THE RHEOLOGICAL PROPERTIES OF THE
ANTERIOR CRUCIATE UGAMENT OF A DOG

Thesis for the Degree of M. S.
MICHIGAN STATE UNIVERSITY
ROGER C. HAUT
1968

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ABSTRACT

THE RHEOLOGICAL PROPERTIES OF THE
ANTERIOR CRUCIATE LIGAMENT OF A DOG

by Roger C. Haut

The purpose of this research was to study the mechani—
cal properties of the anterior cruciate ligament of canines.
The changes in the shape and magnitude of the stress-strain
curves for various strain rates are discussed in detail.

The specimens were tested in tibia-anterior cruciate ligament-
femur preparations which were subjected to environments of
moistening and direct immersion in a saline solution. For
these tests the specimens were constrained in the normal
flexed positions of the hind limbs of canines. Information
dealing with the per cent cumulative stress relaxation was
obtained by allowing the specimens to relax at regular inter-
vals during the loading. By testing each of the specimens
twice, information regarding the possible damage, which
might have occurred during the first test cycle below 10

per cent strain, was obtained.

The stress-strain data fitted extremely well with an
exponential relationship suggested by Y. C. Fung for the

mesentery of the abdomen of a rabbit. The variation in

Roger C. Haut

environments appears to have an effect on the properties

of the ligament. The results of the relaxation tests indi-
cate that for high values of stress the per~cent cumulative
relaxation is linearly proportional to the value of stress.
It was observed that when 2 successive tests below 10 per
cent strain were subjected to the same specimen, the second
resulting test shows an extended initial toe region. How-
ever, in the linear region of the curves the slope is almost

identical with that of the first test.

THE RHEOLOGICAL PROPERTIES OF THE

ANTERIOR CRUCIATE LIGAMENT OF A DOG

BY

Roger C?‘Haut

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Metallurgy, Mechanics, and Material Science

1968

ACKNOWLEDGMENTS

The author wishes to express his appreciation and
graditude to the following people for making this phase of
his graduate work possible:

To Dr. Robert W. Little, his major advisor, for

his encouragement, friendship, and, most of all, for
his valuable assistance in the preparation of this

thesis.

To Dr. Wade 0. Brinker for his constant concern and

medical assistance for this thesis.

To the National Institute of Health for the finan-

cial support required for this study.

To Dr. William Sharpe for his assistance in the de-
sign and Operation of the inStruments needed for this

study.

To his mother for her continued concern and encour-

agement for his academic education.

To his wife's parents for their understanding and

interest in his graduate work.

To his wife, Judy, for her endless patience and

encouragement throughout his graduate work.

ii

TABLE OF

ACKNOWLEDGMENTS . . . . . . .

LIST OF FIGURES . . . . . . .

LIST OF TABLES . . . . . . .

Section
I. INTRODUCTION . . . . .
II. SURVEY OF LITERATURE .
III. EXPERIMENTAL METHODS .
IV. PRESENTATION OF RESULTS
V. DISCUSSION AND CONCLUSIONS

BIBLIOGMPHY O O O O O O O 0

APPENDIX 0 O O O O O O O O 0

CONTENTS
Page
. . . . . . . . . . 11
. . . . . . . . . . iv
. . . . . . . . . . vi
. . . . . . . . . . l
. . . . . . . . . . 9
. . . . . . . . . . 20
. . . . . . . . . . 30
. . . . . . . . . 53
. . . . . . . . . . 60
. . . . . . . . . . 64

iii

Figure

1.1
3.1
3.2
3.3

4.11
4.12
4.13

4.14

LIST OF FIGURES

Anterior Cruciate Ligament . . . . . . . .
A Specimen in Fixtures . . . . . . . . . .
A Ruptured Specimen . . . . . . . . . . . .
Testing Setup . . . . . . . . . . . . . . .
Measurement of Initial Length . . . . . . .
Clip—Gage Calibration . . . . . . . . . . .
Measurement of Area . . . . . . . . . . . .
Stress-Strain Plot for Moistened Specimen .
Stress-Strain Plot for Immersion at 72°F .
Stress-Strain Plot for Immersion at 101°F .
Strain-Strain Rate for Moistened Specimen .
Strain-Strain Rate for Immersion at 72°F .
Strain-Strain Rate for Immersion at 101°F .

Apparent Young's Modulus-Strain Rate
After Toe O O O O O O O O O O O O O O O O O

Stress-Strain Plot for 5%/min. Strain Rate

Stress-Strain Plot for 15%/min. Strain Rate
Static Tests . . . . . . . . . . . . . . .
Stress-Relaxation When Moistened . . . . .
Stress-Relaxation in Saline at 72°F . . . .
Stress-Relaxation in Saline at 101°F . . .

Per Cent Relaxation-Strain Rate When
M018 tened O O O O O O O O O O O O I O O O 0

iv

Page

27
27
28
28
29
29
34
35
36
37
38
39

4O
41
42
43
44
45

46

47

Figure Page

4.15 Per Cent.Relaxation-Strain Rate When
Immersed at 72°F . . . . . . . . . . . . . . 48

4.16 Stress After Relaxation—Strain for
Immersion-at 72°F . . . . . . . . . . . . . . 49

4.17 Stress After Relaxation-Strain for
Moistened Specimen- . . . . . . . . . . . . . 50

4.18 Stress After RelaXation-Strain for
Immersion at 101°F . . . . . . . . . . . . . 51

4.19 Stress-Strain for First and Second
Tests When Moistened . . . . . . . . . . . . 52

LIST OF TABLES
Table Page

1 Values of a and C for Exponential Law . . . . 56

vi

I . INTRODUCTION

Most living organisms are divided into small units
called cells. Robert Hooke, in 1665, first reported and
named cells from an examination of cork. These vacuoles
were not cells but rather depressions that were implanted
in the matrix of intercellular material by the cells of
the cork. In 1839, M. J. Schleiden, a botanist, and Theodor
Schwann, a zoologist, formulated what is now called the
Cell Theory. Their contribution was actually more of a
synthesis of ideas and observations than a discovery. Two
basic principles comprise the Cell Theory.

1. Principle of Cells--That all organisms are formed
from cells.
2. Principle of Biogenesis--That cells are produced
from other cells.
These, coupled with the notion that sperms and eggs are
also cells which unite in the formation of a new individual,
constitute the basis of all life.

When similar cells are joined together, either
closely or loosely, with an intervening non-cellular mate-
rial, the aggregate is called a tissue. In the animal king—
dom there exists 4 primary types of tissues: epithelial,

connective, muscle, and nervous tissue. The study of

tissues is called histology. Epithelial tissue is the
predominate type of tissue in most free internal and ex-
ternal surfaces.‘ The cells of this type of tissue are
closely joined, hence there is a minimum quantity of inter—
cellular material. Epithelial tissue is further subdivided
by its cellular shape. The epithelial surface which lines
the mouth, esophagus, the outer surface of the cornea of
the eye, and most of the female urogenital tract is called
squamous epithelium because of its thin flat plate-like
cells. Columnar, which are rectangular as the name sug-
gests, constitute areas such as the pharynx, soft palate,
and gland ducts. Cuboidal epithelium tissue is found in
the kidneys, the retina of the eye, and on the inner sur-
face of the lens of the eye.

