• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Properties of the system components:...
 Determination of the geometric...
 Modeling of the osteosynthesis
 Experimental verification of the...
 Appendix
 Reference
 Biographical Sketch
 Signature
 Copyright














Title: A biomechanics approach to the interaction of host implant systems
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
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STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00084167/00001
 Material Information
Title: A biomechanics approach to the interaction of host implant systems
Physical Description: xiv, 190 leaves : ill. ; 28 cm.
Language: English
Creator: Miller, Gary Joel, 1947-
Publication Date: 1977
 Subjects
Subject: Bones -- Surgery   ( lcsh )
Mechanical Engineering thesis Ph. D
Dissertations, Academic -- Mechanical Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 184-188.
Statement of Responsibility: by Gary Joel Miller.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00084167
Volume ID: VID00001
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000187982
oclc - 03418761
notis - AAV4586

Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
    Table of Contents
        Page iii
        Page iv
    List of Tables
        Page v
    List of Figures
        Page vi
        Page vii
        Page viii
        Page ix
        Page x
        Page xi
    Abstract
        Page xii
        Page xiii
        Page xiv
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
    Properties of the system components: a review
        Page 8
        Page 9
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    Determination of the geometric propeties of the system components
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    Modeling of the osteosynthesis
        Page 75
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    Experimental verification of the modeling of the osteosynthesis
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    Appendix
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    Reference
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    Biographical Sketch
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    Signature
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    Copyright
        Copyright
Full Text















A BIOMECHANICS APPROACH TO THE INTERACTION OF HOST
IMPLANT SYSTEMS













By

GARY JOEL MILLER


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY








UNIVERSITY OF FLORIDA


1977
















ACKNOWLEDGEMENTS

I would like to thank Dr. George Piotrowski for suggesting

the area of intramedullary fixation for study, and for his guidance

and encouragement throughout this work. I am also indebted to

Dr. Larry L. Hench and Dr. Robert B. Gaither for their continual

support and valuable comments and criticisms.

My special thanks go to Dr. William C. Allen, now at the

University of Missouri Medical School, for his comments and

assistance with the many clinical aspects of this research endeavor;

and to my wife, Suzy, without whose encouragement and understanding

this work would have been impossible.

I would like to acknowledge the technical assistance of

Ms. Karla Smith, Mr. Thomas Carr, and Mr. Mike Hardee. I would

also like to thank Mrs. Loretta Thigpin, Ms. Susan deFalco,

Ms. Dee Schneider, and Ms. Pam Blalock for their assistance in

preparing this manuscript. Thanks are also due Mr. Robert Hagen

for his assistance in preparing the illustrations.

This research was aided by a grant from the Orthopaedic

Research and Education Foundation.

















TABLE OF CONTENTS


ACKNOWLEDGEMENTS . . . .. . ii

LIST OF TABLES . . . . . v

LIST OF FIGURES . . . . .. ... vi

ABSTRACT . . . . .. . . xii

Chapter

I. INTRODUCTION . . . . .... 1

Brief History of Intramedullary Fixation . 1
Previous Work . . . . . 4
The Present Research . . . . 7

II. PROPERTIES OF THE SYSTEM COMPONENTS: A REVIEW. . 8

The Intact Femur. . . . . 8
The Intramedullary Nail . . . ... 16
Summary . . . . . 21

III. DETERMINATION OF THE GEOMETRIC PROPERTIES
OF THE SYSTEM COMPONENTS. . . . ... 23

The Femur . . . . . 23
The Intramedullary Nail . . . .. 69

IV. MODELING OF THE OSTEOSYNTHESIS. . . ... 75

The Preliminary Idealized Model . . .. 75
Intramedullary Interfacial Behavior . .. 83
A General Intramedullary Osteosynthesis Model . 90

V. EXPERIMENTAL VERIFICATION OF THE MODELING
OF THE OSTEOSYNTHESIS . . . . 121

Well Characterized Systems. . . .. 121
In vitro Osteosynthesis . . . .. 134
Conclusions . . . . . 145










VI. BONAIL PREDICTION OF THE OSTEOSYNTHESIS
DEFLECTION. . . . .... .148

APPENDIX A. DEVELOPMENT OF THE DIFFERENTIAL
BEAM DEFLECTION EQUATIONS. . . ... 178

LIST OF REFERENCES ...................... .184

BIOGRAPHICAL SKETCH ................... 189
















LIST OF TABLES
Table Page

3.1 Values for the mean side-to-side difference,
Tp, and standard deviations (in parentheses)
for the paired femurs . . .. . 34

