceps force

increased with dissection of the PCL.

22

ANALYTICAL METHODS

A model was developed for the prediction of ligament forces as a response of
quadriceps muscle contraction and based on a quasi-static representation of the knee joint
mechanics in two-dimensions. Three major ligaments were included in the model, the
anterior and posterior cruciate ligaments and a collateral ligament. Due to the two
dimensionality of the model, the medial collateral and lateral collateral ligaments were
combined into one ligament. The ligaments of the knee joint were modeled as nonlinear
springs to account for the viscoelastic properties of the tissue. Insertion and origin sites
for the modeled ligaments were taken from the literature. Magnetic reasonance imaging
(MRI) slides also were used to determine the anatomic insertion and origin sites for the

modeled ligaments.

I. Tibio-femoral Joint Definition

The geometry of the femur and tibia have been shown to have an effect on the
mechanics Of the knee joint as well as the loads supported by the ligaments (Shelbume et
a], 1997). In the model developed here, the femur was represented as an involute circle
defined by two general parametric equations [1 ] and [2]. These equations describe a
spiral, bounded by the parameter t. The value of t in the parametric equations [1 ] and [2]
needed to be restricted to represent the portion of the involute circle that best fit the
geometric characteristics of the femoral condyle. This restriction is dependent on the
radius (b) of the circle in the two prescribed equations.
Ix(t) = -b - cos(t) - b - t -sin(t) [1]

Iy(t) = b - sin(t) - b - t 'cos(t) [2]

23

 

 

 

A c;
yondtSt .
nation 0
|
defining he

‘ l
anon ri

in union '

 

the tibia def:

rs shoun by

A counterclockwise rotation (o) of 90 degrees applied to the general equations
provides a medial view of the femoral condyle as shown in Figure 6 (Balint, 1998). The
rotation (I) added to the general parametric equations would result in equations [3] and [4],
defining the femoral condyle at the full extension position. To represent knee flexion, an
additional rotational transformation was prescribed to the parametric equations as shown
in equation [5]. The matrix multiplication associated with the rotation of the femur over
the tibia defined a new set of equations dependent on a second rotational component (9)

as shown by equations [6] and [7].

 

 

 

 

Figure 6: Involute circle with a 90 counterclockwise rotation,
representing medial view of the femoral condyle

Ix=-b-cos(¢+t)-b~t-Sin(¢+t) [3]

Iy=b-sin(¢+t)-b-t-cos(¢+t) [4]
Ix” _ cos(0) sin(6) . Ix [5]
Iy” ’ - sin(6) cos(0) Iy

24

Ix'=-b-cos(¢+t+6)-b-t-sin(¢+t+6) [6]

Iy'=b-sin(¢+t+6)-b-t-cos(¢+t+6) [7]
where, o is the prescribed rotation to the original involute,
O is the angle of flexion at the knee joint,

b is the radius of the involute circle

and t is the parameter in which x and y depend and the constraint to the involute.

The tibia was defined as the fixed segment in the model. The curved geometry of
the tibial plateau influenced the distribution of load. tong the knee ligaments. In this
model, the tibial plateau was represented as a flat surface, with a slope of zero degrees. A
flat surface representing the tibial plateau was used due to the difficulty in modeling the
simultaneous contributions of the concave medial and the convex lateral compartments.
This Simplification of the model has been shown to accurately predict the forces
supported by the cruciate ligaments (Chan et al., 1995). Researchers have measured the
slope of the tibial plateau and have found a 10° posterior slope (Meister, Talley,
Horodyski, Indelicato, Hartzel & Batts, 1998; Matsuda, Miura, Nagamine, Urabe,
Ikenoue, Okazaki & Iwamoto, 1999). This slope was assumed to have no direct effect on
the design of this model; however, this assumption Should be examined if the model is to
be expanded into three-dimensions.

Three coordinate systems were used in the definition of the knee geometry and
the joint kinematics. The unprimed coordinate system for the femur was defined at the
full extension position of the knee joint. The location of the unprimed x-axis was derived

from equation [5] at the point where the tangent to the line of the involute is equal to

25

zero. In the femoral condyle representation at full extension, two points with tangent zero
can be found (Figure 7). The second point of slope zero (where t = 1t-21t) is
representative of the contact point and the selected location of the unprimed x-axis of the

femur.

 

Point 1

Point 2

 

 

 

Figure 7: Involute circle with identification of slope zero positions.

The tibial plateau was modeled parallel but not coincidental with the femoral x-
axis. The joint space between the femur and the tibia was taken into consideration in the
development of the model. However, the articular cartilage and menisci were not
included in the model. The tibio-femoral joint was considered frictionless during the
allowed range of motion. The tibia’s unprimed coordinate system had its origin at the
anterior limit of the tibial plateau, and it was set parallel to the unprimed femoral
coordinate system.

The unprimed y-axis of the femur was derived from equation [6]. The axis was

defined at the tangent of the involute that is equal to infinity. For the specified involute

26

circle at the full extension position there were two points where the tangent to the line is
equal to infinity (Figure 8). The second point defined by a t value of rt- 5/21:

corresponded with the location of the unprimed y-axis of the femur.

 

Point 2

Point 1

 

 

 

Figure 8: Involute circle with identification of infinity slope positions.

A second (primed) coordinate system, (x’, y’), was defined at the origin point (0,0)
of the involute at the full extension position. Anterior-posterior displacements of the
femur relative to the tibia were estimated by the translation of the primed coordinate
system in the x-direction relative to the fixed tibia. Position vectors were defined in the
primed coordinate system for the full extension position unless otherwise specified.

Flexion at the knee joint was associated with rotation of the primed coordinate
system of the femur relative to the tibia (Figure 9). The double primed coordinate system
rotated with the body segment, while the unprimed and primed coordinate systems

remained as reference frames.

27

 

 

 

»
x

 

 

 

Figure 9: Representation of the femoral condyle in full extension in the
unprimed and primed coordinate systems.

The insertion locations for the modeled ligaments were estimated from the
unprimed coordinate system at full extension of the knee (Figure 10). The unprimed
coordinate system also was used to estimate the location for the patella and patellar
ligament insertion points. All ligaments in the graphical representation of the knee were
drawn as lines; however, mathematically they were represented as non-linear springs.
Position vectors from the unprimed x and y axes defined the origin and insertion points.
The origin site of the collateral ligament was modeled in the estimated anatomical
location, although the graphical representation of the tibia was not extended to

encompass that area (Figure 11).

28

 

 

Ixacl ,
_' x

 

Iyacl

 

 

 

 

ACL

 

 

Figure 10: Representation of position vectors to locate the insertion of the
ACL on the femoral condyle.

 

 

 

(bl

(c) (d)
(a) Femoral Condyle
(b) Tibial Plateau
(c) ACL

(d) PCL '
(e) CL ('3)

 

 

Figure 11: Graphical representation of the tibiofemoral joint with
ligamentous constraints (Full extension).

29

To estimate the insertion and origin positions of the modeled ligaments during
knee flexion, the relative insertion positions of the ligaments on the femur were
prescribed the same rotational transformation as the involute (Equations 9 a—c). The
positions of the ligament insertion points on the femur, relative to the geometric limits
remained the same following the rotation of the femur. The origin locations of the

modeled ligaments were not affected by the rotation because the tibia was defined as the

fixed segment.

Ixacl' _ cos(6) sin(6) . Ixacl [9 ]
Iyacz' ' - sin(B) cos(0) Iyacl 3
”prcl' _ cos(0) sin(9) . Ltpcl [9b]
_Iypct' ’ -sin(0) cos(6) Iypcl

”Ixcl' _ cos(6) sin(B) . Ixcl [9c]
_ch1' ' -sin(6l) cos(9) chl

 

A representation in the change of orientation of the knee ligaments at 60 degrees
of flexion is presented in Figure 12. Note that the position of the femur relative to the
tibia was arbitrarily set for the purpose of illustration, any linear displacements occurring

as a result of knee flexion are parameters to be investigated in the model.

30

 

 

Rickie.

 

 

 

 

A
Y
yo!
x!
x”
it
(b) ‘» (d)
(a) Femoral Condyle (c) ‘3. X
(b) Tibial Plateau 3
(c) ACL ‘3
(d) PCL (c)
(e) CL

 

 

 

Figure 12: Graphical representation of the tibiofemoral joint at
60 degrees of flexion, and the relative position of
the ACL, PCL and LCL.

II. Patello-femoral Joint Definition

The quadriceps group was the only muscle group included in the model. The four
quadriceps muscles were combined into a single element inserting on the patella. The
patella was modeled as a rectangle with two articulating surfaces. The biarticulating
surfaces allowed the model to predict the patello-femoral mechanics more accurately at
high flexion angles (Gill & O’Connor, 1996). At flexion angles larger than 100°, the
contact surface of the femur shifts from the trochlea to the condyles. The patella sinks
into the condylar groove to maintain contact with the femur. The model simulated this
shift of contact surfaces by the shift in the articulating surface of the patella. The two

articulating surfaces are represented more easily on the simplified rectangular

31

representation of the patellar than on the involute circle representing the geometry of the
distal femur. The dimensions of the patella were taken from MRI slides to match the
tibio-femoral joint geometry.

The normal ratio between the patellar dimensions and the patellar tendon length, a
1:1 ratio, was maintained in the model. Pathological conditions, such as patella alta, can
be represented mathematically by adjusting the ratio between the patella and patellar
tendon length. The insertion site of the patellar tendon in the tibia tuberosity was modeled
according to the estimated anatomical location; however, as in the case of the insertion of
the CL, the tibia was not graphically extended to encompass the insertion site.

With increased rotation of the femur relative to the tibia, the patella displaces
approximately 7 cm proximal over the femoral condyles (Peterson & Frankel, 1986). The
lengths of the quadriceps tendon and the patellar tendon are unchanged during tension.
The inextensible properties of the patellar tendon added a constraint equation to the
motion of the patella relative to the femur, thus defining the location of the anterior-distal
point of the patella about an arc of radius PT. The angle of pull of both tendons changes
as the knee flexion angle increases. The effect is to decrease the mechanical advantage of
the patello—femoral mechanism at greater flexion angles. The angles B and a, as defined
in Figure 13, determined the overall position of the patella relative to the femur at any
given flexion angle. The constraints specified by the inextensible patellar tendon and the
requirement of contact between the patella and the femoral condyles reduced the number

of solutions for B and or angles. A third constraint to the patello-femoral motion was
added by specifying a linear relationship between the angle of flexion (O) and the angle of

inclination of the patella (or). The last constraint necessary to define the patellar motion

32

relative to the femur was given by the orientation of the patello-femoral contact force.
Since the patello-femoral joint was considered frictionless, the contact force had to be
normal to the articulating surfaces of both the patella and the femoral condyle. The
patello-femoral contact point was derived from the solution to the constraints of the

motion and was not constant relative to either surface.
Quadriceps
Tendon

Articularingsurface
for 9>100’

  
   

\

Patellofemoral
contact point

 

Patellar
Tendon

 

Figure 13: Patellar geometry.

Only the insertion position for the quadriceps tendon was of interest in the model.
The origin of the quadriceps muscle group was not a necessary parameter to the model;
however, the angle of pull of the quadriceps tendon relative to the patella must be
included in the input parameters. The quadriceps tendon was modeled parallel to the

longitudinal axis of the femur except at high flexion angles where wrapping of the tendon

33

 

W35 3'

1‘
p331:

Equfl
IS TIC:
lash

quad;

occurs over the femoral condyle. The constraint for the wrapping effect of the quadriceps
was included at flexion angles larger than 90 degrees. The graphical representation of the

patello-femoral and tibio-femoral joints at full extension is shown in Figure 14.

 

 

T a.)

(0

 

 

 

 

 

(d)

‘ (a) Femoral Condyle
1. (C) (b) Tibial Plateau
(c) ACL
(d) PCL
(e) CL
(t) Patella
(g) Patellar Tendon
(h) Quadriceps Force

(8)

 

 

 

Figure 14: Patello-femoral and tibio-femoral joints at full extension.

The patello-femoral joint was not modeled as a simple pulley mechanism.
Equilibrium between the quadriceps tendon force, patellar tendon force and contact force
is necessary throughout the full range of motion (Figure 15). The equilibrium of the

patello-femoral mechanism is dependent on the angles 7, the angle between the

quadriceps tendon’s line of pull and the patello-femoral contact force, and 8, the angle

between the lines of action of the patellar tendon force and the patellO-femoral contact
force. These angles are both dependent on the orientation of the patella relative to the

femur and the angle of flexion of the knee.

QniioqrsTarrhrthe

i. Hello-fem
(Intact Face
PaellarTendeace

Figure 15: Patello-femoral equilibrium mechanism.

Once the inclination of the patella was defined by the angles at and B, the force
equilibrium mechanism could be solved. In addition to the patellar tendon and quadriceps
tendon being treated as inextensible units, they were constrained to carry only tensile
loads. The ultimate goal of the model was the prediction of the force canied by the
quadriceps tendon given the specific constraints determined by the tibio-femoral

kinematics.

35

III. Ligament Forces

The anterior cruciate, posterior cruciate and collateral ligaments provide the
primary constraints to the mechanics of the knee joint. The model was used to estimate
the forces sustained by the ligaments and the force generated by the quadriceps muscle
group to maintain equilibrium throughout the range of motion allowed by the knee joint.
Mechanical properties of the ligaments reported in the literature served as input
parameters in the estimation of ligament forces (Table 1). Equation [10] was used to
calculate the forces sustained by the three ligaments as a function of length change

throughout knee flexion.

Table l: Stiffness coefficient for the modeled ligaments.

 

 

 

 

k
Li gaments (N/mmz)
ACL 3o
PCL 35
CL 15
f = m2 [10]

where k is the stiffness coefficient and A is the change in length as a function of

angle of flexion.

The stiffness coefficient parameter allow for representation of a variety of injury

conditions and the estimation of changes in ligament and muscle forces required to reach

equilibrium. Changes in the stiffness coefficient contribute to a change in the force

36

[Tiff

‘
13:2“

“.uob

V
iMl

llEIIE

I'M!-

til.

cord

Lit!

[pd

produced by the ligaments and affect the mechanics of the knee joint. Adaptations in the
knee joint mechanics, associated with changes in the mechanical properties of the
ligaments, contribute to the better understanding of ligament injuries in the knee joint.
Ligament lengths were calculated by the difference between origin and insertion
locations in the estimated position for any given flexion angle (Equations 11 a-c). The
initial length of ligaments (Lo) was determined from the origin and insertion sites
estimated in the model when the strains in the ligaments were equal to zero. The
condition of 80:0 is not coincidental with full extension but occurs at approximately 35°
of flexion for all ligaments (Adbel-Rahman & Hefzr' l993). Strain values at full
extension for each ligament were obtained from the literature and are shown in Table 2
(Adbel-Rahman et al., 1993). The initial length of each ligament was calculated using

equation [12].

 

Lac! = J(Ixacl' - 0xacl)2 + (Iyacl' — tp)2 [1 la]
where Ixacl” is the insertion of the ACL along the x-axis for the rotated femur.

Oxacl is the origin position of the ACL along the x-axis.

Iyacl' is the insertion of the ACL along the y-axis for the rotated femur.

tp is the location of the tibial plateau along the y-axis.

 

chl = J(0xpcl - Ltpcl')2 + (Iypcl' - tp)2 [1 lb]
where prcl' is the insertion of the PCL along the x-axis for the rotated femur.

Oxpcl is the origin position of the PCL along the x-axis.

37

Iypcl' is the insertion of the PCL along the y-axis for the rotated femur.

 

Let = ,[(rxc1’—0xcz)’ +(ch1"—Oyzc)2 [11c]
where Ixcl' is the insertion of the CL along the x-axis for the rotated femur.

Oxcl is the origin position of the CL along the x-axis.

chl' is the insertion of the CL along the y-axis for the rotated femur.

Oylc is the origin of the CL along the y-axis.

Table 2: Length ratio values for the modeled ligaments at full extension.

 

t
Li gaments (mm/mm)

 

ACL 1.0215
PCL 1.050
CL 1.050

 

 

[12]

where j represents the individual ligaments.

l j is the length ratio of the ligament, L,- is the length of the ligament, and L0]- is

the initial length (resting length) of the ligament.

Li gaments were assumed to carry only tensile forces, which were specified by the

boundary condition given by equation [13]. Changes in length due to wrapping of the

38

11’. NI

331321.
Chergi
The in:

f5111111 a

cruciate ligaments were not taken into consideration in the model. Each ligament’s
ultimate load, taken from the literature (Table 3), was set as the upper boundary
condition. Any motion that would produce a ligament’s force exceeding the ultimate load

was not allowed and a change in the ligament’s stiffness was required.

F =0 for 114.0,- [13]

Table 3: Ultimate load of modeled ligaments and patella tendon.

 

 

Ultimate Load
Li gaments (N)
ACL 1700
PCL 2840
CL 945
PT 2900

 

IV. Mechanical Behavior of the tibio-femoral joint

Because in this model the knee joint was not classified as a hinge joint, the
calculation of the instantaneous joint center more accurately defined the continuously
changing knee center location_due to rolling and sliding motions occurring at the joint.
The instantaneous joint center can be calculated from the relative motion between the

femur and tibia and their respective segment velocities (Figures 16 and 17).

39

   
 

DistalTargetontheThigh
JointCenter
Preximal'l'argetoutheShank

Figure 16: Targeting protocol for joint center calculation.

Distal Thigh

Joint Center

 

Proximal Shank

Figure 17: Position vectors involved in the calculation of
the instantaneous knee joint center.

The thigh and shank segment angular velocities were derived from position
vectors and linear velocities of two thigh targets and two shank targets respectively. The
position vectors and linear velocities were forced to sagittal motion to simplify the
calculation of the instantaneous joint center. The angular velocity for a segment is

defined by equation [l4].