Muscle tissue, known for its contractibility and
irritability, has a cellular structure in which the cells
are elongated into muscle fibers. The contractibility of
this particular type of tissue, as much as 33 per cent of
its initial length, is theorized as being attributed to a
dove-tailing effect of its fibers, myosin and actin.

Nervous tissue is composed of nerve cells, or
neurons, and some supporting cellular structures. Nerve
impulses, according to the Membrane Theory, are conducted
by a depolarization of charge along the tubular membrane
of a neuron. These impulses are then transferred from

one neuron to the next neuron via a synapse. At a synapse

there is no physical contact of the neurons, therefore an
electrical contact between neurons has been theorized.
Connective tissue is by far the most predominate
and diversified tisSue in the entire animal kingdom. With
reference to the human body, connective tissue appears as
bone, cartilage, ligaments, and tendons. Blood is also
classified as a type of connective tissue. The distinguish-
ing characteristic of this type of tissue is the presence
of a large amount of intercellular material. This inter-
cellular material may consist of collagenous or white fibers,
elastic or yellow fibers, reticular fibers, and varying
amounts of an amorphous ground substance. Elastic fibers
are rather long, flat ribbons which contain a substance
called elastin. Elastin is a very~low»modulus material.
The large deformation which is seen in the posterior neck
region of grazing animals results from the stretching of a
large ligament composed ahmost entirely of elastin. This
ligament is called ligamentum nuchae. The fiber giving
the high/modulus effects to tendons and ligaments is col-
Tagenous. The anterior cruciate ligament, which is the
immediate concern at this time, is composed almost entirely
of this type of fiber. These fibers appear as long wavy
ribbons which are not branched. They appear to be the
product of a long, flat, or star-shaped cell called a
fibroblast. The production of collagenous fibers is ap-

parently in the form of a secretion process in which an

aggregate of extracellular material is secreted by many
fibroblasts. The primary building blocks of these fibers
are the collagen molecules: three chains of amino acids
in certain sequences which coil into left-hand helices

and intertwine to form a right-handed superhelix (9,17).
The wavy pattern of a fiber will be considered later with
reference to its effect on the stress—strain curve. Reti—
cular fibers are thought to be immature collagen fibers
which are highly branched.

Connective tissue differs widely throughout the
body. Loose connective tissue contains reticular, elastic,
and collagenous fibers. The matrix of intercellular mate-
rial has no orientation of its fibers, resulting in a low
modulus material such as the mesentery of the abdomen.
Dense connective tissue is essentially the same as loose
connective except for a fewer number of cells and a larger
percentage of collagen fibers. Regular connective tissue
has an abundance of collagenous fibers, and it has a defi-
nite arrangement of its fibers. For example, tendons have
bundles of fibers which lie parallel to the central axis
of the tendon with fibroblasts existing between the bundles.
Ligaments are essentially the same as tendons, but they
have a less regular arrangement of the collagenous and
elastic fibers. These collagen bundles are presumably as
long as the tendons or ligaments (18). The collagen bun-

dles are surrounded by a woven mesh of loose connective

tissue, the peritendineum internum, containing elastic
fibers that tend to draw the bundle into a any formation
in the relaxed condition. This structure, along with the
pattern of the fiber itself, is significant to the stress-
strain curves. Tendon and ligament fibers are continuous
with the fibers of bones and at this point are called the
fibers of Sharpey (24).

Previously it was mentioned that ligaments and ten—
dons had very little histological difference, but func-
tionally they are very different. Ligaments serve as con-
nectors between two bones, whereas tendons attach bones to
muscles. Even though the point of attachment of ligaments
and tendons appears to be continuous, a cellular transition
can be noted in this region. In this region the fibrocytes
undergo a transition to osteocytes (bone cells), and here
they most resemble condrocytes (cells of cartilage) (24).
In early investigations on tendons and ligaments using the
intact bone-ligament-bone or bone-tendon preparations,most
ruptures were encountered in this transition region. This
failure was attributed to a weakness of the specimen in
this transition region. There is reason to believe that
failure might have been encountered in this region because
of the high moments present when testing the Specimen in
an unnatural position. This question is discussed further
during the consideration of a proper testing procedure for

the anterior cruciate ligament of canines.

The stifle joint, located in the lower or hind limb,
is the most complicated joint in the body. The stifle
joint must transmit forces of a much larger magnitude than
any other joint in the body, so that a high degree of sup-
port must be present to stabilize this joint. To comply
with the rigid requirements of stabilization as well as
freedom to flex and extend under large forces, the stifle
joint has the largest articulating surface in the body.
The bones involved with this joint are the upper bone, the
femur, which inserts into the pelvic girdle and the lower
bone, the tibia, which articulates with the talus to form
the ankle joint. Because of the support which is required
in this joint, there are many muscles and tendons surround-
ing this area. Collateral ligaments lie along the medial
and lateral borders of the joint, while the quadriceps
femoris muscle encases the knee cap which protects the
anterior region of the joint. Posteriorly, the biceps
femoris and the gastrocnemius muscles help support the
joint. The entire joint is encircled with hyaline carti-
lage which serves as a cushion and as a capsule for the
encasement of synovial fluid. Synovial fluid is a light
fluid which is present in the joint to reduce friction. In-
terior to the joint are cruciate ligaments, so named because
the two ligaments form a cross in the joint. The posterior
cruciate ligament extends from the posterior intercondylar

fossa of the tibia, a depression between the contact

    
   
   
     

     

Med/'a/ half of /
femur \\ ' /

Pas/en'or lxg \ '
Anferior //g\

T‘ “r em.»

Medial _ _
meniscus

 
    
      
  

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Figure 1.1--Anterior Cruciate Ligament (Reproduced
by courtesy of the J. B. Lippincott Co.. Phil.)

surfaces, upward and forward to the lateral side of the
medial condyle of the femur. The anterior cruciate ligament
extends from the front of the intercondylar eminence of the
tibia upward and backward to the medial side of the lateral
condyle of the femur. Isolation of the anterior cruciate
ligament without damage is possible due~to this central
positioning in the joint.

Injuries to the collateral and cruciate ligaments
of the stifle joint are very common. During hyperextension,
the anterior cruciate ligament and the collateral ligaments
are under large tensile forces. The posterior cruciate
ligament is apparently under a certain degree of tension at
all times. From clinical studies of the stifle joint it is
noted that when flexion with a subsequent medial rotation
of the femur takes place, the anterior cruciate ligament
and the collateral ligaments are under extreme tension.