3.2 Geometric properties of the 14 mm Fluted type
nail, with and without flutes . . .... .72

4.1 Finite difference computational molecules ..... 107

4.2 System properties for optimization evaluation ..... 119

















LIST OF FIGURES


Figure


Page

2


15


17


1.1 Bones of the lower extremity . . .

2.1 The effect of strain rate on the material
properties of bone . . . ..

2.2 Cross-sectional shapes of commonly used
intramedullary nails . . . .

2.3 Structural properties of commonly used
intramedullary nails . . . .

2.4 Elastic modulus for several implant materials. .

3.1 Schematic representation of femoral
length measurements. . . . .

3.2 Serial cross-sections of a representative femur
with size reference marks in place . .

3.3 Definition of axes and distances for geometrical
property determination . . . .

3.4 Mohr's circle construction for locating principal
axes and directions of the cross-sections. .

3.5 Gross cortical bone area as a function of
position for the left femurs . . .

3.6 Gross cortical bone area as a function of
position for the right femurs. . . .

3.7 Area moment of inertia, Ixx, as a function
of position for the left femurs. . . .

3.8 Area moment of inertia, Ixx, as a function
of position for the right femurs . . .

3.9 Area moment of inertia, lyy, as a function
of position for the left femurs. . . .


. 19

. 20


. 24


. 26


. 28


. 30


. 35


. 36


. 37


. 38


. 39









Figure


3.10 Area moment of inertia, I as a function
of position for the right lemurs . . .. 40

3.11 Principal moment of inertia, I as a function
of position for the left femurs. . . ... 41

3.12 Principal moment of inertia, I T, as a function
of position for the right femurs . . .. 42

3.13 Principal moment of inertia, I C, as a function
of position for the left femurs. . .... . 43

3.14 Principal moment of inertia, IC, as a function
of position for the right femurs . . .. 44

3.15 Polar moment of inertia, J, as a function of
position for the left femurs . . ... 45

3.16 Polar moment of inertia, J, as a function of
position for the right femurs. . . ... 46

3.17 Effective polar moment of inertia, Jeff, as a
function of position for the left femurs ...... 47

3.18 Effective polar moment of inertia, Jeff, as a
function of position for the right femurs. . ... 48

3.19 Principal angle orientation and relative
magnitude of the principal moments for
the femurs in group I. . . . ... 51

3.20 Principal angle orientation and relative
magnitude of the principal moments for
the femurs in group II . . . .... 53

3.21 Principal angle orientation and relative
magnitude of the principal moments for
the femurs in group III. . . . ... 55

3.22 Section modulus, Z, as a function of length
along the femur for two sample femurs compared
to data adapted from Koch (1917) . . .. 59

3.23 Graphical representation of overall femoral
length versus average cortical bone area ...... 66

3.24 Graphical representation of lesser trochanter
to femoral condyle length (Llesser) versus
average cortical bone area . . .... 67


Page










Figure


3.25 Graphical representation of the outer midshaft
diameter versus average cortical bone area . .. 68

3.26 Calculated geometric properties for several
intramedullary nail shapes . . .... 71

4.1 Schematic representation of preliminary idealized
osteosynthesis with applied four-point loading
and point contacts . . . . 76

4.2 Free body diagram of the outer tube with external
and point contact reaction loads . . .. 78

4.3 Free body diagram of the inner rod with point
contact reactions. . . . . .. 78

4.4 Idealized angular mismatch representation. . ... 80

4.5 Photograph showing the intramedullary nail in
place in the cast bone section for interfacial
stiffness determination. . . . ... 84

4.6 Schematic drawing of the self-aligning compression
fixture for interfacial stiffness determination. 86

4.7 Load-deflection curves for interfacial behavior
of Kuntscher, Fluted and Schneider nails
in cancellus bone. . . . . ... 87

4.8 Load-deflection curves for interfacial behavior
of Kuntscher, Fluted and Schneider nails
in cortical bone . . . . 88

4.9 Schematic diagram of a segment of the general
osteosynthesis model under distributed load q(x) 92

4.10 Free body diagram of an element of the bone
with external distributed load and interfacial
reaction load. .. . . . .. 92

4.11 Free body diagram of the nail with interfacial
reaction load. . . . .... 92

4.12 Schematic diagram of an osteosynthesis under
equally spaced four-point loading with a
midshaft transverse fracture . . .. 102


viii


Page











4.13 Schematic diagram of a healed osteosynthesis
under equally spaced four-point loading with
the nail remaining in situ . . . .. 105

4.14 Finite difference representation of the
osteosynthesis under four-point loading. . ... 110

4.15 Modified computational molecules for the
describing nondimensional differential
equations of the coupled system. . . ... 114

5.1 Schematic representation of the four-point
loading apparatus used for in vitro verification
studies. . . . . ... ... 122

5.2 Intact human femur with small dimples at 2 cm
intervals to aid in deflection measurement . .. .124

5.3 Intact human femur mounted in four-point loading
fixture for in vitro testing . . .... 125

5.4 Intact femur, with dial indicator in position,
in the four-point loading fixture. . . ... 126

5.5 Graphical comparative deflection results obtained
from the testing and modeling of a uniform steel
beam . . . . ... . . 129

5.6 Graphical comparative deflection results obtained
from the testing and modeling of a tapered
steel beam . . . . ... . 130

5.7 Measured deflections and BONAIL predictions for
an unnailed intact human femur under four-point
load . . . . ... ... .. 131

5.8 Comparative results obtained by measurement and
BONAIL prediction of a four-point loaded intact
femur with 12 mm Fluted nail in place. . ... 132

5.9 Close-up photograph of the fracture site of a
femur with Fluted nail in place. . . ... 136

5.10 Osteosynthesis in the test fixture with the
fracture surfaces distracted to show the
nail and fracture site location. . . ... 138

5.11 Osteosynthesis in the testing fixture for
in vitro verification of BONAIL predicted
deflection response. . . . ... 138


Figure


Page









Figure


5.12 Graphical presentation of actual deflection
measurement and BONAIL prediction for the
12 mm Fluted intramedullary osteosynthesis . .. .140

5.13 Presentation of results of deflection
measurement and BONAIL prediction for a 14 mm
Fluted intramedullary osteosynthesis . ... 141

6.1 Nondimensional midspan bone deflection for the
healed osteosynthesis with intramedullary
nail in situ . . . .... .. '. 151