62' x(‘r;-r-,)=(v,-V.) r141

 

velocity
and ape

1
teen.

ergrlzr

where m, is the angular velocity of the segment, 7’, and 72 are the position vectors of two

targets in the segment and V1 and 172 are the linear velocities of two targets in the

segment.

The instantaneous joint center is defined to be a point in the thigh and shank with
zero relative motion. The same point in the thigh or thigh extended and in the shank or
shank extended will define the knee joint center. To obtain a point with zero relative
velocity, the linear velocity of the knee joint center was calculated as a point in the thigh
and a point in the shank (Equations 15 and 16 respectively). The angular velocity of the
knee joint was defined by the relative angular velocity of the thigh with respect to the

angular velocity of the shank as given by equation [17].

Vjc = 17r: +57): Xch/dt [15]
171.6 =I7PS+CTS X736,“ [16]

where Vic is the linear velocity of the joint center, 17d, and 17p, are the linear
velocities of the targets on the distal thigh and proximal shank respectively, wt

and m, are the angular velocities of the thigh and shank respectively, and

r it I d, and 73c,” are the position vectors from the thigh and shank targets to the
joint center location.
to ,- = w, ’ w s [17]

41

rr‘l '
traf‘

uteri
tenor
Dunn
not];

posit:

From rigid bodies kinematics, the location of the joint center is given by equation
[18]. The direct solution for the position of the joint center can be obtained by expansion
of the vector equation into scalar components, or with a direct vector solution by using a
triple vector product. The position of the joint center was obtained in the inertial
coordinate system and was transformed to the segmental coordinate system of the femur
for the purpose of calculating moments about this point. The ICR location in the
segmental coordinate system was transformed to the primed coordinate system of the

model.

0,- ><r,-.,.. =(vmp. -w. mm.) [18]

The ICR location was used to define the joint kinematics as well as to predict the
anterior displacement of the femur over the tibia. The geometric characteristics of the
femoral condyles influence the rolling and sliding patterns of the femur over the tibia.
During pure rolling of a circular disk, the center of rotation translates in the direction of
motion (Figure 18a). During pure sliding of a disk, the center of rotation will not change

position and coincides with the center of the disk (Figure 18b).

42

center of disk center of disk \

     

\

center of rotation

 

center of rotation

(80 (b)

Figure 18: Rolling and Sliding of a circular disk (a and b respectively).

The geometry of the femoral condyles can be assumed to be similar to two circles
that match with the contact surface of the tibio-femoral joint and patello-femoral joints
respectively (Figure 19). The center of the circle matching the posterior aspect of the
femoral condyles was used as the initial center of rotation of the joint. The selection of
this initial point of rotation was based on the fact that the initial rotation of the femur is
dictated by the geometry of the posterior aspect of the femoral condyle (W ismans et al.,
1980). Rolling of the femoral condyles with respect to the tibia occurred over the initial
25 degrees of knee flexion. The calculated instantaneous center of rotation then was used
to define the change in position of the point of rotation and to define the rolling and

sliding motion of the femur.

43

 

 

 

 

 

Figure 19: Assumption of circular discs representation of the femoral
condyle geometry.

The change of position of the ICR allowed for the estimation of the anterior-
posterior displacement of the femur relative to the tibia. The horizontal displacement of
the femur was estimated from the difference in location of ICR between the flexed and
fully extended positions. The prescribed rotation to the involute, associated with knee
flexion, will cause an anterior-posterior displacement of the contact point from the initial
to the final position. This displacement of the contact point is considered a normal
translation dictated by the geometry of the surfaces in contact. The portion of the
horizontal displacement purely caused by the sliding action of the femur was defined by
equation [19]. The rolling component of the tibio-femoral motion was defined by
equation [20] as the portion of the horizontal displacement that is not involved in the
sliding action. The translation prescribed to the involute, as defined by the ICR, also was

used to transform the location of any points associated with the femoral body segment

(Equation 21).

 

V. Ca

Zn-p
D sliding = J.
Zn-p-o

 

d7.
-dt
,3

where F = 11,0); + If“);

Drolling = (ICRa ' ICRo )' Dsliding [20]
where ICRe is the ICR location at 0 degrees of flexion, and ICR0 is the ICR

location at full extension.

II I
7'. F. _
x1” = Tr ' x]: + ny [21]
’y1 ’y1

where 7'; is the position vector for any given point in the femoral primed CS

Tr is the rotational transformation matrix, and ny is the translation vector (true

ICR displacement).

V. Calculation of TibiO-femoral Contact Force

A contact condition between the femur and the tibia was assumed in the model.
The point of contact was defined by the geometric characteristics of the femur and tibia.
Since the tibial plateau was defined as a flat surface, the contact point at the femoral
condyle was defined by the t value, giving the tangent line with zero slope for a specific
flexion angle (Equation 22). As shown in Figure 7, two points satisfy this condition for

the involute. The point used to define the x-axis of the unprimed coordinate system was

45

 

in: n

U
Cl
y-a

C- Tilt

knew,

l1 Ca

Due to

lid USCI

defined by the t value of Zn. This t value will satisfy the condition Of the contact point for
all flexion angles.
-tan(6+¢+r)=0 [22]

The contact point was derived by solving equations [6] and [7] using t = 21:. The
position vector defined by this solution then was specified, for both the femoral and tibial
articulating surfaces, in equation [23].

a, = Ix(22z)'i°‘ + Iy(27z)'j'

.. .. [23]
F“ = Ix(27z)'i + tpj

where 7;, = 7},

The condition of tibio-femoral contact also must satisfy the assumptions that the
unit norrrrals to the femoral and tibial surfaces must be collinear and the cross product
must be equal to zero (Equation 24).

fifxfi, =0 [24]

The contact forces were defined by the F6, and f0, components acting at point

C. These forces are considered internal or joint forces which influence the joint

kinematics.

VI. Calculation of Internal Moments due to Ligament Forces
The estimated ligament forces produced internal moments about the joint center.
Due to the simplifications of the model, only flexion/extension moments were calculated

and used to estimate the equilibrium position. The internal moments were calculated

about the instantaneous joint center of the knee. The insertions and origins of the
modeled ligaments defined the line of action of the calculated ligament forces. The angle
between the line of action and the tibial plateau was used in the calculation of the
component forces (Figures 20 & 21). In the case of the collateral ligament, a horizontal

line at the insertion site in the tibia was used to define the ligament angle (Figure 22).

 

 

 

 

 

 

 

Figure 20: Representation of the ligament angle for the ACL.

47

 

 

 

 

 

 

Figure 21: Representation of the ligament angle for the PCL.

 

 

 

 

 

 

'\ m...

 

Figure 22: Representation of the ligament angle for the CL.

48

 

 

-,_,

h

u.

Equations [25a—c] were used in the calculation of ligament angles. In the case of
the ACL and PCL, the angles formed with the tibial plateau always are acute. The model
does not allow the femur to displace over the true anatomical constraints dictated by
normal motion of the knee joint. Normal joint kinematics allow the collateral ligament to
restrict excessive motion at both acute and obtuse angles. Given the geometrical
parameters used to estimate the angle of pull and due to the properties Of the sine
function, equation [25c] will not accurately predict obtuse angles. When the sine function
is used for angles greater than 180 degrees, the answer is the complement of the desired
angle. To overcome this limitation, another equation [25d] was developed to use in the
estimation of obtuse angles. A command to check the position of the ligament’s insertion
relative to its origin to determine which equation would give the appropriate CL angle

was included in the model.

ACL - sin'l Racly 25
""8" Length“, [ a]
Rpcl
PCL = sin”1 —’—- 25
angle [ Length”) I: b]
CL , _l Rcly
= 8111 ——
angle Length, [256]
. -r Rcly
Cngk = 180 — srn _-th_ [25d]
"8 d

49

The component forces for each ligament were derived from the estimated
ligament forces (Equation 26 a and b). The x and y components and the perpendicular

force of each component, respectively, were used in the estimation of the intemal

moments (Figures 23-25).

fl}, = Fl}. -cos(ligm,,) [26a]
Fl}, = Fl]. -sin (ligmk) [26b]
where fl}, is the x-component of any 1i gament’s force, Fljy is the y-component

of any ligament’s force, Fl}. is the resultant ligament’s force, and ligmgje is the

angle of pull of any ligament.

 

ICR
dpyxi

dpx

l'r’acly

 

 

 

Facl‘

 

 

 

Figure 23: Graphical representation of the calculations of ACL internal
moments.

50

 

 

ICR

 

 

h dvyE.
dPpr
pcl,
4.—
Fuel.

 

 

Figure 24: Graphical representation of the calculations of PCL internal
moments. '

 

 

d
ICR *3"

 

 

dpxci I Fcl

 

 

 

Fcl

 

 

Figure 25: Graphical representation of the calculations of CL internal
moments.

51

CllCUIa
Wt: de
138m

“if i

The ligament force components (fl;Jr and fl” ) were used in the internal

moment calculations. A flexion/extension internal moment was estimated for each
ligament using equation [27]. The standard sign convention was used in the calculation Of
the moments, where counterclockwise is defined as negative and clockwise is defined as
positive.

17,]. = (fl 1, -dpx,j)+(Fl 1., -dpy,j) [27]
where 117,,- is the flexion/extension moment for ligament j

fl}, is the x-component of the force for ligament j

dpxlj is the perpendicular distance to the ICR from the x-component of the force
of ligament j

fl” is the y-component of the force for ligament j

dpyjj is the perpendicular distance to the ICR from the y-component of the force

of ligament j.

In a similar manner, the internal moment generated by the contact forces was
calculated about the ICR. The moment arms of the two components of the contact force
Was defined by the relative position of the contact point to the ICR, shown in equation

{28}- The moments could then be calculated using equation [27].

RICK/c = ICR - C [28]

52

VII. Calculation of External Moments

External forces applied to the tibia and/or femur result in external moments about
the knee joint. The model allows for input parameters of external forces in the form of
ground reaction forces. Ground reaction force components and ground reaction moment
components were used in the calculation of the flexion/extension external moments about
the ICR. The resolved forces (COP) needed to be computed to obtain the point of
application of the ground reaction forces relative to the ICR. The original moments
obtained from the force platform measurement, and the ground reaction forces based on
measurements about the center of the force platform, were used in this calculation. These
parameters were reduced to a wrench system to determine the point where the line of
application of the resultant ground reaction force crossed the surface of the plate. The
component of the resultant moment parallel to the resultant force vector was first
determined by equation [29]. The perpendicular component of the moment was then
derived using equation [30]. The vector from the center of the force platform to the center

of pressure point was defined by the assumption given by equation [31].

A7,, =Proji -1l7 [29]
iii =fi-Proj,.fi [30]
'r’xrv"=1fii [31]

where 'r' is the vector from the force plate center to the COP.

Equation [31] was expanded into scalar equations. Equations [32 a and b] were

used when solving the scalar equations for the X and Y components of the COP.

53

 

[32a]

 

[32b]

 

where h is the distance from the surface of the plate to the center of the plate.

The center of pressure location calculated as shown above was obtained in the
force plate coordinate system. A transformation of coordinate systems was necessary to
be able to obtain the position of the COP relative to the ICR. This transformation will be
specified in more detail in the experimental methods chapter.

The calculation of the relative position vectors from the COP to the ICR is shown
in equations [33 a—c]. The y-axis component (vertical component) of the COP location
was assumed to be zero because most activities were performed on the surface of the
force plate. Cases when this assumption was not met, will be addressed in the discussion

of experimental methods.

RICK/COB: = COP: T ICRx [333]
Ema,” = COPy — ICRy [33b]
ElCR/COP: = COP: T ICR: [33C]

The external moments were estimated from equations [33a-c] and the ground
reaction forces components were recorded from the force platform. The estimation of the
external moments was performed about the Z-axis of the lab coordinate system, which

matches the out-of-plane axis in the model (Equation 34). To obtain the flexion/extension

54

external moment about the knee joint, the estimated moments were transformed to Euler

8X85.

M2 = Rim/con: ° F) —RICRIC0Py ° 1?: [34]

VIII. Calculation of Patello-femoral Internal Moments

The forces generated within the patello-femoral mechanism produce moments
about the center of the knee joint. All internal moments were calculated about the ICR
point. The component forces were calculated for the quadriceps tendon, patellar tendon

and patellO-femoral contact forces, using equations ’ *‘ 37] respectively.

FQT, = F’QT -sin(I9) [35a]

fQTy = fQT-cos(6) [35b]
where F QT is the resultant force of the quadriceps tendon, O is the angle of
flexion of the knee, and FQTJr and FQTy are the component forces.

FPT, = FPT - sin(fl) [36a]

FM“y = fPT-cosw) [36b]
where 7 PT is the resultant force of the patellar tendon, B is the angle between

the patellar tendon and the vertical, and F PT I and FQT, are the component

forces.
We,r = PFC - cos(a) [37a]
PFcy = Pic-sin(a) [37b]

55

where PFC is the patello—femoral contact force, a is the angle of inclination of

the patella, and [’chIF and PFCy are the component forces.

The moment arms for each of the force components were determined by a
position vector between the point of application of the force and the ICR. The internal
moments about the out-of-plane axis were calculated using equations [38-40]. The
internal moments generated by the patellar tendon, quadriceps tendon and patello-femoral
contact forces resist the sum of the internal moments produced by the ligament and tibio-

femoral contact forces and the external moments produced by external forces.

Mgr; = A70, FQT, - Tiay FQT, [38]
iii-PTz = Tia, -FPT, - Ilia, ~F—PTX [39]
[Tl-PFCz = [Wax - PFC, 4170, - Pfcx [40]

where MaJr and May are the moment arms for the respective component force in

the x and y directions respectively.

IX. Mathematical Solution

The input parameters to the model included the flexion angle, position data for the
two targets on the thigh segment and the two targets on the shank segment that defined
the ICR, and external forces and moments. Given the constraints and geometric
compatibility equations, the number of unknowns was reduced and the equilibrium

equations [41] were solvable.

56

=0
=0 [41]
=0

2

| 53: Fin

The mathematical solution of the model was found by combining the tibio-
femoral forces/moments with the external forces/moments to solve for the requirements
of equilibrium in the patellO-femoral mechanism (Equations 42 a-c). The required
quadriceps force for the specified kinematics of the tibio-femoral joint was obtained as

the single number solution to the model.

FPTJr = F; + ch + Fact: + chlJr + l‘TclJt [42a]
fPT,+ = F; + fa, + Fowly + fpcly + fez, [42b]

Ill—PTz = 11—4- z + MC: + IT'I—aclz + [VI-pd: + 174-ch [42c]

The evaluation of a complete activity, such as squatting, with the quasi-static
model allowed for the comparison of various parameters in terms of flexion angle
changes. Therefore, the output of the model was not limited to a single parameter but,
instead, supplied a number of parameters for the in-depth interpretation of knee joint

kinematics.

57

EXPERIIVIENT AL METHODS

In the past, most experimental testing of knee joint models has been performed
underconstrained conditions that allow only flexion/extension at the joint. Some
experimental testing using cadaver specimens allowed for unconstrained motion of the
knee throughout the full range of motion. The action of squatting was chosen for the
current investigation to experimentally test the accuracy of this model. The knee joint
was unconstrained throughout each trial; therefore, the true mechanics of the knee joint

were measured under normal loading.

1. Subjects and Data Collection

Squatting data collected from two subjects were used to test the accuracy of the
computational model. A male subject, age 55, volunteered for the test. This subject was
considered a control, with no known history of lower extremity injuries that could affect
the knee mechanics. The model also was tested using data from a female subject, age 24.
This subject was tested after having sustained a third degree ACL tear and then later three
months after ACL reconstruction. The testing protocol was approved by the University
Committee on Research Involving Human Subjects (UCRIHS) at Michigan State
University under [RB number 93580.

The data used for the experimental testing was collected using the BTS Elite
system and an AMT I force platform in the Biomechanics Evaluation Laboratory, College
of Engineering at Michigan State University. The working volume was calibrated prior to

data collection using a minimum of 27 known target locations. The calibrated volumes

58

for all tests were of similar dimensions; however, they were not necessarily the same.
The four-camera system, sampling at 100 Hz., was used to follow the position of six
targets (dynamic targets) placed on the lower leg during the squatting (dynamic) trial. A
triad target configuration was used on each segment consisting Of three targets on the
thigh, and three targets on the shank segment respectively (Figure 26). The targets were
covered with retro-reflective tape to allow each target to be recognized by the cameras.
The BTS system calculates the barycentre of each marker and stores the two-dimensional

x and y coordinates for each camera view in a file for later analysis.

11. Targeting Protocol

The dynamic targeting protocol, displayed in Figure 26, allowed for the definition
of segmental coordinate systems in the thigh and shank respectively. Position vectors in
the inertial (lab) coordinate system to each target in the thigh and shank segments were
obtained from three-dimensional kinematic analysis. Relative positions between segment
markers were used to define a plane on the body segment and subsequently construct a

segmental coordinate system (Equations 43).

Figure 26: DynamicTargeting configuration for the thigh and
shank segments.

59

2‘: 0/2 X's/2
I’i/z X’3/2l

 

I; : 0/2 X’
Ir“, x? [431
j=£xi

where 71,2 is the position vector from target 2 to target 1.

In addition to the dynamic trials, position data were collected during a standing
file in which additional targets (virtual) were placed on anatomical landmarks. Virtual
markers were used to define the anatomical axes and segment planes along the thigh and
shank. The relative positions between the virtual markers and the dynamic targets in the
segmental coordinate system are constant in magnitude and direction and are independent
of the segment position in the inertial coordinate system. Therefore, it only is necessary
to have both sets of targets (anatomical and dynamic) on the segment at the same time for
one frame of data. Once the relative position was established in the segmental coordinate
system, the location of the virtual targets were derived using an inverse translational

transformation followed by a rotational transformation (Equation 44).

