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This is to certify that the

dissertation entitled

A CRITERION T0 PREDICT DAMAGE

IN ARTICULAR CARTILAGE DUE TO BLUNT IMPACT

presented by

Xiaowei L1

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Mechanics

 

 

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February 20, 1995

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A CRITERION TO PREDICT DAMAGE
IN ARTICULAR CARTILAGE DUE TO BLUNT IMPACT

By

Xiaowei Li

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Materials Science and Mechanics

1994

ABSTRACT

A CRITERION TO PREDICT DAMAGE
IN ARTICULAR CARTILAGE DUE TO BLUNT IMPACT

By

Xiaowei Li

Excessive mechanical load to the knee during a blunt impact can initiate a
degenerative disease process. Laboratory experiments have shown that the stresses and
strains generated in the knee joint during impact can cause gross mechanical tissue
damage. There are two basic types of damage reported in the knee joint under impact
loading: fissures on the surface of the cartilage and/or microcracks in the subchondral
bone or calcified cartilage. A mechanical damage criterion is developed in this
dissertation using new experimental data, old observations and some new analytical
results using the finite element method. A Coulomb-Mohr damage criterion is applied to
study mechanics of biological tissue trauma. The failure envelop reveals that the damage
of cartilage from the tibial plateau is controlled by two variables: shear stress and
hydraulic pressure. In fact, the effect of static hydraulic pressure is significant and can not
be ignored. The criterion is robust and can be used to help explain other types of

biological tissue damage during blunt impact loading.

Dedication

To my parents: Zhou Li and L. Zheng. To my husband XianLi Huang and my
daughters: SiRui and Rachel.

iii

Acknowledgments

I would like to express my sincere appreciation to my advisors, Professor Nicholas
J. Altiero and Professor Roger C. Haut for their guidance, friendship and assistance all the
way through the dissertation and for their training on me in professional faith, skills and
insights of engineering.

I would like to thank my committee members Professors D. Liu and D. Yen for
their advice, comment and help during my studies.

I would like to thank the following people: Jane Walsh for her histological
preparations, Tammy Haut for her processing the data base by FEM models, Jean
Atkinson for her work with the animals and Bill Newberry for his logistic analysis.

This research has been supported by a grant from the Centers for Disease Control

and Prevention (R494CCR-503607).

iv

TABLE OF CONTENTS

TABLE OF CONTENTS ................................................................................................ v
LIST OF TABLES ........................................................................................................... viii
LIST OF FIGURES ......................................................................................................... ix
Chapter 1 Introduction .................................................................................................... 1
1.1 Osteoarthritis in Our Society ........................................................................ 1
1.2 Nature of Impacted Articular Cartilage ........................................................ 2
1.3 Mechanics Models ....................................................................................... S
1.4 Statement of the Problem .............................................................................. 6
Chapter 2 Background and Objectives ........................................................................... 7
2.1 Experimental Results for Impact of Rabbit Knee Joints .............................. 7
2.2 Candidate Failure Criteria ............................................................................. 11
2.3 Physical Justification for Use of a Coulomb-Mohr Failure Criterion .......... 16
2.4 The Study Objectives .................................................................................... 18
Chapter 3 Computational Modeling Assumptions ......................................................... 20
3.1 Linear Elastic Behavior of Cartilage and Bone ............................................ 20
Viscoelastic Model .................................................................................. 20
Biphasic Model ........................................................................................ 21
3.2 Justification for the Assumption of a Quasi-Static Load .............................. 24
Quasi-static State Solution ....................................................................... 24
The Effect of Stress Wave Propagation ................................................... 25

3.3 Plane Strain Modeling of the Rabbit Knee Joint and the Associated

Boundary Conditions .................................................................................... 32

vi

Chapter 4 Estimation of the Unknown Material Properties of Cartilage Using an

Inverse Approach ............................................................................................................. 36
4.1 Experimental Methods and a FEM Model of the Rabbit Knee Joint ........... 36
4.2 Estimation of Material Properties ................................................................. 41

Chapter 5 Application of the FEM Model and Probability Analysis of Damage

Prediction ....................................................................................................... 45

5.1 FEM Analysis of the Rabbit Knee Joint Impact Tests ................................. 45

FEM Model and Boundary Conditions .................................................... 45

Analytical Results .................................................................................... 47

5.2 Probability Analysis ...................................................................................... 53
Chapter 6 Determination and Application of a Coulomb-Mohr Failure Criterion ......... 55
6.1 Impact Tests on the Rabbit Tibial Plateau .................................................... 55

The Experimental Method ....................................................................... 55

Results of the Indentation Experiments .................................................. 59

Histological Observations ........................................................................ 62

6.2 A FEM Model of Impact on the Rabbit Tibial Plateau ................. 64
Analytical Results .................................................................................... 66

6.3 A Coulomb-Mohr Failure Envelop for the Rabbit Tibial Cartilage ............. 74

6.4 Application of the Criterion to Ide's Experimental Data .............................. 76
Chapter 7 Conclusions .................................................................................................... 78

APPENDIX A The Effect of Loading Location on the Distribution of Stress and
Strain in the Rabbit Knee joint ............................................................... 83
APPENDIX B Estimation of Material Properties and Stress Analysis for the

Human Knee Joint .................................................................................. 85
(1) The FEM Model .............................................................................. 85
(2) Estimation of the Material Properties .............................................. 87

(3) The Stress Distribution in the Human Knee Joint ........................... 92

vii

APPENDIX C Application of the Coulomb-Mohr Criterion to Human Knee Joint ...... 94
(1) The Damage of Cartilage ................................................................. 94
(2) The Damage of Subchondral Bone .................................................. 97

BIBLIOGRAPHY ............................................................................................................ 100

LIST OF TABLES

Table 3.1 Material properties of cartilage in biphasic model (Armstrong, et a1,

1984) ................................................................................................................ 22

Table 3.2 Comparison 0f 0 711/80, of the elastic solution to the Viscoelastic and

biphasic SOIUIIOHS ............................................................................................ 23

Table 3.3 Material properties of rabbit cartilage and bone ............................................. 26

Table 5.1 Analytical results from the FEM model based on Ide's experimental

data .................................................................................................................. 51
Table 5.2 Logistic regression results .............................................................................. 53
Table 6.1 Maximum shear stress and their ratio of shear in three cases ........................ 67
Table A] Maximum shear Stress caused by different loading locations ....................... 84

Table CI The maximum values of shear stress and the corresponding mean

stresses on the cartilage surface ....................................................................... 97

Table C.2 Experimental observation and data ................................................................ 99

viii

LIST OF FIGURES

Figure 1.1 Surface fissures extend deeply into the radial zone in the rabbit tibial

plateau .......................................................................................................... 4

Figure 1.2 Horizontal splitting at the tidemark was frequent in tibial cartilage from the

experiments ( Radin, at. al. , 1984) 4

Figure 2.1 A Typical distribution of contact pressure recorded on Fuji Film by

impact ........................................................................................................... 8
Figure 2.2 The profile of contact pressure in the cut section .......................................... 8
Figure 2.3 Average pressures data from 34 impact experiments (Ide, 1992) ................. 9
Figure 2.4 Fissure location in the impacted cartilage and its orientation ....................... 10
Figure 2.5 The Tresca failure criterion ........................................................................... 1 1
Figure 2.6 The Failure criterion of Mohr ........................................................................ 14
Figure 2.7 The model of Coulomb-Mohr ....................................................................... 15
Figure 2.8 The Failure criterion of Coulomb .................................................................. 15

Figure 2.9 (a) Scanning a section of cartilage, (b) Scanning a section of the

damage zone on the impacted sample from a rabbit .................................... 17

Figure 2.10 Fissure growth in the soft tissue along collagen fibers ............................... 18

ix

X

Figure 3.1 Forces at the interface in a one dimensional discontinuous medium ............ 26

Figure 3.2 (a) Stress wave propagation from cartilage to bone, (b) Stress wave

propagation from cartilage to cartilage surface ............................................ 27
Figure 3.3 Reflected force at the cartilage-bone interface vs. time ................................ 28

Figure 3.4 The effect of the reflected force on experimental impact load (cartilage

to bone) ......................................................................................................... 29

Figure 3.5 (a) Stress wave propagation from bone to cartilage, (b) Stress wave

propagation from bone to bone surface ........................................................ 30
Figure 3.6 Reflected force at the cartilage-bone interface vs. time ................................ 31

Figure 3.7 The effect of the reflected force on experimental impact load (bone to

cartilage) ....................................................................................................... 32

Figure 3.8 (a) Plane strain model of the patella, (b) Impact induced surface

fissuring of cartilage ................... 33

Figure 3.9 The contact pressure recorded with Fuji film on the anterior surface of

the patella and the PF contact surface .......................................................... 34

Figure 3.10 The profile of impact pressure on the anterior surface at the cut

section ........................................................................................................... 34

Figure 3.11 (a) The cross-section of the rabbit knee joint, (b) The 2-D FEM model

and boundary conditions .............................................................................. 35

Figure 4.1 Schematic of the restraint system used for experiments #1 and #2.
Note that the femur was constrained with an intramedullary pin

inserted via the greater trochanter ................................................................ 37

XI
Figure 4.2 Schematic of the restraint system used in experiment #3. The hind

limb was constrained with spring-loaded clamps and a strap across the

pelvis ............................................................................................................ 38

Figure 4.3 The curve of impact load vs. time in a typical experiment on rabbit

knee joint ...................................................................................................... 38

Figure 4.4 (a) A section cut through the patello-femoral joint. (b) Typical FE
mesh for 2-D analysis of the blunt impact response of the rabbit PF

joint ............................................................................................................... 40

Figure 4.5 (a) A plot showing the experimental contact pressure distribution and
the corresponding theoretical distributions for various values of
Poisson's ratio at Young's modulus 2 MPa. Note how well the results
at 0.49 compare to the results from experiments #3, (b) The residuals

decreased continuously as Poisson's ratio approached 0.49 ......................... 42

Figure 4.6 (a) The effect of variations in the elastic modulus of cartilage on the
residuals computed between the experimental and the theoretical
contact pressures. A minimum was obtained for an elastic modulus
near 2 MP3 for the cartilage at Poisson's ratio of 0.49, (b) The residuals
reached minimum at a Young's modulus of 1 to 2 MPa at a Poisson's

ratio of 0.49 .................................................................................................. 43

Figure 5.1 Typical FE mesh for the 2-D analysis of the blunt impact response of
the rabbit patella for Ide's experiments (1992) ............................................. 46

Figure 5.2 Contact pressures recorded by Fuji film during Ide's experiments ............... 47

Figure 5.3 A 2-D profile of these PF contact pressures .................................................. 47

XII
Figure 5.4 A typical location of surface fissuring measured often a severe blunt

insult ............................................................................................................. 48

Figure 5.5 Typical contour plots for the stress and strain distributions in cartilage:
(a) the minimum principal stress distribution, (b) the maximum shear
stress distribution, (c) the shear stress distribution, txy, and (d) the

maximum tensile strain distribution ........................................................... 49

Figure 5.6 A correlation between the maximum values of each indicated
parameter and the occurrence of surface fissures on the cartilage: (a)
maximum shear stress, (b) mean stress, (c) maximum strain and ((1)

peak contact pressure .................................................................................... 52

Figure 5.7 The comparison of the prediction power of each variable given in Table

5.2 ................................................................................................................. 54
Figure 6.1 Surface view of the rabbit's tibial plateau ..................................................... 56
Figure 6.2 The impact device ......................................................................................... 57

Figure 6.3 Surfaces fissures located in the contact area after impacts. These are
enhanced with India ink. The diameter of this mark was
approximately 2.0 mm. (a) before initial fissuring, (b) initial fissuring

and (c) fissures after severe impact .............................................................. 58

Figure 6.4 Load vs. time recorded with a 2.0 mm radius spherical indenter impact

test ................................................................................................................ 59

Figure 6.5 The experimental results from group 1 impact tests .................................... 60

Figure 6.6 The experimental results from group 2 impact tests .................................... 61

XIII

Figure 6.7 The experimental results from group 3 impact tests ..................................... 61
Figure 6.8 Typical fissures from a group 1 specimen .................................................... 62
Figure 6.9 Typical fissures from a group 2 experiment .................................................. 63
Figure 6.10 A deeper microcrack generated in a group 2 experiment ............................ 63
Figure 6.11 A typical surface fissure from a group 3 experiment .................................. 64

Figure 6.12 Microcracks generated in a test specimen from group 3. Note the

India ink on the cartilage surface ................................................................. 64
Figure 6.13 3-D tibia impact model ................................................................................ 66
Figure 6.14 Axisymmetric model ................................................................................... 66

Figure 6.15 Minimum principal stresses and directions for the different indenters

(Pa) (a) R=2.0 mm, (b) R=4.0 mm, and (c) R=20.0 mm. ............................. 68

Figure 6.16 Maximum shear stresses and directions for the different indenters (a)

22.0 mm, (b) R=4.0 mm, and (c) R=20.0 mm ........................................... 69

Figure 6.17 Maximum principal stresses and directions impacted by different

indenters (a) R=2.0 mm, (b) R=4.0 mm, and (c) R=20.0 mm ...................... 70

Figure 6.18 Shear stresses in the xy plane for different indenters (a) R=2.0 mm, (b)

R=4.0 mm, and (c) R=20.0 mm ................................................................... 71

Figure 6.19 Mises' stresses for the different indenters (a) R=2.0 mm, (b) R=4.0
mm, and (c) R=20.0 mm .............................................................................. 71

Figure 6.20 The distribution of maximum shear stresses at the top of the cartilage

layer .............................................................................................................. 72

XIV
Figure 6.21 The distribution of maximum shear stresses at the interface between

cartilage and bone for various indenters ....................................................... 73

Figure 6.22 The distribution of maximum shear stress in the bone just below the

cartilage-bone interface ................................................................................. 74
Figure 6.23 The failure envelop for tibial cartilage under impact loading ..................... 75

Figure 6.24 Application of the tibial failure criterion to Ide's data. Solid dot
indicates a fissure, whole open circle indicates that no surface damage

was observed experimentally by Ide ............................................................ 77

Figure 6.25 Application of the non linear tibial failure criterion to Ide's data. Solid
dot indicates a fissure, whole open circle indicates that no surface

damage was observed experimentally by Ide. .............................................. 77

Figure A.1 This figure shows the effect of impact location on the distribution of
contact pressures generated in the PF joint. More bimodal types of
distributions were generated with more medial and lateral oriented
impacts on the patella. A central located impact on the patella
generated a more uniform profile of contact pressures over the medial

and lateral facets ........................................................................................... 84
Figure B.1 The cross section of the human knee joint ................................................... 85

Figure B2 A typical patella from the human knee joint and the corresponding Fuji
film. The scalpel locates the location of the cross-section used for the

FEM model. .................................................................................................. 86
Figure B.3 FEM model of human knee joint .................................................................. 87

Figure B.4 (a) A plot showing the experimental contact pressure distribution and

XV
corresponding theoretical distributions for various values of Poisson's

ratio in Knee-l model. Note how well the results at 0.45 compare to
the results from the experiments. (b) The residuals decreased

continuously as Poisson's ratio approached 0.45 .......................................... 88