6.2 Nondimensional interfacial loading profiles for a
healed osteosynthesis with interfacial stiffness
of 5 x 106 N/m/m at various rigidity ratios. ... .152

6.3 Nondimensional interfacial loading profiles for a
healed osteosynthesis with interfacial stiffness
of 5 x 107 N/m/m at various rigidity ratios. ... .153

6.4 Nondimensional interfacial loading profiles for a
healed osteosynthesis with interfacial stiffness
of 2 x 108 N/m/m at various rigidity ratios. ... .154

6.5 Nondimensional interfacial loading profiles for a
healed osteosynthesis at a modulus ratio of 13.0
for various interfacial stiffnesses. . ... 155

6.6 Nondimensional interfacial loading profiles
using different properties and stiffnesses
to produce equal midspan bone deflections. . ... 156

6.7 BONAIL predicted midspan deflections for a fractured
osteosynthesis as a function of modulus ratio for
various interfacial stiffnesses. . . ... 158

6.8 BONAIL predicted interfacial reaction profiles for
fractured osteosyntheses at a modulus ratio of
4.34 for several interfacial stiffnesses . ... 160

6.9 BONAIL predicted interfacial reaction profiles for
fractured osteosyntheses at a modulus ratio of
8.7 for several interfacial stiffnesses. . .. .161

6.10 BONAIL predicted interfacial reaction profiles for
fractured osteosyntheses at a modulus ratio of
13.0 for several interfacial stiffnesses 162


Page









Figure


6.11 Nondimensional fracture surface mismatch
as a function of modulus ratio for various
interfacial stiffness values ... . . .163

6.12 Fracture surface angulation as a function of
modulus ratio for various interfacial
stiffness values . . . . .. 164

A.1 A segment of a beam having been curved by
an applied moment Mb . . . .. .179

A.2 Stress distribution profiles developed in
the curved beam. . . . . .. 179

A.3 A segment of a beam under distributed
load, q(x) . . . . ... . 182

A.4 A differential element of the beam with
distributed load, q(x) and reaction
moments and shears . . . .... 182


Page
















Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy


A BIOMECHANICS APPROACH TO THE INTERACTION OF HOST
IMPLANT SYSTEMS

By

Gary Joel Miller

June 1977

Chairman: Robert B. Gaither
Cochairman: George Piotrowski
Major Department: Mechanical Engineering

A general model of an intramedullary osteosynthesis of the

femur is postulated. The model allows for variable geometric

properties along the length of the bone, and allows homogeneous

and isotropic material properties to be chosen. The model includes

an intramedullary nail of arbitrary material and geometric proper-

ties running the full length of the bone and unvarying with position.

The nail and bone are assumed to be separated by a linearly elastic

layer which transmits the nail-bone interfacial loading.

Nondimensional multicomponent beam equations which relate

external loading to deflection are presented. Boundary conditions

which describe four-point loaded osteosyntheses with transverse

midshaft fractures and osteosyntheses consisting of healed bones

with intramedullary nails remaining in situ are included. The

resulting equations are cast into finite difference form and are









solved with the aid of a digital computer program (BONAIL) using

numerical relaxation techniques.

Results of in vitro verification studies involving steel

beams, intact femurs, and femurs with intramedullary fixation are

presented and serve to substantiate the accuracy of the BONAIL

model. Both the effect of relative flexural rigidity of the com-

ponents and the character of the interfacial layer are found to

play a major role in the solutions obtained.

Graphical results are presented for the nondimensional

midspan bone deflection, the transverse deflection mismatch and

angulation of the fracture surfaces, and the nail-bone interfacial

loading profiles. These results are plotted as a function of the

relative nail-bone elastic modulus ratios for various interfacial

layer stiffnesses.

These data indicate that the more flexurally rigid devices

reduce the overall midspan deflection, deflection mismatch, and

angulation. However, the use of different combinations of modulus

ratios and interfacial stiffnesses are found to result in comparable

deflections. This is caused by the different interfacial reaction

distributions and intensities which develop. Nonetheless, the

overall results appear to support using nails of high flexural

rigidity to effect lower deflections and therefore enhance primary

healing of the fracture. However, healed bones, with more rigid

nails remaining in situ, are subjected to significant load sharing

between the components, and the bone is partially protected from

stress. By being so protected, the functional remodeling at the


xiii









fracture may be hampered and result in a weakened bone which is

liable to refracture upon removal of the intramedullary nail.

The lack of information in the literature concerning the

geometric properties of the femur and the character of the nail-bone

interface for use in modeling the system led to a parallel experi-

mental study. Ten pairs of human femurs were processed to determine

the gross cortical area, area moments of inertia, principal moments

of inertia and principal angles, and polar moments of inertia at

serial points from the level of the lesser trochanter to the distal

metaphysis.

The results of this experimental study are presented graphi-

cally as a function of position along the femoral length. While

considerable variation in properties is found from one pair to the

next ( 25% or more), no statistically significant right-left bias

was observed.

In vitro experimental results of the interfacial stiffness

behavior of several nail types were also obtained in different bone

material. These results ranged over several orders of magnitude

and were dependent on both the nail cross-sectional shape and the

nature of the medullary bone, i.e. cancellus or cortical.















CHAPTER I

INTRODUCTION


The femur is one of the major load carrying members in the

human body. It is the single bone which joins the knee to the hip

(Figure 1.1) and is therefore called upon to support essentially

the entire weight of the body during standing. During walking or

strenuous activity, this bone is subjected to loads several times

larger than the weight of the body. Fractures of the femur are

not uncommon, and present a management problem to the orthopaedic

surgeon. One common mode of treatment for transverse femoral

fractures involves the insertion of an intramedullary nail through

the marrow cavity of the bone, using this metal rod as a strut to

hold the two pieces of bone in alignment while they heal together.