The most common type of injury is when the tibia is rotated
laterally, as the joint is flexed. This type of injury is
frequently encountered in football and skiing accidents (1).
It has been suggested that similar movements may be the
cause of the ruptures of the anterior cruciate ligaments

in canines.

II. SURVEY OF LITERATURE

Many investigations have been conducted in an at-
tempt to describe the mechanical properties of biological
materials. More specifically, results dealing with regu-
lar connective tissue such as ligaments and tendons were
considered as relative background for the study of the
mechanical properties of collagenous tissues. Correlation
was extremely difficult and the results were sometimes
contradictory because of the large number of various meth-
ods of testing. For this reason a chronological survey
of the results dealing with regular connective tissue,
specifically tendons and ligaments, was chosen as the most
appropriate description of earlier work.

Some of the first bioengineers were Harvey, Helm-
holtz, Poiseuille, Hooke, Galilei, Boyle, Euler, and Young.
William Harvey was credited with the discovery of blood
circulation in 1615. To von Helmholtz might go the title
of "Father of Bioengineering." He discovered the focusing
mechanism of the eye, and invented the Helmholtz resonator
'for the study of hearing. Other accomplishments were
the invention of the phakoscope to view the retina, the
ophthalmometer for measurement of eye dimensions, and the

stereoscope with interpupillary distance adjustments for

10

stereo vision. Poiseuille invented the mercury manometer
to measure-the blood pressure in the aorta of a dog while
he was a medical student, and he discovered Poiseuille's
law of viscous flow upon graduation. As previously men-
tioned, Robert Hooke gave us the word "cell." Galileo
Galilei discovered the constancy of the period of a pendu-
lum, and used it to measure the pulse rate of humans. He
invented the thermometer, and was also the first one to
design a microscope in the modern sense. Robert Boyle
studied the lung, and discussed the function of air-in
water with respect to fish respiration. Leonhard Euler
studied the propagation of waves in arteries. Thomas Young,
who gave us Young's Modulus of Elasticity, was a physician
in London. He worked on the wave theory of light while he
was concerned with astigmatism in lenses and in color vision.

One of the earliest investigations of ligaments
was Annovazzi's work in 1928 (3). Although his work can-
not be obtained, the literature states that he studied the
properties of the anterior cruciate ligament of dogs. The
only information known by most investigators is that he
found that the mechanical properties of the ligaments
changed drastically within the first hour after the death
of the animal, but there is no indication as to how the
specimen was preserved before or during the tests.

In 1935, A. E. Cronkite conducted tests on almost

every tendon in the human body (6). He fixed the tendons

11

in what was called "a normal laboratory manner“ for as long
as 18 months. He reported that there was no significant
difference in the fixed or non-fixed fresh tendons. Cron-
kite reported that there was no one tendon which was
stronger than another. He did note that tendons with a
larger cross-sectional area seemed to have a little higher
tensile strength, and suggested that this might be due to
a greater density of collagen fibers in the larger area.
Cronkite indicated that the tensile strengths of various
tendons in the human body ranged from 2.8 x 108 dynes per
square centimeter 1x) 2.1 x 109 dynes per square centi-
meter’with variations in different cadavers being 6.0 x
108 dynes per square centimeter to 1.2 x 109 dynes per
square centimeter. No correlation could be noted between
the strengths of tendons and the age of the individuals.
Also, tendons which normally are required to transmit
large forces, such as flexors, tended to be no stronger
than those which normally are required to transmit lower
magnitudes of forces, such as extensors.

In 1935, C. M. Gratz prepared 3 types of tendons
and subjected them to loads under 500 kilograms per square
centimeter, which he states as the maximum stress for
which the specimens were still in the elastic range (10).
The specimens were cut to test size and.moistened with
Ringer's solution.until the test time. For the Achilles

tendon he found a maximum tensile strength of 6.8 x 108

12

dynes per square centimeter with a Young's Modulus of 3.9
x 108 dynes per square centimeter. Similarly for the
flexor digitorum longus, his results indicate a tensile
strength of 7.3 x 108 dynes per square centimeter and a
modulus of 3.4 x 108 dynes per square centimeter, and for
the erector spinae tendon 6.7 x 108 dynes per square centi-
meter, respectively. He noted that the Ringer's solution
made no significant difference from the control group for
3 hours of testing.

In 1951, R. H. Hardy conducted a series of tests
in which ligamentum flavum and plantar calcaneo-navicular
ligaments were subjected to various loads (12). Ligamentum
flavum is a ligament which histologically is composed of
elastic fibers whereas the plantar ligament is entirely
collagenous. Hardy indicated that the plantar ligament
displayed no deformation when a stress of approximately
70.4 kilograms per square centimeter was applied to it.
On the other hand the ligamentum flavum strained approxi-
mately 140 per cent under the same loading conditions.
These results indicate that collagen fibers have an ex-
tremely large modulus, whereas ligaments composed of elastic
(or elastin) fibers display a very low modulus. In each
of the above tests the specimens were fixed for 24 hours
in a 10 per cent formol-saline solution. Since Hardy did

not indicate his manner of testing the specimens, his work

cannot be fully justified.

13

In 1954, J. W. Smith published a paper in which he
examined the properties of anterior cruciate ligaments of
rabbits (22). Histological tests show that anterior cruciate
ligaments of rabbits are entirely collagenous. Smith re-
ported that considerable histological changes took place
within the first hour after the death of the animal. More
precisely, he noted an increase in Young's modulus and in
the elastic limit. Therefore, he performed all of his
tests within 30 minutes after death. His results indicate
that the rupture load of the anterior cruciate ligament is
proportional to the cube of the body weight of the animal,
although subsequent investigations have not been able to
substantiate these results. Smith further noted that for
a 5 pound rabbit the stress-strain curve was linear when
loads below 15 pounds were applied to the specimen. With
no reported specifications other than a uniform and rapid
stress rate, Smith strained the ligament as much as 20 per
cent in the elastic range. His testing apparatus suspended
the cruciate ligament in a position such that the femur,
tibia, and anterior cruciate ligament were in a line coin-
ciding with that of the applied load. All the ruptures
occurred at the tibial insertion of the ligament.

In 1959, B. J. Rigby studied the mechanical proper—
ties of rat~tail tendon (20). He found that the mechanical
properties of rat tail tendons were reproducible within a

strain range of 0-4 per cent. The tendons were rinsed in

14

normal saline (0.9 per cent NaCl), then refrigerated at

3°C in the same medium. A number of the tendons under study
were frozen at -35°C. No significant differences in the
mechanical behavior between the frozen or immersed speci-
mens and the fresh specimens was noticed. Also, the mechan-
ical properties did not show a temperature dependence
within the 0-37°C range. As the strain rate was increased,
the resulting stress-strain curves were identical except

for a slight shift toward the stress axis. The average
maximum slope of a stress-strain plot at a strain rate of

10 per cent per minute was 8.0 i 2.0 x 109 dynes per square
centimeter. If the strain rate is held to approximately

1 per cent per minute, strains as large as 20 per cent can
be obtained without any apparent breakdown of collagen
fibers.