Figure B.5 (a) A plot showing the experimental contact pressure distribution and
corresponding theoretical distributions for various values of Poisson's
ratio in Knee-2 model. Note how well the results at 0.475 compare to
the results from experiments, (b) The residuals decreased continuously

as Poisson's ratio approached 0.475 ............................................................ 89

Figure B.6 (a) The effect of variations in the elastic modulus of cartilage on the
residuals computed between the experimental and the theoretical
contact pressures. A minimum was obtained for an elastic modulus
near 2 MPa for the cartilage at Poisson's ratio less than 0.475 in Knee-
1 model, (b) This plot shows the residuals for Young's modulus from 1

to 20 MPa ................................................................................... . .................. 90

Figure B.7 (a) The effect of variations in the elastic modulus of cartilage on the
residuals computed between the experimental and the theoretical
contact pressures. A minimum was obtained for an elastic modulus
near 2 MPa for the cartilage at Poisson's ratio less than 0.475 in Knee-
2 model, (b) This plot shows the residuals for Young's modulus from 1
to 20 MPa. .................................................................................................... 91

Figure B8 The contour plots of the minimum principal stress and the maximum

shear stresses in the Knee-l model ............................................................... 93

Figure CI The 2-D patellar FEM model for the human data analysis of the

boundary conditions ..................................................................................... 95

XVI
Figure C 2 Cartilage damage of human patella .............................................................. 96

Figure C.3 Application of Coulomb-Mohr criterion to the human cartilage data .......... 96

Figure C.4 Results of the stress analysis in the subchondral bone for the fifteen

human knee joint experiments ...................................................................... 98

Figure C.5 Crack propagation along subchondral bone in this typical section of

human patella ............................................................................................... 99

Chapter 1 Introduction

1.1 Osteoarthritis in Our Society

Arthritis in all its forms is perhaps the most prevalent cause of disability in the
United States (Brown, 1988). Chronic disability caused by osteoarthritis (DA) is second
only to that caused by cardiovascular disease. Diagnosis of this disease is primarily based
upon radiographic assessment and clinical examination of the features (Altman, et al.,
1986; and 1987). Although the states of this disease have been described in detail
(Mankin, 1974), many questions still exist about its pathogenesis and the sequence of
events leading to its formation. Traumatic joint injury has been speculated as a possible
triggering mechanism for the onset of osteoarthritis. It has been hypothesized that
articular cartilage damage sustained during impact may lead to altered joint loading and
progressive degeneration (Repo and Finlay, 1977). While the mechanisms responsible for
this disease are largely unknown, mechanical insult is speculated as one potential factor
(Insall, 1976). A primary complication associated with lower extremity injury in motor
vehicle accidents is post-traumatic osteoarthrosis (States, 1970). A direct association,
however, between mechanical insult and the development of osteoarthritis has been
difficult because radiographic evidence of the disease often does not show up for 2-5 years
(Wright, 1990). On the other hand, only about 12% of patients with secondary
osteoarthritis from a mechanical insult are free of pain in the interval immediately
following the event (Chapchal, 1978 ).

Osteoarthritis (DA) is a degenerative disease which involves deterioration of the
articular cartilage and subchondral bone (Meachim, and Brooke, 1984). This
degeneration process is believed to result from a combination of mechanical loading and

biological weakening of the cartilage matrix. An understanding of the mechanisms of
1

2
macro trauma to the structures and tissues of the knee requires a simple understanding of

their normal function.

Articulating joints are the functional connections between different bones of the
skeleton. In synovial, or freely moving joints, the articulating bone ends are covered with
a soft white layer, a low friction gliding layer of connective tissue: the articular cartilage.
The main functions of this articular cartilage are: (1) to spread loads to the joint so that it
is transmitted over a large area, and the contact stresses are decreased and (2) to allow
relative movement of the opposing surfaces with minimum friction and wear.

There are two main components in articular cartilage. The solid matrix of
cartilage, which accounts for 20 to 40 % of the tissue's wet weight, is composed of
collagen fibers. The remaining 80 to 60 % of the tissue is water, most of which can be
squeezed out under load.

Since articular cartilage is highly resistant to fluid flow (i.e., low permeability), its
material behavior is very much dependent on the rate at which the load is applied and
removed. Thus in a rapid loading situation, when there is not time for the fluid to be
squeezed out, the tissue will behave more or less like an elastic, single-phase material,
deforming instantaneously upon loading and recovering instantaneously upon unloading.
However, if the load is applied slowly to the tissue, as for example during prolonged
standing, the deformation will continue to increase in time as the fluid is squeezed.

The two types of cartilage mechanical behavior described above are: time
independent or elastic material behavior (the deformation may be recoverable); and time
dependent or Viscoelastic material behavior. A discussion of both materials of cartilage
follow.

1.2 Nature of Impacted Articular Cartilage

Several impact models have been used to study the in vivo processes leading to the

development of osteoarthritis. These models typically involve a single, concentrated blow

to the test joint in order to inflict trauma. The trauma induced by transarticular impact can

3
be divided into two categories, based on the extent of the damage. These categories

include (1) articular cartilage damage without disruption of the underlying bone or
calcified cartilage and (2) subchondral bone fractures (often including damage to the
overlying articular cartilage).

The type 1 defects are limited to the articular cartilage, do not involve the
subchondral bone and do not elicit an acute inflammatory reaction (Glowacki, 1986; and
Chueng, 1878). Acute fissures are a frequently reported form of impact damage to
articular cartilage (Figure 1.1). These fissures most often begin at the surface and extend
downward into the middle/deep zones of the cartilage (Repo, 1977; Broom, 1986; Haut,
1986; Thompson, 1990; Silyn-Roberts, 1990; Ide, 1990; Thompson, et al 1990; Tomatsu,
1992; Thompson, 1993). Repo and Finlay (1977) examined the response of human
cartilage-on-bone samples under several levels of impact loading. They documented
radial fissures in the center of the contact zone at the cartilage surface extending to the
deep zone, and death of chondrocytes when average normal contact pressures reached 25
MPa. Silyn-Roberts, et a1, (1990) reported that impact induced failures of cartilage, in
vivo, were usually in the center of the contact zone and oriented at 45 degrees to the
articular surface and extended into the mid-zone.

The type 2 defects penetrate the subchondral bone and participate an inflarmnatory
response and the attempted regeneration of the tissue (Glowaki, 1986; Pritzker, 1991; and
Bland, 1983). The fractures in the calcified cartilage and subchondral bone are often seen
in conjunction with fissures that extend up into the cartilage, but do not reach the articular
surface (Thompson, 1991; Vener, 1992). Using a rabbit model and cyclic impulse
loading, Radin (1973, 1978, 1984)) was able to measure an increase in the subchondral
bone stiffness as well as progressive decreases in the proteoglycan content of cartilage in
the traumatized knee( Figure 1.2). Separation of the cartilage from the underlying bone in
the zone of calcified cartilage has also been reported in response to blunt impact

(Armstrong, et al, 1982). Vener, et al, (1991) have observed cracks in the zone of

4
calcified cartilage after impact of canine joints. In a more recent study, increased bone

volume was reported to have a correlation with the severity of osteoarthritis observed in
human articular cartilage (Shimizu, 1993). The results of studies such as these have
prompted many investigators to believe that osteoarthritis is mediated by initial changes in

the subchondral bone.

 

Figure 1.1 Surface fissures extend deeply into the radial zone in the rabbit tibial
plateau.

 

Figure 1.2 Horizontal splitting at the tidemark was frequent in tibial cartilage from
the experiments ( Radin, at. al. , 1984).

1.3 Mechanics Models

Many researchers believe that fissuring of the articular surface ultimately leads to
the development of osteoarthritis. A few different mechanics models have been developed
to address the impact contact problem that yields these fissures.

Askew and Mow (1978), using elastic models subjected to impact profiles, argued
that tensile strains, rather than stresses, may be a more reliable predictor of where fissures
are likely to form. The model predicted large shear stresses at the cartilage and bone
interface, which may account for separations seen experimentally (Askew and Mow,
1978; Chin, 1986). Eberhardt, et a1 (1990) modeled joint contact as a layered elastic
sphere against a layered elastic cavity. They discussed the effects of cartilage thickness
and stiffness, bone stiffness and curvature on the distribution of stresses in cartilage and
the underlying bone. A later study (Eberhardt, et al, 1991) utilized a solution based on
normal contact of two elastic spheres. The authors suggested that cartilage stresses are not
significantly affected by stiffness of the underlying bone. Tensile radial strains in the
cartilage approach one-third the normal compressive strains, and increase with softening
of the cartilage. The model by Askew and Mow (1978) bonds the elastic. layer to a high
modulus half-space of subchondral bone and introduces a parabolically distributed
pressure to the surface. They note that stresses and strains in the articular layer strongly
depend on the contact aspect ratio. The also documented large tensile strains parallel to
the surface near the center of the loading area. Armstrong, C. G., et al (1984) indicated
this coincides with the area where cartilage delaminates during blunt impact experiments .
Large tensile strains also develop in the cartilage near the location of surface fissures
(Haut, 1992). These fissures are typically oriented at 45 degrees to the surface (Silyn-
Roberts, H. 1990), so the authors suggest that tissue failure in the zone of compression is
primarily due to excessive shear stress.

G. A. Ateshian (1994) performed an analysis of cartilage based on its biphasic

material behavior. This analysis provides strong support for the hypothesis that, if sudden

6
loading causes shear failure within the cartilage-bone layer structure, failure would take

place at the cartilage-bone interface, and the plane of failure would be either parallel or
perpendicular to this interface.
1.4 Statement of the Problem

The degree of mechanical damage to a knee under impact loading initiates a
degenerative disease process. There are many factors affecting degenerative process. It
has been shown by experiments that the stresses and strains generated in the knee joint
cause fissures on the cartilage surface and cracks in the bones. The damage mechanism of
tissue trauma leading to osteoarthritis is not yet fully understood, even though there have
been a large number of contributions to this field of study. There are two types of damage
in the knee joint under impact loading shown by experiments: fissures occurring on the
surface of the cartilage, and cracks in the subchondral bone or calcified cartilage. The
question arises: what causes these fissures and cracks? To answer this question, a
quantitative analysis is required to determine which factors cause the damage and a
quantitative damage criterion is needed to predict tissue damage. A quantitative analysis
of the tissue fissures and cracks can be accomplished by a finite element model of the knee

joint to simulate the damage process.

Chapter 2 Background and Objectives

A lot of experimental data will be needed to develop a quantitative damage
criterion. The experimental data and observations from impacted knee joints of rabbits
performed by Ide (1992), will be used in this dissertation. It will be necessary to review
these experimental results. Furthermore, a damage criterion will be developed in this
dissertation on the basis of a quantitative analysis of these data and relevant damage
theories. Damage theories and their application to tissue injury mechanisms will be
reviewed.

2.1 Experimental Results for Impact of Rabbit Knee Joints

Alterations in the mechanical properties of the rabbit's patellar cartilage were measured
with a single blunt impact (Haut et a1, 1992). The impacts were delivered to the flexed
hind limb of thirty-four anesthetized Flemish Giant rabbits. The patellar cartilage of the
rabbit was traumatized at one of the energy levels: 0.9], 4.2J, or 6.3 J. COntact pressure
distributions were recorded in the patello-femoral joint with Fuji Prescale film (Figure
2.1). The peak contact pressures over the lateral patellar facet statistically exceeded those
over the medial facet by approximately 8 MPa (Figure 2.2). The experimental study
indicated that the existence of a biomodal distribution in the contact pressures had a strong
correlation with cartilage damage. The correlation of fissuring with the magnitude of the

average contact pressure, per se, is shown in Figure 2.3.

Cutting
section

mhzmmvhm

 

Figure 2.1 A Typical distribution of contact pressure recorded on Fuji Film by impact

       

Medial side

Lateral side

 

 

33% 238.5 63.80

Ize (mm)

Contact 5

Figure 2.2 The profile of contact pressure in the cut section

 

 

 

 

 

 

 

 

 

 

28
. I
1; I I
§ 24- I n I. U
v I I I.
20 El "I {3
. [J
2 I I D n n
3 16 I
w n I
3 D u E] I
a ‘2 u U n
9 El
3 8
h d
9
< 4
O ' 1 V f t u fl

 

 

 

 

 

 

 

 

 

 

 

 

0 4 8 12 16 20 24 28 32 36 40

I Sample with fissure
T9511 NO- D Sample without fissure

Figure 2.3 Average pressures data from 34 impact experiments (Ide, 1992)

Another important experimental observation was that fissures in the cartilage layer
were oriented longitudinally on the lateral facet. The fissures were generally located at the
periphery of the lateral contact zone nearest to the center line (Figure 2.4 a). Histological
sections (Haut, 1992 and 1991) showed that the fissures follow the line of chondrocytes
(cells), orienting themselves parallel to the major collagen bundles in the cartilage layer

(Figure 2.4 (b)).

 

(a) (b)

Figure 2.4 Fissure location in the impacted cartilage and its orientation

These experimental data and subsequent observations gave us important
information needed to help understand injury to soft tissue by blunt impact, even though
the results only show a rough relationship of damage to contact pressure. In this
dissertation, the stress fields from Ide's experimental data will be estimated by a patella
FEM model. The damage of cartilage, i.e. fissures, and its correlated stresses obtained by
the FEM model will be further analyzed using based on engineering damage criteria.
Although a number of criteria have been applied to normal engineering materials, none to

date have been applied to biological tissues.

11

2.2 Candidate Failure Criteria

Starting from these experimental data I will provide a reasonable explanation about
the mechanics of damage and develop a failure criterion to give an explanation for this
kind of impact trauma. In order to explain these damage phenomena, it is necessary to
review the existing failure criteria in engineering applications before developing an
effective criterion for knee joint damage.

1. The Tresca criterion, also known as the maximum shear theory, states that
yielding of an isotropic material will begin when the maximum shear stress reaches the
yield limit for the material. This theory is suggested by the observation that failure
surfaces in ductile materials in static tension are at 45° to the axis of loading. That is,
failure surfaces are planes that carry maximum shear stress. The Tresca criterion can be

written as:

Tmax S IS (21)

where Tmax is the maximum shear stress, and 1: S is a constant related to material yield.

AT

Tresca Failure Criterion

/

    

     

     

 

 

Figure 2.5 The Tresca failure criterion

12
This criterion has been applied to engineering materials that fail by yielding, such

as metals and alloys.
2. The Von Mises criterion states that yielding of an isotropic material will begin
when the effective stress reaches the yield limit for the material. The Von Mises criterion

can be written:

 

2

2
y, +1. )Sos

o*- oz+oz+oz-oo —oo —oo +3(t 2+1
— x y z x Z Z 1' xy J (22)

y y
where 6* is the effective stress and as is the yield constant for the material in unaxial

tension. Equation (2.2) can be written in another form in terms of the principal stresses:

 

* 1
G =\/§{(51‘02)2+(02-G3)2+(03—01)2}50's (2.3)

From equation (2.3) it is clear that the effective yield stress is not affected by the level of
hydraulic pressure in the material.