Ra = Rd + [TI-lfa/d [44]

where Rd is the position vector to the anatomical target in inertial

coordinates, Rd is the position vector to the dynamic target in inertial

coordinates, T is the transformation matrix, and Fa , d is the relative

60

position vector between the anatomical and dynamic targets in segmental

coordinates.

The greater trochanter and the medial and lateral epicondyles were used to define
the mechanical axis and segmental plane for the thigh. The medial and lateral femoral
epicondyles defined the medial-lateral axis, or Yf axis. The longitudinal axis of the femur
was defined using the relative position vector from the lateral femoral epicondyle to the
greater trochanter. The Yf axis and the longitudinal axis of the femur formed the
segmental plane necessary to define the femoral coordinate system (Equations 45). X,, Y,

and Z defined the coordinate axes fixed to the thigh segment.

medmm, — lat

 

 

 

_ (and
Y: _
lmedcond - latcond
, . r -lat H
Longitudinalm. = g ”0‘ “"1

 

lgrtroc _ latcond
[45]

_ K x Longitudinalam

' 11’, x Longitudinalmsl

 

Z, =X,xY,

The shank coordinate system was defined in a similar manner. The tibial crest was
used for the definition of the'longitudinal axis of the tibia (2,), and a posterior target was

used to define the para-sagittal plane of the lower leg. A position vector between the two
targets on the tibial crest and a position vector between the distal target on the tibial crest

and the posterior target on the leg defined the segmental plane. The development of the

61

coordinate system for the shank segment is shown by the set Of equations 46. The fixed

coordinate axes of the shank segment are X5, Y5 and Z5 (Figure 27).

_ proxk,g —dist

5

leg

 

 

proxkg — dist“g

 

dist,“ - post“,g

 

 

 

 

Postm =
ldistkg - pasta,g
[46]
__ Z; x Pastam
5 Z; x Postml
X s = Y3 x Z5

Y

S
X, dist”:g

Figure 27: Thigh and shank segmental coordinate systems.

Euler angles were used to determine the angle of knee flexion throughout the
Squatting trial. Even though this method provided a three-dimensional analysis of knee

lOint motion, only the sagittal plane rotations were used as input parameters in the model.

62

The three-dimensional kinematic analysis of the knee joint motion provided a more
accurate estimation of the flexion angle, as most rotations occurring at the knee joint are
coupled. Knowledge of out of plane motion helped determine the validity of the
simplifications made in the model.

The sequence of transformations required for the specification of Euler angles of
the knee were the same as that reported by Grood and Suntay (1983). For the purpose of
this calculation, and to maintain consistency with the clinical community, the calculation
of Euler angles was performed for the motion of the tibia relative to the femur. This
definition of relative motion did not influence the input parameter to the model or any
other calculations performed within the model because 0gc will be the same in cases when
either the femur or tibia is taken as the fixed segment. The first rotation was performed
about the medial-lateral axis of the femur and specified knee flexion/extension. The
second rotation was about the floating axis and defined adduction/abduction of the knee.
The last rotation was about the inferior-superior axis of the tibia thus establishing

internal/external rotation of the knee (Figure 28).

63

 

 

 

Figure 28: Rotational sequence for calculation of Euler angles.

The joint coordinate system was defined by two base vectors (é2 and 53 ). that

corresponded to fixed axes, and a third mutually perpendicular base vector (é, ). The

relationship between the unit vectors and the corresponding segmental axis are illustrated

in Equation [47] (Figure 29).

82: r
83:23
. é xé
6]. 2 3

[47]

 

 

 

Figure 29: Joint coordinate system.

The calculation of the flexion/extension angle is shown in equation [48]. The
rotation angles, as well as the abduction/adduction angles, have a side dependency. Right
and left sides carry opposite signs, where external rotation and adduction are positive
(negative) for a right (left) leg. The internal/external rotation angle and the
abduction/adduction angle calculations are shown in equations [49] and [50] respectively.
The flexion-extension angular motion, as defined by equation [48], was used as the input

parameter (0) in the model.

0=-sin"(1€, -é,) [48]
t9 = —sin’1 (.23l - j,) [49]
6’ = —sin ‘1 (é2 45,) [50]

65

III. Computational Model

The rest Of the experimental methods will be presented in relationship to the
computational model written in Matlab (version 6.0). The computer program (k_exper.m)
is shown in its entirety in Appendix A. In this section, the functions called within the
main program are identified and explained in more detail as necessary. Comments have
been added throughout the program in Appendix A to aid in the understanding of the
commands and to reduce the explanation necessary in this chapter. Mathcad (version
2000) was used in the computation of the ICR, in the calculation Of the patella’s position
relative to the femur and in the final solution to the equilibrium equations. The templates
used for these purposes are presented in Appendix B and Appendix C, respectively.

The instantaneous center of rotation was calculated using the position coordinates
of two targets on the thigh and two targets on the Shank segment. The calculation method
was explained in detail in the analytical methods section. The ICR location was estimated
relative to the superiOr target on the shank segment, which was treated as the fixed
segment in the model. Linear parameters obtained from the experimental'data and used as
input variables in the model, such as the ICR location relative to the shank target, were
converted to model units to allow for an anatomically correct representation of the knee
motion in the model.

Model units were estimated using the true dimensions of the subject’s knee as
calculated with relative vectors between targets in the standing file. The depth of the
knee, as defined by the anterior and posterior contours of the femoral condyles, was

estimated using the anatomical ratio between depth and width of the epicondyles (Figure

66

j 30). The model calls the function kneesize to calculate the conversion variable. The
relative position between the medial and lateral epicondyles in the standing file defined
the width of the subject’s knee. From this parameter, and with the known width to depth

ratio, the depth was estimated and used in the representation of the knee model.

 

Figure 30: Graphical representation of the transverse view of
the femoral condyles and depth-width
relationship.

The anterior-posterior linear displacement of the femur over the tibia was defined
using the location of the ICR relative to the superior target on the fixed tibia. Pure sliding
of the femur relative to the tibia would occur about the center of curvature of the femur
(Figure 19), which was estimated to be approximately 24 mm anterior to the posterior
aspect of the femoral epicondyle. Any difference in the position between the ICR and the
center of curvature of the femur would indicate the existence of rolling in the joint
motion. The model calls the contact _point and tct _pt functions for the estimation of the
location of the contact points of the femur and tibia respectively. Equations [19-21] are
used to define the contact point between the femur and the tibia in pure rolling and pure

sliding conditions.

67

Once the position of the femur relative to the tibia‘was estimated using the ICR
position, the ligament length was calculated using the difference between the insertion
and origin locations. As explained in the analytical methods, the ligaments were treated
as extensible units, but wrapping was not considered in the calculation. The change in
length of each ligament was taken as the difference between the ligament length at the
specific flexion angle and the original ligament length. The function change_length is
called to calculate Aacr, Ape. , Ale and ligaments strains. All three ligaments were assumed
to be taut at full extension. The original length was calculated from known strain
parameters of the ligaments at full extension. According to the literature, the ligaments
would be under no strain at approximately 30° of flexion, assuming no imposed
displacement of the femur over the tibia.

The stiffness coefficient (k) for each ligament was treated as an input parameter to
the model. In the non-inj ured case, k was assigned normal values taken from the literature
(W insman et al., 1980). In the experimental case of an injured ligament, the k variable
was adjusted accordingly to represent the pathology. The changes in this parameter did
not affect the estimated ligament length, however, changes were noticeable in the forces
sustained by the ligaments during the activity. The ligjorce function was called to
calculate the forces sustained by the ligaments as a function of knee flexion. The
experimental model included a constraint check to limit all forces within the
physiological range of each ligament. If the force estimated at any ligament exceeded the
ultimate failure value for the ligament, the model reduced the stiffness coefficient by a

. value of 0.01 until the force generated was within the normal range. In cases when the

68

ligament is assumed to be completely torn, this constraint check would be omitted, and
the k value would be zero.

Internal moments were calculated about the ICR using the estimated ligament
forces and ligament angles. The function moments calculates the ligament moments about

the ICR. The moments were calculated about the 22 base vector of the joint coordinate

system. The transformation of the M2 moment about the Euler axis defined the moment
about the true flexion/extension axis and not the Z-axis of the inertial coordinate system.
Internal moment values were converted to N*m units and normalized to percent body
weight, to allow for easy comparison across subjects.

The ground reaction forces and moments acquired with the AMT I force platform
sampling at 1000 Hz. were used as the external input parameters in the model. The
assumption that in a quasi-static Situation all internal forces equal all external forces had
to be met for equilibrium to exist. The model approached this assumption from the stand
point of the known external forces generated during the squatting trial and the estimated
ligament forces. The unknown forces of interest were the ones necessary to create
equilibrium by the patello-femoral mechanism. The function e_moments was called to
calculate the external moments about the ICR. The ICR location in the inertial coordinate
system was used for the calculation of the external knee moments. The center of pressure
calculation, as explained in the analytical methods, was included in this function. The
moments were calculated about the x, y, and z axes of the inertial coordinate system.
These external moments were then transformed to the Euler axes defined by the joint

coordinate system. Only the flexion/extension external moment was used in the model.

69

The external moment was normalized to percent body weight for purpose of data
analysis.

TibiO-femoral contact forces were estimated in the Mathcad template during the
solution of the equilibrium equations. The moments about the flexion/extension axis
passing through the ICR were also calculated within this template. The tibio-femoral joint
was assumed to be frictionless, therefore the contact force must satisfy the condition of
normality to the surfaces in contact. The horizontal force was also calculated with the
purpose of analysis and interpretation of the internal forces generated on the ligaments.

Prior to the estimation of the quadriceps group force necessary to satisfy the
equilibrium condition, the position of the patella relative to the femur needed to be
known. The patella position is dependent on the overall position of the femur relative to
the tibia as defined by the flexion angle and the antero—posterior displacement. The
inextensibility of the patellar tendon was used as another constraint to the motion of the
patella. Appendix B contains the Mathcad template used in the solution of the
simultaneous set of equations as defined by the constraints for the motion Of the patello-
femoral joint. The patella was assumed to rotate about the trochlear groove. The Center Of
rotation is approximately 22 mm from the anterior aspect of the trochlear grove of the
femur. The center of rotation Of the patella was defined as a constant parameter in the
model. The biarticulatin g patella model allowed the shift of contact of the patella from
the trochlear groove to the condyles occurring at flexion angles greater than 100 degrees.

The solution to the quasi-static model was Obtained with the Mathcad template

presented in Appendix C. Parameters from the patella position were used to solve the

equilibrium mechanism of the patello-femoral joint and then used in the estimation of the

70

quadriceps forces necessary to balance the external and internal forces and moments
calculated in the model. The position of the patella as defined by the angles or and B was
used in the estimation of the unit vectors for the patellar tendon, quadriceps tendon and

the patello—femoral contact force as shown in Figures 31 and 32.

     
  

Quadriceps
Tendon

Patellar-femoral
contact force

 

Patellar
Tendon

 

Figure 31: PatellO-femoral equilibrium mechanism.

71

     
  

Quadriceps
Tendon

Patellar-femoral
contact force

Patellar
Tendon

Figure 32: Unit vector representation in the patellO-femoral
mechanism.

72

RESULTS AND DISCUSSION

Many mathematical models of the knee joint have been developed to predict the
mechanical behavior of the joint. Most experimental testing of these models has been
performed on cadaver specimens under constrained conditions. The mathematical model
presented in the analytical methods was tested using biomechanical data obtained during
squatting. The results of the experimental tests for the initial half squat cycle, are
presented in this chapter and some comparisons with other proposed models are included.

Researchers have demonstrated that the knee joint is not a true hinge joint.
Normal knee mechanics include a combination of rolling and sliding of the femur over
the tibia. These two actions are influenced by the geometry of the femoral and tibial
condyles and are believed to be constrained primarily by the cruciate ligaments. Even
though the knee joint is no longer classified as a hinge joint, the quantification of rolling
and sliding has been difficult. As presented in the review of literature a variety of models
have been used to define the knee joint mechanics; however, very few models have been
tested in-vivo while performing unconstrained movements.

In this model the ligaments were treated as extensible units allowing a measure
of ligament laxity as determined by joint motion. The instantaneous center of rotation
was calculated and used in the estimation of the rolling and sliding characteristics of the
joint. Following the determination of joint position, the equilibrium equations were used
to estimate the muscle and internal forces necessary to maintain the joint in that position.

Differences in the mechanical behavior of the knee joint were evident in the

comparison of the data for the three subjects tested. These results also showed some

73

variations when compared to results presented in the literature. However, the differences
in values and patterns of the data were found to be consistent with the known mechanical
characteristics of the tested knees. Only the more important parameters are discussed in
this chapter.

The equilibrium condition that needed to be satisfied by the system is dependent
on the relative position between the femur and the tibia. As the position changes, the
constraints to the motion change, influencing the internal forces of the system. The
predicted displacement of the femur relative to the tibia illustrated by the change in
position of the contact point of the femoral condyle is shown in Figure 33. The
comparison in the displacement pattern of the contact point for the three tested subjects
shows that in all cases the knee behaves under a combination of rolling and sliding

motions. The pure rolling pattern is included in Figure 33 as a point of comparison.

 

Contact Point Displacement

 

 

OTNOD-F
7
>
:3

ill
55?
3?-

- anterior/ + posterior
f

Displacement (model units)

 

 

 

 

Angle of Flexion (degrees)

 

 

 

Figure 33: Contact point displacement comparison.

74

The displacement pattern for the ACL deficient patient gave an indication of the
instability present at the knee joint. The increase in the amount of translation allowed in
the joint demonstrates the importance of the cruciate ligaments acting as the primary
constraints to rolling and sliding of the knee joint. One of the objectives of ACL
reconstruction is to return the joint to its normal mechanical behavior. The comparison
between contact point displacement for the ACL deficient knee and the ACL
reconstructed knee indicates that the reconstruction was successful in meeting this
objective. The translational motion of the femur relative to the tibia was decreased
following the ACL reconstruction; and the pattern was similar to that of the normal knee,
except for that observed during the initial 20° of flexion.

The differences in pattern observed between the displacements presented in
Figure 33 and those previously presented in the literature are due to the unconstrained
motion of the joint under the current testing conditions. A more predictable rolling and
sliding pattern can be expected when the knee joint motion is not influenced by external
forces and is moved through a controlled range of motion. For equilibrium to exist, the
internal and external forces and moments have to cancel out. The internal forces and
moments are dependent on the relative position of the femur and the tibia. Therefore, the
mechanics of the joint cannot be compared between constrained and unconstrained
motions.

The amount of laxity of the joint is a direct indicator of the mechanical behavior
of the constraints. An anterior displacement Of the femur relative to the tibia has to be

controlled by the PCL, while a posterior displacement of the femur relative to the tibia is

75

controlled by the ACL. The cruciate ligament angles determine the amount of horizontal
force contributed by each ligament in the constraint of the horizontal translation at the
knee joint. Due to the orientation of the collateral ligaments, the contribution to anterior-
posterior constraint of motion by this ligament is limited when compared to that of the
cruciate pair.

The laxity curves for each subject are presented in Figures 34-36. The horizontal
forces of the ACL, PCL and CL were summed and graphed as a single horizontal force
which acted as the constraint to the motion. A negative horizontal force would represent a

force acting anteriorly on the joint, and a positive horizontal force acted posteriorly on

the joint.

 

Laxity CUM
Normal
200

 

150-
100-

504

 

   

 

 

.50 l’

 
 

Horizontal Forces (N)

-100 .
-150 r

 

 

 

+ Ligament - Fx
4'99

Displacement (Model Units)
(- Anterior! + Posterior)

 

 

 

 

 

Figure 34: Laxity Curve: Ligament horizontal force vs. Anterior-posterior
displacement for a normal knee.

76

 

 

Horizontal Force (N)

Laxity Cum
ACL - Deficient
409

 

150‘
1001

 

 

 

 

+ Ligament - Fx

 

 

 

 

Displacement (Model Units)
(- Anterior/ + Posterior)

 

 

Figure 35: Laxity Curve: Ligament horizontal force vs. Anterior-posterior
displacement for an ACL deficient knee.

 

 

Horizontal Force (N)

 

 

Laxity Cunle
ACL - Reconstnrcted
201}
150 A
100 d

 

 

q
u

 

 

 

 

—o— Ligament- Fx

 

ace
Displacement (Model Units)
(- Anterior/ + Posterior)

 

 

Figure 36: Laxity Curve: Ligament horizontal force vs. Anterior-posterior
displacement for an ACL reconstructed knee.

77

The tighter the pattern represented by the laxity curve, the less motion that is
allowed in the joint; and, therefore, the lower the amount of force necessary to constrain
the motion. Figure 35 is a representation of the amount of horizontal force produced by
the ligaments in an unstable knee. The amount of laxity in the unstable knee due to the
deficient ACL, was considerably greater than that observed in the normal knee. This
increase in laxity was primarily in the posterior direction, which is regularly controlled by
the ACL. There was also an apparent increase in the laxity in the anterior direction.
However, when comparing the ACL-deficient and ACL-reconstructed graphs (Figures 35
and 36 respectively), the same anterior displacement can be observed. The laxity in the
anterior direction suggests that the anterior laxity was not a result of the ACL injury, but
a natural difference in ligament laxity between this subject and the subject used as the
norm.

Following the ACL reconstruction, the laxity of the knee was decreased and the
forces sustained at the ligaments also were decreased. The tightness of the laxity curve
demonstrated the ability of the new tissue to act as a good constraint to posterior motion
of the knee joint. Differences between the laxity curve from the uninjured subject and
that of the post reconstruction subject can be explained by the differences in contact point
displacement presented earlier.