Brief History of Intramedullary Fixation

Lambotte of Belgium (1913) has been credited with using the

first intramedullary nail in a clavicle in 1907. It was not until

1918, however, that Hey Groves of England used intramedullary nailing

for fixation of a fracture of the femur. Rush and Rush (1937, 1939)

presented the first cases of intramedullary fixation in the United

States using nails of round cross-section. Kuntscher's (1940)

classic treatise, "The Intramedullary Nailing of Fractures," served

to lend strong support to intramedullary nailing as a viable method














'~~~r~r ': *T .. I


Figure 1.1. Bones of the lower extremity









of fixation of fractured long bones. At that time, Kuntscher intro-

duced his V-shaped cross-section nail which was said to be more

easily driven into the intramedullary canal than a solid nail because

its open cross-section allowed it to compress on insertion. This

shape was subsequently modified to a cloverleaf pattern which fol-

lowed a guide pin more easily and provided greater nail strength.

Street et al. (1947) reported the use of the diamond

cross-section nail which had been developed by Dr. Hansen in

1945 to provide greater flexural and torsional rigidity. Schneider

designed a four flanged nail in 1950, which could be driven into

the intramedullary canal and engaged the endosteal surfaces with

its flanges to maintain torsional stability. Its cross-section was

also said to provide higher flexural rigidity than previous designs.

(Schneider, 1968). In 1967, the Fluted nail was used for the first

time (Allen, 1977). This nail was designed to provide increased

torsional and bending rigidity and better torsional gripping

of the intramedullary surfaces.

The reader gets the impression from a cursory review of these

clinical reports on the evolution of intramedullary devices that the

author's primary concern in redesign is to increase the torsional

and bending rigidity of the device in order to affect a better

osteosynthesis. There is, however, no experimental verification

that continued increases in rigidity will, in fact, produce this

desired result.









Previous Work

McKeever (1953) has summarized the general criteria for

design and application of an effective intramedullary device.

His conclusions, and those of the above investigators, can be

summarized as follows:

1. The nail should maintain alignment and apposition
of the fracture surfaces, i.e. close fit at the
fracture site.

2. It should permit compression of the apposing fracture
surfaces by allowing axial sliding of the bone seg-
ments along the nail.

3. It should prevent torsion of the fractured bone
segments with respect to each other.

4. The nail must be constructed from materials which
meet biocompatibility requirements.

5. It should be able to withstand.the synamic bending
stresses applied by the bone segments and their
related muscle groups.

6. The nail must be rigid enough to provide suitable
bending stabilization so that new bone can form.

Many authors have directed their attention to meeting these

criteria which are generally accepted as necessary for the attainment

of good results using intramedullary nails in fracture fixation.

Lindahl (1962, 1967) stated that rigidity (6) and immobilization (1 and

3) are of the highest priority. Braden and Brinker (1973), Andersen

(1965), Andersen et al. (1962), and Hutzschenreuter et al. (1969) have

supported Lindahl and reiterate the need for "rigid fixation" to

affect proper healing using intramedullary nailing. Taylor (1963) has

also noted that the use of intramedullary nails which do not produce

complete rotational fixation (3) will lead to a higher incidence of

complications such as malunions and nonunions.









In clinical use, variable success has been reported in the

literature regarding intramedullary fixation. Wickstrom et al. (1968),

in a review of 324 total cases, cited many of the complications that

occur in the use of intramedullary fracture fixation. While many of

these complications were biological in nature (i.e. infections, etc.),

others were the result of improper implant choice or application.

These biological and biomechanical complications led to 3 1/2% of

the patients having delayed or nonunion of their fractures. These

data support the work of Nichols (1963), who also found a 3% delayed

union rate in intramedullary osteosyntheses.

Success or failure of this surgical procedure depends on the

mechanical interaction between the nail and the bone. Peters (1972)

has put the problem into perspective.

Studies of the most appropriate material for
immobilization of fractures must be correlated with
determination of the biomechanical factors which act
to increase and decrease fracture healing. The
material and method of application can then be de-
signed to minimize the disruptive stress at the
fracture site and enhance those that increase
healing... If the healing process can be materially
speeded and non-union prevented, months and years of
patient disability could be eliminated...(p.18)

Attempts to improve intramedullary fracture fixation have

focused on increasing bending rigidity, torsional rigidity, ultimate

strength, and biocompatibility of the nail, by experimenting with

variations in cross-sectional geometry, length, and material com-

position of the nails. Results reported to date are, however,

qualitative and have not been correlated with mechanical factors.

Both good and bad results have been reported using similar surgical

techniques and intramedullary nails.









The above review of intramedullary fixation leads to two

major questions. Do the relative flexural rigidities of the nail

and bone have a significant effect on the displacements of the

fracture surfaces of the loaded osteosynthesis? Are there other

parameters which affect the performance of the system?

Allen et al. (1968) underscored the need for understanding

this interrelationship of the geometric, material, and structural

properties of the nail and bone in an intramedullary osteosynthesis.

However, quantification of the deflection at the fracture interface

of an osteosynthesis has not been published to date. Bourgois and

Burny (1972) took a first step in describing this complex problem.