This criterion has been applied to engineering materials which fail by yielding,
such as metals and alloys. The Von Mises criterion gives a more accurate analysis of
failure than does the Tresca criterion for many engineering applications.

3. The maximum principal strain criterion states that tensile fracture surfaces will
form in previously uncracked isotropic materials when the maximum principal strain

reaches a limiting value in tension.

Emax 5 elimit (2.4)
where 8max is the maximum tensile strain, 3mm: is the limiting constant value of material

in tension. This criterion is usually applied to brittle engineering materials in tension.
These often fail by fracture and not by yielding. Glass is one type of material that fails by
this mechanism.

4. Mohr criterion states that failure of an isotropic material, either by fracture or by
the onset of yielding, will occur when the largest Mohr circle (having a diameter

determined by the difference between the maximum principal stress and the minimum

13

principal stress) reaches a failure envelop. Experiments show that the envelop, which is
tangent to all the failure circles and bounds the safe region, is usually curved convex
outward.

This criterion is usually applied to some brittle engineering materials in
compression, which fail by fracture, instead of yielding. Rock, cast metal, soil, brick and
concrete are typical examples. Such materials exhibit a compressive strength that is much
higher than the tensile strength. The media is usually discontinuous with a lot of voids
and flaws, or micro cracks. Cartilage and bone have this kind of characters and typically
function under high levels of compression, especially during blunt impact. Furthermore,
the compressive strength of these tissues is more than two times higher than the tensile
strength (H. Yamada, 1970). Thus, a Mohr criterion is considered in this study. A more
detailed discussion of this failure criterion will be given below.

Mohr proposed a criterion of the form

I‘rl = f (on) (2.5)

Where T and on are the shear stress and the normal stress on the failure plane, and

f (on)is the failure function for the envelop. The failure envelop for a Mohr criterion can

be determined experimentally from a series of triaxial compression tests. This criterion
includes the effect of hydraulic pressure. Mohr theory is usually used for brittle materials,
which are much stronger in compression than in tension. When compression is dominant,
failure occurs along shear planes inclined at an angle of 450+1/2¢ to the direction of
minimum principal stress (Figure 2.6). Mohr circle describing the state of stress at
fracture is plotted for each test. The function f (on) is then defined by the envelop which
is tangent to the set of these Mohr circles. The failure envelop function f ( on), which is

called Mohr criterion, is usually non-linear (Figure 2.6).

14

J

f

i / 7) Mohr‘s Circle
K 2 24>
- + -o
|

 

TA / Failure Envelope f( O'n )

 

 

 

 

n

 

 

_l_

|
Plane of Failure Test 1

 

T t 2
es Test 3

Figure 2.6 The failure criterion of Mohr

A failure criterion due to Coulomb in 1776 states that fracture or damage occurs in
a solid body when shear stresses on some plane in that body are large enough to overcome
the sum of the "cohesive strength", C, and the frictional stress, 110m caused by the normal

stress on that plane on (Figure 2.7). The failure envelop, which is called the Coulomb-

Mohr Criterion, is linear and can be written as:

I = C + #6,. (2.6)

where C is the intrinsic shear strength of the material (cohesion) and on is the normal
stress on the plane, it = tanq) , 4) is the angle of internal friction, and 1: is the shear stress.

A simple way to approximate the envelop of Equation (2.6) is to draw a pair of
straight lines tangent to two Mohr circles (Figure 2.8). One circle represents the ultimate
uniaxial tensile stress, and the other represents ultimate uniaxial compressive stress. The
materials constants C and u can be determined from curve fitting, based on these two

Mohr circles.

in. .

    

 
  

9
9

    
 

         
    

  

    

     

99 Ԥ
9 94
‘9 Q 9
999999994
$§§flflflflfifl J9
999999999 99
.QOQOOQOQ4 .Q.‘
999999999 9999
999999991 9999
999999999 9999
0.....OQ4 .9.
999999999 999
999999994 9999
999999999 9999
999999994 9999
999999999 9999c
999999999 9999
999999999 9994
999999999 9999
999999999 9999
999999991 9999
999999999 9999
999999991 9999
999999999 9999
99999999< 9999
999999999 9999
999999991 9999
999999999 9994
99999999< 9999
999999999 9994
999999999 9999
999999999 9999
999999994 9999
999999999 9999
999999991 9999
flflfiflflfiflfi 3%”
OO‘O..‘.. .0...‘

Figure 2.7 The model of Coulomb-Mohr

With a straight-line envelop, the Coulomb—Mohr theory is also known as the
internal-friction theory because it can be derived by accounting for friction along slip
planes. The Coulomb-Mohr criterion specifies maximum shear stress on any plane. That

is, the resistance to failure of a material is a constant shear strength plus a friction-like
force. HG" (Figure 2.7).

—Intemal Friction Theory of Coulomb
T: C +O'ntan ¢

 

0
V
3'6

 

 

On Om

 

——Ultimate stress by unaxial tension test

Ultimate stress by unaxial compression test

Figure 2.8 The failure criterion of Coulomb

16

Unlike the one-parameter failure model (Tresca criterion or maximum strain
failure criterion), the Coulomb or Mohr failure theories include two or more material
parameters and the effect of hydraulic stress on material failure.

2.3 Physical Justification for Use of a Coulomb-Mohr Failure Criterion

Several theoretical studies have addressed the problem of determining the stress
and strain fields in a cartilage layer in order to explain the remarkable longlasting quality
of this tissue, as well as to identify those mechanical factors which may lead to cartilage
degeneration. However, applicable damage criterion have not been well studied.

There are several reasons why a Coulomb-Mohr failure criterion may be applied to
articular cartilage. First, articular cartilage is a discontinuous material with a lot of micro
flaws, voids, and fibers. When high normal pressures act on these flaws, the frictional
resistance reduces shear stresses and helps resist micro crack growth. The cartilage in the
knee joint, typically withstands high hydraulic pressures. The convex and concave
curvature contact surfaces contain this soft tissue so as to bear high pressures under short
time loading. The fluid phase of cartilage does not have time to move out and therefore
shares a significant percentage of load (Ateshion et. al, 1994).

Second, the damage phenomenon in cartilage is similar to that of brittle materials.
There are a lot of microcracks and fiber debonding before fissures actually form. Surface
images of non damaged and damaged areas scanned by electronic microscopy are shown
in Figures 2.9 (a) and 2.9 (b), respectively. Comparing these two pictures, we can see that
the distribution of fibers is nearly uniform in the non damaged zone and there is a zone of
micro-cracks along the crack tip in the damage zone. Clark (1971) also showed that many
micro cracks occur before a fissure grows. The network of fibers is strong in tension, but
the matrix of soft tissue is very easily damaged by shear deformation.

Third, the loading rate can change a material's behavior from ductile to brittle,

particularly if the material is under hydraulic compression. Brittle materials initially failed

17
by many micro-cracking (Altiero, 1974), and then failure by macro-crack growth occurs

(Figure 2.10). The resistance to shear deformation increases with an increase in level of
hydraulic stress. However, the 'failure surface' in such conditions depends not only on the

deviation (shear) stresses, as in the Von Mises law, but also on the magnitude of the

average compressive stress.

 

Figure 2.9 (a) Scanning a section of cartilage (b) Scanning a section of the damage zone
on the impacted sample from a rabbit

 

Figure 2.10 Fissure growth in the soft tissue along collagen fibers

2.4 The Study Objectives

The exact mechanism of impact induced cartilage fissuring is not known. A
number of predictive parameters for articular cartilage failure have been proposed. The
common factors in these proposals and associated models have not been proved by
experiments, so they have not given a quantitative analysis for establishing a damage
criterion for bio-tissue under impact. The objective of this dissertation is to conduct
experiments and develop a theoretical model to help understand the physics behind the
development of cartilage fissures during blunt impact. This development of a criterion for
the impact-induced micro damage of articular cartilage has involved in several stages.

(1) A finite element model of the knee joint was employed to estimate the
unknown material properties of cartilage under impact by correlation with the
experimental data using an inverse approach.

(2) Data from knee joint impact experiments (Ide, 1992) were then input into the
FEM model to analyze the stress fields in the impacted knee joints. The results of the

stress analyses were compared with Ide's experimental observations to develop a damage

19
criterion. A logistic regression analysis was used to find out which mechanical parameters

best predicted cartilage damage.

(3) Since the regression analysis indicated that a combination of two variables
(shear and mean stress) best predicted damage, a Coulomb-Mohr criterion was employed.
The material properties needed for an effective failure criterion were then determined in
independent experiments. The rabbit tibial plateau are flatter and have a better quality
cartilage than do the patella, so they were used in this experimental tissue.

(4) The failure criterion was then applied to the earlier data of Ide. It is shown in
that this dissertation that this new tissue failure criterion predicts the onset of surface

fissures in the rabbit knee joint well.

Chapter 3 Computational Modeling Assumptions

Computational models of rabbit knee joints can quantitatively generate the stress
and strain fields under impact loading. The stress and strain fields can then help determine
relevant variables around fissure locations. The computational models require the material
constitutive relations, the geometry of the joint, and impact loading conditions. For
simplicity of the computational models, some assumptions were made and they are
explained below.

3.1 Linear Elastic Behavior of Cartilage and Bone

The mechanical behavior of cartilage is very complicated and time dependent,
particularly as related to free fluid flow under loading. Mow and his colleagues developed
a biphasic continuum description to model the behavior of cartilage (Mow, et a1, 1980).
Others have applied a single phase Viscoelastic model to describe the material response
(Parsons, and Black, 1977). In this previous work, the attention has largely been focused
on relatively long time scales. Our experiments are conducted under loading in a short
time. Under this test condition, the fluid in the cartilage may not have time to move.
Therefore, it may be reasonable to assume elastic behavior. We show that the available
data support the use of an elastic behavior at rapid loading rates, and the effect of time
dependence can be ignored in the following. The two typical solutions of cartilage
constitutive models, Viscoelastic and biphasic, are now compared to those of the elastic

model.

Viscoelastic Model

Simon and his colleagues (1984) presented cartilage as a Viscoelastic material.

The normalized relaxation function G*(t) is given in the following form.
20

21

* _ —0.92:
G (t)—0.23+0.77e (3.1)

The function G*(t) in equation (3.1) allows one to determine the normalized

stress history, 0*(1) , for any prescribed strain history by a hereditary integral formulation

(Fung, 1965)
G (t): I G (t _ T)_d€(1)d(3.2)
where 8(t) is the prescribed time-dependent strain. For a constant strain rate:
80) =21 (3.3)

a
where Ta is the time of load application. Substituting (3.1) and (3.3) into (3.2) and

integrating yields the normalized stress
0*(t) = 8—0[0.23t + 084(1— $193)] (3.4)
T0

Taking a Taylor expansion and ignoring higher order terms yields an approximate

expression for the normalized stress:

o'm = [1.0 — 0.3542t](§-T9—’-)

a

(3.5)

It can be seen from (3.5) that the Viscoelastic solution consists of two terms. The
first term is the elastic solution and the second term is the viscous effects. It indicates that
when the loading period of application is small the solution is approximately an elastic
solutions and when the loading period of application is long the Viscoelastic solution
dominates.

Biphasic Model
The stress expression history for a cylindrical biphasic material in unconfined

compression was derived by Armstrong, (1984). For t—>0, the normalized relaxation

22

function, G*(t) can be written as

at: 4112 HA,“
G I =1-——S- —— 0 .6
() 3HA (/ m2 + (t) (3 )

If the higher order terms, O(t), are neglected, G*(t) becomes:

. 4112 HAkt
G t=l——S— —
( ) 3H,, naz (3'7)

where k is the permeability of the material, a is the radius of the specimen, 145 is the shear

modulus of the solid matrix and H A is the aggregate modulus. HA and Us can be

expressed as follows:

 

_ (l—vs)
HA _ S(1+vs)(l—2vs) (3.7 a)
ES
”3 __—2(l+vs) (3.7 b)

where ES and vs is Young's modulus and Poisson's ratio of the solid matrix, respectively.
Substituting the following values into equations (3.7 a and b): ES :07 MPa and vs=0.45
and 0.0, respectively, H A and us can be calculated and are listed in Table 3.1.

Table 3.1 Material properties of cartilage in biphasic model (Armstrong, et al, 1984)

 

vs=0.45 vs=0.0

HA (MPa) 2.66 0.70
it, (MPa) 0.24 0.35

 

 

 

 

 

 

 

Substituting these values of HA, us, k=0.2x10'l4 m4/Nsm, and a=3.17 mm into

equation (3.7) yields the expressions for G*(t) .

23

The function G*(t) in equation (3.7) allows one to determine the normalized
stress history, 0*(1) , for any prescribed strain history by a hereditary integral formulation
using equation (3.2). Substituting G*(t) into equation (3.2) and integrating, the

normalized stress expressions of the biphasic model can be obtained:

1
— 801

o*(z)=[1.0—0.00104(t)2](T for v,=0.45 (3.8)
1 8t
6‘0):[1.0-0.00300>21(—TQ—) for 950.0 (3.9)

a

 

*
f 6 Ta
801

The values 0 determined using equations (3.8), (3.9) and (3.5) are given

in Table 3.2 for various times.

Table 3.2 Comparison of OT“ /80t of the elastic solution to the Viscoelastic and biphasic

 

 

 

 

 

 

solutions
Equation # t=5 (ms) t=50 (ms) t=200 (ms)
Armstrong vs=0'45 (3.8) 0.9999 0.9997 0.9995
Armstrong vs=0.0 (3.9) 0.9998 0.9993 0.9986
Simon (3.5) 0.9980 0.9820 0.9290

 

 

 

 

 

It can be seen from Table 3.2 that both models deviate from the elastic model by a
very small amount within a loading time of 50 ms. The biphasic model stress varies from
the elastic by 0.07 % at 50 ms. The Viscoelastic model variation is 1.8 % at 50 ms.

Therefore, the elastic constitutive law can be used for our model, since our impact loading

time is less than 20 ms.

24

3.2 J ustification for the Assumption of a Quasi-Static Load

A quasi—static state solution implies the use of a static solution to approximate a
dynamic model. Since our experiments involve the free dropping of indenters on the knee
joint, dynamic modeling would be appropriate. However, in this section, justification for
a quasi-static approximation is examined. It is proposed that tensile stress waves occur
only if the impulse load rate is larger than the speed of tensile wave propagation. In our
impact model, stress wave propagation can be ignored. The simple proof of this impact

model is presented in the following section.

Quasi-static State Solution
The total impact load, including indenter weight and dynamic load by acceleration,
is input into our finite element model of the knee joint. It is equivalent to applying a
d'Alembert force to a static-analysis model if the stress wave propagation is ignored. The

equilibrium statement, written as the virtual work principle, is

j/ozoDdV = £r’.5vd8+£fT.8vdv (3.10)

where 0 is the stress tensor, SD is the virtual strain rate tensor, IT is a surface force

vector, f T is a body force vector, 8v is the virtual velocity of deformation, V is the body

volume and S is the body surface.