In 1964, L. B. Walker performed a series of eXperi-
ments in which he tested plantaris tendons in the foot of
human cadavers (30). The tendons were dissected from hu-
man cadavers which were embalmed and stored in a wetting
solution of 5 per cent phenol. Tensile strengths of 7.3
x 108 dynes per square centimeter to 1.5 x 109 dynes per
square centimeter were reported for tendons in the wetting
solution. If the tendons were dried before testing 5 per
cent higher strengths than the wetted specimen were re-

corded. Walker indicated that there was no substantial re-

laxation of the stress when the specimen-were held at a

15

specified strain level during each test. The stress-strain
curves for 2 successive runs on the same specimen at the
same strain rate for strain levels below 10 per cent dis-
played no significant differences.

Tests performed by J. D. VanBrocklin in 1965 on
the extensor digitorum tendons of the human foot indicated
no variations in the mechanical properties between the
frozen or the fresh specimens (25). The data obtained
shows, however, a marked increase in the elastic modulus
as the rate of stress is increased. Furthermore, when the
Specimens were loaded with a saw-tooth loading function,
the energy dissipated for cycles of comparable amplitude
remained essentially constant as the rate of stress appli—
cation was increased.

In 1966, E. H. Harris performed a series of tests
on the tendons of the upper extremity of the human body
and obtained results contradictory to those of Cronkite
(14). Harris reported that the elastic modulus for tendons
of the muscle flexor digitorum was significantly smaller
than those of the extensor digitorum muscle. Clearly, the
flexors in normal physiological activity are subjected to
larger loads than the extensors. He reported that the av-
erage Young's Modulus for embalmed flexor digitorum tendons
was approximately 7.6 x 109 dynes per square centimeter.
This indicates that if a certain tendon is required to

withstand large loads it has the ability to deform much

16

more than those which are normally subjected to smaller
loads.

In 1967, M. Abrahams performed a series of tests
relating the mechanical behavior of horse and human tendons
to load, strain cycling, and strain rate (2). Human ten-
dons were tested in Ringer's solution at 37°C within 36
hours after autOpsy, and the horse tendons were tested at
38°C in Ringer's solution within 48 hours after autopsy.
All specimens were refrigerated until the time for testing.
The tendons were placed in shelf-tightening spring loaded
wedge-shaped jaws, and strain measurements were recorded
by a 1 inch Instron extensometer attached to the specimen.
The stress-strain curves described 3 distinct regions.

The primary region was that of 0-5 per cent strain in
which there was a considerable increase in length with
only a slight increase in stress. In this region it is
thought that the collagen fibers begin to straighten their
wavy pattern. The secondary region was that of 1.5-3.0
per cent strain where the collagen fibers are thought to
become fully oriented and begin to assume most of the load.
In the final region, that of 3.0-5.0 per cent strain, it
is thought that the entire response is due to the collagen
fibers in pure tension. The stress-strain curve is a
straight line in this region. Visible ruptures of the
collagen fibers begin at 5.0 per cent strain level. In

tests involving human Achilles tendons, human toe extensor

17

tendons, and horse extensor and flexor tendons, the maxi-
mum stress level recorded was 352 kilograms per square
centimeter for human tendons and 563 kilograms per square
centimeter for horse tendons. Investigations in the
cycling effects display complete recovery when the speci-
mens were cycled to 2.0 per cent strain with a maximum
stress of 113 kilograms per square centimeter. When the
specimens were cycled to 3 per cent strain for 10 cycles,
the specimens did not return to the same initial 1ength
but acquired a 0.5 per cent "residual" strain. As the
"residual" strain increased, the peak stress decreased
from 228 kilograms per square centimeter to 187 kilograms
per square centimeter. The last 10 cycles to a 4 per cent
strain level indicated "residual" strains of l per cent
with a maximum peak stress of 317 kilograms per square
centimeter. These tests verify that when a specimen is
strained beyond the 2.0-3.0 per cent strain level in cy-
clic testing, a permanent "residual" strain will occur.
The effect of strain rate on the stress-strain curve was
to change its shape and magnitude. Tests were conducted
on Achilles tendons in the range of l per cent strain per
minute to 50 per cent strain per minute. Results indicated
that as the strain rate was increased to reach a given
strain level, the magnitude of the induced stress increased.
This explains the ability of tendons to transmit large
forces through bones and through joints in a very short

time.

18

A. Viidik, in 1967, studied the properties of an-
terior cruciate ligaments in rabbits (26). The specimens
were kept in a saline moistened gauze for two hours before
the tests, and then moistened periodically during the tests
with saline solution. The general shape of the load-
elongation curves was similar to that of Abraham's. How-
ever, the second load-elongation curve exhibited a longer
and more accentuated toe part than the corresponding one
from the first loading. Also, the slope of the load-
elongation curve was slightly higher for the unloading than
for the loading cycle. The relative relaxation in per cent
of initial load was a constant for any load level. The
relaxation of load at a specified load level was largest
for the first test and decreased in the following tests on
the same specimen. The ligament experienced an asymptotic
load relaxation value after an interval of ten minutes.

Viidik also studied the effects of postmortal stor-
age on the tensile strength characteristics of the anterior
cruciate ligament of rabbits (28,29). He considered four
types of storage:

1. (Direct immersion in a saline bath-at 20°C for 5
hours.

2. Refrigeration in a saline solution for 24 hours
at 4°C.

3. Deep-freezing in a saline solution.

. 4. Storage in 10 per cent formaldehyde.

19

The parameters he considered were: the gross shape of the
load-elongation curve, its slope, its failure energy, fail-
ure load, elongation at failure, and failure site on the
bone-ligament-bone preparation. Viidik reported that none
of the above parameters were comparable to that of the con-
trol or fresh group. Viidik concluded his paper by sugges-
ting that all tests on collagenous structures should be

run immediately after death in an environment such as blood
plasma or synovial fluid.

Y. C. Fung has reported results dealing with the
effects of strain rate on the mechanical behavior of the
mesentery of the abdomen, which is a thin connective tissue
membrane connecting the intestines to the abdominal wall
(8). Fung noted that as much as one hundred per cent strain
could be encountered before reaching the state at which nor-
mal physiological observations are made. Fung also indicated
that hysteresis loops did not depend significantly on the
strain rate. The total amount of stress relaxation was lin-
early proportional to the value of the load at which the re-
laxation test had begun. Fung relates a term called the
elastic tension, which is the stress value after the material
has completely relaxed, to the slope of the elastic tension
versus deformed length per initial length. Approximating
this as a straight line, he integrated to get an exponential
type of constitutive equation describing the mesentery of

the abdomen.