The position of the femur relative to the tibia and the internal forces produced by
the ligaments determined the position of the patella relative to the femur and
consequently the patello-femoral equilibrium mechanism. Two angles described by the

patello-femoral mechanism are of particular interest in this model: B, which is the angle

between the patellar tendon and the vertical, and or, which is the angle between the

78

anterior aspect of the patella and the vertical. These two angles affect the forces
necessary for equilibrium which are produced by the quadriceps and are transmitted
through the patellar tendon. The differences in a and B between the normal, ACL-

deficient and ACL-reconstructed conditions can be observed in Figures 37 -39,

respectively.

 

Patella-Femoral Mechanism
Normal

 

 

Angle (Degrees)

 

 

 

 

 

Angle of Flexion (Degrees)

 

 

Figure 37: PatellO-femoral Mechanism angles vs. Knee flexion. Normal
knee.

79

 

 

Angle (Degrees)

Patello-Fernoral Mechanism
ACL- Deficient

 

 

 

 

 

 

Angle of Flexion (Degrees)

 

Figure 38: Patello-femoral Mechanism angles vs. Knee flexion. ACL-

deficient knee

 

 

Angle (Degrees)

Patella-Femoral Mechanism
ACL - Reconstnrcted

 

 

-o-- Beta

__.Alpha

 

 

 

 

Angle of Flexion (Degrees)

 

Figure 39: PatellO-femoral Mechanism angles vs. Knee flexion. ACL-

reconstructed knee.

80

 

 

 

 

The magnitude of or, or degree of inclination of the patella, increased with knee
flexion. This pattern agreed with the results presented by Gill et. al. (1996). However, the
overall range of or in the normal knee condition was lower for the current study. The
sharp decrease in magnitude seen at 100° of flexion corresponded with the change in
contact surface that was allowed by the biarticulating model. In Figure 38 there was an
apparent dependency between a and B magnitudes. In the ACL-deficient knee, the angle
of inclination of the patella was not dependent on the angle of flexion of the knee joint
but was dependent on the position of the patellar tendon. Following ACL reconstruction
(Figure 39), the dependency between or and B disappeared for most of the range of
motion, and there was a return to the normal increase in or with an increase in knee
flexion. The initial 20° of knee flexion still showed a dependency between or and B
angles in the ACL-reconstructed condition. This pattern matched the difference in the
displacement pattern observed in Figure 33 when the normal and ACL-reconstructed
conditions were compared.

When examining the pattern defined by the line of action of the patellar tendon,
the results of the present study did not agree with previous research. The results presented
by Gill et al. (1996) showed a constant decrease of B from 20° to -10° with increased
knee flexion. The results presented in Figures 37-39 showed a greater dependency
between the position of the patellar tendon and the position of the femur relative to the
tibia. The differences in results can be explained by assuming that the dependency of the
line of pull of the patellar tendon on the angle of knee flexion has to be accompanied by a
similar dependency with the relative position of the femur in the horizontal direction.

This dependency was particularly evident in Figure 38 for the ACL-deficient knee. As

81

was previously shown, the knee joint with a deficient ACL was more lax. This increased
motion of the femur over the tibia affected the overall position of the patella relative to
the femur.

One of the constraints used in the solution of the patello-femoral equilibrium
equations was the requirement of contact between the femur and the patella throughout
the range of motion. Adjustments in the angles B and or contributed to partially satisfying
this constraint. The greater adjustments necessary were accomplished by changing the
orientation of the patellar tendon, which changed the overall position of the patella
relative to the femur; then smaller adjustments to alpha were made to satisfy the contact
point condition.

In the ACL reconstructed condition, the angle B (Figure 39) followed a similar
pattern to that of the displacement of the femur over the tibia; however, or followed a
pattern that was considered normal. This dependency can be compared to the ACL
deficient condition (Figure 38) where both or and B were dependent on the displacement
pattern. Greater adjustments in the position of the patella were necessary to satisfy the
condition of contact between the patella and the femur.

The forces produced by the quadriceps are influenced by the internal and external
forces at the tibio-femoral joint as well as by the condition of equilibrium in the patello-
femoral mechanism. Changes in the orientation of the patella will influence the
transmission of forces from the quadriceps tendon to the patellar tendon. The ratio
between patellar tendon (PT) and quadriceps tendon (QT) forces is a simple indication of

the mechanical behavior of the patello-femoral mechanism. This ratio is illustrated in

82

Figures 40-42 for the normal, ACL deficient and ACL reconstructed conditions,

 

 

 

 

 

 

 

 

 

 

 

 

 

respectively.
PT/QT
Normal
1.2
1 a
’2‘
2 0.8 -
'9 6
a 0. -
3 0 4 -
LE .
0.2 -
O . T . T T
0 20 4O 60 80 100 120
Angle of Flexion (Degrees)
Figure 40: Patellar Tendon to Quadriceps Tendon force ratio for the
normal knee.
PT/QT
ACL- Deficient
1.2
1 .
’2T
2 0.8 .
.9
d? 0.6 .
3 0 4 -
o .
L
0.2 «
O 4 *= e 9. i i i i : fi‘ :
0 20 40 60 80 100 120
Angle of Flexion (Degrees)

 

 

 

Figure 41: Patellar Tendon to Quadriceps Tendon force ratio for the ACL-
deficient knee.

83

 

 

 

 

 

PT/QT
ACL-Reconstruted
1.2
1 J
’2
2 0.8 i
.9
‘6 0.6 i
tr:
0
g 0.4 -
u.
0.2 «
0 . t T i e i t i i i e
0 20 40 60 80 100 120
Angle of Flexion (Degrees)

 

 

 

Figure 42: Patellar Tendon to Quadriceps Tendon force ratio for the ACL-
reconstructed knee.

The ratio between the patellar tendon force and the quadriceps tendon force was
similar in all tested conditions and followed closely that described by Gill et al. (1996).
The two points of interest in Figure 40 were the start of the wrap of the quadriceps tendon
and the switch in contact surface between the patella and the femur. As the quadriceps
tendon started to wrap around the femoral trochlea at approximately 75° of flexion, there
was a marked decreased in the ratio between the PT and QT forces. At 100° of flexion
when the contact between the patella moved from the trochlea to the condyles, there was
another marked decrease in the ratio of forces between the two tendons.

Even though the ratio between the PT and QT forces followed a similar pattern
for the ACL-deficient knee followed a pattern similar to that observed in the normal

knee, there were some additional small fluctuations that could be observed (Figure 41).

The smaller changes in the slope of the line could be matched with the large changes
Observed in the translation of the femur over the tibia (Figure 38). As a result of the more

predominant changes in the a and B angles, the line of pull of both the patellar tendon

and the quadriceps tendon were affected. The mechanical advantage of each of these
tendons was affected at different times throughout the range of flexion, which affected
the ratio of forces.

The purpose of this mathematical model was to develop a tool for the prediction
of ligament strains and forces during dynamic activities without the need for an invasive
technique. The strains on the ligaments were calculated using the difference in length of
the fibers throughout the range of motion of the knee. Change in ligament length was
directly affected by the position of the femur relative to the tibia and by the distance
between origin and insertion sites. The forces generated by the ligaments are presented in
Appendix D, and more attention is given to the strain patterns obtained from the model.
Since the displacement characteristics of the joint were found to be different fi'om
previous studies, the strain patterns shown in Figures 43-45 also showed differences

when compared to previously reported data.

85

 

 

Strain (mm/mm)

oooo Add—L
orchards-sneeze»

ACL Strains

 

 

 

 

 

Corn partson
' — Normal
— ACL-Def.
O 20 40 60 80 100 120
Angle of Flexion (Degrees)

 

 

Figure 43: Strain comparison for the ACL.

 

 

Strain (mm/mm)

1.8

PCL Strains
Comparison

 

1.6-
1.4-
1.2~

1.
0.8-
0.6-
0,4.

 

— Normal
— ACL-Def.
—ACL-Rec. I] i

 

 

 

 

 

 

20 40 60 80 100 120
Angle of Harden (Degrees)

 

Figure 44: Strain comparison for the PCL.

86

 

 

CL Strains
Comparison

 

0.8

—Norrnal
0.7 « —ACL-Det.
—ACL-Fiec.
0.6 «
0.5 A
0.4 «

0.3 -

Strain (mm/mm)

0.2 «
0.1 -

 

 

 

0 20 40 60 80 100 120
Angle ot Flexion (Degrees)

 

 

 

Figure 45: Strain comparison for the CL.

Winsman et al. (1980) indicated that the ACL is under-strain from O to 20° of
knee flexion. The ACL is considered to be slack from approximately 20° to 40° of
flexion, at which point there is a constant increase in strain of the ligament with increased
knee flexion. The pattern for ACL strain obtained with the current model for the normal
knee (Figure 43) demonstrated the lowest strain level at approximately 15° of flexion and
a constant increase in strain with increased flexion. The absence of the period when the
ligament is unstrained, expected in the normal pattern, can be explained by the slight
posterior displacement of the femur over the tibia which occurred from 20° to 45° of
flexion (Figure 33). This posterior displacement of the-femur relative to the tibia very

likely produced the observed strain on the ACL.

87

The overall strain pattern for the other two conditions, ACL-deficient and ACL-
reconstructed, was similar to that obtained for the normal subject (Figure 43). The
expected peaks in strain were observed for the ACL-deficient condition where the femur
was abnormally displaced over the tibia. A higher knee flexion angle combined with a
larger posterior displacement of the femur relative to the tibia, generated a larger strain
when compared to similar translations at lower flexion angles. When designing
rehabilitation exercises for ACL-deficient patients, the range of motion allowed at the
knee joint should be taken into consideration due to the strains placed on the ACL with
increased knee flexion.

Following ACL-reconstruction, the strain pattern at the ACL was similar to that
obtained for the normal knee. The differences in the displacement patterns of the femur
relative to the tibia, between the two conditions, did not influence the strain patterns for
the ACL. This fact also was apparent when comparing the laxity curves for the two
conditions.

Winsman et. al (1980) showed that the PCL was strained from 40° of flexion to
full flexion. The data obtained for the normal knee with the present model showed an
increase in the PCL strain from 5° to full flexion (Figure 44). The differences between
strains obtained with the present model and those previously reported data are believed to
be predominantly due to of possible errors associated with the insertion and origin
locations of the ligament. The selected insertion and origin sites for the present model
nright not have been representative of the overall behavior of the ligament fibers.

Deviations from the normal pattern for the PCL strains obtained from the ACL-

deficient knee data could be explained in the same manner as the differences obtained

88

with the ACL strain patterns. The instability of the knee joint affects not only the injured
tissue, but also the surrounding structures. The anterior laxity observed in the data for the
ACL-reconstructed knee placed increased strains on the PCL during the initial 30 degrees
of knee flexion. Slightly higher strains were observed for the ACL-reconstructed knee
throughout the range of flexion than for the normal knee when both were tested with the
present model.

There was a constant increase in strain for the CL throughout the range of flexion
for the normal knee (Figure 45). Increases in strain obtained for the ACL-deficient and
ACL-reconstructed knees were due to the displacement patterns that accompanied knee
flexion. At lower flexion angles, an anterior displacement of the femur over the tibia
increased the strain on the CL. The opposite was true at higher flexion angles where a
posterior translation had a greater influence on the strain pattern of the CL.

Following the calculation of the relative position between the femur and the tibia
and the calculation of ligament strains and forces, the solution to the equilibrium
equations was obtained by calculation of the quadriceps force necessary to balance the
system. Equilibrium of the patello-femoral mechanism, quadriceps tendon, patellar
tendon and patellO-femoral contact forces had to be satisfied before the final solution
could be obtained. The forces obtained for the QT and PT were dependent on the angle of
flexion as well as the internal and external forces applied to the system. Due to the
influence of the displacement of the femur, relative to the tibia, to the angle of pull of the
patellar tendon, the horizontal and vertical components of the QT and PT forces are
presented separately. The QT and PT forces obtained for the normal knee are presented in

Figures 46a-b.

89

 

 

 

 

 

 

 

 

 

 

 

 

 

 

QT and PT Horizontal Forces
Normal
500
+ PT
400 -- —OT
A 300
a l
A
g 200 / U
0
LL
100
o -
9 20 40 60 80 11F0
-100
Angle of Flexion (Degrees)
(3)
OT and PT Vertical Forces
Normal

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 20 40 60 80 100 120
Angle of Flexion (Degrees)
(b)

Figure 46; Quadriceps and Patellar Tendon forces for the normal knee.
(a) Horizontal Forces (b) Vertical Forces.

The resultant QT and PT forces increase with an increase in knee flexion angle.
The horizontal force transmitted to the PT is limited when compared to the horizontal
force generated by the QT. As Observed by changes in B, the range of the angle of pull of

the patellar tendon was limited in the normal knee when compared to the results of
previous studies. The greatest contribution to knee motion for the PT comes from the
vertical force.

In the ACL-deficient knee, the quadriceps was found to be more active in the
initial period of knee flexion (Figure 47). This contribution can be explained by the
stabilization role that the muscle has to perform in the absence of healthy primary
constraints. When compared to the normal knee, higher forces for the QT and PT were
found in both the horizontal and vertical components. The horizontal forces transmitted
through the PT followed the pattern of displacement of the femur over the tibia (Figure
47a). The horizontal pull of the PI‘ being in the same direction as the displacement of the
femur relative to the tibia supported the role of the quadriceps muscle as a stabilizer of
the joint. An anterior pull of the PT on the tibia, combined with an anterior displacement
of the femur over the tibia, reduced the relative displacement between the two bones. The '
increase in the vertical forces of both the PI‘ and QT was a result of the increase in the

resultant force.

91

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

QT and PT Horizontal Force
ACL- Deficient
500
--0— PT
400 i- —QT '
300
2
“g 200 I l
u.
100 T AV
0 i i W4 i
? 20 40 60 80 100 1‘20
-100
Angle of Flexion (Degrees)
(2!)
OT and PT Vertical Force
ACL-Deficient
400
350 ._ —0— PT
—QT
300
g 250
g 200
IL 150
100
50
0 . i i
0 20 100 120
Angle of Flexion (Degrees)
(b)

 

 

Figure 47: Quadriceps and Patellar Tendon forces for the ACL-deficient
knee. (a) Horizontal Forces (b) Vertical Forces.

92

The forces obtained for the ACL-reconstructed knee were of particular interest in
the analysis of the results (Figure 48). A common adaptation during gait for subjects with
ACL-deficiency, or following an ACL-reconstruction, is known as quadriceps avoidance
gait. When quadriceps avoidance gait is present there is a reduction in the quadriceps
muscle activity. This reduction in muscular contraction is believed to be due to the
possible anterior displacement of the tibia relative to the femur caused by excessive force
transmitted to the PT. The reduction in QT and PT forces observed in the ACL-
reconstructed knee could be interpreted as a similar type of adaptation being present
during the squat. The quadriceps contraction is not needed as a secondary stabilizer to the
knee following reconstruction of the ligament. However, the subject might still feel
cautious about contracting the quadriceps for fear of causing an excessive translation of
the tibia over the femur. A weaker quadriceps group might also influence the forces

generated during the squat.

 

 

 

 

 

 

 

 

 

 

 

 

 

OT and PT Horizontal Forces
ACL-Reconstnrcted
500
+PT
400 ‘* —OT
300
2
g 200
o
u.
100
0 -w t . . .
f 20 40 60 80 100 1:20
-100
Angle of Flexion (Degrees)
(a)

93

 

QT and PT Vertical Forces
ACL- Reconstnrcted

 

 

 

 

 

 

 

 

 

 

 

 

 

 

400

+PT

35° " —QT
300
g 250
g 200
LL 150
100
50

O -M i i i e
0 20 40 60 80 100 120
Angle of Flexion (Degrees)
(b)

Figure 48: Quadriceps and Patellar Tendon forces for the ACL-
reconstructed knee. (a) Horizontal Forces (b) Vertical Forces.

The results obtained from the mathematical model followed the expected trends
for the data sets used in the experimental testing. Subjects with known differences in the
mechanical behavior of the knee joint were selected to test the accuracy of the model in
predicting such differences. Data from the uninjured subject were used as the norm for
comparison against the other two conditions tested with the model as well as for
comparison with previously reported data.

The ICR was determined to be a good estimator for the relative position between
the femur and the tibia. The linear translation between the two bones was found to be
consistent with the known laxity of the knee joint. Even though the rolling and sliding
pattern did not directly match those of previous research, the differences in the testing

protocols supported the differences in results.

94

Laxity of the joint and ligament response was measured and illustrated using
laxity curves. The results of the forces sustained by the ligaments in response to the laxity
of the knee joint were accurate. The expected differences between the norm, ACL-
deficient and ACL-reconstructed conditions were shown in the data from the model.
These results add to the usefulness of the model in the rehabilitation and injury
prevention fields.

In combination with the laxity curves, information about ligament strain versus
angle of flexion and quadriceps force versus angle of flexion provide important tools that
will add to the better development of rehabilitation exercises. Estimation of quadriceps
forces and ligament forces during dynamic activities also can help identify possible injury

situations.

95

CONCLUSIONS

The knee joint has been one of the most researched joints in the human body. The
complicated joint mechanics have led to the development of a number of mathematical
models. In the review of literature, a twoodimensional quasi-static mathematical model
was presented for the estimation of ligament and quadriceps forces as a function of knee
flexion. The model used the ICR for the estimation of the amount of rolling and sliding
present at the knee joint. The anterior cruciate, posterior cruciate and collateral ligaments
were modeled as extensible units. The patella was modeled as a rectangle allowing the
transmission of the quadriceps force to the patellar tendon assuming equilibrium of the

patello-femoral mechanism.