The body force at a point, f T, can be written as the resultant of the externally prescribed

body force, F and a d'Alembert force as follows.

f = F - P“ (3.11)
where p is the current density of the material at this point and ii is the acceleration at the

point. Inserting the body force term into the virtual work, equation (3.10) becomes:

[(5:8de = jtT.8vdS +1 (F— pa)T.5vdV (3.12)
V S V

25
If wave propagation is neglected, equation (3.12) represents the quasi-static state solution.
The Effect of Stress Wave Propagation

Sudden loading to the knee joint or the cartilage surface of tibial plateau by impact
is essentially a dynamic problem. When the load is first applied, its action is not
transmitted instantaneously to all parts of the body. Waves of stress and deformation
radiate from the loaded region with finite velocities of propagation. Let us consider a
simple one-dimensional discontinuous-medium model, as shown in Figure 3.1.

In a simple one-dimensional discontinuous-medium model, the velocities of
longitudinal wave propagation in medium 1 and medium 2, i.e., Ci i=1, 2, are given by

Clove (1975):

Pi i=1, 2 (3.13)

where E.- and Pi are Young's modulus and the density of the ith medium, respectively.
The reflected force, N b, and the refracted force, NC, can be expressed as fractions

of the input force, Na, at the interface of medium 1 and medium 2 (Clove ( 1975)):

N =N ——— 3.14
b a “+1 ( )
201
N =N — 3.15
c a 01+ 1 ( )
where 0t is the following material constant :
9252

0‘ = —P\/1—?1 (3.16)

26
p

Compressive incident wave Na

\\\\ p\\ Compressive reflected force N b
,C\

 

\

\\\ \\\Ԥ\\\\\\\

/

 

\\\\\\\\ II I IIOIIIIIII
I!!! I!
\\\\\\\\
III/(III
\\\\\\\\‘
I III/II
\\\ \\\

Compressive refracted force wave N 0

I’ll

\H’\\\\\\\\\\

 

 

 

////////////

Figure 3.1 Forces at the interface in a one dimensional discontinuous medium
In this study, the Young's modulus and density of bone were obtained from S. C.
Cowin (1987), and the properties of cartilage are based on estimations made in Chapter 4

of this dissertation. These properties are given in Table 3.3.

Table 3.3 Material properties of rabbit cartilage and bone '

 

 

 

Material E1 (N/mm pi (kg/mA3)
Cartilage 2 x 106 IO.8x 106
Bone 2 x 109 1.8 x 106

 

 

 

 

 

Now let us apply this one-dimensional wave propagation model to two
possibilities: (1) impact load transmission at the interface from cartilage to bone (similar
to our indenter impact test of tibial plateau), and (2) impact load transmission at the
interface from bone to cartilage (similar to our impact test of knee joint).

Let medium 1 be cartilage and medium 2 be bone (Figure 3.2 (a)). Using the

material properties given in Table 3.3 yields a=47.34 by Equation (3.16). The first

27
reflected force, N},, can be obtained by Equation (3. 14) and is equal to 0.954 N21, and the
first refracted force, N2, can be obtained by Equation (3.15) and is equal to 1.954 NA.
Both N; and N; have the same sign as the input load, N}, (i.e., they are compression
forces).

Now let us calculate the first reflected force transmitted back to the cartilage
surface. In this case, the input load, N3, is equal to N;. Let medium 1 be cartilage and
medium 2 be air (Figure 3.2 (b)). The Young's modulus of air is 0, so the second
reflected force at the cartilage surface, NZ, can be obtained by Equation (3.14) and is

equal to - N11,. The refracted force, N3, can be obtained by Equation (3. 15) and is equal to

0.

 

 

 

 

 

P Impact force
1 Reflection force
Na
Ai" N1-0954N1
Cartilage ‘ Nit-N;
' Cartilage-bone
interface N2 = N 1
b b
l- 54 Nl
NC— 1.9 a
Refraction force
(a) (b)

Figure 3.2 (a) Stress wave propagation from cartilage to bone. (b) Stress wave
propagation from cartilage to cartilage surface

We can repeat the calculation to determine subsequent wave magnitudes of the

28
reflected force at the cartilage to bone interface. These magnitudes are shown in Figure

3.3.

 

 

 

 

 

 

Time (ms)
Figure 3.3 Reflected force at the cartilage-bone interface vs. time

The wave propagation velocity in the cartilage can be calculatedrusing equation
(3.13) and is equal to 1.58 mm/ms. If the thickness of cartilage is 0.5 mm, the time of
wave propagation through the cartilage to the interface, tp, can be simply calculated by
dividing the bone thickness by wave velocity, c 1. Thus, tp is equal to 0.316 ms/per half
wave. Thus N}, is equal to the value of the impact load at t=0.316 ms (Figure 3.4).

The results of the curve of experimental impact load vs. time and reflected force at
the cartilage-bone interface are shown in Figure 3.4. The magnitude of stress wave is 6.6
% of the peak impact load. We have neglected damping of the material in this calculation

and thus our estimates are liberal.

29

22-

x Reflected Force

9 Experimental Load (N)
from indenter impact test

 
 
  

Force (N)

 

 

. 4 . . . .
0.0 05¢ 1.0 1.5 2.0 2.5
t =O'316 Tensile Time (ms)

Figure 3.4 The effect of the reflected force on experimental impact load (cartilage to
bone) '

Let us now examine the second case. Let mediums 1 be bone and medium 2 be

cartilage (Figure 3.5). Using the material properties given in Table 3.3, 01:0.024 by
Equation (3. 16). The refracted force, NC]. , can be obtained by Equation (3.15) and is equal

to 0.046 Ni, and the reflected force, N11,, can be obtained by Equation (3.14) and is equal
to -0.954 N; . Note that the negative sign of the reflected force means the opposite of the
sign of the input force. So, in this case, there exists a tensile reflected stress wave at the

bone-cartilage interface. These forces are shown in Figure 3.5 .

Now, let us calculate the first reflected force transmitted back to the bone surface.

In this case, the input load, N3, is equal to N11,. Let medium 1 be bone and medium 2 be

air (Figure 3.5 (b)). The Young's modulus of air is 0, so the second reflected force at the

30
bone surface, NE, can be obtained by Equation (3.14) and is equal to - N11. The refracted

force, N}, can be obtained by Equation (3.15) and is equal to 0.

     
  
  

 

Impact force
I
N Reflection force

a
1 1
Nb: 'O.954 Na

1 1
N c =0.046N a
Refraction force

Bone -cartilage
interface

   
   

Cartilage

Figure 3.5 (a) Stress wave propagation from bone to cartilage. (b) Stress wave
propagation from bone to bone surface.

We can repeat the calculation to determine subsequent wave magnitudes of the
reflected force at the bone to cartilage interface. These magnitudes are shown in Figure

3.6.

31

 

 

 

IN;
>
>
>-
>
>
>

E

 

 

 

 

 

 

 

 

 

 

Time (ms)

 

 

 

0
N4
&
O)
on

10

Figure 3.6 Reflected force at the cartilage-bone interface vs. time

The wave propagation velocity in the bone can be calculated using equation (3.13)
and is equal to 33.33 mm/ms. If the thickness of bone is 10.0 mm, the time of wave
propagation through the bone to the interface, tp, can be simply calculated by dividing the
bone thickness by wave velocity, c 1. Thus, tp is equal to 0.3 ms/per half wave. Thus N},
is equal to the value of the impact load at t=0.3 ms ( Figure 3.7).

These curves of experimental impact load vs. time and reflected force at the bone-
cartilage interface are shown in Figure 3.7. The magnitude of stress wave is 2 % of the
peak impact load. Again, we have neglected damping of the material in this calculation,

and our estimates are thus liberal.

32

-600 __
I‘ ‘
-5001. ,’ g
I t
I \
I \
I \
-4001 ,’ ‘,
I \
E I I \‘
V I \
O '300 -I- ’ \
e I, ‘\
.2 ’1 Experimental Load‘

 

 

100 .. 1 {61181011

' Time (ms)
t=0.3

Figure 3.7 The effect of the reflected force on experimental impact load (bone to

cartilage)

3.3 Plane Strain Modeling of the Rabbit Knee Joint and the Associated Boundary
Conditions

The actual geometry of the animal knee joint and the loading locations are our
focus in this study. For simplicity, a two-dimensional analysis was performed with plane
strain conditions assumed in order to represent a confinement of the cross section.

The two-dimension plane strain model may be approximately taken place of the 3
dimension geometry, because of the character of the geometry character and the character

of the fissures in the impacted cartilage (Figure 3.8 (b)). The size in the z direction of the

33
patella is larger than in the x and y directions, and the thickness of the patella cartilage is

much smaller than the patella' 8 length in the z direction (Figure 3.11 (a)). The fissures are
generally along with the z direction (Figure 3.8 (b)) so it seems that the stress field is
independent of the z direction. Thus the actual geometry of the animal knee joint may be
modeled by plane strain conditions. Figure 3.8 (a) shows our strain plane model

assumption.

Z Cutting section

    

Cutting section

Patella
Cartilage

 

(a) (b)

Figure 3.8 (a) Plane strain model of the patella. (b) Impact induced surface fissuring of
cartilage

The boundary loads were obtained experimentally by recording pressures from
Fuji film along a 0.5 mm thick zone at the cross section of interest on the anterior surface
of the patella or PF contact pressures that were generated during blunt impact (Figure 3.9).
The input loading profile on the anterior surface of the patella for the 2 Dimension model

is plotted in Figure 3.10, and is used as the input loading condition of the FEM model.

34
The bottom of the femur of the rabbit knee joint is restrained in the x and y directions. A

2-D plane strain FEM model, the boundary conditions of the input load and the restraints

are shown in Figure 3.11.

Cutting section of 2-D model

 

Pin hole

    

 

231;
‘8‘

Impact pressure on the anterior surface of patella '

 

PF contact pressure

Figure 3.9 The contact pressure recorded with Fuji film on the anterior surface of the
patella and the PF contact surface

35
Pin hcle

Contact Pressure (MPa)

' Medial side Lateral side
0 ' I ' l f 1 ‘7 I - I .

0 1 2 3 4 5 3
Contact size (mm)

 

 

 

Figure 3.10 The profile of impact pressure on the anterior surface at the cut section

Pin hole ———>

 

(a) __.. (b)

Figure 3.11 (a) The cross-section of the rabbit knee joint. (b) The 2-D FEM model and
boundary conditions

Chapter 4 Estimation of the Unknown Material Properties of Cartilage

Using an Inverse Approach ‘

A primary aspect of this effort has been to develop a model of the joint to help
explain injury mechanisms and to use in vivo results from the animal studies to predict in
vivo human experiments. A problem that must be dealt with in any analytical model is
material properties. Cartilage itself is a complicated biphasic material (Mow, et al, 1980),
so the mechanical properties are difficult to define, especially for the analysis of blunt
impact response. We justified (in Section 3.1) the use of an elastic model for situations in
which impact load duration does not exceed 50 ms. This is indeed the case in a typical
blunt impact to the knee (Haut, 1985). Eberhardt et a1 (Eberhardt, A. W., 1991), for
example, uses mechanical properties obtained from indentation experiments and confined
compression tests on cartilage performed by other investigators. However these data are
documented for time duration longer than one second, which is much greater than the
typical duration of a blunt impact.

4.1 Experimental Methods and a FEM Model of the Rabbit Knee Joint

Experimental data were collected from two mature (6-8 months of age) Flemish Giant
rabbits. Impact experiments were performed on euthanized specimens within 1 hour of
death. All experimental protocols were approved by the MSU All-University Committee
on Animal Use and Care and followed NIH guidelines. Experiments were performed on
the right (Experiment #1) and left (Experiment #2) limbs of one animal. The animal was
held in position by insertion of a 3.2 mm diameter pin into the medullary canal via the
greater trochanter (Haut, 1983). A bed of PMMA was placed at the proximal end of the

femur to help distribute axial loads generated during knee impact (Figure 4.1). A 0.43 kg

36

37
mass was dropped onto the flexed limb from 0.2 meters.

STRAINT BELTS
INTRAMEDULLARY PIN /
(f—xl/K SPECIMEN
W-

 

a.
SUPPORT
FIXTURE

 

Figure 4.1 Schematic of the restraint system used for experiments #1 and #2. Note that
the femur was constrained with an intramedullary pin inserted via the greater
trochanter.

For a second animal (Experiment #3) the protocol followed that which has been
used in previous in vivo studies (Haut, 1992). The animal was positioned'supine with the
right limb hyperflexed to align the femur vertically (Figure 4.2). The flexed limb was held
in place against a rigid seat with spring-loaded clamps. The pelvis was also restrained
with a strap to prevent rotation during blunt impact on the knee. A one-inch diameter, flat
aluminum impact interface was attached to a load transducer and dropped on the patella
from 0.46 meters. The total impact mass was 1.33 kg. This test protocol was known to

produce surface fissures on patellar cartilage in a previous study (Haut, 1992).

 

Figure 4.2 Schematic of the restraint system used in experiment #3. The hind limb was
constrained with spring-loaded clamps and a strap across the pelvis

Impact force-time data were collected on a personal computer equipped with an

analog-digital board sampling at 10,000 per second, A typical history of impact loading

vs. time is shown in Figure 4.3.

120

8 8 g

Impact load (Lb)

8

20

0.0 0.0

 

 

 

 

 

\/\ A

I l l M

10.0 20.0 30.0 40.0
Time (ms)

Figure 4.3 The curve of impact load vs. time in a typical experiment on rabbit knee joint

39

Two 1.0 mm diameter holes were drilled through the patella prior to blunt impact.
Pressure sensitive film was inserted in the patello-femoral joint to record contact pressure
distribution generated during blunt impact on the knee (Haut, 1989, Huberti, 1982). The
film was encased in polyethylene and inserted under the patella and folded over the
anterior surface of the patella. A hemostat was used to hold the film in place during
impact, and no movement of the film was evident during impact. After each experiment,
needle probes were inserted through each hole in the patella piercing the film and
therefore locating the film with respect to the patella. The pressure sensitive film
(Prescale, medium range, mono-layer, Fuji ) was calibrated with a servo-hydraulic testing
machine using a 100 ms haversine load pulse (Haut, 1989). The film was digitized with a
video scanner and a personal computer-based analysis program (Image 1.44). The scanner
sensitivity was 11.8 pixels per mm. The pressure data were documented across the entire
contact area and also scanned medially to laterally across the patella at a cross section
approximately midway between the two pin holes. This cross section was selected
because it was the area which typically encompassed the largest contact pressures. This 2-
D pressureprofile served as the experimental data to compare with the theoretical results.

After the impact experiments, the hind limbs were removed intact by sectioning at
the lumbar spine. The joint flexion angle was preserved while the limbs were immersed in
a 10% formalin solution for approximately 3 weeks. After incubation, a single transverse
cut was made with a diamond bladed saw through the patello-femoral joint at the location
chosen for pressure analysis. Photographs of the joint geometry were enlarged and served
as input to a finite element model (Figure 4.4 (a)). The surfaces of the cartilage on the
patella were wiped with India ink and examined under a microscope at 25X for surface

fissures.