III. EXPERIMENTAL METHODS

In order to study the mechanical properties of the
anterior cruciate ligament of canines, a femur-anterior
cruciate ligament-tibia preparation was chosen. Using this
type of preparation the end effects caused by direct clamping
to the ligament can be neglected. The fixtures which were
used for these tests were designed such that the preparation
could be flexed for the tests. This flexing of the prepara-
tion was necessary so that the anterior cruciate ligament
could be tested in a_natural position. The angle formed by
the fixtures was set at approximately 140° for all of the
tests. In the natural standing position, a dog flexes the
stifle joint from approximately 135° to 140°.(Fig. 3.1).f

Contradictions in the literature as to the effects
of various testing environments suggested three different
modes Which were considered:

1. Moistened slightly during the test with a saline
solution.

2. Immersion in a saline bath at room temperature.

3. Immersion in a saline bath at the body temperature
of 101°F.
The saline solution used throughout all of the tests was
Lock's solution. The solution was prepared in eighteen

liter quantities according to the following formula:

20

21

l. 7.56 grams of KCl per 18 liters of water.

2. 2.70 grams of NaHCO3 per 18 liters of water.

3. 4.32 grams of CaCl2 per 18 liters of water.

4. 165.5 grams of NaCl per 18 liters of water.
The water used in this formula was de-ionized with an Illco-
Way Universal Ion-Exchanger.

In order to maintain a prOper temperature bath, a
"Chill-Chaser" electric immersion heater with a Type 316
stainless steel heating element was used. This particular
type of instrument was capable of maintaining a temperature
of 101°F to within i l°F throughout all of the tests.

The specimens were tested within 6 hours after the
death of the animals. Initial tests were started within
2 hours after death of the animals with individual tests
requiring approximately 1 hour. While one specimen was
being tested, the remaining specimens were kept at room
temperature with the knee capsule left intact. Each spec-
imen was subjected to the temperature bath for approximately
30 minutes before the beginning of each test.

Due to its central location in the stifle joint,
the anterior cruciate ligament was easily isolated. All
interior cartilage and surrounding tendons and ligaments
were removed from the joint area before testing,and the
specimen was clamped to a table where the measurement of

area was made. The cross-section of the anterior cruciate

ligament was assumed to be elliptical. Since the anterior

22

cruciate ligament has a 90—degree twist as it passes from
the femur to the tibia, it was necessary to adjust the
clamps so that the ligament was as uniform as possible
when the measurements of the major and minor diameters
were being taken. A special set of micrometer probes were
designed in order that easy access into the intact joint
was possible. The probes were made from steel pins that
were ground to a diameter of 1/32 of an inch and had an
overall length of 1 inch. The probes were then attached
.to a 2—3 inch micrometer (Fig. 3.6).

The testing machine used for these tests was an
Instron Model TT-CM metric with a 0-50 kilogram load cell
inserted into the head. A cell range of 0-10 kilogram
was sufficient for these tests (Fig. 3.3). As described
earlier, the testing fixtures held the specimen in a flexed
position. The fixtures were also designed such that either
the femur or the tibia could be rotated to allow for better
aligmment of the ligament during the tests. Attached to
the stationary upper fixture was a system of coordinate
axes. Using a pair of highly versatile calipers, 3 radii
were drawn from the coordinate axes to the points of inser-
tion of the anterior cruciate ligament on the tibia and

on the femur (Fig. 3.4).

23

(019:0)

 

( x ',o,o) R3

r’jiL’I””/,///’[3(mgg)

(0,0,2 )

 

 

In order to solve for the point P(x,y,z), a system of 3

simultaneous quadratic equations must be solved.

x2 + (y - y')2 + 22 = R

NM

(x - x')2 + y2 + 22

II
:0
UlN

x2 + y2 + (z - z')2 = R

paw

Knowing R1, R2, R3, x', y', and z', the above system of
equations can be solved for the points of insertion of the
ligament into the tibia and the femur.* Therefore, after

calculating for the tibial and femoral insertions, the

 

*See Appendix.

24

length of the ligament was calculated using the following

equation.

L = {(xt - xf)2 + (yt - yf) + (zt - zf)

t' yt, zt is the insertion point of the tibia,

and xf, yf, zf is the insertion point of the femur.

Where x

The anterior cruciate ligament assumes approximately

a vertical position in the test fixture. Due to this posi-
tioning of the joint, the lateral displacement during the
tests was minimal. Therefore the z-component of displace-
ment was the only displacement that was considered for the
calculation of the strain induced in the ligament. The

new length of the ligament was calculated by a correspond-
ing change in the z-component of the tibial insertion as

follows:

1/2

2 2

L' = {(xt - xf) + (yt - yf) + (zt - zf + Az)2}

The strain in the ligament was calculated by using the

Lagrangian definition of strain.
a = (L' — L)/L

In order to eliminate false strains due to slipping
of the femur and the tibia in the fixture, a clip gage was
inserted into the joint (Fig. 3.1). The clip gage was con-

structed from a strip of spring steel 0.01 inches thick

25

and 0.25 inches wide. Two SR-4 strain gages were attached
to either side of the clip gage. The strain gages, Micro-
Measurements Model EA-06-125AD-120, were bonded with East-
man 910 cement and the lead wires were coated with W. T.
Bean Gagekote #2. The strain gages were coated with Gage-
kote #5, which is a waterproofing designed for direct im-
mersion in oil or water. The output was recorded on a
Sanborn Model 60-1300 Twin Viso Recorder with a Sanborn
Model 64-500A Strain Gage Amplifier. To provide sufficient
sensitivity and temperature compensation, the 2 SR-4 gages
were attached to adjacent arms of the bridge network in
the recorder. Slipping of the clip gage in the joint was
avoided by the puncture of holes in the femoral and tibial
intercondyles to receive the pointed tips of the clip gage.
The calibration fixture used for calibration of the clip
gage consisted of a Scherr Tumico dial gage graduated in
0.001 inch intervals attached to a sliding frame (Fig. 3.5).
With this particular design for the clip gage, the cali-
bration curve was linear throughout the range of interest.
The procedure used for testing the anterior
cruciate ligament was one of a steprloading pattern.
Applying a constant strain rate, the load was allowed to
reach a predetermined level and then there was a subse-
quent stress relaxation at this particular strain level.
This type of loading was applied until the strain level

reached approximately 10 per cent where the loading was

26

reversed back to the point of a zero strain level. At
each particular load level the specimen was reiaxed for
4 minutes at which time the amount of relaxation approached
an asymptotic value. After the completion of the first

test cycle, the specimen was subjected to a second identi-

cal test.