Although some of the results did not match the trends presented in the literature,
the model was based on some underlying differences which might have led to such
differences in results. The model was developed with the purpose of being used in
conjunction with kinematic and kinetic data obtained during an unconstrained activity. To
test the accuracy of the model, data from squatting trials were used as the input
parameters. Most of the previously reported data were from tests performed in-vitro and
under constrained motions. Therefore, any comparison of results has to be done keeping
these differences in mind.

The simplifications of the model were found to be valid by the accuracy of the
results obtained. The model was able to predict the expected differences in mechanical
behavior between the three tested conditions without being affected by the simplifications

imposed in the mathematical develOpment. Limiting the model to two-dimensions did not

96

have an influence on the accuracy of the results. Further validation of the model might be

supported with multiple tests for similar conditions.

1. Limitations

The following limitations need to be taken into consideration when analyzing the
results of the model. First, due to the use of two different mathematical software
packages, the solution of the equilibrium equations was performed on a frame-by-frame
basis thus making the process time consuming, especially for long files. This led to the
decision only to present one half cycle of the squat for each tested condition.
Programming the model in one software package would allow a faster execution of the
code and the ability to use the entire data set. Second, due to the characteristics of the
data and the low number of subjects used to test the model, a statistical comparison of the
results was not performed. The results Obtained were reported in a descriptive manner.
Third, the model was designed to take into consideration only the forces generated by the
quadriceps muscle. The importance of the hamstrings group and the gastrocnerrrious as
contributors to the knee mechanics was ignored. Strain and joint displacement patterns
might have been affected by the contraction of the other muscle groups which were

omitted during the model’s use and interpretation of the results.

11. Suggestions for future studies

Further investigation in this area is recommended. The model could be modified

to calculate the forces of the hamstrings instead of the quadriceps and to study the effects

97

of the knee flexors on the joint mechanics. Further improvement of the model could lead
to the inclusion of both muscle groups.

Further research would be beneficial in determining the exact attachment sites for
the modeled ligaments. More accurate results also might be obtained by modeling the
ligaments as multi-bundle fibers. The ability to calculate strains for more than one fiber
per ligament would give a better indication of the mechanical behavior of the ligament
bundle.

The mathematical approach to the calculation of the internal forces sustained by
the soft tissue of the knee joint allowed the examination of different mechanical and
structural adaptations to the knee joint to be performed in a non-invasive manner.
Kinematic and kinetic data collected during any activity could be used as the input to the
model, with the ability to mathematically induce changes in the structural and mechanical
properties of the knee joint. The model offers a unique contribution to the fields of
orthOpaedics and sports medicine. The results obtained with the model will delineate the
mechanical behavior of the knee joint and could be used to address treatment or surgical
questions. The model also provides insight as to the behavior of the ligaments and
quadriceps muscles, which could be applied to the development of safe and efficacious

rehabilitation protocols.

98

APPENDIX A

 

APPENDIX A

Main Program Code

This appendix presents the code written in Matlab computational software.
Comment lines have been included in the program to clarify the definition of variables, or
guide the reader in the flow of the program. These comments are enclosed on percentage
signs (%). Any wording preceded by a % is not an executable line in the program. The
name of functions called by the main program have been highlighted. These functions

have been added to the end of the main code.

% °/o°/o °/o °/o °/o °/o °/o °/o °/o °/o °/o°/o °/o°/o°/o°/o°/o°/o°/o °/o°/o °/o°/o°/o °/o°/o °/o % % °/o °/o °/o °/o °/o °/o °/o °/o °/o °/o

°/o k_exper Matlab script - Uses the Instantaneous %
"/0 center of rotation to determine the relative position °/o
°/o between the femur and tibia, calculates the change in °/o
°/o length of the ACL, PCL and LCL, and calculated the Force %
°/o on each ligament. °/°

o/Oo/Oo/Oo/OO/OO/Oo/Oo/Oo/Oo/Oo/Oo/OO/OO/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/OO/OO/Oo/Oo/Oo/OO/O

"/0 The program has been tested for the "normal" motion of the
"/0 femur relative to the fixed tibia using made up parameters
°/o for the position of 2 points in the femur and 2 points in

°/o the tibia for the calculation of the ICR. The calculation

°/o was made on Mathcad and the values are input parameters to
"/0 this program. The Experimental testing of this program

°/o will be tested later using data collected during squatting

% trials.

%

% Author: Claudia A. Angeli

O

°/o Model reads ICR file for theta, ICR r.t.shank and ICR in

°/o lab C.S. Output file has: theta, APdisp, roll and dispx

"/0 parameters

% The position of the Sa target relative to the center of

% the tibial plateau and the ICR location in model units

% taken from the normal ratio between the length and the

°/o width Of the femoral condyles.

100

 

%%°/o %°/o°/o °/o °/o %%%%%% %%%%%% °/o %%% %%% °/o°/o°/o °/o °/o °/o % °/o °/o °/o % % °/o
% Definition of model parameters taken from MRl images

"/0 values in mm (MRI image sizes)

%%% ‘Vo‘Vo‘Vo%°/o°/o°/o°/o°/o°/o°/o °/o°/o°/o°/o°/o°/o°/o°/o %%% °/o°/o °/o°/o% °/o %% % °/o °/o % % °/o %

Tratio=1 .644;

addline=19-5;
heightcond=19;
widthcond=37;
mecaxis=20;

tbwdthratio=3/3.5;
maxy=2*pi;
minx=5*pi/2;
maxx=3*pi/2;
miny=pi;

midx=2*pi;

midy=1.5*pi;

°/o True ratio - ratio between the MRI image and the

"normal" knee

°/o Distal segment to adductor line

°/o Height of the condyle

°/o Width of the condyle

°/o Distance from anterior aspect Of the condyle to the
mechanical axis of the femur

"/0 Width ratio of the tibial plateau

°/o I value - defining the distal limit of the femur

°/o I value - defining the anterior limit of the femur
% t value - defining the posterior limit of the femur
°/o I value - defining the proximal limit

of the femur femur

"/0 t value - used to specify the location

Of the Tb target

°/o I value - used to specify the location Of the Tb
target

°/o °/o °/o°/o% °/o % % %% %%%%% %%% °/o°/o°/o°/o°/o°/o°/o%‘yo‘yo %%%%% °/o °/o %% % °/o °/o
"/0 Read file with ICR location and theta angle
°/o°/o°/o°/o°/o%%°/o °/o % % % °/o°/o°/o °/o% °/o°/o% °/o°/o°A;°/o°/o°/o‘yo°/o °/o°/o °/o°/o°/o % °/o °/o%°/o °/o °/o

lCR_t=topen('lCR_m.pm');

[p,pf]=fscanf(|CR_f,'°/of',[6,498]);

pc=p/1 0;

tmax=p(1,:);

lCRsx=p(3,:);
|CRsy=p(4,:);
lCRix=pc(5,:);
ICRiy=pc(6,:);

lOl

%%%%%%%%%%%%%%%°/o°/o°/o°/o°/o%°/o°/o°/o% %%%°/o°/o°/o°/o°/o°/o%°/o°/o°/o°/o %
% Definition of the involute circle
%°/o%°/o% % °/o°/o%%%%°/o%%°/o‘Vo‘Vo°/o°/o°/o°/o% °/o% °/o°/o°/o °/o °/o °/o °/o °/o % % °/o °/o °/o °/o %

b=1.5; °/o radius

phi=pi; % phi value

for it=1:max(size(tmax))
d=tmax(it); % loop for range of knee flexion
theta=d*(pi/180); °/o Convert angles from radians to degrees

Tr=[cos(theta) sin(theta); -sin(theta) cos(theta)];
°/o Transformation matrix

for i=1 :1 % Loop to define the involute (all x and y values
within the involute)
T(i)=(i-1)*0.002;
tc(1)=T(i):
11:0;
tf=5*pi/2-T(1)-phi;
tftrunc=str2num(num28tr(tf));
=ti:0.001 :tftrunc;

o/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/OOADO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O(yoo/Oo/Oo/Oo/Oo/Oo/Oo/O0A)
°/o Definition Of femoral geometric surface
°/o°/o °/o % °/o °/o%°/o °/o °/o°/o %% °/o°/o % °/o°/o°/o°/o°/o°/o%°/o°/o°/o % %°/o°/o °/o % °/o°/o°/o°/o % % °/o °/o

lx=-b.*cos(T(1 )+phi+t+theta)-b.*t.*sin(T(1 )+phi+t+theta);
°/o X values of involute for rotated femur (at angle theta)

ly=b.*sin(T(1 )+phi+t+theta)-b.*t.*cos(T (1 )+phi+t+theta);
°/o Y values Of involute for rotated femur (at angle theta)

lxo=-b.*cos(T(1 )+phi+t)-b.*t.*sin(T(1 )+phi+t);
% X values of involute at full extension

lyo=b.*sin(T(1 )+phi+t)-b.*t.*cos(T (1 )+phi+t);
% Y values of involute at full extension

°/o°/o°/o°/o°/o°/o°/o °/o %%%%%%%°/o°/o°/o°/o°/o%°/o°Aa°/o°/o°/o%%%%%%%°Aa°/o°/o°/o°/o°/o%
°/o Definition of limits of condyles
°/o%°/o°/o%°/o°/o°/o°/o%%°/o%°/o°/o°/o% °/o%°/o%°/o%%%%%% % °/o °/o °/o°/o°/o°/o%°/o°/o%°/o

lydist=b.*sin(T(1 )+phi+(-phi-theta+maxy)+theta)-b.*(-phi-theta+maxy).*cos(T( 1 )+

phi+(-phi-theta+maxy)+theta);
% Distal limit for rotated femur

102

lyprox=b.*sin(T(1 )+phi+(-phi-theta+miny)+theta)-b.*(-phi-theta+miny).‘cos(T(1 )+
phi+(-phi-theta+miny)+theta);
°/o Proximal limit for rotated femur

lxant(it)=-b.*cos(T(1 )+phi+(-phi-theta+minx)+theta)-b.*(-phi-theta+minx).*sin(T(1 )
+ phi+(-phi-theta+minx)+theta);
°/o Anterior limit for rotated femur

lxpost=-b.*cos(T(1)+phi+(-phi-theta+maxx)+theta)-b.*(-phi-theta+maxx).*sin(T(1 )
+ phi+(-phi-theta+maxx)+theta);
°/o Posterior limit for rotated femur

o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O
°/o Definition of fixed limits
o/OO/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O

lyfdt=b.*sin(T(1)+phi+(-phi+maxy))-b.*(-phi+maxy).* cos(T (1 )+phi+(-phi+maxy));
% Distal limit for femur at full extension

lyfpx=b.*sin(T(1)+phi+(-phi+miny))-b.*(-phi+miny).* cos(T(1)+phi+(-phi+miny));
°/o Proximal limit for femur at full extension

leant=-b.*cos(T (1 )+phi+(-phi+minx))-b.*(-phi+minx).* sin(T(1)+phi+(-phi+minx));
°/o Anterior limit for femur at full extension

lepst=-b.*cos('l' (1 )+phi+(-phi+maxx))-b.*(—phi+maxx).* sin(T(1)+phi+(-
phi+maxx));
°/o Posterior limit for femur at full extension

lyptc=b.*sin(T (1 )+phi+(-phi+minx))-b.*(-phi+minx).* cos(T(1)+phi+(-phi+minx));
% Y location of patellO-femoral contact point at full extension

%%%% °/o % % % °/o°/o°/o°/o°/o°/o°/o %%% °/o°/o°/o %%%% %% % °/o % % % °/o %% % %% % %
°/o Definition of the Initial point of rotation. A circle with

% a 24mm radius was taken to fit the posterior aspect of the
% femoral condyle and taken as the initial rotation point

% when the knee starts to flex. The Tb target (or point)

% will be define as this initial point of rotation. The 24mm

% is converted to model units (modelratio=6.6285) and then
°/o the point is found by subtracting/adding this value to the
°/o previously specified limits of the femoral condyle. During
% knee flexion this point is located by rotating the Tb

°/o (original) point with the transformation matrix, ONLY to

103

% specify the coordinated of the point as given by the

°/o transformation of the involute. The true coordinates of

% this point are found using the ICR calculations.

%% °/o°/o% %%%°/o°/o%%°/o%%°/o°/o %%°/o°/o°/o°As°/o % % °/o% % % °/o °/o°/o °/o °/o °/o % °/o % °/o

lxtbo=lepst-3.62; % X coordinate of the Tb target at full
extension
lytbo=lyfdt+3.62; % Y coordinate of the Tb target at full
extension
ltb=Tr*[lxtbo; Iytbo]; % Transformation of the Tb target position to
define the MODELS rotated position
lxtb=ltb(1); % Separation of the X-Y coordinates
lytb=ltb(2);
end
for m=0:100
C='l;
end
if 0::
d=11;
else
end
lxo=lxo(1); "/0 Definition of the first X value in the involute at FULL
EXTENSION
lyo=lyo(1); % Definition Of the first Y value in the involute at FULL
EXTENSION

%°/a°/o%°/o%%%°/o%%°/o%°/o°/o%‘7o°/o%%°/o%%%%%°/o°/o°/o°/o°/o%%%%°/o°/o°/o%%
% Call function to find the true ratio between the subject's

°/o knee and the model
%°/o%%°/o%°/o%%%%%°/o°/o°/o°/o‘7o%°/o°/o°/0°Ai%%°/o%%%%%%%%%%%%°/o°/o°/o

[pmratio, Smax, Smayl=Eh§ésté
%%%%%%%%°/o°/o°/o%%%%%%%%%%%°/o%%%%%%%%%%%%%%°/o%%
% Definition of model parameters dependent on IX and ly
%%%%%°/o%°/o°/o°/o%%%%%°/o%°/o%%%°/o%°/o%%%%%%%%%%%%%%%°/o
modelratio=heightcond*Tratio/norrn(lyfpx-lyfdt);

% Model ratio using the height of the condyle and the
True ratio from MRI images

104

 

wdthtibia=widthcond*Tratio*tbwdthratio/modelratio;
% Definition of the width of the tibial plateau

jtspace=0.3*Tratio/modelratio;
°/oDefinition of the joint space taken from MRI images

o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Ocyoo/Oo/Oo/Oo/O
°/o Model parameters for the tibia
°/o °/o %%% °/o°/o °/o °/o%% % %°/o °/o °/o°/o°/o %% °/o%%%% °/o°/o % °/o °/o °/o °/o°/o°/o % % % % °/o %

lxotibia=leant; °/o Anterior limit of the tibial plateau at full extension
lyotibia=lyfdt; % Y position of the tibia relative to the femoral condyle distal
limit

tibia=[lxotibia; lxotibia+wdthtibia];
°/o definition of the tibial plateau

jp=lyotibia; °/o jp variable

°/o °/o°/o% °/o°/o°/o°/o% °/o°/o % °/o°/o°/o°/o°/o°/o°/o°/o°/o°/o°/o°/o °/o°/o °/o °/o%% °/o°/o°/o°/o°/o °/o °/o °/o °/o °/o
°/o Calculate relative position between ICR and Involute

°/o origin
o/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/OO/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O

o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/OO/Oo/OO/OO/Oo/Oo/OOA)O/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/O
°/o Calling function to calculate contact point at rotated

% position of the femur (from full extension --> transformed

°/o by Tr)
O/OO/OO/o0/00/00/00/00/00/00/9OY/PO/oO/qO/oO/OO/oO/oO/OO/OO/oO/oO/oO/oO/oO/oO/oO/oO/oO/OO/OO/OO/OO/OO/OO/oO/OO/OO/oO/OO/o
[Rfcp, lxct, purer]=§onja§tj§ffit(theta, T, phi, maxy, maxx, minx, miny, b, Tr);

Xc=Rfcp(1); °/o Definition of the rotated contact point (from full
extension transformed by Tr)

Yc=Rfcp(2);

lyct=lydist; °/o lxct & lyct are the contact points at theta degrees of

flexion.
°/o lxct is used in the definition of the displacement of
the contact point.

105

°/o°/o°/o°/o°/o% %%%%%°/o %% %%% %%%% °/o°/o°/o°/o°/o°/o°/o °/o°/o°/o % °/o % °/o % °/o % °/o %
°/o Calling function to calculate contact point at rotated

% position of the femur.
%°/o°/o°/o °/o°/o °/o % %°/o °/o°/o%°/o°A>°/o°/o°/o°/o°/o°/o°/o°/o %% °/o% °/o °/o °/o % °/o °/o °/o °/o % °/o °/o °/o °/o

thpfifipjfi, phi, maxy, maxx, minx, miny, b, jtspace, lydist);

Xtc=thp(1); % Contact point at full extension, prior to the
application of the translation of disp x.
th=thp(2);

% % %°/o °/o°/o °/o °/o % % % °/o°/o °/o °/o°/o°/o°/o°/o°A>°/o% °/o°/o °/o °/o°/o °/o°/o% %% %% °/o °/o % °/o % °/o
% Instantaneous Joint center relative to the SA shank target
o/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O

lCRx=-28.9757; %Using values from Mathcad pure rolling/sliding and/or
combination files

lCRy=26.6107; °/o (rjcs)
°/o Convertion of the ICR relative to SA to model units

lCFixsa=lCRsx(it)/pmratio;
lCRysa=ICRsy(it)/pmratlo;

Sax=lxotibia+(wdthtibia/2)+Smax;
% location of the Sa target calculated from the
difference in the X direction between femoral condyle
and the Sa target position in the standing file. Units
have been converted to model units using a 9.641
convertion factor. This factor was calculated using a
“normal” .8636 ratio between the length/width of the
femoral condyles and the known width of the femoral
condyle from the standing file. The modelratio
parameter specified in this program only applies to
the convertion between the MRI image estimated
parameters to model units.

Say=lyotibia-Smay; °/o location of the Sa target relative to the tibial plateau
in the Y-direction.