40

 

Figure 4.4 (a) A section cut through the patello-femoral joint. (b) Typical FE mesh for 2-
D analysis of the blunt impact response of the rabbit PF joint

A 2—dimensional finite element model of the joint was constructed using 1597
plane strain elements to model experiment 1, 1406 plane strain elements to model
experiment 2 and 2577 plane strain elements to model experiment 3. The computer code
employed was I-DEAS, Structural Dynamics Research Corporation, Milford, OH. .
Assuming a frictionless contact surface, fifty to sixty gap elements 0.1 mm thick were
used to simulate the joint space between the patella and the femur. The work followed
that of Eberhardt et al (1990) in assuming that a linear elastic model was suitable for the
analysis. The elastic modulus and Poisson's ratio of bone were assumed to be 2 GPa and
0.3, respectively (Cowin, S, 1987). The study involved correlation with the elastic
modulus of cartilage varying from 1-20 MPa, and parameter analyses in which Poisson's
ratio of cartilage was varied from 0.01 to 0.5. It was assumed that the bone and cartilage
properties in the patella and femur were identical. This was based on an earlier study by
Jurvelin, et a1 (1990) which found the "500 ms response" (shear modulus) of the canine
femoral and patellar cartilage to be statistically identical. The boundary loads were

obtained experimentally by recording pressures on a 0.5 mm thick zone at the cross

41
section of interest that were generated during blunt impact in the anterior surface of the

patella (See Chapter 3.4). A mathematical routine was developed to compute the sum of
the squared differences (residuals) between the computed and measured contact pressures,
based on data at 50 locations in a single slice of the patello-femoral surface. This
parameter helped quantify the material properties of the cartilage.

4.2 Estimation of Material Properties

The degree of correlation between experimental and theoretical contact pressures
in the PF joint depended on the material properties chosen for the cartilage (Figure 4.5 a).
As the value of Poisson's ratio approached 0.5 for the cartilage, the residuals decreased
(Figure 4.5 b). We also noted that the residuals were a minimum for an elastic modulus of
approximately 2 MPa for the cartilage at Poisson's ratio 0.49 (Figure 4.5 a). The residuals
were significantly higher for all values of elastic modulus when Poisson's ratio of the
cartilage was chosen to be less than approximately 0.49, for example (Figure 4.5b). The
degree of correlation did not change as the modulus of bone was altered from 1 GPa to 20
GPa.

The calculated results of Experiment #3, by the method of minimization residuals,
are shown in Figure 4.5 and 4.6. The same processes were used for Experiment #1 and
#2. The minimum residual of PF contact pressure between the FEM model and the
experiment occurred at a Young's modulus equal to 2 MPa and Poisson's ratio equal to

0.49. The details of these analysis are presented in a recent paper (Li, X., et a], 1994).

 

42

 

 

 

 

 

 

 

 

’5 .
E
a it
:1
8
Q
l-
D-
H
0
£3
:1
e
O
in
a.
7
.5 (a)
:1
2% 500
or ..‘T_'.'_.:..‘_.irrr..—..:....-... .....................................................
.8 400 "‘.~.~
3., . . ‘1“- .
g ...... .‘..>._....; ..................
an 300 s s 2 “ex.
3 ¥ 5 2 s 3 ‘--
g 200 ... ........ (K...‘:....
I I I I x
m 1m ...............................................................................................
O I I I I
0 0.1 0.2 0.3 0.4 0.5
Poisson's Ratio
('3)

Figure 4.5 (a) A plot showing the experimental contact pressure distribution and the
corresponding theoretical distributions for various values of Poisson's ratio at
Young's modulus 2 MPa. Note how well the results at 0.49 compare to the
results from experiments #3, (b) The residuals decreased continuously as
Poisson's ratio approached 0.49

 

 

 

 

Contact Pressure on the Cartilage (MPa)

 

(a)

 

Sum of Squared Residauls

 

 

 

 

Young's Modulus
(b)

Figure 4.6 (a) The effect of variations in the elastic modulus of cartilage on the residuals
computed between the experimental and the theoretical contact pressures. A
minimum was obtained for an elastic modulus near 2 MPa for the cartilage at
Poisson's ratio of 0.49, (b) The residuals reached minimum at a Young's
modulus of 1 to 2 MPa at a Poisson's ratio of 0.49

The primary objective of this study was to develop an analytical model of the

rabbit's patello-femoral joint that could be used to study the stresses and strains generated

44
during a blunt impact. The differences between the model generated contact pressures and

experiment were minimized when Poisson's ratio was 0.49 for the layer of cartilage. This
result was consistent with the biphasic model of Mow (1980), and agreed with Eberhardt
et a1 (1990) and Askew and Mow (1978) who selected values of 0.475 and 0.45,
respectively, in their elastic models. In our study the difference between theory and
experiment was also minimized when an elastic tensile modulus of 2 MPa was chosen for
the layer of cartilage. In the joint model of Askew and Mow (1980), a value of 12 MPa
was chosen for the analysis. Eberhardt et al (1990) chose a value of 1.5 MPa for the
elastic shear modulus of the cartilage (which corresponds to a tensile modulus of
approximately 4.4 MPa if one assumes isotropy of the layer). Our laboratory is also
studying the mechanical properties of patellar cartilage using an indentation relaxation
experiment (Haut, 1992). The analysis follows that presented by Hayes et a1 (1972) and is
based on the elastic solution of a layer bonded to a rigid half-space. The patellar cartilage
is indented 0.1 m using a rigid probe. An instant shear modulus of the layer is evaluated
from the analysis approximately 50 ms after its deformation. The experiment includes a
period of relaxation in which the rate of load relaxation and the equilibrium responses are
documented (Haut, 1992). The instant shear modulus of the rabbit's patellar cartilage
determined by this technique was approximately 0.46 +/- 0.17 MPa. The optimum value
of tensile modulus from the current study (i.e. 2 MPa) corresponds to an elastic shear
modulus of approximately 0.67 MPa. We see, therefore, that there is good agreement
between the elastic shear modulus determined via the whole joint model and the results

obtained in the short-time indentation experiment using an elastic analysis.

Chapter 5 Application of the FEM Model and Probability Analysis of
Damage Prediction

In order to study the mechanisms of cartilage damage in Ide's experiments, I have
used the previously discussed FEM model and examined the distribution of stress and
strain in an attempt to correlate these parameters with injury. This correlation has used a
logistic regression analysis correlating various mechanical parameters with the incidence
of the surface fissures. In Ide's experiments, contact loads on the anterior surface of the
patella were not recorded. Therefore, the FEM model used to analyze Ide's experiments
had to be changed. In order to study the distribution of stress and strain in the patella, I
used the measured contact pressures in the patello-femoral joint as the input for loading
boundary conditions (Figure 5.1). The analysis was performed using the geometry and
assumptions discussed in Section 3.4.

5.1 FEM Analysis of the Rabbit Knee Joint Impact Tests
FEM Model and Boundary Conditions

The geometry for the 2-dimensional finite element model was copied from an
actual section of a typical patella. This FEM model was constructed using 800 plane
strain elements (Figure 5.1). The Young's modulus and Poisson's ratio for bone were
assumed to be 2 GPa and 0.3, respectively (Cowin, 1987 ). The Young's modulus and
Poisson's ratio for cartilage were assumed to be 2.0 MPa and 0.49, respectively. The

values were obtained by the inverse method discussed in the Chapter 4.

45

46

// /////
5,1)--. ...:

   
   

5.1 Typical FE mesh for the 2-D analysis of the blunt impact response of the rabbit
patella for Ide's experiments (1992) —

The measured contact pressures shown in Figure 5.2 (Ide, 1992) were used as the
input boundary conditions at the cartilage surface (Figure 5.1). The contact pressure
distribution was documented across the lateral and medial facets of the patella by taking a
0.5 mm thick slice of the distribution and generating a 2-dimensional pressure profile
(Figure 5.3). The anterior aspect of the patella was rigidly fixed (Figure 5.1). From Ide's
experimental data, thirty-four different input loads were used in the FEM model to obtain

the related stress fields.

47

 

Medial Lateral

Figure 5.2 Contact pressures recorded by Fuji film during Ide's experiments

Contact Pressure (MPa)

Medial side Lateral side

 

 

 

0 ' l ' l ' l ' I ' l T I
o 1 2 3 4 5 6
Contact size (mm)

Figure 5.3 A 2-D profile of these PF contact pressures.
Analytical Results

Ide's experiments described in Section 2.1 indicated that the incidence of surface

fissures correlated well with the occurrence of a bimodal distribution of contact pressures

48
on the patellar facets. The contact pressures distributions were input into the patella FEM

model which employed the program IDEAS. The resultant distributions of stress and
strain varied with these contact pressures.

The results showed that largest values of stress and strain occurred near the center
of the patellar cartilage on the lateral facet, when the input loading was a biomodal
distribution. The area of large stresses was associated with the area where Ide observed
surface fissures after blunt impact (Figure 5.4). Typical contour plots of the stress and
strain in the cartilage layer are shown in Figures 5.5 (a), (b), (c) and (d). The maximum

values of each parameter along the surface of the lateral facet are also given in Table 5.1.

    

Medial : Lateral

Fissure

Figure 5.4 A typical location of surface fissuring measured often a severe blunt insult.

49

3.0.4.0.

 
 

(a)

 

(C)

 

(b)

(d)

 

.OOOIOC
...I’OI
...III.

1
...lill I}...

1 :1

-.OIIII .z‘"
...

....Yy.
o... a. X

Figure 5.5 Typical contour plots for the stress and strain distributions in cartilage, (a) the

minimum principal stress distribution,

(b) the maximum shear stress

distribution, (c) the shear stress distribution, txy, and (d) the maximum

tensile strain distribution

50
The results show that a few maximum values for maximum shear stress, tensile

strain and mean stress appear near the damage zone. In order to correlate each
mechanical parameters with the cartilage damage, the maximum values of each variable
along the lateral patellar surface are listed in Table 5.1, and plotted in Figure 5.6 (a), (b),
(c ), and ((1). These data were analyzed using a logistic regression statistical model to
determine the best predicted articular cartilage damage.

The data shown in Figure 5.6 indicate that there is a correlation between the
magnitudes of contact pressure, maximum shear stress, maximum strain, peak contact
pressure and the occurrence of surface fissures. In order to quantify the predictive power

of each parameter, a logistic regression analysis was conducted in the next section.

51

Table 5.1 Analytical results from the FEM model based on Ide's experimental data

 

Fissure Strain % Mean Max. C0.05 C01 C02 C0..3 C04 C05
stress Shear
stress

(MPa) (MPar (MPa) (MPa) (MPa) (MPa) (MPa) (MPa)
0.115 30.14 3.69 5.19 6.7 9.71 12.7 15.74 18.76

0.1 15 17.25 1.9 2.76 3.62 5.35 7.07 8.8 10.52
0.13 23.65 2.25 3.43 4.61 6.98 9.34 1 1.71 14.07
0.179 21.79 2.78 3.86 4.95 7.13 9.31 11.49 13.67
0.271 34.47 3.62 5.34 7.07 10.5 14 17.41 20.86
0.261 31.78 4.14 5.73 7.32 10.5 13.7 16.85 20.03
0.232 23.87 3.6 4.79 5.99 8.37 10.8 13.15 15.54
0.213 30.19 3.45 4.96 6.47 9.49 12.5 15.53 18.55
0.172 19.72 2.7 3.69 4.68 6.65 8.62 10.59 12.57
0.234 25.2 3.22 4.48 5.74 8.26 10.8 13.3 15.82
0.22 27.15 3.56 4.91 6.27 8.99 1 1.7 14.42 17.13
0.24 30.85 3.82 5.36 6.9 9.99 13.1 16.16 19.24
0.266 29.05 3.53 4.98 6.43 9.34 12.2 15.15 18.05
0.234 20.87 3.55 4.59 5.63 7.72 9.81 1 1.89 13.98
0.054 8.59 0.9 1.33 1.76 2.62 3.48 4.34 5.2
0.21 21.76 3.26 4.35 5.44 7.61 9.79 1 1.97 14.14
0.257 29.79 4.05 5.54 7.03 10 13 15.96 1 8.94
0.15 18.57 2.54 3.47 4.4 6.26 8.11 9.97 1 1.83
0.208 22.31 3.25 4.36 5.48 7.71 9.94 12.17 14.4
0.205 23.78 3.21 4.4 5.58 7.96 10.3 12.72 ' 15.1
0.1 12 17.21 1.86 2.72 3.58 5 .3 7.02 8.74 10.46
0.217 27.24 3.45 4.82 6.18 8.9 1 1.6 14.35 17.07
0.17 28.93 2.83 4.27 5.72 8.61 l 1.5 14.4 17.29
0.059 9.71 0.99 1.47 1 .96 2.93 3.9 4.87 5.84
0.136 15.3 2.11 2.88 3.64 5.17 6.7 8.23 9.76
0.196 30.65 3.22 4.75 6.29 9.35 12.4 15.48 18.55
0.292 33.81 4.6 6.29 7.98 11.4 14.8 18.13 21.51
0.279 30.99 4.38 5.93 7.48 10.6 13.7 16.77 19.87
0.262 26.34 4.04 5.36 6.68 9.31 12 14.58 17.21
0.361 32.33 5.51 7.12 8.74 12 15.2 18.44 21.67
0.282 25.13 4.29 5.55 6.8 9.32 11.8 14.34 16.86
0.177 21.02 2.8 3.85 4.9 7 9.1 11.21 13.31
0.172 22.21 2.71 3.82 4.93 7.15 9.37 11.6 13.82
0.226 21.64 3.476 4.56 5.64 7.8 9.97 12.13 14.3

 

*
p—i

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

-NN—v—t—t—-NN~HNNNNNNNNN-~NN~——N—-N

 

 

 

 

 

 

 

 

 

 

 

 

* H H ' II I!

. 1 means the spec1men fissured and 2 means no surface fissures were recorded. In
Table 5.1, c0.05, c0.1, 00.2, c0.3, c0.4 or c0.5 is a combination stress of shear stress with
5%, 10%, 20%, 30%, 40% or 50% mean stress.

(MPa)

Max. shear

%

Max. Strain

52

 

 

3' Oi

Mean Stress (MPa)

 

 

040

035

030

025

020

015

0J0

005

' I ’ l ' l ' 1 ’ l ' 1
5 1o 15 20 25 30
Test No

(a)

 

0 5

(C)

V f

35

1015 20 25 30 35
Test No.

Peak contact pressure (MPa)

30

 

25"

 

 

 

'1 '1 '1 '1 '1 x1 ‘1
5 10 15 20 25 30-35

Test No.

(b)

 

Q5_W

 

_35__.,,‘
-4o~~

oi. ..
, 1. .

: ‘ '
,Hrwflgmhfimqg

 

 

 

 

45

0

1 '1 '1 '1 '1 '1 '1
5 10 15 20 25 30 35

1

Test No

(d)

Figure 5.6 A correlation between the maximum values of each indicated parameter and
the occurrence of surface fissures on the cartilage. (a) maximum shear stress,
(b) mean stress, (c) maximum strain, and (d) peak contact pressure.