27

 

Figure 3.2--A Ruptured Specimen

28

 

Figure 3.4--Measurement of Initial Length

29

 

Figure 3.5--Clip Gage Calibration

 

Figure 3.6--Measurement of Area

IV. PRESENTATION OF RESULTS

The general shape of the stress-strain curves can
be seen in Figures 4.1—3. The initial portion of the curves
is called the toe region. In this region the magnitude of
the strain increases much more rapidly than does the magni-
tude of the stress. The next region, which is not so clearly
visible, is called the transition region. This is the region
in which the specimen begins to accumulate load quite rapidly.
In the third and final region the curves become almost linear.
As stated earlier in Section III, Abrahams described these
three regions in terms of the response of the collagen fibers
in the specimen (2). In the initial region the fibers have
been said to organize along the axis of stress. In the tran-
sition region the fibers have organized and begun to assume
a tensile force. In the final region the response is due
entirely to the collagen fibers in tension.

The effect of strain rate on the shape and magnitude
of the curves can also be seen in these figures. As the
strain rate is increased, the toe region tends to decrease
until at high strain rates the region is pushed to the bot-
tom of the curves. This effect along with the increase in
the apparent Young's Modulus, as seen in Fig. 4.7, tends to

shift the curves toward the stress axis. Using Fig. 4.1 as

30

31

an example, for a change in strain rate from 2-40 per cent
per minute the toe region decreases from approximately 6
per cent strain to a value of l.5_per cent strain- The
apparent Young's Modulus increases from approximately 1.0

x 109 dynes per square centimeter to 2.0 x 109 dynes per
square centimeter for the specimen which was moistened with
saline solution throughout each test.

Curves can be made from the graphs in Figures 4.1-3
which will relate the effects of strain rate for both the
immersed and the moistened specimens at room temperature.
Such curves are plotted in Figures 4.4-6. In these graphs
the strain is plotted versus the strain rate for varying
stress levels for each of the environments. It is clear
that for an increasing strain rate, the stress required to
induce a specified strain increases. Also, from these curves
the approximate stress-strain curves for various strain rates
can be plotted for each environment in order to compare them.
By chosing a couple of representative examples, such as Fig-
ures 4.8 and 4.9, the difference arising from these two en—
vironments for shmilar strain rates can be determined. In
these figures one can notice that the toe regions have essen-
tially the same order of magnitude and the same shape for
each of the environments. The major difference can be seen
by observing the linear portion of the curves. This differ-
ence stems from a variation in the slopes of the curves, as
seen in Fig. 4.7, between the moistened and the immersed

specimens.

32

By extending the curves in Figures 4.4-6 a zero
strain rate or static test can be approxflmated, as given
in Fig. 4.10. Here one can note that the slopes of the
linear portions of the curves are just about identical.
However, a difference in the toe regions can also be noticed.

In Figures 4.11-l3 results dealing with the cumula-
tive per cent relaxation of the initial stress versus the
value of initial stress for varying strain rates for each
of the environments are presented. In the region of 0-20
kilograms per square centimeter, the cumulative per cent
relaxation changes very rapidly with initial stress. Above
this level the curves become linear. There also appears to
be a tendency for the curves to be shifted toward the stress
axis as the strain rate is reduced, however, this result is
not entirely consistent throughout all of the tests. The
actual magnitudes of per cent relaxation fall between 10
and 30 per cent for all of the tests. If one sketches a
plot of strain rate versus the cumulative per cent relaxa-
tion for various stress values between the strain rates of
5 and 25 per cent per minute, Figures 4.14 and 4.15, the
magnitudes of stress relaxation are almost identical for
each of the environments. Outside of this range there is
an insufficient number of data to determine the pattern.

In the Figures 4.16-18 the value of stress after
relaxation is plotted versus the strain for each of the

strain rates. These curves follow the same shape as the

33

initial stress versus strain curves except for a decrease
in each of the stress values. As previously mentioned, at
each stress level the relaxation period lasted for 4 minutes,
during which the stress reached an asymptotic value.

The variation of results between the first and sec-
ond tests on the same specimen are shown in Fig. 4.19.
Three different strain rates for the moistened specimen at
room temperature were used as representative examples of the
existing trends apparent in all of the tests. As seen in
this plot, the slopes of the linear portions of the curves
are approximately identical for the two tests applied suc-
cessively to the same specimen. The extended toe region

in the second test was prevalent in all of the tests.

34

70

a
0
6

50

N

.EU\.mM .o .mmmuum

 

20

 

8.0

6.0
e, %

Strain,
4.1, Stress-Strain Plot for Moistened Specimen

4.0

2.0

Fig.

35

 

 

0

-8

0%
6
I
n
.1
a
u

0

r as

4

0

.2

0

 

70-

60-

50-

N

1 q

0 0
4 3

.EU\.mM .o .mmmuum

20'

10‘

Stress-Strain Plot for Immersion at 72°F

4.2

Fig.

36

2.0

f0

10.

 

 

N

o§\omVH s0 smmmHum

 

Strain,

%

Er

4.3 Stress-Strain Plot for Immersion at 101°F

Fig.

37

 

cofiwommm Uocmumfloz How uoam ovum camnumugflmupm v.v .wwm

m .w .cflmuum
o.v

 

0.00 Om. Os Om Gm .o_ -
«.Eo\$x mmumem

 

0H

ON

om

ow

om

'eqeu urexqs

’3

'UTW/%

moms. HM .GOHMHGEEH .HOM “GHQ 0.....mm GflflHflMICHMHDm

a .0 .aamuum

mé .mE

 

~.Eu.mox mmmmbw

 

 

,

o.m 0.x ooh Com com Dov oom O.N O.H o
- L|F r p p L P p — o
//
O// Q
/
o x q .oH
.ON
x q ,
8
3
.0m
10¢
Tom
0 X

0 X 4 q

Axumv mxum Aumxunxumw mXUmw nxU_

 

‘urw/g '3 'aqeg uterus

moaoa um coamumssH now uoam 60mm camuumucamuum v.4 .mam
m .w .aamuum

0.0 o.m 0.x. o.m o.m 03v on o.m o...” o
p h h n o

.OH

 

// / / ’ wON

 

2

, E, dynes/cm.

-8

Young's Modulus x 10

301

N
O
l

10-

—’

 

,0 /_—————-—"/’ 0
O
o Immersion 72°F.
/ o
0
0
/00/0

40

   
  

Immersion lOl°F.

O

Moistened

’

 

0
0 10 20 30 40 5'0
Strain Rate, é, %/Min.
Fig. 4.7 Apparent Young's Modulus Versus Strain Rate

After Toe

41

 

 

 

70.

60-

50-

a

0
3

.mmmuum

20 .

10

10.0

Strain,

%

E:

Strain Rate

Stress—Strain Plot for 5%/Min.

4.8

Fig.

2

Stress, 0, Kg./cm.