In analyzing various loading configurations, they developed relation-

ships between fracture callus rigidity and percentage healing. The

model used consisted of a cantilever beam (the intramedullary nail)

surrounded by bone, with a fracture callus present at the midsection.

Deflections were calculated for this type of system under a trans-

verse end load. To carry out these calculations, an assumption of

complete bonding of the nail to the bone was implicitly made (i.e.

the interface between bone and nail transmitted shear stresses).

This transmission of shear stresses at the interface affects the

deflection results significantly and is not seen clinically with

intramedullary nails. However, first approximations on the effects

of variations in geometry, material, and fracture callus properties

on deflections were obtained. No other quantitative studies which

deal with intramedullary fixation have been found, and the following

investigation was therefore undertaken.









The Present Research

The present work is aimed at quantification of the nail-bone

interactions and the relative displacements of the fracture surfaces

present during the bending of an osteosynthesis. Utilizing engine-

ering techniques and analyses, supported by in vitro experimental

verification, the ability of different nail shapes and sizes to

maintain alignment and fracture surface apposition are presented.

A model of the femur is used which involves varying geometric

properties and constant material properties along its length. An

intramedullary nail of constant material and cross-sectional

properties is within the bone. It is assumed that a four-point

bending load is applied; consequently, interfacial loading is

assumed to develop at the nail-bone interface due to relative

differences in displacements of the nail and bone. Osteosyntheses

with and without fractures present are considered. Emphasis is

placed on the effects of the observed differences of structural

properties of currently used intramedullary nails on the inter-

facial deflections at the fracture site and the expected contact

forces between the nail and the bone in the medullary canal.
















CHAPTER II

PROPERTIES OF THE SYSTEM COMPONENTS: A REVIEW


The engineer is often faced with the need to analyze the

deflection behavior of load carrying machine elements. In order

to proceed, it is important that the engineer consider the elements

in a two-fold manner. The first involves understanding of the

components as geometrical entities, i.e. their shape and/or dis-

tribution of material. The second involves the appreciation of

the properties of the materials from which the elements or devices

are fabricated. By combining these two views, a quantitative

evaluation of the performance and interaction of the devices can

be performed. It is believed that understanding and modeling the

intramedullary osteosynthesis requires the same approach. There-

fore, this thesis begins with an examination of the existing

information on the structural and material properties of the

components of the intramedullary osteosynthesis.


The Intact Femur

Geometrical Properties

Koch (1917) was one of the first to apply engineering

principles to the determination of the stress response of the

loaded femur. Using time consuming graphical methods, he analyzed

the shape and distribution of bone in a single human femur. He









showed that the external form of the bone, and its internal

structure, were proportioned in such a way as to resist axial,

bending and shear loads applied during normal activity with a

minimum use of material.

Photoelastic techniques (Pauwels, 1951) and stress lacquers

(Evans and Lissner, 1948, and Frankel, 1960) have also been used

to determine stress levels in bones under axial and bending loading.

More recently, Toridis (1969) and Rybicki et al. (1972) have pre-

sented analyses using engineering beam theory to predict stress

levels throughout the femur. Their results indicate that engine-

ering beam theory is applicable to analysis of the long bones in

the midshaft region. (The bone ends, with their flared structures,

are not adequately modeled using these methods.) Piziali et al.

(1976) analytically examined several assumptions inherent in the use

of classical engineering beam theory to long bones (specifically

tibias), and found that deflections due to shear deformations are

negligible, and that the effects of initial curvature are not

significant. The mismatch between the neutral axis of the

cross-section and the applied moment, however, introduces de-

flection components perpendicular to applied loads. Their assumption

of linearly elastic properties led to predicted deflections which

were inversely proportional to the modulus of elasticity of the

bone with shear deformations causing negligible deviation in this

behavior. Yamada and Evans (1970) presented empirical load-deflection

curves for long bones but did not relate these data to the actual

geometry of the bones involved.









Long bones have an irregular cross-sectional geometry

which varies considerably along their length. The determination

of geometric properties, such as the area moments of inertia, is

necessary to apply the above cited strength results to other bones.

As Evans (1957) pointed out, much of the work on structural per-

formance of long bones requires approximation of the geometric

properties of bones. Allen et al. (1968), realizing this problem,

predicted the structural rigidity of a femur in bending and torsion.

They used tedious numerical integration techniques which yielded

the moments of inertia necessary for calculation of the bending and

torsional rigidity of the femur. This work was, however, limited

to only three cross-sections. Piotrowski and Wilcox (1971)

developed a computer program which performed the numerical

determination of the geometric properties and stresses in solid

(singly connected) cross-sections and allowed the analysis of

over 200 cross-sections of canine fibulae (Piotrowski, 1975).

The program was updated (Piotrowski and Kellman, 1973) to take

into account the presence of internal holes (the intramedullary

canal). Minns et al. (1975) used finite element techniques to

analyze four left tibias and reported variations in geometrical

properties as a function of length. They also found that geometrical

properties between bones can differ by 30% or more. Piziali et al.

(1976) determined the geometric properties of two femurs, two

fibulae and four tibiae. However, they presented only the tibial

information, which corresponded to that reported by Minns et al.

(1975).









It is important to note that adequate understanding of the

femur as a structure required the knowledge of the geometric

properties and response to load of the system under consideration.

Knowledge of only the load response is not sufficient to allow

for application of the results to the modeling of other loading

situations or bone configurations.