53
5.2 Probability Analysis

A logistic regression program (Crunch 4) was used to perform the analysis. The
data in Table 5.1 was used as the data base to find out the best predictor of surface
damage. The results of this analysis are presented in Table 5.2 and plotted in Figure 5.7.
The occurrence of a surface fissure on the lateral patella facet in Ide's experiments
correlated best with the magnitude of contact pressure (82.1%). Maximum shear and
maximum strain were not as good of predictors; being 76.5%. The analysis also allows an
analysis of combining various effects. Keeping in mind the discussion given in Section
2.3 for the use of a Coulomb-Mohr type of criterion, the values of this combination of
shear stress and the part of mean stress for each experiment are also shown in Table 5.1.
The logistic regression analysis indicated that the best predictor of surface fissuring was
85.3% , when c was equal to 0.1. The predictability of this combination stress exceeded
that simply using contact pressure, per se. (Figure 5.7 and Table 5.2)

These general results indicate that a Coulomb-Mohr type of the failure criterion is
reasonable for predicting cartilage damage. In the following sections, an experimental

approach will be presented to independently determine the value of c.

Table 5.2 Logistic regression results

 

 

 

 

 

 

 

 

 

Variable S.E p Value Optimal point Dverall Probability
Max. Strain 7.49 0.0149 0.445 76:0
Mean Stress 0.114 0.0026 0.587 82.4
Max. Shear 0.76 0.0085 0.356 76.5
C 0.05 0.67 0.0063 0.524 79.4
C 0.1 0.55 0.0051 0.559 85.3
C 0.2 0.39 0.0039 0.51 82.4
C 0.4 0.23 0.0034 0.507 82.4
verage Pressure (Ide) 0.11 0.0027 0.587 82.4

 

 

 

 

 

 

 

54
Note, in Table 5.2, SE is the estimated standard error; p value is associated with the chi-

square; the optimal point is the cutpoint to obtain the derived dichotomous variable, and

the last column given the overall probability.

 

Overall Probability %

 

Max. Strain Ave. Press. Mean Stress C04 C02 C01 C005 Max. Shear
Variable

Figure 5.7 The comparison of the prediction power of each variable given in Table 5.2

Chapter 6 Determination and Application of a Coulomb-Mohr Failure Criterion

The probability analysis given in Chapter 5 was conducted for those factors that
may produce damage of cartilage. The probability analysis was an important factor in
generating a qualitative explanation of tissue damage. The results showed that the
interaction of two variables, shear stress and hydraulic pressure, yielded the best predictor
of tissue damage. This lends some degree of support for the use of a Coulomb-Mohr type
of failure criterion for cartilage. A quantitative criterion for the prediction of cartilage
damage during a blunt impact can be quite useful in studies designed to investigate ways
of mitigating the injury in our society. In order to determine more precisely the material
parameters of the fracture criterion, experiments were conducted on the rabbit tibial
plateau. An FEM model was developed, and histological damage of cartilage was
correlated with the resultant stress field. A failure envelop was developed based on these
experimental results. The failure criterion was then applied to the earlier experimental
data of Ide (1992).

6.1 Impact Tests on the Rabbit Tibial Plateau
The Experimental Method

The objective of the earlier impact tests on rabbit knee joints (Haut and Ide, 1992)
was to study the impact parameters useful in predicting surface fissures on cartilage,
document fissure location, and correlate contact pressures with cartilage damage. It is
hard to use these data directly to determine the reason of initial damage of cartilage and
bone, because tissues damages may be affected by many experimental factors; such as
loading location, rate of loading and the geometry of each specimen. In order to
determine the critical parameters of cartilage damage, I chose a simplified geometry to

help with repeatability of experimental conditions and generate a more controlled loading
55

56
condition. The rabbit tibial plateau was selected for these experiments, because the

surface is flat and smooth, and its thickness is nearly constant. Tibia plateaus from 20
Flemish Giant rabbits (6-8 months of age) were used for the impact experiments. The
animals were being used in other studies. The tibia plateau was fixed in a plastic tube
with PMMA (Figure 6.1), and covered with a saline moistened cloth prior to each
experiment. The quality of the cartilage surface was checked before each test to make
sure there were no pre-existing fissures or other pathologies on the cartilage surface.

Experimental data were collected from impact experiments within 1 hour of death.

 

Figure 6.1 Surface view of the rabbit's tibial plateau.

In order to document initial damage of cartilage on the tibial plateau, an impact
device was developed. The device had an adjustable grip which could be adjusted to
orient the surfaces so they would be perpendicular to the indenters (Figure 6.2). A
sensitive load cell (50 Pound) was mounted on the indenter. The indenter head was
spherical with radii of 2.0, 4.0 or 20.0 mm. The indenter masses were 27.4, 28.6 and

30.0 grams, respectively.

 

Figure 6.2 The impact device

In order to locate the contact area during impact, India ink was painted on the
indenter before each experiment. The contact area was marked clearly, and some of the
ink penetrated the surface, depending on the extent of surface damage. This helped
document surface fissures using a microscope. A typical example of surface fissuring is

shown in Figure 6.3.

58

 

(C)

Figure 6.3 Surfaces fissures located in the contact area after impacts. These are
enhanced with India ink. The diameter of this mark was approximately 2.0
mm. (a) before initial fissuring, (b) initial fissuring and (c) fissures after severe
impact

The impact force-time data were collected on a personal computer equipment with
an analog-digital board sampling at 10,000 per second. The typical loading time was less
than 1.0 ms. A typical plot of load vs. time is shown in Figure 6.4.

59

0 Experimental load

20‘

Experiment Load (N)

 

 

 

0.0 0.5 1.0 1.5 2.0 2.5
Tlme (ms)

Figure 6.4 Load vs. time recorded with a 2.0 mm radius spherical indenter impact test

The contact surface was examined with a microscope at 50 power after each
impact (Figure 6.3). Histological slides were prepared using standard methods. The
histological sections were made across each impact location. These sections were cut at 6
micron, stained with safranin 0 -fast (green) and examined at 400 power. A histological
technician (J .W.) also examined each section to rule out preparation artifacts.

Results of the Indentation Experiments

Impact tests were conducted using different three impact indenters. The first
group, for which a 2.0 mm radius indenter was used, consisted of seventy-six
experiments. Thirty-four of the specimens suffered obvious damage on the cartilage
surface. When an impact load reached 30 N, there was a 84.3 % probability of generating

a fissure on the cartilage. The peak load, the drop height and the resultant cartilage

60
damage for this group of experiments is shown in Figure 6.5. The critical load (30 N)

which resulted in cartilage fissuring was used as input for FEM model discussed in next
Section. In the second group a 4.0 mm radius impact indenter was used in fourteen
experiments. Seven of these specimens had surface damage. When the impact load
reached 45 N, there was a 87.5 % probability of observing a fissure. The results of
experiments are shown in Figure 6.6. In the third group, a spherical indenter of 20.0 mm
radius was dropped onto 16 specimens. In this case when the impact load reached 50 N,
there was an 85.7 % probability of observing damage to the surface of the cartilage.

Seven samples showed damaged. The results of impact experiments are shown in Figure

6.7.

40-
35-1
... 30- A? “AA“ AAA
E ‘ 5
g 25- MmAAAA
g 20'""""”'"'M'A‘A"§""A"A"""'A"“A"""A ---------------------------- A
.5 .1 g
I 15- AAMA 5 AA
‘8 - a
a. 10- AM A A
g . .
5-
0 ‘ l ' I ' I ' I ' I ' I I I 1 I ' I r l

 

 

0 10 20 30 40 50 60 70 80 90 100

Peak Load (N) ‘ Fissure
A No fissure

Figure 6.5 The experimental results from impact tests in group 1

61

 

 

70"
1 a
60" O
A . o E
E a
5 50' 5° 0 o
E ‘ i c
'9 0 1 o c
Q ------- -----------------: ......... - ..........................................
1: 40 E
*5 o :o
8 i
E 30'° °° s
20 I ' 4: ' I - r . 1
20 40 60 80 100
Impact Peak Load (N) oFissure

0 No fissure

Figure 6.6 The experimental results from impact tests in group 2

 

 

70-
- o E o o
60 :
A 1
E E
g - a 1e n on
.. 50 """"""""""""""""" ‘ """"""""""""""""""
5» - GD 5
.51-:1 o o o o E
... 40 E
0 .
a ‘ 1
g- E
- o:
20 ‘ I r I '1 I ' I ' I ' I
0 20 40 60 80 100 120

Impact Peak Load (N) 3:15:11“?
0 ISSUTC

Figure 6.7 The experimental results from impact tests in group 3

62
Histological Observations

The damaged specimens from test in group 1 only showed surface injuries at the
center of the contact zone. The fissures (splits) were oriented approximately either 40 or
90° to the cartilage surface(Figure 6.8) . No residual deformations were observed at 400

power.

 

Figure 6.8 Typical fissures from specimen in group 1

In tests in group 2, the initial damage in six specimens occurred on the cartilage
surface. The fissures were oriented approximately either 40 or 90° to the cartilage
surface, and most fissures were located in the center of the contact zone (See Figure 6.9).
Damage was recorded in one specimen deeper in the zone of calcified cartilage. This
fissure (or crack) occurred along the tide mark and above the subchondral bone ( Figure

6.10).

63

 

Figure 6.9 Typical fissures from specimen in group 2

 

Figure 6.10 A deeper microcrack generated in the specimen of group 2

In tests of group 3, 4 out of 7 specimens suffered surface damage. These fissures
again had an angle of approximately either 40° or 900 to the cartilage surface and most
fissures were located in the center of the contact zone (Figure 6.11). In three specimens

damage was recorded in the calcified cartilage below the tide mark (Figure 6.12).

 

Figure 6.11 A typical surface fissure from specimen of group 3

 

Figure 6.12 Microcracks generated in a test specimen from group 3. Note the India ink
on the cartilage surface

6.2 A FEM Model of Impact on the Rabbit Tibial Plateau
In this Section, a quasi-static, elastic, finite deformation, axisymmetric FEM
model of the tibial plateau was developed using ABAQUS (Figure 6.13). Model analysis

was performed for three different indenters of radii 2.0, 4.0 and 20.0 mm. These

65
aluminum indenters are assumed to be rigid, smooth spheres. For simplicity of geometry,

a constant thickness of the cartilage layer was assumed. This thickness was assumed to
be 0.5 mm, which is an average value recorded on one tibial plateau. The thickness of
bone was assumed to be 1.5 mm, which is an average value measured from the PMMA
base to the top of the plateau in one experiment. A total of 1165 axisymmetric elements
were used in this model. Fifteen of these were frictionless interface elements between the
indenter and the cartilage surface.

Young's modulus and Poisson's ratio for the cartilage were assumed to be 2 MPa
and 0.49, based on the rabbit patella study using the results discussed in Chapter 4. The
Young's modulus and Poisson's ratio of bone were assumed to be 2 GPa and 0.3,
respectively (Cowin, 1987).

The tibial bone was restrained in the x and y directions, and a quasi-static load
was applied to the cartilage surface by a half sphere (Figure 6.14). This quasi-static load
was approximately equal to the measured impact loads (Figure 6.4) which caused initial
damage of cartilage in the tests discussed in the Section 6.1. The input load history was
linear related to time. At time 1 ms, the input peak loads in the FEM model were 30 N,

45N, and 50 N for the 2.0, 4.0 and 20.0 mm radius indenters, respectively.

66

1P
1 Rigid body

  

 

 

&

Figure 6.13 3-D tibia impact model Figure 6.14 Axisymmetric model

Analytical Results

In all three loading cases, the maximum compressive principal stress occurred in
the center of the contact area (Figure 6.15). They were 8.4, 11.4 and 21.4 MPa for the
cases of the 2.0, 4.0 and 20.0 mm radius indenters, respectively.

The highest maximum shear stresses occurred in two areas: at the center of the
contact zone and in the bone just below the cartilage-bone interface (Figure 6.16). The
magnitudes of maximum shear stress at the center of the contact area in the 2.0, 4.0 and
20.0 mm indenter cases were are 2.82, 3.16 and 3.71 MPa, respectively. The direction
associated with the maximum shear stresses was in the range of 35° to 45° (Figure 6.16).
The maximum shear stresses in the bone for the 2.0, 4.0 and 20.0 mm indenter cases were
approximately 4.08, 5.35 and 7.54 MPa, respectively. The directions of these maximum

shear stresses were in the range of 20° to 45° (Figure 6.16). The ratio of maximum shear

67
in the bone layer to that on the cartilage surface increased with the radius of the impact

indenter (Table 6.1).

Table 6.1 Maximum shear stress and their ratio of shear in three cases

 

 

 

 

Indenter radius Max. Shear in Max. Shear in Ratio of shear in bone to
bone cartilage cartilage
(mm (MPa) (MPa)
2.0 4.048 2.821 1.435
4.0 5.351 3.163 1.679
20.0 7.544 3.712 2.032

 

 

 

 

 

 

The maximum tensile stresses were located on the edge of the impact indenter
(Figure 6.17) and had magnitudes of 0.6 to 0.1 MPa for the three indenter cases. The
maximum tensile strain was approximately 14.2%, 13.3%, and 10.2% in the 2.0, 4.0, and
20.0 mm indenter cases, respectively and it was located in the center of the contact zone

directed parallel to the cartilage surface.

The shear stress distributions, txy, are shown in Figure 6.18 for the three loading
cases. In all cases, the maximum magnitude of txy occurred at a distance from the
symmetric line equal to approximately 60% of the radius of the contact area. The
magnitudes of these maximum shear stresses were 1.6, 2.4 and 2.7 MPa in the 2.0, 4.0,
and 20.0 mm indenter cases, respectively. The locations of these maximum shear values
correlated very well with those found in the biphasic model of Ateshian (1994), who also
obtained these values at a distance from the symmetric line equal to approximately 60 %
of radius length of the contact area.

The Mises stresses distributions for the three models are shown in Figure 6.19 and

are similar to the maximum shear stress distributions.

68

-IIIII.II

 

 

(a)

 

 

 

 

(C)

Figure 6.15 Minimum =principal stresses and directions for the 0different indenters (Pa) (a)
.0 mm, (b) R=4. 0 mm, and (c) R=20 0mm.

 

 

 

Figure 6.16 Maximum shear stresses and directions for the different indenters (a) R=2.0
mm, (b) R=4.0 mm, and (c) R=20.0 mm

55555555

 

 

 

 

 

 

 

IL

 

(C)

Figure 6.17 Maximum principal stresses and directions impacted by different indenters
(a) R=2.0 mm, (b) R=4.0 mm, and (c) R=20.0 mm

71

1.1-:50: 0.51-oo-

0.0u-ocu

5.5-no.0
I.Ic-oco
|.-0locu

I.o'-~nc

 

 

 

n.00noov

l.lllol1

1.00....

1,-0.001

-.oc.coc

 

 

 

1.1-.cnv

1......v

 

 

   

Y I.
(c) Ann-u.

Figure 6.18 Shear stresses in the xy plane for different indenters (a) R=2.0 mm, (b) R=4.0
mm, and (c) R=20.0 mm

Figure 6.19 Mises' stresses for the different indenters (a) R=2.0 mm, (b) R=4.0 mm, and
(c) R=20.0 mm

72
The maximum shear stresses at the top of the cartilage for the all cases are shown

in Figure 6.20. Note that the shear stresses for the flattest indenter were 1.33 times higher

than those caused by the small radius indenter.