 

42

 

 

70+
60 ‘ O o
2‘"

T \0

50 o
<\
\0
‘o
K
e
8e
\
40‘ o
30- o O
20 ‘ O o o
10‘ o 00
0 r . 4 r .
0 1.03 2-0 3.0 4.0 5.0
Strain e, % '
Fig. 4.9 Stress-Strain Plot for 15%/Min. Strain Rate

43

 

 

 

0
fi.
0
.l.
0
c
To
0
fl.
6
0
r.
4
O
0
I.
2
a d d 0
0 0 o O 0 0 0 0
7 6 5 4 3 2 l

.EO\.mM .b .mmoupm

Strain, e,

%

Static Test

4.10

Fig.

44

 

30.0

 

 

70

60

503

N

d

o o
4 3

.E0\.mM .o .mmmuum

20

10

0

Relaxation, %0

Relaxation When Moistened

4.11 Stress vs.

Fig.

45

 

 

 

7O

60

50

N

a -
0
4. 3

.E0\.mM .p .mmmuum

20

10

50.0

 

Relaxation,

%0

Relaxation in Saline at 72°F

Stress VS .

4.12

Fig.

46

 

 

 

80

J

70

60 '

fi

0
5

N

0
4

.EO\.mM .o .mmmuum

30

20

10

50.

40

Relaxation,

%0

Relaxation in Saline at 101°F

4.13 Stress vs.

Fig.

47

omcmumfloz cons mumm cflmnum momnm> soflummemm ucmu mom va.v .mHm

.cHE\w .m .mumm camuum
ov om oh 0H 0

LEO x 0.0. 335 o

0.0m

 

no“
0.0m m“
0\

.oa

r0N

 

 

.0m

0% ’uorqexeteu

moms um cmmHmEEH coax oumm camuum mdmuo> cowumxmaom ucmo mom ma.e .mam
.cflz\w .w..oumm.cflmuum

om 0N 0H 0

h! D

O

 

O
O
O
.Ei? 0.9 3,25 o
- o T|\\\\\\\\\\\ o

0.0 m T \
0.0m \

0.0¢
000

0000

00000
0

0.00

 

.OH

.oN

.om

’uoraexeteu

0%

2

Stress, quKg./cm.

 

 

49

 

Fig.

 

4.16

4'. 0 6T. 0 8'. 0
Strain, e, %

Stress After Relaxation Versus Strain
for Immersion at 72°F

2

Stress, 0R, Kg./cm.

50

7O ‘

60 J

50

40

30

20

10 <

 

 

 

43 610 8.0 10.0

Strain, e, %

Fig. 4.17 Stress After Relaxation Versus Strain for
Moistened Specimen

51

 

 

.0.
1.
o
MJW o
/ O
0/0 0 0 8
IIIIIIIIIIo //////
N.n /0 0
/ /o / o.
O /O O .6
0/// /// /////
«.0 o o/
o
Ill/Io/l/ ////d7/// 0 0 4M“
/0 v. 0/ /0 /
/o W\ 0/0 / o
/O / O
/O m/.@Mv/O 0/0 /0 / O 0
IIIIIIIIb OOIIIIIIIIo Il/Io ////o o .L
oon/\ at.
QCNQkkm 00/0 07 0 O
IIJWIToTIIIImHWI. ,/

. . TTT . . o T. O
0 0 0 0 0 0 m0 0 nu
8 7 6 5 4 3 2 1

.Eo\.mM .mb .mmmuum

N

Strain, e,

%

Stress After Relaxation Versus Strain

for Immersion at 101°F

4.18

Fig.

52

O
/
O
.///O
/ N\ / /0/
. / O
/ 0
O /0/// O/
O O [/0
I/O/.I[O/ /
’0

——Firs’r
——Sec0nd

 

 

70¢

60-

50 .

TIT

0 0
4 3

20 «

.E0\.mM .o .mmmuum

10 -

10.

8.0

 

Strain,e,

%

Stress Versus Strain for First and Second

Tests When Moistened

4.19

Fig.

V. DISCUSSION AND CONCLUSIONS

The shape of the stress-strain curves follow the
same pattern given most often in the literature. As seen
in Figures 4.1-3, the shape of the stress-strain curves
is not altered by various strain rates, except for high
strain rates where the toe region appears at an early
strain level. Unlike the range of 0-l.5 per cent strain
for the toe regions of tendons given by Abrahams (2), the
anterior cruciate ligament of dogs displays a toe region
which varies with strain rate up to 6 per cent strain.
Also, in these figures the increase in the apparent Young's
Modulus with increasing strain rate is quite pronounced.
For flow strain rates the modulus changes rapidly with
strain rate, but for higher values of strain rate only a
gradual increase can be seen. Results similar to these
were described by VanBrocklin (25) and Abrahams (2). The
average apparent Young's Modulus given by Frasher (11) for
collagen fibers in blood vessels ranged from 3.0 x 107 to
1.0 x 109 dynes per square centimeter. Values of approxi-
mately 2.0 x 109 dynes per square centimeter were obtained

in these experiments on the anterior cruciate ligaments

of dogs.

53

54

Y. C. Fung (12) described the elastic properties
of the mesentery of the abdomen of rabbits by an exponen-
tial function. He used the value of stress after a com-
plete relaxation of the specimen at each stress level and
plotted this value versus the value of strain. This rela-
tionship describes the so-called "elastic" test condition
corresponding to an infinitely slow application of load.
By plotting the slope of the tension-deflection curve
against the elastic tension, he was able to approximate

the following linear relationship.

=bT,

arm
>‘6

where T equals the elastic tension and 1 equals the exten-
sion ratio (the deformed length divided by the relaxed

length). An integration gives

a
u
and
m

where c is an integration constant. This, however, does not
give the required zero tension when the eXtension ratio is 1.
Using the fact that the strain energy function W(ll,12,l3)

of an isotrOpic compressible elastic body is expressible

in terms of the strain invariants I1 and 12, the Lagrangian

stress in simple elongation in given by

55

The zero factor must be of the form (A - £2] if the strain
1

energy has no singularity at the undeformed state.

Insertion of this factor yields

where D is a constant.

If we describe the Lagrangian strain by

then,

1 eae
e + 28 + l

 

£2 + 28 as
e
e + 26 + 1

 

By dividing the quantity in brackets and keeping terms to
the third order, one gets
4 2 as
c=C[e[l-e+§-e)]e
where C and a are undetermined constants.
This form of Fung's Law was plotted for various values
of a and C in Figures 4.1-3. In Table l the constants

which were used to draw these curves are given. It can

be seen that Fung's Law very closely represents the

56

Table 1.-—Values of a and C for Exponential Law

 

 

 

 

 

Strain Rate a C
Environment
%/Min. Kg./cm.2
0.0 20 147.7
1.7 18 167.6
Moistened 2.6 18 213.4
With saline 10.8 20 263.8
15.3 28 367.1
36.8 40 508.1
0.00 18 118.5
1.55 20 132.8
2$m$§§;°n 4.75 16 277.6
9.75 16 333.1
21.50 15 430.5
53.80 21 491.4
0.0 24 35.6
2.5 20 96.0
5.7 14 282.1
Immersion 9.7 1 1000.0
at 101°F 15.4 0 1600.0
16.2 0 2210.0
19.7 0 3070.0

 

57

results which were obtained in the tests. If one examines
Table 1, it may be noted that the value of the constant a
depends on the strain rate. In the tests for the moistened
specimens the constant a increases with the strain rate,
whereas for the immersed specimens the constant decreases
with an increase in the strain rate. With the limited
amount of data available at this time it is difficult to
determine the exact nature of this dependence. Also in
Table l the constants for the static test are given.