Material Properties

The material properties of bone have received more attention

than the geometric properties. Many hypotheses have been offered

to explain the microscopic behavior of bone. Currey (1964), Katz

(1970), and Kraus (1968) discuss the many theoretical lumped

parameter models used to describe the mechanical behavior of bone

substance on a microscopic scale. The present work, however, is

concerned with the macroscopic behavior of bone, and the interested

reader is referred to the above review articles for further dis-

cussion of microscopic behavior and modeling.

As Kummer (1972) has pointed out in his review of the bio-

mechanics of bone, bone is neither morphologically homogeneous nor

isotropic. In fact, Reilly and Burstein (1974) indicate that a

transversely isotropic model would be more accurate than the

homogeneous and isotropic representations which are often used in

bone modeling. Whether or not bone can be regarded as homogeneous

with respect to its macroscopic behavior, however, remains to be

seen.










Reilly (1974) and Reilly and Burstein (1974) have presented

a comprehensive, critical review of the literature on the mechanical

properties of human cortical bone. Their review gives a detailed

discussion of the many variations in mechanical properties of bone

presented throughout the literature. These include the effects

of specimen handling, and testing procedures for determination of

these elastic properties. Several articles referred to are dis-

cussed below.

Reilly (1974) measured the following average properties for

freshly frozen human bone tested at high strain rates:


E' = 17.1 x 109 N/m2


E = 11.5 x 109 N/m2

v = 0.46

v' = 0.58


G' = 3.28 x 109 N/m2

where E is the modulus of elasticity, v is the Poisson's ratio, and

G is the shear modulus.- The primed values refer to properties

found parallel to the long axis of the bone, and the unprimed values

refer to the radial and transverse properties. There are large dif-

ferences in properties in these two orthogonal directions. The

values reported for E and E' are associated with standard deviations

of approximately 15%-20% of the modulus if determined in tension,

and 7%-10% if determined in compression. The Poisson's ratios had a

standard deviation which represented 30% of their mean values.

Standard deviations for the shear modulus were typically 10%.









These results for the variations to be expected in determin-

ation of the material properties correlate well with the results

cited by Evans and Lebow (1951), who made additional observations

concerning the local variations in properties along the length of

a given bone. They found 2%-8% variations in the modulus of

elasticity and similar variations in the ultimate strength and

hardness as well. These results indicate that, while there is

some variation to be expected in the properties measured at dif-

ferent locations along a single bone, these variations are not

statistically significant when considering the overall scatter in

results obtained at several locations on several different femurs

as indicated by Reilly.

As Reilly (1974) noted, the average results he obtained in

measuring material properties were for freshly frozen, nonembalmed

bone. Many authors have dealt in considerable detail with the

effects of different treatment on the mechanical properties of

human cortical bone (Evans and Lebow, 1951; Evans, 1957; McElhaney

et al. 1964; Sedlin and Hirsch, 1966; Kraus, 1968; Kummer, 1972; and

Burstein et al. 1972). They made several salient observations:

1. There was no significant temperature effect on the
maximum failure stress (210 370). The modulus
of elasticity was found to increase slightly along
with maximum deflection at 370.

2. Air drying alters the physical properties in only
a few seconds, and a marked increase, as much as
25% or more, in the modulus of elasticity is observed.
A decrease in the observed plastic deformation occurs
as well.

3. While freezing increases the strength of the specimen
slightly, it does not alter any other properties.










4. Fixation in formalin does not appear to alter the
modulus of elasticity.

5. Significant differences were found in all the prop-
erties of bone from individual to individual and
these are not correlated with age.

The results on the effects of embalming on the properties of

bone are equivocal. It has been suggested (Allen, 1977) that

embalming may have a highly variable effect on the mechanical

properties of the bone due to the differences in vascularity and,

therefore, disposition of the embalming medium.

McElhaney and Byars (1965), testing bovine bone in compression,

Panjabi et aZ. (1973), testing rabbit femora in torsion, and Burstein

and Frankel (1968), testing human bone in torsion, also point out

that the physical properties of bone may vary with changes in load

or strain application rates. Figure 2.1 indicates the large

variations in properties which may be expected.

Thus, as Reilly and Burstein (1974) have pointed out, data

can be extracted from the literature to support or contradict many

hypotheses. This is not due to any inherent ambiguities in the

data, but is rather due to the absence of standardization or unifi-

cation of experimental approaches and goals. These large variations

in the reported data indicate the need for more work before the

intrinsic and extrinsic variations of bone properties are clearly

understood.
























HUMAN

'max -Suc


BOVINE


A Elastic Modulus

* Ultimate Strain


i I I I L L `


10-4 10-3 10-2 10-1 100


102 10j 10*


STRAIN RATE


Figure 2.1.


The effect of strain rate on the material
properties of bone (Bovine data from McElhaney
and Byars, 1965; Human data from Burstein
and Frankel, 1968)


Co
(0
0-

C\J

CC
)



C

Cu X
E

t-










The Intramedullary Nail


Geometrical Properties

Some of the more commonly used nail shapes are shown in

Figure 2.2. The American Society for Testing and Materials (1976)

has developed standard specifications for the geometry of Kuntscher,

or cloverleaf nails (F339), and for solid cross-section intramedullary

nails (F455) which include the Schneider and Hansen-Street configura-

tions. There has been no specification adopted to date for the
**
newly developed Fluted nail. It is important to note that these

are voluntary standards and that the manufacturers are not bound

by them in any way. In actuality, the dimensions obtained from

the implants themselves appear to vary considerably from nail to

nail and manufacturer to manufacturer for a given cross-sectional

configuration. It is, therefore, important to consider these

devices on an individual basis for determining their geometric

properties. There has been little done in this area in the way of

characterizing the important geometric properties of intramedullary

devices.