4.
283.5.
5..
$2.5-
a
1. 2-
is“
{01.5-
a 1-
2

0.5-

0

 

 

 

0 0.5 1 1.5 2 2.5
Distance from the line of symmetry (mm)

Figure 6.20 The distribution of maximum shear stresses at the top of the cartilage layer

The maximum shear stresses at the cartilage-bone interface are shown in Figure
6.21. The shear stresses generated for group 3 indenters were 1.79 times higher than

those generated in test using the smallest radius indenter.

73

 

 

 

 

é
r72
III
8
u:
U)
E
2

0 1 : : ‘-~"E—“"'T:

0 0.5 1 1.5 2 2.5

Distance from the line of symmetry (mm)

Figure 6.21 The distribution of maximum shear stresses at the interface between cartilage
and bone for various indenters

The maximum shear stresses in the layer of bone just below the cartilage-bone
interface are shown in Figure 6.22. The shear stresses resulting for impact the indenter
with the largest radius were 1.88 times higher than those used by smallest radius indenter.
The highest shear stress was 7.79 MPa, which may help explain the incident of bone

damage in these experiments.

74

8 .. _ ..... R=2.0 mm
7 -- ..........
‘~.‘ —'—'—— R=4.0mm
6 -- “~.‘
‘~. -------- =20.0mm

Max. Shear Stress (MPa)

 

 

 

0 0.5 1 1.5 2 2.5

Distance from the line of symmetry (mm)

Figure 6.22 The distribution of maximum shear stress in the bone just below the
cartilage-bone interface

6.3 A Coulomb-Mohr Failure Envelop for the Rabbit Tibial Cartilage

In Section 6.1, the locations of damage and the critical impact loads were
described for the experiments. In the Section 6.2, the stress fields for these critical loads
were determined by an axisymmetric FEM model. In this Section, a failure criterion has
been developed to explain the mechanical cartilage damage.

By comparing the stress fields of Section 6.2 with the experimental results of
Section 6.1, I could determine the maximum shear stresses and the mean stresses at the
fissure location for each case. In all cases, the location of fissuring was at the center of
contact.

The Coulomb-Mohr's circles associated with the three critical load conditions are
plotted in Figure 6.23. The center point and the radius for the Coulomb-Mohr's circles
are the mean failure stress and the maximum failure shear stress, respectively. These
were 7.02 MPa and 2.82 MPa for the 2.0 mm indenter case, 8.90 MPa and 3.16 MPa for

the 4.0 mm indenter case, and 15.6 MPa and 3.712 MPa for the 20.0 mm indenter case.

75

The ultimate tensile stress of cartilage, as determined by H. Yamada (1970), has also

been plotted in Figure 6.23. The ultimate tensile stress was 3.3 MPa. According to H.

Yamada, the ultimate shear stress is 65% of the ultimate tensile stress. This circle is also

plotted in Figure 6.23. A failure envelop was determined by linear curve fitting for these

three cases. The equation for these data was

 

 

 

 

 

 

 

 

 

 

 

”t = 2.2251 + 0.0976920“ . (6.1)

R2 = 0.962 (6.1 a)

Failure stress state

_Data from H. Yamada for 2.0 mm radius indenter

for the 4.0 mm radius indenter
A Tresca

Criterion
for the 20.0 mm

radius indenter

 

 

 

 

 

4-

76%

n.

g o

to

In 0 >

2!

*- 1

(I)

3.25“" ......................................................

0

.1:

'4 I T j T j T If I I I I I I T I r I I I I

-4 2 4 6 8 10 12 14 1O 18 20

Normal Stress (MPa)

Figure 6.23 The failure envelop for tibial cartilage under impact loading

76

The failure envelop suggests that the failure shear stress increases with mean
stress. Thus cartilage damage may be dependent upon two variables: the shear stress and
the hydraulic pressure. A maximum shear stress theory, such as that suggested by Tresca,
has also been shown in Figure 6.23 (dash line).

6.4 Application of the Criterion to Ide's Experimental Data

The experimental results from the thirty-four animals from Ide (1992) were
outlined in Section 2.1. The corresponding stress analyses associated with these
experiments were performed using the model of the patella described in Sections 5.1 and
5.2. In this Section, the failure criterion developed in Section 6.3 was applied to these
data.

Cartilage damage always occurred on the lateral facet of the patella. The
maximum shear stresses and mean stresses in that area was given in Table 5.1. Each pair
of maximum shear stress and mean stress have been plotted in Figure 6.24 and 6.25. The
solid dots represent those samples with observed fissures on the patellar cartilage surface,
and the hollow dots represent those samples without fissures. The linear failure envelop
developed from experiments on the tibial plateau has also been plotted in Figure 6.24.
The non linear failure envelop developed from experiments and Yamada's data on the
tibial plateau has been plotted in Figure 6.25. Note that most of the data associated with
fissured specimens appear above or near the failure envelop. The non linear failure
envelop shows better classification for these experimental data. A Tresca criterion,
represented by the dashed line, would appear to be too conservative and not for the

experimental data as well as the Coulomb-Mohr criterion.

77

Failure criterion

Tresca criterion based on ”us study

(MPa)
0

4.. O . -.'Po‘ °
:1, // 0°11) '\0
280/0 /1 /

..l...,...,.,.,...,..
-4 -2 0 2 4 6 8 1012 14 16 18 20 22 24 26 28

Normal Stress (MPa)

 

Stress

 

v°

 

 

Shear

 

Figure 6. 24 Application of the linear tibial failure criterion to Ide‘s data. Solid dot
indicates a fissure, whole open circle indicates that no surface damage was
observed experimentally by Ide.

Tresca failure criterion ‘ ‘1 1 Failure envelope based on this study
1

 

 

 

 

 

 

 

 

 

5 7 77 1 ‘C 7 ‘1 7 o '
4 ' i ‘ I ° .
\ O 1
A 1 / ~:\ \ O /0”' w a O .00 1
N A \ A
E 2 X ‘ O ’0 v ‘
x/ ' "\ ix , \
V 11 1', 11 \ ‘l O O/ G 1
*2 . 1 1 1 1 , 277*7 'j,‘
E 0 11 11 ,1 1\ 1v 11 1 I
s \ x . 1 1
.5 g n “7 1r] / \ / \
CID-2 ‘ \J / /
=41 // \ 1
1:1... 1
-4 1 1 1 1 1 1 1

-4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28
Normal Stress (MPa)
Figure 6.25 Application of the non linear tibial failure criterion to Ide's data. Solid dot

indicates a fissure, whole open circle indicates that no surface damage was
observed experimentally by Ide.

Chapter 7 Conclusions

The objective of the research in this dissertation was to develop a criterion to
predict damage of articular cartilage caused by impact loading. A number of impact tests
were performed on tibial plateaus to determine the critical states of initial damage. An
axisymmetric finite element analysis model was set up to help document the distribution
of the stresses and strains during blunt loading with various impact indenters. By
comparing the damage location with the associated stress distribution, the correlated
maximum shear stresses and the normal stresses in all three cases were used to develop a
failure criterion. The failure envelop was a function of normal stress and maximum shear
stress. It had been proposed that both maximum shear stress and hydraulic pressure
affect fissure creation and the effect of hydraulic pressure can not be ignored. That is, the
maximum shear stresses associated with the initial damage of cartilage are not a constant.
Thus, a Tresca criterion is not applicable for the damage of cartilage during blunt impact.
The criterion is robust and can be applied to other experimental data. The application of
the failure criterion to Ide's experimental data shows a good ability to predict initial
damage of cartilage during blunt impact on the bone.

In our indenter impact tests, most damaged specimens showed initial surface
injuries at the center of the contact zone, and fissures (splits) were oriented approximately
either 40 or 900 to the cartilage surface. Comparing the zone of damage with the
correlated stress field from the FEM model, I found that the maximum values of the
maximum shear stresses occurred in the observed damage area, and the fissures were
oriented parallel to the directions of maximum shear stresses. So I believe the initial

fissures on the cartilage surface were due to excessive shear stress yet, the actual critical
78

79
stresses may be influenced by the properties used for the cartilage on the tibial plateau,

which were actually taken from studies in patella cartilage. A few experimental
observations are in agreement with my experimental observations (Repo, 1977; Broom,
1986; Haut, 1986; Thompson, 1990; Silyn-Roberts, 1990; Ide, 1990; Thompson, et al
1990; Tomatsu, 1992; Thompson, 1993). Repo and Finally (1977) documented radial
fissures of human cartilage in the center of the contact zone at the cartilage surface
extending to the deep zone, and death of chondrocytes when average normal contact
pressures reached 25 MPa. Silyn-Roberts, et al, (1990) reported that impact induced
failures of cartilage, in vitro, were usually in the center of the contact zone and oriented at
45 degrees to the articular surface and extended into the middle zone. They did not study
the stress distribution inside the cartilage, but their experimental observation supports our
results.

A few stress and strain analyses of the joint (Eberhardt, et al, 1990, Ateshian,
1994) show that the maximum shear stresses, 1”,, are located at the bone cartilage
interface away from the center of contact at 60 % of the contact radius. They suggest

that the documented splits at the interface could be due to shear stress. Our solution also

showed that the maximum values of 12x), occurred in that location. But 1: W is much

smaller than Tmax. It is true, as we discussed in Section 1.2, there exits another type of

damage phenomenon. The separation of the cartilage from the underlying bone in the
zone of calcified cartilage has been reported in the previous model during blunt impact
(Armstrong, 1982). Vener, et al, (1991) also observed cracks in the zone of calcified
cartilage after impact of the canine joint. We also observed cracks at the interface in a
few tests, but they occurred at the line of symmetry under the indenter and prOpagated
along the tide mark in the calcified cartilage. When the tibia was impacted by a flat
indenter, the FEM model showed two areas of the high shear stress: one at the contact
surface, and one in the bone just below the cartilage-bone interface. My model results

suggest that when the contact radii are large, such as in the human knee joint, there is a

80
possibility of initial damage occurring at the interface with bone versus the cartilage

surface. On the other hand, we do not know the damage criterion and properties of
calcified cartilage, per se. This is an area of future work.

This study suggested that the damage of cartilage on the tibial plateau due to blunt
impact is controlled by two variables: shear stress and hydraulic pressure. In fact, the
effect of static hydraulic pressure is significant and can not be ignored. So a Coulomb-
Mohr damage criterion has been applied to study mechanics of biological tissue trauma.
The difference between Coulomb-Mohr criterion and a Tresca theory is the effect of
hydraulic pressure on material failure. There are three factors which could produce high
hydraulic pressures in the articular cartilage of the knee joint. Firstly, a flat surface of
articular cartilage and a low permeability would provide ideal conditions to keep the fluid
inside and generate high hydraulic pressure during compressive loads. Secondly, if the
loading occurs in a short time, the cartilage would generate a high hydraulic pressure
because the water does not have time to move out. Thus, cartilage could bear very high
loads without damage in this situation. The hydraulic pressure acts to help support the
cartilage making it more resistant to shear damage. This may be the reason why the
cartilage can withstand 3 or 4 times body weight during jumping and running, yet also be
damaged simply by long periods of standing. Thirdly, the hydraulic pressure and shear
stresses are affected by the directions of impact loading. The results in Appendix A
showed that the contact pressures were uniform when impact was directed more centrally
or a little more laterally in the rabbit model. In these cases, maximum values of shear
stress at the interface of cartilage and bone were higher than those of shear stress at the
cartilage surface. On the other hand, contact pressures were more bimodal when impact
occurred on the medial side of the patella. In this case the maximum values of shear
stress at the cartilage surface were higher than those at the interface of cartilage and bone.
This may explain why surface damage was more prevalent in Ide's experiments. These

results may be useful in the study of designs that could help prevent cartilage damage

81
during blunt impact. One must still develop a criterion for interface injuries of the

cartilage.

Another important aspect of the work in this dissertation was to develop a method
of estimating properties of the cartilage during impact tissues. An inverse method was
employed to estimate the material properties of cartilage by Comparing the distribution of
experimental contact pressures to that generated by the FEM model. The residuals were
minimized when Poisson's ratio was 0.49 for the layer of cartilage. This result was
consistent with the biphasic model of Mow, et a1 ( 1980), and agreed with Eberhardt et a1
(1991) and Askew and Mow (1980) who selected values of 0.475 and 0.45, respectively,
in their elastic models. In our study the difference between theory and experiment was
also minimized when an elastic tensile modulus of 2 MPa was chosen for the layer of
cartilage. In the joint model of Askew and Mow (1984), a value of 12 MPa was chosen
for the analysis. Eberhardt et a] (9) chose a value of 1.5 MPa for the elastic shear
modulus of the cartilage (which corresponds to a tensile modulus of approximately 4.4
MPa if one assumes isotropy of the layer). The above discussion proposes that as a first
order approximation, response of the rabbit joint under a blunt impact can be determined
by assuming the layer of cartilage to be approximately incompressible, i.e. Poisson's ratio
0.49 with an elastic modulus of 2 MPa and a reasonable value of elastic modulus for the
underlying bone. Since some of the experiments with the large radius indenter showed
damage of cartilage in the zone of calcified cartilage, it will be necessary to develop a
scheme to estimate properties of calcified cartilage and subchondral bone during impact.
This is an area for future study that may have direct application to the human knee. A
more complex description of joint tissues may give better estimates of the failure stress
distributions. In the lower layers of cartilage and bone it may also be important to set up
a three dimensional model of the knee for a better understanding of the effects of load
direction and load location. This would be especially true for the human model, where

transverse fractures of the patella are often observed (Shelp, et al, 1994). The current two

82
dimensional model can not show stress distributions in the z direction. Another area of

future study would be to deal with the problem of stress waves developed during impact.
Realistic impacts on the human knee during a car crash occur at speeds of 30 mph and
higher. To study the stress distributions in cartilage and bone a dynamic FEM model

would be necessary.

APPENDIX A The Effect of Loading Location on the Distribution of Stress and
Strain in the Rabbit Knee joint

Stress analysis of the rabbit knee joint (X. Li, et al, 1993) suggested that the location
of initial impact trauma may be dependent on the direction of the impacting load. A
typical joint geometry was selected for calculating the effects of loading using the FEM
model (Figure 4.4). The same input load was located at the three different sites on the
anterior surface of patella; the medial side, the centers side and the lateral side,
respectively. The contact pressures generated in the patello-femoral joint were calculated
by the FEM model (Figure A. 1). Loading on the more medial side of the anterior surface
of patella (similar to the in vivo experiments of Ide) yielded a strong bimodal
distribution of contact pressure. The contact pressures on the lateral facet were higher
than those on the medial facet. When loading was on the center of the patella or the
lateral side of the patella, contact pressures on the medial facet were higher than those on
the lateral facet.

The location of loading affected the patello-femoral contact pressures and the
resulting stress field. The maximum values of maximum shear for the three loading
locations are listed in Table A.1. For loads acting on the medial facet of the patella (most
of Ide's experiments) the highest shear stresses occurred on the cartilage surface. In this
case the ratio of shear stress in the bone layer to that in the cartilage was 1.21. When
loading acted on the center or the lateral facet, the ratio of the shear stress in the bone
layer to the cartilage was up to 3.0 and 3.47 times, respectively. In Ide's experiments
most impacts were directed in the more medial facet of the patella. The above results
help explain why the mechanism of injury was surface fissures on the cartilage, without

fracture of the calcified cartilage or underlying subchondral bone (Haut, 1991).