JCx(it)=Sax+lCRxsa; % Estimation of the Joint center position. Taken from
the Sa target and adding the

106

JCy(it)=Say+lCRysa; % relative positon of the ICR.

% °/o°/o°/o°/o% %%% °/o°/o °/o°/o % °/o°A>°/o°/o°/o°/o°/o°/o %%°/o°/o°/o°/o % % °/o %% °/o % % °/o % °/o %
°/o Estimation of the Tb target given the position of the

% joint center (previously defined) and subtracting the

% relative position of the ICR (femoral component). This

% value is the predicted location of the Tb target as

°/o ESTIMATED from TRUE data, not the simple rotation of the

°/o involute

°/o°/o°/o°/o°/o°/o%°/o°/o°/o % °/o°/o % °/o %%% %%% %%% °/o %% °/o °/o °/o °/o % °/o % °/o % % % °/o %

diff2=-purer-Xtc; % difference between the Contact Point at full
extension and the contact point of the theta rotated
femur.

diff3=lx(1)-lxo; °/o Difference between the origin location of the

rotated femur and the femur at full extension.
diff1=lxtb-lxtbo;

diff4=JCx(it)-Ixtbo;

°/o %% °/o %%% °/o %°/o°/o°/o°/o°/o°/o°/o %%%%%% °/o%°/o%‘Vo°/o%°/o°/o°/o°/o°/o°/o %%%°/o%
°/o Calculation of the difference between the ESTIMATED Tb

% target position and the CALCULATED tb target position

°/o given the rotation of the involute. The dispx and dispy

% parameters specify the translation of the femoral condyle,

% at point Tb between the two methods of calculation.
o/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OOA)O/OO/Oo/Oo/O0A)o/Oo/O%O/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O

di=diff4-diff1; °/o The difference between the distance given by the
true motion (lCR-lxtb) and the distance between the
two contact point parameters. This defines the
necessary displacement of the femoral condyle over
the tibial plateau specified by the true motion of the
femur over the tibia.

dispy(it)=-(|yo-IY(1));
°/o Translation of the involute to the CORRECTED
position. The difference calculated above is used to
translate the entire involute. Rigid body mechanics
prescribes the translation of the entire involute given
the especified translation of one point.

dispx(it)=di;

107

lx=lx+dispx(it); °/o Translation of the involute by parameter dispx
Iy=ly+dispy(it); °/o Translation of the involute by parameter dISpy

|CPx=Xtc; °/o definition of new variable to allow the untranslated
value of the contact point at full extension to be used
in subsequent calculations.

Xtc=Xtc+dispx(it); °/o Translation of the contact point according to the
involute translation
th=th+dispy(lt);

Xc=Xc+dispx(it); %Translation of the contact point according to the
involute translation
Yc=Yc+dispy(it);

roll_disp=purer; °/o Definition of true rolling given by the x-distance
between the rotated contact point and the contact
point at full extension.

roll(it)=roll_disp*pmratio;
°/o Definition of pure rolling displacement in mm units.
Converted back.

APdisp(it)=(Xc+roll_disp)*pmratio;
°/o APdisp is define as the slipping component of the
translation of the femur over the tibia. The
displacement of the contact point is subtracted
from the pure rolling component associated with knee
flexion. Parameter in True mm units.

end

% % °/o %% %% °/o % °/o % %% °/o % %% %%% °/o°/o %% °/o°/o %%% % % °/o°/o % °/o °/o % °/o % °/o
% Definition of the Ligament parametes taken from MRI

% images. Ratio parameters are values taken from the MRI

% images. Insertion parameters are calculated using the

"/0 model values (limits specified earlier) to define the x

"/0 and y coordinates of the insertion points.

°/o % °/o °/o°/o°/o%% °/o°/o %%% °/o°/o °/o°/o°/o%°/o % °/o °/o °/o °/o°/o°/o°/o % °/o °/o °/o°/o %%% °/o°/o % °/o
ratioaclt=1 5/modelratio;

ratiolcltx=23*Tratio/modelratio;

ratiolclty=1 1*Tratio/modelratio;

lnsertiontacl=lxotibia+ratioaclt;

108

Insertiontpcl=lxotibia+wdthtibia;
lnsertiontlcl=lxotibia+ratiolcltx;
lnsertiontlcly=jp-ratiolclty;

o/Oo/Oo/O°/Oo/(DO/Oo/Cio/Oo/OO/OO/OO/OO/Oo/Oo/OO/tno/Oo/Oo/Oo/Oo/O0/0o/Oo/Oo/OO/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O
% Model parameter for the femur

°/o Definition of the Ligament parameters taken from MRI

°/o images. Ratio parameters are values taken from the MRI

°/o images. Insertion parameters are calculated using the

°/o model values (limits specified earlier) to define the x

% and y coordinates of the insertion points.
o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/OOADO/OO/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/O

lnsacl=3.226;
lnspcl=2.2081 ;
ratioacl=1 .6477;
ratiopcl=2.631 5;
ratioxlcl=1 .76294;
ratioylcl=6.5794;

Insertionxacl=lnsertiontacl+ratioacl;
lnsertionxpcl=lnsertiontpcl-ratiopcl;
lnsertionxlcl=lnsertiontch-ratioxlcl;

lnsertionyacl=jp+lnsacl;
lnsertionypcl=jp+lnspcl;
lnsertionylcl=lnsertiontlcly+ratioylcl;

o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O
°/o Call function to calculate of external moments
O/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OGAO/Oo/Oo/OO/Oo/Oo/OOADO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O

[M_flex, M_flex2, BW, 90, Fy, Fz]=§fiffif;efi:t§(ICRix, ICRiy);

e3y_all=ec(:,2); °/o Definition of column vector with 93 values to be
used in calculation of internal moments.

for it=1:max(size(tmax))
d=tmax(it); °/o Loop (same as defined earlier) to go through range
of flexion
theta=d*(pi/180);
Tr=[cos(theta) sin(theta); -sin(theta) cos(theta)];
°/o Redefinition of the transformation matrix

109

 

o/Oo/OOAO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O%°/O%%o/O°/Oo/O°/O°/O°/O°/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O
°/o Transformation of ligament insertions on the femur to

°/o follow the rotation of the knee. ALL previously

% defined values were taken from the full extension

% position.

°/o°/o%°/o°/o°/o°/o°/o%°/o°/o°/o%°/o°/o%%%%%°/o°/o°/o°/o°/o°/o°/o °/o °/o °/o % °/o % °/o % % °/o % % °/o

[Rlpcl]=Tr’[lnsertionxpcl; lnsertionypcl];

[Rlacl]=Tr*[lnsertionxacl; lnsertionyacl];
[RIIcl]=Tr*[lnsertionxlcl; lnsertionylcl];

o/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/OGAO/Oo/Oo/Oo/Oo/Oo/oo/oo/Oo/O
% Separation of the matrix components for each ligament

°/o insertion, and translation of the points according to

°/o the previously specified displacement of the entire

°/o involute.
o/OOADO/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O

Rlpcl(1)=Rlpcl(1)+dispx(it); Rlpcl(2)=Rlpcl(2)+dispy(it);
Rlacl(1)=Rlacl(1)+dispx(it); Rlacl(2)=Rlacl(2)+dispy(it);

Rllcl(1)=Rllcl(1)+dispx(it); RI|cl(2)=Rllcl(2)+dispy(it);

%°/o %°/o°/o°/o°/o%% °/o%°/o%% °/o°/o%°/o°/o°/o °/o% °/o°/o°/o°/o°/o%%°/o°/o°/o°/o°/o°/o°/o°/o°/o°/o °/o
% Definition of the Length of the ligament (AT THE

% SPECIFIED FLEXION ANGLE). Pythagorian Theorem
o/O(yoo/OO/OOADO/OO/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OOADO/Oo/Oo/Oo/O

chl=sqrt((lnsertiontpcl-Rlpcl(1))’\2+(Rlpcl(2)-jp)’\2);
Lacl=sqrt((Rlacl(1)-lnsertiontacl)’\2+(R|acl(2)—jp)’\2);
Llcl=sqrt((lnsertiontlcl-Rllcl(1))’\2+(Rllcl(2)-lnsertiontlcly)"2);

% °/o °/o°/o°/o°/o°/o°/o°/o°Aa %°/o %%% °/o°/o°/o%%°/o°/o%%%%% %% % %%%%%% % °/o % %
"/0 Definition of the angle between the ligament and the

°/o horizontal. All angles were taken as acute angles

°/o during FULL FLEXION. The following values are during

°/o the specified theta.

%% °/o °/o%%°/o%% °/o°/o°/o°/o%°/o°/o°/o°/o°A:°/o%°/o%%% %% % °/o°/o°/o%°/o°/o °/o °/o °/o %% %

110

 

angacl=asin((Rlacl(2)-jp)/Lacl);

if Rlacl(1)<lnsertiontac|
% Necessary to control for angle greater than 90

angacl=pi-asin((Rlacl(2)-jp)/Lacl);

end

angpcl=asin((Rlpcl(2)-jp)/chl);

if Rlpcl(1)>lnsertiontpcl
% Necessary to control for angle greater than 90
angpcl=pi-asin((Rlpcl(2)-jp)/chl);
end

anglcl=asin((Rllcl(2)—lnsertiontlcly)/Llcl);
if Rllcl(1)<lnsertiontlcl
°/o Necessary to control for angle greater than 90
anglcl=pi-asin((Rllcl(2)-lnsertiontlcly)/Llcl);
end

°/o °/o °/o°/o°/o°/o°/o°/o°/o°/o°/o °/o %°/o%%°/o°/o%°/o%°/o%%%%%%%% °/o°/o%°/o%%%°/o°/o°/o
°/o Conversion of ligament angle values to degrees
o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OOADO/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/OO/Oo/OOADO/Oo/Oo/Oo/Oo/Oo/O

ta(it)=angacl*180/pi;

tp(it)=angpcl*180/pi;
tl(it)=anglcl*180/pi;

111

°/o % °/o %% %% °/o °/o°/o°/o°/o°/o°/o°/o %%%%% %°/o %% %%%% °/o °/o °/o °/o % % % °/o % % °/o %
°/o Calculate ligament length change using the

°/o change_length FUNCTION. The output of the function is

% the lchg (ligament length change) and strain (the

% calculated strain for each ligament). Following the

°/o calling of the function the output matrices are split

°/o into the 3 ligament components.
o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/OO/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/OO/Oo/Oo/OO/Oo/Oo/o

u‘-w~n;mw-¢

[lchg, strain]=§hanmge lo M( Lacl, chl, Llcl);

Dacl=lchg(1);
Dpcl=lchg(2);
chl=lchg(3);

Sacl(it)=strain(1);

Spcl(it)=strain(2);

Slcl(it)=strain(3);
o/Oo/OO/Oo/OO/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/OO/C)o/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/C)o/Oo/C)o/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O
°/o Calculation of ligament forces using the lig_force

°/o FUNCTION. The output contains the forces for each

% of the 3 ligamens.

°/o °/o °/o°/o °/o °/o °/o % °/o °/o°/o%°/o°/o % % °/o °/o°/o°/o°/o°/o °/o°/o°/o°/o°/o°/o°/o %°/o °/o°/o °/o °/o % °/o
[Facl, chl, Flcl]=[fi:f§fic§2(lchg, modelratio);

Fa(it)=Facl;

Fp(it)=chl;

Fl(it)=Flcl;

e3y=e3y_all(it); % Definition of Z axis for internal moments to be
calculated about same true axis as external moments.
See before the loop for definition of e3y_all variable.

112

°/o % % °/o % °/o°/o %°/o °/o% %%% °/o °/o °/o °/o °/o°/o % °/o °/o °/o °/o % °/o % °/o % °/o % % °/o % °/o %
% Calculation of internal moments (call function)
°/o°/o%°/o%°/o% %°/o °/o°/o %%% %% % %%%%% °/o °/o °/o°/o °/o % %%% °/o°/o % °/o % %

[Maz, Mpz, Mlz, le, Fly]=fi%eg{§(93y,BW,modelratio,JCx, JCy, angacl,
angpcl, anglcl, Rlacl, Rlpcl, Rllcl, jp, Insertiontacl, lnsertiontpcl, lnsertiontlcl,
Insertiontlcly, Facl,chI,Fch);

Mia(it)=Maz(it);

Mip(it)=Mpz(it):
Mil(it)=Mlz(it);

Faclx(it)=le(1 );
chlx(it)=le(2);
Flclx(it)=le(3);

Facly(it)=Fly(1 );
chly(it)=Fly(2);
Flcly(it)=Fly(3);

Fyi=Fy(it); °/o Definition of Fy and F2 forces for the given it value
°/o to be used in

in=Fz(it); °/o the calculation of the contact force.

end

%%% % °/o % °/o%°/o°/o % °/o °/o °/o °/o °/o°/o°/o%%°/o °/o %%% °/o%°/o°/o°/o % °/o°/o % %% °/o % % %
°/o Writing output files.

°/o Parameters will be read by Mathcad templates for the

°/o calculation of patello-femoral equilibrium mechanism and

% find the solution for the quasi-static model of the knee

°/o°/o%°/o°/o°/o°/o °/o %%%% °/o °/o °/o°/o°/o°/o°A>°/o°/o % %%% % % %%°/o% %%% % °/o °/o °/o % %

°/o Output file 1
fout=fopen('disp_mdl2n.out','w')
for it=1:max(size(tmax)),

fprintf(fout, '%f %f %f %f %f %f %f %f\n',tmax(it), APdisp(it), roll(it),
lxant(it),dispx(it),dispy(it), JCx(it), JCy(it))

113

 

 

end

°/o Output file 2
fout=fopen('momsZn.out','w')

for it=1:max(size(tmax)),
fprintf(fout, '%f %f %f %f %f %f %f %f %f %f %f %f\n',tmax(it), Mia(it), Mip(it),
Mil(it), M_flex(it), Fa(it), Fp(it), Fl(it),SacI(it), Spcl(it), Slcl(it),e3y_all(it))
end
°/o Output file 3
fout=fopen('forces_mdl2n.out','w')
for it=1:max(size(tmax)),
fprintf(fout, 'o/of o/of °/of 0/of 0/of o/of o/of o/of %f\n',tmax(it), Faclx(it), FpC|X(it),

Flclx(it),Facly(it),chly(it), Flcly(it), Fy(it), Fz(it))
end

fclose all;

o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O
"/0 Definition of the ligament parameters for GRAPHING

°/o purposes only. X and Y coordinates are given for the

°/o origin and insertion points. tb is defined as the Y

°/o parameter for the tibia plateau.

°/o°/o%% %% °/o°/o°/o°/o°/o°/o°/o%°/o% % °/o °/o°/o °/o °/o °/o °/o°/o % °/o °/o % °/o % °/o°/o °/o °/o °/o °/o % °/o %

aclligx=[lnsertiontacl; Rlacl(1)];
aclligy=UP; Rlacl(2)];
pclligx=[lnsertiontpcl; Rlpcl(1)];
pc||i9y=lip; Rlpcl(2)l:

lclligx=[lnsertiontlcl; Rllcl(1)];
lclligy=[lnsertiontlcly; Rllcl(2)];

tb=lip; ID];

114

°/o Patella constant paramenters

 

[LPTe, PTalphae, PTbetae, PTi, gama]
modelratio, jtspace, lxotibia,jp);

' lyfdt, leant, lypfc, Tratio,

o/oPatella

[cp, Ppa,” Pd ,fpp, Pda, Ps2d, PsZp, PTo, PTi, Pcy,
ch]=E§tEeInltlon(theta,dispx, dispy,lxant, Tratio, modelratio, lxotibia,jp);

 

PTox=PTo(1 ); PToy=PTo(2);

PTix=PTi(1); PTiy=PTi(2);

comp=90*pi/180; °/o To define theta from the vertical

QTx=Ppa(1 )+(cos(comp-theta)*3);
°/o Quadriceps Tendon definition (length of 3 -->
arbitrarily picked)

QTy=Ppa(2)+(sin(comp-theta)*3);

% Definition of lines for graphing purposes only

Pdpx=[Ppa(1); Ppp(1)]; °/o Top X

Pdpy=lea(2); Ppp(2)l; % Top y

Pdax=[Pda(1); Pdp(1)]; °/o Bottom x
Pday=[Pda(2); Pdp(2)]; °/o Bottom y
Pantbx=[Pda(1); Ppa(1)]; °/o Anterior x
Pantby=[Pda(2), Ppa(2)]; °/o Anterior y
Ppostx=[Pdp(1) Ppp(1)]; °/o Posterior x
Pposty=[Pdp(2) Ppp(2)]; °/o Posterior y
PTx=[PTix; PTox]; % PT x

PTy=[PTiy; PToy]; °/o PT y

Pst=[P32d(1) P52p(1)]; % 2nd Art. Surface x
P32y=[Ps2d(2) P32p(2)]; °/o 2nd Art. Surface y
Ouadx=[Ppa(1) QTx]; °/o Quadriceps tendon x
Quady=[Ppa(2) QTy]; °/o Quadriceps tendon y

115

°/o Plot of knee
line(aclligx, aclligy) °/o ACL
title('Squatting - F267 (Norm) 110 Degrees of Flexion (Trial #10- Right)')
axis([-12 4 -10 6])

hold on
line(pclligx, pclligy) %PCL
hold on
line(lclligx, Iclligy) %LCL
hold on
plot(lx,ly) °/o Involute
hold on
line(tibia, tb,'color‘,'r,'markersize',2)
% Tibia Plateau
hold on
plot(Sax,Say, 'o','color','y')
°/o Sa target
hold on

plot(lxo,lyo, 'o','color,'r')
°/o First point of involute at full extension

hold on
plot(lxtb,lytb, 'o','color','b')