These constants are used in the curves which are plotted
in Figure 4.10. Again, Fung's Law represents the extrap-
olated data very well. The constant C increases with an
increase in strain rate and there is very little differ-
ence between the moistened and immersed at 72°F tests.

The difference between the first and second tests
on the same specimen was noted (Fig. 4.19) to be an ex-
tended toe region for the second test. Rigby first re-
ported this result in his tests for the mechanical prop-
erties of tendons (24). He "preconditioned" the specimen
by straining it to a given value of strain before each
test. He suggested that material other than collagen
fibers was ruptured in this conditioning. After this
procedure, he claimed that the tendons were elastic beyond
the strain level of 4 per cent. If one considers the de-
scription of the toe region given by Abrahams (2), it

could be possible that the rupturing of these other

58

constituents might cause the toe region to become extended
during the second test of a specimen. Furthermore, the
fact that the lepes of the linear portions of the curves
are identical for both of the tests shows that the actual
collagen fiber response is not affected by the testing of
the same specimen twice in the range considered.

In Figures 4.11-13 the cumulative relaxation in
per cent of initial stress versus the initial stress for
varying strain rates is plotted. For lower strain rates
it can be seen that the per cent relaxation changes rapidly
with the value of stress. At higher values of stress,
above 20 kilograms per square centimeter, the curves be-

come linear.

%0 = C0
relax

or A0 = C02 ,

where Ac equals total amount of stress relaxation and 0
equals the initial stress. In the results from Figures
4.11-13 there is a trend for the curves to become shifted
toward the stress axis as the strain rate is reduced, but-
as stated previously this trend is not entirely consistent
throughout all of the tests.

Throughout this discussion of results there has
been no reference to the results of the tests run at the
elevated temperature of 101°F. If one looks through the

results of these tests, the temperature of the saline bath

59

has rather drastic effects on the mechanical prOperties
of the anterior cruciate ligaments of dogs. Due to the
irregularities in the results of the tests conducted, a
complete discussion is not possible at this time. In
order to determine the actual effects of temperature on
the mechanical prOperties, a series of varying temperature
tests must be made to complete the investigation. The re-
sults obtained for the single elevated temperature indicate
that this study should definitely be made and that the ef-
fect of using other bath fluids should be investigated.

A further observation noticed by the author can
be seen in Fig. 3.2. In this picture an apparent organized
pattern of the fibers may be seen which took place while
the specimen was being strained. Even though this was
only an observation which was not dealt with in any de-
tail, there is a possibility that a recrystalization might
take place under tensile forces similar to that which has
been seen to occur at the so-called "recrystalization

temperature" of biological materials.

BIBLIOGRAPHY

10.

11.

12.

BIBLIOGRAPHY

Abbott, L. C., "Injuries to the Knee Joint," Journal
of Bone and Joint Surgery, 1944, pp. 503-517.

 

Abrahams, M., "Mechanical Behavior of Tendon in Vitro,
A Preliminary Report," Medical and Biological Engineer-
ing, 1967, V01. 5, pp. 433-443.

 

Annovazzi, G., "Osservazioni sulla elasticita dei
legamenti," Arch. Sci. Biol. (Napoli), 1928, Vol. 11,

 

Bloom, W., and D. W. Fawcett, A Textbook of Histology,
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APPENDIX

APPENDIX

Ineorder to solve for the initial length of the
anterior cruciate ligament before each test, 3 radii
were drawn to the points of insertion of the ligament

into the femur and the tibia.

 

 

 

(0.y'.0)
Pf(X.y.z)
Cruciate ligament
(X. 7010)
X Z

The point of intersection of the 3 spheres is the point
Pf(x,y,z). The equations to be solved form a system of

3 simultaneous quadratic equations.

(x - x')2 + y2 + 22 = R3 A-1
\

2 ,2 2 2

x+(y—y)+z=R2 A-2

x2 + y2 + (z — z')2 = Ri A-3

65

 

 

66

From A-l, A-2 and A-3 one obtains

-2xx' + x'2 + 2yy' - y'2 = R3 - R: , A-4
-2yy' + y'2 + 222' — 2'2 = R: - Ri , A-5
-222' + 2'2 + 2xx' - x'2 = Ri - R: . A-6
Using A-5 and A-6, yields
_ l ,2 _ ,2 2 _ 2 _ z' _
y-ffr-{y z +Zzz'+Rl R2}—A2+T-z A7
_ l ,2 _ I2 , 2 _ 2 _ z' _
X - 5‘1- {X Z + ZZZ + R1 R3} - A1 + x-—'- Z , A 8
where
_ l I,2 _ ,2 2 _- 2
A2 - 75,-1- {y Z + R1 R2}
_ l , 2_ , 2 2__ 2
A]. - 23?.- {(X ) (Z ) + R1 R3} 0

Substituting these results into A-BV yields

2
2 z' z' 2 z' z' 2 , 2 _ 2
A]. + 2A1 i-r Z + (fr) + A2 + 2A2 if Z + (SIT) Z + (2-2 ) — R1
2 2 2 z' z' { z' 2 2'32 J} 2 2 ,+ ,2__ 2
Al + A2 + [A]. Err '1' A2 y-T) Z + (SET) 4" (yr) + Z - 22 Z - R1

 

67

Rearranging gives

2 2 A A
I |
{(Er) +T§r) +1}zz+2z'T;}-+§-2r- 112 + {21% + A: + 2'2 mi} = 0

Solving this quadratic equation for 2 produces the final

 

result,
.1.
A1132 2A1A2 2 22 22 2'22'2 2
- Z'{l-}Tr,-y—r}i [Z' (x-fi-j' fir-1) -{A1+A2+Z' ‘R1}{(?)L+[§r)+n]
z— ' 2 '2 c A-10

{(ET) + (gr) + 1}

From equations A-7 and A-8 the remaining coordinates are
obtained. Since this system of equations yields 2 solutions,
it was necessary to determine which solution was applicable.
This choice was easily made by determining the approximate
coordinates during the tests. The point of insertion of

the ligament into the tibia was calculated in a similar

manner 0

 

436

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