83

84

3O

 

Load at medial
Load at cente . : : 3 :
25Load‘at'late‘ral” ........... ..... . ..... 3 ............ .3 .............

20 ..... ............ ............

 

 

 

 

Contact Pressure on the cartilage (MPa)

15 .......................................................................
10. ................................................................................
5+

1 ......................... g ....................................
. Medial Lateral ,

0 1 2 3 4 5 7' 7

Location (mm)

Figure A.1 This figure shows the effect of impact location on the distribution of contact
pressures generated in the PF joint. More bimodal types of distributions were
generated with more medial and lateral oriented impacts on the patella. A
central located impact on the patella generated a more uniform profile of
contact pressures over the medial and lateral facets.

Table A.1 Maximum shear Stress caused by different loading locations

 

 

 

 

 

Model Name Load direction Max. shear Max. shear Ratio of shear in the
in subchondral In cartilage subchondral bone/cartilage
bone (MPa) surface (MPa)

#3-medial medial 6.0 5.76 1.21

#3.center center 7.5 2.50 3.00

#3-Iateral lateral 8.0 2.88 3.47

 

 

 

 

 

 

APPENDIX B Estimation of Material Properties and Stress Analysis for the
Human Knee Joint

(1) The FEM Model

The methods for the human knee joint experiments were discussed by Shelp
(1994). The basic assumptions used in the human knee joint FEM model were essentially
the same as for the rabbit FEM model (discussed in Chapter 3). The geometry for the 2
dimensional FEM model of the human knee joint was set up by sectioning one of the
experimental samples and photographing its cross—section (Figure 3.1). The section was
located between two pin holes at a location coinciding with the zone of high contact

pressure generated during impact (Figure B.2).

 

Figure B.1 The cross section of the human knee joint

85

86

 

Figure B2 A typical patella from the human knee joint and the corresponding Fuji
film. The scalpel locates the location of the cross-section used for the FEM model.

Based on the geometry of the human knee joint shown in Figure B.1, an elastic
finite element model was developed. For simplicity, a two dimensional analysis was
performed with plane strain conditions assumed in order to represent a confinement of the
cross section. The patella and distal femur were modeled with 3000 elements (See Figure
B.3). Approximately 100 node-to-node gap elements were used to model the contact
zone between the patellar and femoral cartilage surfaces. These were used in my
assumption that no friction existed between contact surfaces on the patella and the femur
during impacted loading.

The distal end of the femur was fixed at an unspecified distance from the patello-
femoral joint, but the length was chosen to be quite removed from the joint proper. As in
the animal model, the impact loading was applied to the human model based on the

experimental pressure data from the anterior surface of the patella. Contact pressures

87
generated at the interface of the patella and the femur were measured in experiments with

pressure sensitive film. Correlation between the contact pressures obtained from the
model and those from the experiments were used to determine material properties of the

cartilage using the inverse method discussed in this dissertation.

-
I
l
a

.
it?-

va’flm‘z

{fauna wag}.

Yr:

 

Figure B.3 FEM model of human knee joint

(2) Estimation of the Material Properties

The degree of correlation between the experimental and theoretical contact
pressures in the PF joint depended on the material properties chosen for the cartilage.
The calculated results from experiments Knee-1 and Knee-2, using the method of

minimum residuals, are shown in Figures B4 to B7. As the value of Poisson's ratio

88
approached 0.475 for the cartilage, the residuals decreased (Figure 8.4. and B5). We

also noted that the residuals were a minimum for an elastic modulus of the cartilage of

approximately 1 to 2 MPa and a Poisson's ratio of 0.475 (Figure B6 and B7).

 

KNEE_1, E=2 MPa, v=0.01‘~ 0.49

§ v=O.49

  

 

PF Contact Pressure (MPa)

 

Experiment

 

 

 

5 10 15 2o 25 3'0 35 40

Location (mm)

 

‘é‘é‘

 

Sum of Squared Residuals
U1
8

 

 

 

400

300

200

100o 01.1 01.2 013 014 0.5
Poisson Ratio

Figure B.4 (a) A plot showing the experimental contact pressure distribution and
corresponding theoretical distributions for various values of Poisson's ratio in Knee-1
model. Note how well the results at 0.45 compare to the results from the experiments.
(b) The residuals decreased continuously as Poisson's ratio approached 0.45.

89

3O

 

KNEE_2, E=2 MPa, v=0.01 ~ 0.49

  

............................

1 __Experiment
\‘ :

...................................................

  

PF Contact Pressure (MPa)

 

 

 

 

 

Sum of Squared Residuals

 

 

 

0 O. 1 0.2 0.3 0.4 0.5
Poisson Ratio

Figure B.5 (a) A plot showing the experimental contact pressure distribution and
corresponding theoretical distributions for various values of Poisson's ratio in Knee-2
model. Note how well the results at 0.475 compare to the results from experiments. (b)
The residuals decreased continuously as Poisson's ratio approached 0.475.

90
KNEE_1, E=1,2,4.20 MPa, v=0.475

 

 

 

 

 

r; 25 1 . :
E 1 —2o ;
v . . - ----- -!-~. E=f . i
a: 20W.-. .5...._;...;..._f..1;.;_..f-..—..—..-.:ffr;§:2.g .......... g ............ g ...........
a ’1. ’.¢.,gv "...-4....” , ‘ Experiment
E I ’1'. ' I ...-u I'Ic ~.
I. / : . ' .\ . {S '
8 15 ...... I.. . ........... .......... ........... 1 ....................... gng. ...............
s I ’0: : : 1 3 . '5‘
fl :' I ' ‘. ‘-
8 .~‘ . ; ; . \, a}
fin 10 ...... ........... g ............ g ........... 1 .............................. 121‘ .........
Gut :' : : : 1 \O'fi.‘
5 1 3‘,
5 . i . {'1‘
5 :' : : : 1 : : : '2
O 5 10 15 20 25 3O 35 40
m Location (mm)
732
E
8
a: 1500
'8
§
c,,1000
m
h
o
E 500
:1
m

 

 

 

 

Youngs Modulus (MPa)

Figure B.6 (a) The effect of variations in the elastic modulus of cartilage on the
residuals computed between the experimental and the theoretical contact pressures. A
minimum was obtained for an elastic modulus near 2 MP3 for the cartilage at Poisson's
ratio less than 0.475 in Knee-l model. (b) This plot shows the residuals for Young's
modulus from 1 to 20 MPa.

91

 

 

 

 

 

 

 

 

 

 

 

A O KNEE_2, E=1,2,4.20 MPa, v=0.475
«1 3 = s = s
E 3E=20
V 3 f : : 1'”; : .
o 25 ........... 3 ............ ....... f ........... 3.,.',....°.; ...... 3 .......... I ......... .
:5 5 3 5 -- ; - ; Experiment
g S
a: : : : : : I ..
H 20 ........... 3 ............ 3 ............ ........... 3 ............ ..........
v s s s s 2 2 -
S
1::
8 '. . . . : : : :
[a 15 ......... 3: ......... ~.... ........... g ............ 3 ............ g ......
m ' I I I I 2
Lateral
10 . I I I '. '. '. I
O 5 10 15 20 25 3O 35 40

w Location (mm)
2 400 ,
'1: E
g 350 ;
a: E

300 :
E» a
£250
m 200 f
6-1 .
o .
E 150 3
= I
W 100 3

5 E

00 5 10 15 20

Youngs Modulus (MPa)

Figure B.7 (a) The effect of variations in the elastic modulus of cartilage on the
residuals computed between the experimental and the theoretical contact pressures. A
minimum was obtained for an elastic modulus near 2 MPa for the cartilage at Poisson's
ratio less than 0.475 in Knee-2 model. (b) This plot shows the residuals for Young's
modulus from 1 to 20 MPa.

92
(3) The Stress Distribution in the Human Knee Joint

Using these optimized values for the material properties of the cartilage and the
contact pressures measured experimentally on the anterior surface of patella , I obtained
the stress distributions from the FEM knee joint model. The contour plots of the
minimum principal stresses and the maximum shear stresses in the knee joint model
(Knee-2) are shown in Figures B.8. There were two areas of high shear stress in the knee
joint; one at the cartilage surface and another at the subchondral bone layer. The
maximum shear stresses developed within the subchondral bone were 23 MPa in the
knee-2 model and 12 MPa in knee-1 model, respectively. The maximum shear stresses
developed on the cartilage surface were 5.9 MPa and 5.7 MPa, respectively.

The distributions of the maximum shear stresses and the minimum principal
stresses for the Knee-2 FEM model are plotted in Figure B.8. Since the surfaces of
cartilage in both knee joints were fissured or damaged before impact, we could not tell if
impact loading caused surface damage. (See a typical human cartilage in Figure B.2).
But in some experiments reported by Haut, et al, (1994) cracks were observed lying in

the subchondral bone.

93

 

Minimum principal stress Maximum shear stress

-l|.lI ’~“ 1.)! ...-s

Al'IIl-I unsun. v.1: ...-.... gun... I...

Figure B8 The contour plots of the minimum principal stress and the maximum shear
stresses in the Knee-l model

 

Figure 8.9 Crack propagation along subchondral bone in the typical section of human
knee patella (Knee-2 specimen).

APPENDIX C Application of the Coulomb-Mohr Criterion to Human Knee Joint
(1) The Damage of Cartilage

A total of fifteen human knee joint impact tests were performed (Haut, et al,
1994). In order to study the mechanisms of cartilage and bone damage for these
experiments, I used the previously discussed FEM model, and examined the distribution
of stress and strain in an attempt to correlate these parameters with injury. For these
fifteen impact experiments, the contact loads on the anterior surface of the patella were
not recorded. Therefore, the FEM model used to analyze these experiments had to be
changed. In order to study the distribution of stress and strain in the patella, I used the
measured contact pressures in the patello-femoral joint as the input loading boundary
conditions (C. 1).

The geometry for the 2-dimensional finite element model used for modeling these
experiments was taken from specimen knee-2. The FEM model was constructed using
1300 plane strain elements (Figure CI). The Young's modulus and Poisson's ratio for
bone were assumed to be 2 GPa and 0.3, respectively (Cowin, S., 1987 ). The Young's
modulus and Poisson's ratio for cartilage were assumed to be 2.0 MPa and 0.475,

respectively. The values were obtained by the inverse method discussed in Appendix B

94

 

Figure CI The 2—D patellar FEM model for the human data analysis of the
boundary conditions

The contact pressure profiles measured from the Fuji film (Figure B.2) were used
as the input boundary conditions at the cartilage surface (Figure CI). The contact
pressure distribution was documented across the lateral and medial facets of the patella
by taking a 0.5 mm thick slice of the distribution and generating a 2-dimensional pressure
profile. The anterior aspect of the patella was rigidly fixed (Figure C.1). From these
experimental data, fifteen different input loads were used in the FEM model to obtain the
related stress fields.

Analyzed by the human FEM model, the maximum shear stresses and mean
stresses on the cartilage surfaces were obtained. Since cartilage surfaces were fissured or
damaged before impacted for all specimens due to pre—existing pathology, I could not tell
if the impact loading caused damage, (See a typical human cartilage in Figure C2). The
maximum shear and mean stresses are plotted in Figure C.3 for these experiments. The
Coulomb-Mohr criterion for the rabbit is also plotted. It is clear that most of human data
lies below the failure envelope developed from the rabbit experiments. Maybe the flat
contact surfaces in the human knee joint and the center impact loading produced higher

stress in the bone, and produced the microcracks shown in Figure C.2.

 

Figure C 2 Cartilage damage of human patella

 

 

 

6 - . .
O
in
m
o
I-
#
m
I-
«I
o
.c
In
x'
a
2 d
O
1 u
The failure criterion for Rabbit's cartilage
0 ' I ' l ' I ' l

' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I
-2 o 2 4 6 a 10 12 14 16 18 20 22 24 26 28
Normal Stress (MPa)

Figure C.3 Application of Coulomb-Mohr criterion to the human cartilage data

97

Table CI The maximum values of shear stress and the corresponding mean stresses on
the cartilage surface

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Exp. Name Mean Stress (MPa) Max. Shear Stress (MPa)
hoZ-ro3 10.57 2.32
9793-r5 15.16 3.78
9793-11 18.94 1 5.49
8093-r1 12.93 2.78
8093-15 27.29 5.88
7093-11 15.43 2.84
6893-r1 0.46 l .22
6893-14 1 1.87 2.49
4393-12 15.64 3.45

28091-le 14.99 2.96
26993-r1 20 4.53
26993-15 19.44 3.96
1021-16 22.21 4.52
1021-11 9.91 2.24
7093-knee-2 15.31 5.79

 

 

 

 

 

(2) The Damage of Subchondral Bone

The FEM model has been utilized to study the bone damages reported in Table
C.2. By using the human patellar FEM model and the 15 different pressure distributions
as input, the stress distributions in the patella were obtained. The maximum shear stress
and the mean stress in the bone just below the cartilage-bone interface (in the subchondral
bone layer) are plotted in Figure C4. The solid dots represent the samples with
subchondral bone cracked, and the hollow dots represent those with no damage observed
by experiment. We found eight out of fifteen specimens had cracks in the subchondral
bone. A typical crack is shown in Figure C.5. Since I could not find any information
about the ultimate tensile or compression stress for human subchondral bone, 1 could not
develop a failure envelope. However these experimental data do show an increasing state
of hydrostatic stress as shear stress increases in the subchondral bone. Hence, I could

imagine that both parameters will be involved in any future criterion developed for

98

 

 

subchondral bone.

35 _

30 ' . .
A
E
E 25 -
V c
U)
3 20 _ o o o
4': o o
m o o o
‘5 15 -
.8 ° 0
(D 1 0
x- 10 ‘
«I
5 1

5 ‘ O
1
0 - I - l ' l ' I ' I ' I
o 2 4 6 8 10 12
Mean Stress (MPa)

0 fissure
0 without fissure

Figure C.4 Results of the stress analysis in the subchondral bone for the fifteen human
knee joint experiments ., _ 1

  

Figure C.5 Crack propagation along subchondral bone in this typical section of human
patella

99

Table C.2 Experimental observation and data

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Test No. Crack in subchondral bone Mean Stress (MPa) Max. Shear Stress (MPa)
h02-ro3 no 1.61 , 12.88
9793-r5 no 0.56 13.90
9793-11 no 1.50 16.35
8093-r1 yes-center 4.22 16.29
8093-15 yes 8.07 30.57
7093-1 1 yes-center 6.96 20.84
6893-r1 yes-medial 1.14 4.63
6893-14 no 4.84 16.43
4393-12 no 1.60 17.47
28091-le no 5.40 18.95
26993-r1 yes 3.98 18.16
26993-15 yes 4.86 20.83
1021-r6 yes 8.55 29.55
1021-11 no 3.37 11.79
7093-5 yes 6.34 22.92

 

 

 

 

 

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