°/o Initial point of rotation at full extension
hold on

plot(JCx,JCy, '*','color,'g')
°/o Instantaneous center of rotation

hold on
plot(lxotibia, lyotibia, '*','color‘,'r)
"/0 Most anterior point in the tibial plateau (constant)

hold on

plot(Xtc, th, '+','color','b')
%Tibial plateau contact point

hold on
plot(ch, Pcy, '*','color, 'r’)
°/o Patello-femoral contact point

hold on
plot(Xc, Yc, '+', 'color'. '9')

116

°/o Tibio-femoral contact point at full extension

line(Pdpx, Pdpy)

hold on

line (Pdax, Pday)

hold on

line (Pantbx, Pantby)
hold on

line (Ppostx, Pposty)
hold on

line (Pst, PsZy, 'LineStyle',:)
hold on

line (Quadx, Quady)
hold on

line (PTx, PTy, 'color’,'g')
hold on

% %% %% % °/o °/o °/o °/o°/o %°/o %%% °/o°/o°/o°/o%°/o°/o°/o%°/o°/o°/o°/0°Aa % °/o°/o °/o % °/o °/o°/o °/o °/o
% Kneesize Function Code "/0
o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/OOADO/Oo/Oo/Oo/Oo/O

function [pmratio, Smax, Smay]=kneesize

"/0 User input to define the true ratio between the knee and
°/o the model. Also this values will be use to define the

°/o position of the Sa target used in the

°/o calculation of the ICR and in the graphical

°/o representation of the model

disp('Enter the following target locations in mm from the standing file');
disp(' ');

disp('Enter the X, Y, 2 location of the fl-cond target (one at a time)');
flcondx=input('X: ');

flcondy=input('Y: ');

flcondz=input('Z: ');

disp('Enter the 2 location of the fm-cond target');

fmcondz=input('Z: ');

disp('Enter the X, Y location of the Sa-prox target (one at a time)');
Saproxx=input('X: ');

Saproxy=input('Y: ');

width=norm(fmcondz-flcondz);

°/o definition of the true knee width
trueratio=0.8636; % ratio between "normal" knee depth and knee width

117

 

Modellength=9.4248; % parameter establishing the model's knee depth
(from lxant to lxpost)

length=trueratio*width;
°/o Definition of the depth of the patient's knee using
the established true ratio between depth and width
pmratio=lengtthodellength;
% Calculation of the patient's model ratio to be used
in any conversion between linear parameters
established in inertial or segmental coordinate
systems.
Smax=(Saproxx-flcondx)/pmratio;
°/o Position of the Sa target relative to the mid tibial
plateau (x)

Smay=(flcondy-Saproxy)/pmratio;
°/o Position of the Sa target relative to the tibial
plateau (y)

% °/o °/o % °/o °/o % °/o % °/o % °/o %%% °/o °/o °/o °/o°/o °/o % °/o % %%% °/o% °/o °/o °/o°/o °/o % °/o °/o % °/o °/o
% Contact_Point Function Code °/o
o/Oo/OO/OO/Oo/Oo/Oo/Oo/Oo/OO/Oo/OO/Oo/Oo/Oo/OO/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/O

function [Rfcp, lxct, purer]=contact_point(theta, T, phi, maxy, maxx, minx, miny,
b, Tr)

% Function CONTACT_PO|NT will define the initial contact
°/o point of the femur/tibia at full extension. lyfdt is the

% distal limit of the femur during full extension Xc is the

"/0 intersection with lyfdt at full extension The Contact

°/o point will the transformed in the rotated coordinate

% system to follow the movement relative to the rotation of
°/o the femur.

% The (lxc, lyc) will be used to calculate anterior

°/o displacement of the femur relative to the tibia (see

°/o tct_pt function)

lyfdt=b.*sin(T (1 )+phi+(-phi+maxy))-b.*(-phi+maxy).* cos(T(1)+phi+(-phi+maxy));
Ixc=-b.*cos(T(1)+phi+(-phi+maxy))-b.*(-phi+maxy).* sin(T (1 )+phi+(-phi+maxy));

lxct=-b.*cos(T(1 )+phi+(-phi-theta+maxy)+theta)-b.*(-phi-
theta+maxy).*sin(T(1 )+phi+(-phi-theta+maxy)+theta);

118

purer=-(1/2.*(b’\2.*(maxy-phi-theta)’\2)"(1/2).*(maxy-phi-theta)-1/2.*(b"2.*(maxy-

phi)’\2)A(1/2).*(maxy-phi)):
Rfcp=Tr*[lxc; lyfdt];

°/o °/o % °/o°/o %%%%% °/o %%%% %% °/o°/o°/o°/o°/o°/o °/o °/o °/o°/o°/o°/o °/o °/o °/o°/o °/o % °/o °/o °/o °/o %
°/o tcp_pt Function Code °/o
0A)o/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/O

function [thp]=tct_pt(T, phi, maxy, maxx, minx, miny, b, jtspace, lydist)

% Function to calculate the tibial contact point at the full

% extension position Iydist is the distal femur limit at

°/o full extension, defined by the maxy variable and dependent
% on the angle of flexion lxc will define the X point

% relative to lydist.

°/o No transformation is necessary since the TIBIA is taken as
% the FIXED segment the jtspace is the adjustment due to the
°/o shifting when graphing different angles.

lxc=-b.*cos(phi+(-phi+maxy))-b.*(-phi+maxy).*sin(phi+(-phi+maxy));
thp=[lxc; (lydist-jtspace)];

°/o % % °/o °/o °/o % °/o °/o % °/o °/o °/o °/o %%% °/o°/o°/o% %°/o °/o°/o °/o %°/o°/o%°/o°/o% °/o °/o°/o°/o°/o % °/o
% e_moments Function Code °/o
°/o °/o °/o % °/o °/o °/o °/o°/o°/o°/o°/o °/o °/o % °/o°/o %%% %%%°/o°/o°/o°/o°/o°/o°/o %%% °/o°/o°/o°/o°/o°/o°/o

function [M_flex, M_flex2, BW, ec, Fy, Fz]=e_moments(ICRix, lCRiy)

°/o Calculation of External moments using as input paraments 3
% components of Forces and 3 componet of Moments as measured
"/0 by the force plate. Need the X0 and 20 offsets from the

"/0 Lab OS to the FP OS. All 8 numbers are input parameters in
°/o the function. The ICR location should be read in the LAB

°/o CS. The transformation is written for T to be transformed

°/o from FP to LAB.

% The output of the moments will be in N/m about the ICR all
% in LAB CS.

% Input parameters necessary in the function. These should
% be modified to read the force file to be able to calculate

°/o the e-moments for the complete motion trial.

119

"/0 Read force file
force=fopen('a1 0c52.frc');
[f,ff]=fscanf(force,'%f',[7,236]);

°/o Read condyle location file
cond=fopen('virtual.pm');
[p,pf]=fscanf(cond, '°/of',[12,236]);
P=pl10;

°/o Assign variables to force file columns
Fx=f(3,:);

Fy=f(4,:);

Fz=f(2,:);

mx=f(6,:);

my=f(7,:);

mz=f(5,:);

% Used input of X and Z offsets and BW
Xo=input('Enter the X0 offset: ');
Zo=input('Enter the 20 offset: ');
BW=input('Enter BW in Newtons: ');

% Transpose force variables
Fx=Fx';

Fy=Fy‘;

Fz=Fz';

mx=mx';

my=my':

mz=mz';

% Assign variables to virtual targets
fm_condx=p(4,:);
fm_condy=p(5,:);
fm_condz=p(6,:);

fl_condx=p(1 ,:);
fl_condy=p(2,:);
fl_condz=p(3,:);

PVOXX=P(7.I);

proxy=p(8.:);
prOXZ=p(9.:):

distx=p(10,:);

disty=p(11,:);
distz=p(12,:);

120

°/o Resultant Force Vector
r=sqrt(Fx.’\2+Fy.’\2+Fz.A2);

% Unit vectors
rx=Fx./r;
ry=Fy./r;
rz=Fz./r;

w=mx.*rx+my.*ry+mz.*rz;

qx=mx-w.*rx;
qy=my-W-*ry:
qz=mz-w.*rz;

%Position of x and y intercepts

pz=0.0405; °/o to center of forceplate in z direction
py=(QX+pz*Fy)./Fz;

px=(pz*Fx-qy)./Fz;

"/0 Change of units from mm to cm
px=100*px;
Py=100*Py;

°/o Change units to N for torque vectors
tx=100*w.*rx;
ty=100*w.*ry;
tz=100*w.*rz;

%Transformation from force plate CS to Lab CS
T1=[0 1 0; 0 0 -1; -1 O O]; % Transformation matrix

COPx=1 .0*py+Xo;
COPz=-1 .O*px+Zo;
COPy=O;

Rf=-[Fx Fy Fz]*T1';
Rt=-[tx ty tz]*T1';

Fx=Rf(:,1); Fy=Rf(:,2); Fz=Rf(:.3);
tx=Rt(:,1); ty=Rt(:,2); tz=Flt(:.3);

% Define the 2 variable of ICR by the middle of the 2 condyles
KJCx=(fm_condx'+fl_condx‘)/2;

121

 

KJCy=(fm_condy'+fl_condy')/2;
KJCz=(fm_condz'+fl_condz')/2;

°/o COP to ICR Position Vector
chx=COPx-lCRix';
chy=COPy-ICRiy';
chz=COPz-KJCz;

chx2=COPx-KJCx;
chy2=COPy-KJCy;

%Calculation of Knee moments in Lab CS
Mx=chy.*Fz-chz.*Fy+tx;
My=chz.*Fy-chx.*Fz+tz;
Mz=chx.*Fy-chy.*Fx+tz;

Mx2=chy2.*Fz-chz.*Fy+tx;
My2=chz.*Fy-chx2.*Fz+tz;
M22=chx2.*Fy-chy2.*Fx+tz;

for i=1:max(size(Fx))

°/o Create vectors containing x,y,z
fl=[fl_condx(i)' fl_condy(i)' fl_condz(i)'];
fm=[fm_condx(i)' fm_condy(i)' fm_condz(i)'];

prox=[proxx(i)' proxy(i)' proxz(i)'];
dist=[distx(i)' disty(i)' distz(i)'];

% Create Y axis of Thigh and Z axis of Shank
Yt=(fm-fl)/norm(fm-fl);
Zs=(prox-dist)/nomi(prox-dist);

% Create axis of rotations

e3=Zs';

eZ=Yt';
e1=cross(Yt',Zs')/nonn(cross(Yt',Zs'));

T=[el'; 92'; e3'];
M=[MX(i) MW) MZ(i)1;
M2=[Mx2(i) My2(i) M22(i)l;

Mm=inv(T)*M';
Mm2=inv(T)*M2';

122

 

m_flex(i)=Mm(2);
m_flex2(i)=Mm2(2);

ecx(i)=93(1);
ecy(i)=63(2):
ecz(i)=e3(3);

end
M_flex=m_flex/(1 00*BW/9.81 );
M_flex2=m_flex2/(1 00*BW/9.81 );

ec=[ecx' ecy' ecz'];

O/Oo/Oo/Oo/Oo/OO/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/O
% change_length2 Function Code %
°/o °/o°/o °/o°/o % °/o °/o°/o °/o%°/o%%°/o %%% %% %%% %% °/o %% °/o°/o % °/o°/o % %%% °/o°/o %

function [lchg, strain]=change_length2( Lacl, chl, Llcl)

°/o The ligaments are strained at full extension, therefore
°/o these position should not be taken as the Lo position.
°/o The strain values were reported in the article, and used
"/0 to calculate the Lo position for all ligaments. The Length
°/o at full extension was used in combination with the strain
°/o at full extension to obtain the correct Lo values.

Eacl=.031; °/o strain value at full extension (average between anterior and
posterior fibers)

Epcl=.05; °/o strain value at full extension

Elcl=.05; °/o strain value at full extension

Lfeacl=3.620726; °/o value taken from Lacl at 0 degrees of flexion
Lfepcl=3.4352; °/o value taken from chl at 0 degrees of flexion
Lfelcl=6.811496; °/o value taken from Llcl at 0 degrees of flexion

Loacl=LfeacI/(Eacl+1); °/o Calculation of the original “unstrained” length
Lopcl=Lfepcl/(Epcl+1 );
Lolcl=LfeIcV(Elcl+1);

Dacl=Lacl-Loacl; °/o Calculation of the change in length
Dpcl=chl-Lopcl;

123

 

chl=Llcl-Lolcl;

if Dacl> 0
Sacl=DacVLoacl;
else
Sacl=0;

end

if Dpcl>0
Spcl=Dpcl/Lopcl;
else
SpCI=0;
end

if chl>0
Slcl=chVLolcl;
else
SICI=O;
end

lchg=[Dacl; Dpcl; chl];
strain=[Sacl; Spcl; Slcl];

o/Oo/Oo/OO/OO/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/OO/Oo/Oo/Oo/Oo/OO/OOADO/O
°/o lig_forceZ Function Code °/o
°/o °/o °/o% °/o °/o % °/o°/o%% °/o °/o °/o °/o °/o°/o %%% %%% °/o°/o °/o % °/o °/o % °/o °/o°/o °/o °/o °/o °/o %% %

function [Facl, chl, Flcl]=|ig_forceZ(lchg, modelratio)

°/o Function lig_force calculated the resolved force at the
°/o ACL, PCL and LCL. The stiffness parameter is constant
°/o Change in length is read from the main script

°/o A negative change in length of the ligament indicates

°/o buckling resulting in F=0 Force Units will be in Newtons.

kacl=20; % Stiffness units are in N/mm2
kpcl=17.5;
klcl=15;

Dacl=lchg(1)*modelratio/10; °/o Change in length units are in "Real" units (mm)
Dpcl=lchg(2)*modelratio/1 O;
chl=lchg(3)*modelratio/10;

124

if Dacl> 0
Facl=kacl*(Dacl/\2);
while Facl>1700

kacl=kacl-0.01 ;
Facl=kacl*(Dacl"2);
end

else
Facl=O;

end

if Dpcl>0
chl=kpcl*(Dpcl’\2);
while chl>2840
kpcl=kpcl-0.01 ;
chl=kpcl*(Dpcl’\2);
end

else
chl=0;
end

if chl>0
Flcl=klcl*(chl’\2);
while Flcl>1000
klcl=klcl-0.01 ;
Flcl=klcl*(chl’\2);
end

else
Flcl=0;
end

°/o°/o °/o°/o°/o°/o°/o°/o°/o°/o°/o°/o°/o°Aa°/o°/o°/o°/o°/o% °/o°/o°/o°/o°/o°/o°/o°/o °/o°/o %%% °/o °/o°/o°/o%%°/o
°/o moments Function Code °/o
°/o °/o°/o°/o°/o°/o °/o %°/o°/o°/o %%% °/o°/o °/o%°/o% %%% °/o°/o°/o°/o%°/o%°/o%%°/o %% °/o %%%

function [Maz, Mpz, Mlz, le,Fly]=moments (e3y,BW, modelratio,JCx, JCy,

angacl, angpcl, anglcl, Rlacl, Rlpcl, Rllcl, jp, Insertiontacl, lnsertiontpcl,
lnsertiontlcl, Insertiontlcly, Facl,chl,Flcl)

°/o Set X-parameters for the point of application of the
% ligament force at the tibial plateau

125

“I

 

 

aclpx=lnsertiontacl;
pclpx=lnsertiontpcl;
lclpx=|nsertiontlcl;

% Estimation of the perpendicular distance in the X-

% direction from the Iigament's force point of application

°/o to the JC. Multiply by modelratio to convert to real

% units (mm). Divide by 10 to convert to cm. (same units as
% e-moments calculation)

dpxacl=(JCx-aclpx)*(modelratio/10);
dpxpcl=(pclpx-JCx)*(modelratio/10);
dpxlcl=(lclpx-JCx)*(modelratio/10);

°/o Set Y-parameters for the point of application of the
°/o ligament force at the tibial plateau for the ACL and PCL.
% The LCL has point of application lower on the tibia.

aclpy=jp§
PC'PY=lPi
lclpy=lnsertiontlcly;

°/o Estimation of the perpendicular distance in the Y-
°/o direction from the ligament to the JC.

dpyacl=(JCy-aclpy)*(modelratio/10);
dpypcl=(JCy-pclpy)*(modelratio/10);
dpylcl=(JCy-lclpy)*(modelratio/10);

°/o + Fy --> Superior pull from the tibia (origin of forces

"/0 taken at tibia plateau)

°/o + Fx --> Posterior pull from the tibia for AOL and LCL --
°/o Anterior Pull for PCL (origin of forces taken at tibia

°/o plateau)

Faclx=Facl*(cos(angacl)); Facly=Facl*(sin(angacl));
chlx=chl*(cos(angpcl)); chly=chl*(sin(angpcl));
Flclx=Flcl*(cos(anglcl)); Flcly=Flcl*(sin(anglcl));

% - CW and + CCW
°/o Units for the moments Nmm

126

 

Macl=-(Facly*dpxacl)+(Faclx*dpyacl);
Mpcl=(chly*dpxpcl)-(chlx*dpypcl);
Mlcl=(Flcly*dpxlcl)-(Flclx‘dpylcl);

% Transformation of the calculated moments about the true
"/0 medial-lateral axis of the knee joint.

Maclz=Macl*e3y;
Mpclz=Mpcl*63y;
Mlclz=Mlcl*63y;

%Nonnalize to %BW for comparison
Maz=Maclz/(100*BW/9.81);
Mpz=Mpclz/(100*BW/9.81);
MIz=Mlclz/(100*BW/9.81);

le=[Faclx chlx Flclx];
Fly=[Facly chly Flcly];

127

 

 

 

APPENDIX B

128

 

APPENDIX B

Mathcad Template 1

Mathcad Template used for the calculation of the ICR

Reading Thigh and Shank Target location from KTA file (modified).