Allen et aZ. (1968) examined the overall structural per-

formance of various nail types in torsion and bending. These

included the Kuntscher, Hansen-Street, and Schneider nails. They

neglected to note, however, the nail size being considered. The



F339, Specification for Cloverleaf Intramedullary Pins;
F455 Specification for Solid Cross-Section Intramedullary Nails.

Sampson Corporation, Pittsburgh, PA.
Sampson Corporation, Pittsburgh, PA.































































Figure 2.2. Cross-sectional shapes of commonly
used intramedullary nails









Sampson Corporation (1976) used proposed ASTM standard test methods,

which appeared to be F383, to evaluate the bending rigidity, bending

strength, and torsional rigidity of these devices. Their comparative

results appear in Figure 2.3 a-c. In this test method, however, the

geometrical properties were not explicitly considered. The data

needed to further model nail use in an osteosynthesis is, therefore,

only partially known.


Material Properties

The ASTM has also devised several recommended standard

specifications for the materials commonly used in implant manufacture

(F55, F56, F67, F75, F90, F136, F138, F139). These include the

chemical requirements, and the minimum acceptable mechanical

properties, including ultimate tensile strength, yield strength,

elongation, and reduction of area. Because of the effects of

machining, cold working, and processing techniques in fabrication

of implants from the annealed metals, these values for the unworked


*
F55, Specification for stainless steel bars and wire for
surgical implants; F56, Specification for stainless steel sheet
and strip for surgical implants; F67, Specification for unalloyed
titanium for surgical implants; F75, Specification for cast
cobalt-chromium-molybdenum base alloy for surgical implants;
F90, Specification for wrought cobalt-chromium alloy for surgical
implants; F136, Specification for titanium 6A1-4V ELI alloy for
use in clinical evaluation of surgical implant materials; F138,
Specification for stainless steel bars and wire for surgical
implants (Special quality); F139, Specification for stainless
steel sheet and strip for surgical implants (Special quality).


















BONE 316SS INTRAMEDULLARY
{inside by outside DEVICES (diamelers in mm)
diameters in mm) 0 ( >

17X 29.,
16x28 (0I) BONE
S} SAMPSON
iSX27T. I KUNTSCHER

14X26 SCHNEIDER

13X25 *ANSESTREET
APPROPRIATE
12X24 \ROD SIZE
X2 \ due to FLUTES
II 23
OX 22
I16
14 7



13 .*1
:12 .11
11-111\0 .8.9
<) ^"7y ^"~9


Figure 2.3.


Structural properties of commonly used intramedullary nails
(Data taken from Technical Memorandum 1-76, Sampson Corp.
Pittsburgh, Pa.)


600
soo


s
P 500
it



(3
Z 400



W
I--
S300

I.-
U)
200
z
a
a 100


0


BONE 316SS INTRAMEDULLARY
(inside by outside DEVICES diameters in mm
diameters in mm) 8 ( '


() BONE
a SAMPSON
400
o () KUNTSCHER
SCHNEIDER
S(0 UNSEN-STR EET
APPROPRIATE
SIE \ ROD SIZE
S6 due to FLUTES


Q
.- /

20 200



14X26
16X 25 2
S 4 -X-26 12
100 12X 24
11X23
tOX 22
1 2 -10 .13

I1\ V2 12. I .^ 1*II
9 I7 5 o .O
0 4~ 1




























a
0
C)
25


u 20


S15



10
"o
0
2


Cr Co Mo Alloy

cold
worked
annealed
Stainless n a
Steel


Ti6A14V


Bone


Figure 2.4. Elastic modulus for several implant materials










material have little use in evaluating the performance of the

devices themselves. Figure 2.4 gives the average values for the

moduli of elasticity of several alloys used in implants. These

values have been shown as ranges due to the effects of fabrication

techniques on the mechanical properties of the nails.


Summary

The review of the literature on the mechanical and geometric

properties of the components of an intramedullary osteosynthesis

of the femur, as presented above, has led to several observations.



1. While many investigations have been carried out in an
attempt to characterize the mechanical properties of
the femur, the large variations in results indicate
that careful consideration must be given to the use
of these results in the modeling of an osteosynthesis.
Indiscriminate use can lead to significant differences
in the results obtained.

2. Determination of the geometric properties of the femur
has received little attention. The information reviewed
above concerns itself with the rigidity and ultimate
properties of the nail and the bone. While the flexural
rigidity is related to the modulus of elasticity and
the area moment of inertia, the nonuniform geometry and
difficulty in specifying the modulus of elasticity of
the bone does not allow for the direct calculation of
the moment of inertia or other geometric properties.

3. The material properties of intramedullary nails have
been found to be adequate. Here again, however, care
must be exercised in the use of these properties so
that the effects of machining and other processing are
taken into account (Utah Biomedical Test Laboratory,
1975).

4. The load-deflection behavior of several intramedullary
nails has been carried-out, and the geometric properties
necessary for modeling their use in an osteosynthesis
can be derived from this information.







22

Before a general modeling of the osteosynthesis can be carried

out, quantitative characterization of the geometric properties of the

femur must be performed. The first portion of the experimental work

presented here concerns itself with this characterization.
