Thigh Targets
T3 3: ' Tb 1: I Tc :=
E E Q
C:\..\aS.xls C:\..\a5.xls C:\..\a5.xls
Shank Targets
Sa := ' Sb := ' Sc =
Q Q
C:\..\a5.xls C:\..\aS.xls C:\..\aS.xIs

Transpose Matrix to allow long vector format

Tar := TaT Tbr := TbT Tor := TCT Sar := SaT Sbr := SbT Scr := ScT
Calculations for Frame 1
Definition of i and delta t
Define the first frame to calculate Sampling rate of

At :=0.01

the second forward difference cameras (100 Hz.)

Calculation of Linear Velocity for 2 targets per segment
using Forward Difference Equations - (Burington, 1973)

Vtc(i) := (4) '(‘25'T6f6> + 48 -'I‘cr<i+1> - 36-Tcr<'+2> +- 16'Tcr<i+3> — 3 .Tcr<i+4>)

12 'At

 

Vlbm 1" ; -(-25'11>r<i> + 4811f“) - 36-1'or<‘+2> +16-Tbr<‘+3> - 3.1‘or<'+‘>)
12m

Vsa(i) := —1—)-(-25-Salr<i> +48 ,Sa,<+1> - 36°Sar<i+2> +16-Sar<‘+3> - 35.15””)
12m

Vsb(i) := (L) (-25-Sbr<i> + 118-sum"> - 315-5135“) + 16-Sbr<‘+3> _ 3 5135””)
12.25

H

129

Definition of position vectors between targets in the same segment

G>

r<i> rsba“) := Sbr<i> - Sat

tuna) :=Tcr<i> - Tb
Definition of relative velocity between the velocities of targets in the dame segment
vtba(i) :=Vtc(i)-th(i) vsba(i) :=Vsb(i)-Vsa(i)
Calculation of angular velocity of the thigh and shank respectively

'tba“) X vtba(i) rsba“) X vsba(i)
(D ((i) 3: . . (0 5(1) :=
rtba(‘)'rtba(‘)

 

 

r sba(i)'r sba(i)

Calculation of Joint’s angular velocity

wj(i) :=(n s(i)- u) t(i)

Definition of relative vectors and linear velocities between targets in different segments
vts(i) :=th(i)-Vsa(i) rts(i):='1'br<i>-Sar<>

vst(i) :=Vsa(i) — th(i) r stm := Sax<> - m“)

Calculation of position vector from thigh and shank targets to ICR
(Class Notes MSM442, 1998)

 

rjct(i): ’ (“)1“) x vtS(i)) +r‘5(i)°(w1(i)m 5(i))]..l

(“’j‘Wj‘”)

 

ro (i):=
”5 wjumjri)

 

lull-(i) x vst(i)+rs,(i)-(wj(i)m t(0)]

Transformation of ICR into Lab Coordinate System

ICR(i) mm“) + r jam ICR 20 :=Sar<> + r jcsm

130

 

Calculations for Frame 2

i :=1 Identification of Frame 2

Calculation of Linear Velocity for 2 targets per segment
using Fomard Difference Equations - (Burington, 1973)

VIC“) == (—-1—)-(-3-Tcr<‘"> - lo-Tcr<i> +18-Tcr<‘+‘> - 6~Tcr<i+2> + MG”)
12m

th(i) :=(—1—)-(-3-Tbr<“"> - IO-Tbr<i> +18-‘I‘br<i+l> — 6-Tbr<+2> + 1135””)

12 'At

Vsa(i) := (-1—) -(—3 ~Sar<" ‘> - 10821:“) +18-Sar<‘1"> - 6-Sar<‘+2> + 36””)
12A:

Vsb(i) :=
lZ-At

 

_1_) (Ls-51:5“ '> - lO-Sbr<i> + 113-3135“"> - 6-Sbr<+2> + Sbr<+3> ))

Definition of position vectors between targets in the same segment

r tbs“) :='I‘cr<i> - 'l'br<i> r sbaU) := Sbr<i> - Sat<i>

Definition of relative velocity between the velocities of targets in the dame segment
vtba(i) :=Vtc(i)- th(i) vsba(i) :=Vsb(i)- Vsa(i)

Calculation of angular velocity of the thigh and shank respectively

'tbafi)" vtba(i) rsba(i)x vsba(i)
(Dt(1):= _ . (05(1)::
rtba(‘)'rtba(')

 

 

r sba( i ) -r sba( i)
Calculation of Joint's angular velocity

raj-(i) :=o) S(i)- (o t(i)

Definition of relative vectors and linear velocities between targets in different segments

vts(i) :=th(i)-Vsa(i) ”8(a) :=Tbr<i>— Sax”

min) := Vsa(i) - th(i) r stm :=Sar<‘> - Tbr<i>

Calculation of position vector from thigh and shank targets to ICR
131

(Class Notes MSM442, 1998)

0 (wjuw vts(i))+r,,(i)-(mj(i)w sm) 1
r- I 2- ..
J“ (%“WJW

 

mj(i) x vst(i)+rst(i)-(u)j(i)-m tm)

 

r- (i) :=
”5 wjumju)

Transformation of ICR into Lab Coordinate System

ICR(i) :=Tbr<'>+rjc,(i) ICR 21 :=Sar<‘> +rjcs(i)

Calculations for Rest of the file

i :=2,3 .. 500 Definition of frames - Upper limit can be adjusted according
to file length.

Calculation of Linear Velocity for 2 targets per segment
using Forward Difference Equations - (Burington, 1973)

Vtc(i) I: (;) '(Tqu-b - 8-Tcrq- l) + 8—'l‘cr<i+l> - Tcr<i+2>)

thh) := ___l_ .(Tbrfi-D _ 8-Tbr<i- 1> + 8-Tbr<i+l> _ Tbr<3+2>>
12m

Vsa(i) := _l_ (Sal-(1’2) .. 3.536- D + 8-Sar<i+l> _ Sal_<i-+-2>)
lZ-At

Vsb(i) := (—1-)-(Sbr<‘2> — 851:5" ‘> + 8-5br<'+'> — 5135””)
lZ-At

Definition of position vectors between targets in the same segment
rtbafi) :=Tcr<‘> — m0 rsba(i) := $er) — Sax<>
Definition of relative velocity between the velocities of targets in the dame segment

vtba(i) :=Vtc(i)-th(i) vsba(i) :=Vsb(i)-Vsa(i)

132

 

 

Calculation of angular velocity of the thigh and shank respectrv‘ ely
. x ,
l'u,;,((i)"vtba(i) (”s“) :grsba“) vsba(i)

u) I“) := ’sba(i)"sba(i)

 

’wa(i)"tba(i)

Calculation of Joint’s angular velocity
wj(i) :=w s(i)- u) t(i)

Definition of relative vectors and linear velocities between targets in different segments

vts(i) :=th(i)- Vsa(i) rm(i):='1‘br<i> - Sm<i>
vst(i) :=Vsa(i) - thm r 5,0) := Sal:<i> - Tbr<i>

Calculation of position vector from thigh and shank targets to ICR
(Class Notes MSM442, 1998)

 

rum .= (00,0) x mm) +r,s(i)-(w,-(i)w sm) -1
J (“’j‘Wj‘”)

 

wj(i) x vst(i)+rs,(i)-(wj(i)m t(0)]

r- (i) :=
JCS . .
[ (.0j(1)‘0)j(1)

Transformation of ICR into Lab Coordinate System

ICR(i) :=Tbr<‘>+rjc,(i) 1CR2(i) :=Sa:<‘>+rjcs(i)

133

 

 

 

 

Mathcad Template 2

Mathcad Template used in the Transformation of ICR to the Shank Coordinate
System

Read ICR (In Lab CS) from previous Mathcad file
E] Referencezc:\Dissertation\1CR-LCS.mcd

Reading target locations from modified KTA file

Thigh Targets

Ta := ' Tb := ' To := I

B H U
C:\..\a5n.xls C:\..\a5n.xls C:\..\a5n.xls
Shank Targets
Saz= ' Sb := ' Sc :=
E U E
C:\..\a5n.xls C:\..\a5n.xls C:\..\a5n.xls

Transpose of matrix

Tar := TaT Tbr := TbT Tcr := TcT Sar := SaT Sbr := SbT Scr := ScT

Reading Force values from FRC file

F:= '

e E
C:\..\a5cs.frc
Transpose of Matric
F := F

CI 6

Definition of frames in the file
i :=0, 1 .. 499

Definition of position vectors between segment targets

rtab(i) mm“) — Tb:<> rsba(i) :=31>r<‘> - Sm<i>
rtcb(i) := Tcr<i> - 'I'br<i> rsca(i) := Scr<i> - Sar<i>

134

 

“-1

 

Definition of segmental coordinate system for thigh and shank respectively

 

 

 

 

1‘“): rtab(i)xrtcb(i) k (i):= rsba(i)xrsca(i) .
t lrtab(i)xncb(i)l 5 Irsba(i)xrsca(i)l “medial
rtab(i)xkt(i) ks(i)xrsba(i)
it“): . . is(i):= - - i+ osterior
lnabn)xk,(i)| lks(l)xrsba(i)| P
j,(i):=kt(i)Xit(i) js(i)1=ks(i)xis(i) j+superior
Definition of Transformation matrix
Thigh Shank
”(1)0 “('); 1.0)2 15(00 15(1)l 15(1)2
Tt(i);= jt(i)o jt“), jt(i)2 Ts“); js(i)o 15(5)l 15(02
_kt(i)0 kt“)! kt(i)2‘ _ks(l)0 k5“)! ks“):d

 

 

 

 

Calculation of Relative position between ICR
and
Shank Target.

_ ICR(i)o- (32.5”)o .
R i530) := ICR(i)‘- (Sarto),

(Sar6>)2— (Sar<i>)2‘

 

 

l.

Transformation of ICR location to Shank CS

ICR Sm :=Ts(i)-R isa(i)

Break down of components of ICR location for export to excel file

ICR x(i) i=1CRS(i) ICR y(i) :=ICR s(i) ICR z(i) i=ICR 5(i)2
0 1

lCRt x(i) :=ICR(i)0 ICR! y(i) :=1CR(i)1

135

 

 

 

Transformation of Force values to Shank CS
' <i> ‘
(F .. ),

I:ema) := (F“<i>)o

 

 

ch(i) :=Ts(i)-ch(i)

ch(i) ==F¢s(i)0 Fcyfi) :=F¢$(i)l Fan) :=ch(i)2

Variables read in excel.

. |_ . l_
{CRX(1) — 1CR y(1) —
ICR: (i)'= ICRty(i)'=
I x I
Fax“).= Fey“).=

136

APPENDIX C

137

APPENDIX C

Mathcad Template 3

Mathcad Template used in the solution of patello-femoral equilibrium, and in the
final solution of the quasi-static model

Input Variables - (Taken from Matlab script)

dis_values 1: 1i g_forcc_v :=

H H
C:\.\disp_mdl2n.oul C:\.llorces_mdl2n.oul
dis_vr := dis_valuesT 1i g_vr := lig_force_vT
ext_force :=

Ii
C:\.\forces_scs.)ds

. T
eforce_vr := ext_torce

mom_values := ‘ cont_va1ue :=
R R
C:\.lmomsZnout C:\.\Cont_mdl.out
.. .T ._ T
mom_vr .- mom_values cont_vr .- cont_value
i :=25 Frame from motion file - Calculations can be executed 1 frame at a time

Definition of input variables

 

Constant Variables of the Tibio-femoral and Patello-femoral models

 

 

 

 

Ptxt=4fl Pty2=15fl w2:= 7
6.6285 6.6285 6.6285
Ptyrnser :=-4.7124 - Pty Ptxmser :=-7.0686- Ptx Tryf := 1.5
ptwidth :=9 L644 ptlength := 39'4 ptwidth2 :=ptwidth - w2
6.6285 6.6285
trochr := 22 Trx :=I -7.0686 I— trochr Trxtf := lxant+ trochr
6.6285

138

47.3
6.6285

LPTe :=

 

Trxt := Trxtf + dispx
Try := Tryf+ dispy

Knee Flexion Angle

 

Ligament forces
L am := (lig_w<‘>)l
L alcly:=(1ig_w<‘>)4
Ligament Moments

,_ <i>)
M acl .- (mom_vr 1

External Forces
F ch := (eforce_vr<i > ) 0

External Moment

,_ <i>)
M e .- (mom_vr 4

Input paramenters - From Matlab script

jp :=-4.7124

F ev := (eforce_vr<i>)

1

“v g 'P . sp- ‘ 1'"

Ppax :=- 10.1846 ; Cx"

fl ._'._,... _4-
v-A. ~. rs

139

 
 
  

L ICIX 1: (lig_VTq>)3

(Iig_vr<>)6

L lcly :

,_ (1))
M lcl .- (mom_vr 3

Body Weight
,_ BW
1101111 I-—

  

Initial Values for Calculated parameters

beta :=5 deg anga :=l
alpha :== .1 angc :1
a:=l cc :=1 pl :=O.1

Set of Simultaneous equations

Due to Margins limitations the set of similtaneous equations used at this point
are presented at the end of the Appendix.

Calculations beyond this point can only be performed after obtaining values for
beta,p1,alpha, a, cc,anga, angc after solving the simultaneous equations.

y := (Ptyinser + cos( beta) -I..P'1'e)

x := ( Ptxinser - sin( beta) ~LPTe)

C x :=(x- sin(anga- alpha)-cc) C y :=(y+ cos( anga- alpha)-cc)

l

Definition of unit vectors for the 3 patello-femoral forces

(alpha+ (90deg - theta)) if 90-deg — theta>alpha 5 := (90-deg - beta - alpha)
((alpha)) otherwise

 

 

 

 

9 xi i: X+ Dammit-sin( alpha) q yi ==y+ ptlength -cos( alpha) Insertion of QT on the Patella
Unit vectors
qxi-Cx‘ thxinser—Cx' 'PFCx-cx'
qyi‘cy li‘tyinser-Cy Pl-‘r'Cy-Cy
.- . 0 . .- . 0 . .- . 0 J
cq-- - cm." . °n°"* ,
qxi'cx Ptxmser-Cx PFCx-Cx
qyi’cy Ptymser-Cy PFCy-Cy
0 . . O . . 0

 

 

 

 

 

 

140

Perpendicular Distances used in the calculation of moments

PTpery :=(Ptytnser-1Cy)0.66285 PTperx :=(Ptxmser- 100066285
FC pery := (jp— JCy) 0. 66285 FC perx := (Xct- JCx) 0.66285
mperx :=( .7972- JCx) 0.66285

Inverse Matrix solution method - solving for Fpt and th

 

 

 

 

jzrly+zpcy Pp,
2F +21= Fc
R = 1x ex X:=
-2M¢+ZM1 Fin
0 . .th.
cos(beta) -1 0 0
sin(beta) O -1 0
Ct: .
(srn(beta))-P'I‘pery-(cos(beta))-P'I'pcrx ’Fcperx ’Fcpery 0
. (sin(5)) 0 0 sin(y).

 

 

 

 

PFCF := (- cos(y) -F qt - cos( 8) F pt)

 

 

[thx .- qut|°q0 PFCFX .. IPFCFlcno [Fptx]_ IFPllePto
thy ’thl-eql PFCFy |1>1=c1=|enl Fm letltpt,
. theta . theta
rtb.=— na.=
beta alpha

141

Solutions to the simultaneous equations

x - 8.683

y - - 1.324
Position of distal-anterior
comer of patella

Position of the
[PFCX] = Patello-femoral contact
PFCy point
_8_ = 79.27
deg
Angle between PT force and PFCF
239.. = Angle of PT from vertical
deg
F
F my
F
F qty
PFCF = PFCFx
PFCFy —
F = Tibia-femoral Contact Force

l§:=1ll::::

Point of intersection of PF forces

8. 089 _
qu=“4.5[9] Pornt of insertion of OT
q yi

_Y_ =
deg
Angle between OT force and PFCF
”J1“- =.5 73
deg Angle of patella inclination

Patello-femoral mechanism forces

Patella-femoral mechanism forces

Patello-femoral mechanism forces

F hx = Horizontal Force

pl = 0.1 Position of Patello-femoral contact point r.t. patellar length

Me = 7.136 Length of Patellar Tendon

M c ‘= (FC perX'F 6) " (Fe very'F hX)

 

nb _ ratios of beta and
' alpha relative to
m = the angle of
flexion

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143

APPENDIX D

144

APPENDIX D

Ligament Forces

Presented in this appendix are the graphs of ligament forces obtained with the
model. The graphs (Figure 1A-3B) are presented for each tested condition to allow
comparison of the interaction between the ligaments throughout the range of motion. The
pattern of tensile forces sustained by the ligaments closely resembles the pattern of

ligament strain.

 

I Ligament Forces

 

I Normal l
; 300 —ACL l
l 250 — —PCL l
5 —CL :
l

 

 

 

 

 

 

Angle of Flexion (Degrees)

 

 

 

Figure 1A: Ligament forces for the normal knee.

145

 

 

r"—

Force (N)

Ligament Forces
ACL-Deficient

 

 

 

 

300
—ACL
250 . — PCL
200 «
150 -
100 1
50 .
O 4 1 1gp 1
_50 20

 

 

Angle of Flexion (Degrees)

Figure 2A: Ligament Force for the ACL-deficient knee.

 

 

 

Force (N)

Ligament Forces
ACL-Reconstructed

 

300
we
250 - CL

200 -

150*
100 4

 

 

80 1 00
Angle of Flexion (Degrees)

 

120

 

 

Figure 3A: Ligament forces for the ACL-reconstructed knee.

146

REFERENCES

147

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