CHAPTER III

DETERMINATION OF THE GEOMETRIC PROPERTIES
OF THE SYSTEM COMPONENTS


The Femur


Materials and Methods

Ten pairs of embalmed human femurs were obtained and cleaned

of all soft tissue and periosteum. The femurs were then allowed

to air dry for one day. Information regarding age, sex, height,

and weight of the subjects was not available. Two reference

lengths were measured as shown in Figure 3.1. The first, L r
maj or'
was defined as the overall length of the femur as measured from

the most proximal portion of the femoral head to the most distal

portion of the femoral condyles. The second length, Llesser was

defined as the distance from the midpoint of the lesser trochanter

to the most distal portion of the femoral condyles. With the femur

resting on the posterior aspect of the femoral condyles and greater

trochanter, a small hole, approximately 3 mm diameter, running

approximately anterior to posterior, was drilled in the femur at

the level of the midpoint of the lesser trochanter. A 3 mm dowel

was inserted to act as a reference for position measurements along

the femur. The bone was placed in a plywood mold, with the dowel

marker in a vertical orientation, and plaster poured over it to

make a rectangular slab. After the plaster had set, 20 mm thick



























































Figure 3.1.


Schematic representation of the femoral
length measurements: Llesser, the lesser
trochanter to femoral condyle length, and
Major, the femoral head to condyle length









sections were marked off using the lesser trochanter marker as the

starting point. Approximately 16 serial sections were obtained

from each bone. These represented positions (measured proximal to

distal) from approximately 20% of the total femoral length, the

level of the lesser trochanter, to 80%, the level of the distal

metaphysis.

After sectioning and numbering, each section was carefully

cleaned of bone marrow and a size reference marked directly on the

surface of the plaster blocks (Figure 3.2). The size reference

marks were made so that the magnification and the location,

relative to a fixed axis, of each cross-sectional image could

be maintained. Each block was projected on a large sheet of paper

using an opaque projector. A tracing of the cortical bone of each

cross-section and its reference line was produced. These tracings

were used to generate a set of coordinate points on computer punch

cards, representing the outside and inside boundaries of each

section, using a digitizing machine. The punched data deck of the

boundary coordinate points became the input information for the

SCADS computer program (Piotrowski and Kellman, 1973; Piotrowski

and Wilcox, 1971). The SCADS program was used to calculate the

geometrical properties of the cross-section as defined below.

Cortical bone area. The computer program determines the

area of cortical bone in each cross-section by integrating all



Department of Radiation Therapy, J. Hillis Miller Health
Center, Gainesville, Florida





















2 6 10 14



3 7 11 15
99 '

4 12 16


I _
S5 9 13 17







Figure 3.2. Serial cross-sections of a representative
femur with size reference marks in place









the differential area elements such that


AORT = dA = AA (3.1)
(CORT3
fA i


Area moments of inertia. The area moments of inertia, or

second moments of the area, are generally used in the calculation

of bending stresses and deflections and are a measure of the

distribution of the load-carrying material about a set of orthogonal

axes. For this work the orthogonal reference axes were chosen to

be the medio-lateral (x-axis), and the antero-posterior (y-axis)

planes. The moments of inertia are defined, referring to Figure

3.3, as


I = x2 dA = x dA. (3.2)
xx A i I


I = y2 dA = E y? dA. (3.3)
yy i 1 1


I = xy dA = xiy. dA. (3.4)
xy J 1 1
i


Principal angle and moments of inertia. Another set of

orthogonal axes, n and C, whose origin is at the centroid of the

cross-section can be determined such that


I = n dA = 0 (3.5)
JA

These axes are known as the principal axes, and correspond with

the axes of symmetry in cross-sections which are symmetrical. For

nonsymmetrical cross-sections these axes represent the least and















ANTERIOR


LATERAL


POSTERIOR


Figure 3.3.


Definition of axes and distances for
geometrical property determination


MEDIAL









most rigid directions of the cross-section. The principal moments

are then defined with reference to the principal axes or directions

as


n = n2 dA (3.6)


and


I = Cf 2 dA (3.7)

These quantities can be obtained from a Mohr's circle construction

(Beer and Johnston, 1962) as shown in Figure 3.4. The principal

angle, 0, between the x-axis and the n-axis is calculated as


0 = arctan ( xy ) (3.8)
yy xx

The principal area moments of inertia are also readily computed

from I I and I as
xx yy xy

I +1 I -I
I xx yy + 12 + xx yy (39)
nn 2 xy 2 J

and
I +1 / I I 2
I x yy 2 xx y (3.10)
2 2 xy 2


Polar moment of inertia. Used in determination of torsional

shear stresses and deflections of round cross-sections, the polar

moment of inertia, or polar second moment of area, is defined as


J = fr2 dA (3.11)
-/A




























xx,-Ixy


lIlnn l ii


Iyy,l


Figure 3.4.


Mohr's circle construction for locating
principal moment magnitudes and directions
of the cross-sections










where r is.measured from the centroid of the cross-section. The

polar moment may be calculated using the previously determined

values of the moments of inertia, I and I Using trigonometry
xx yy
and referring to Figure 3.3


r2 = x2 + y2 (3.12)


Therefore, equation 3.11 can be rewritten as


J = f (x2 + y2)dA = x2 dA + f y2 dA (3.13)


and substituting equations 3.2 and 3.3 yields


J = I + I (3.14)
xx yy

The effective polar moment of inertia. The last property to

be considered is the "effective" polar moment of inertia, Jeff. As
eff
den Hartog (1952) has shown in his discussion of the torsion of

machine elements, a torsional stiffness, C, may be defined as the

torque, T, per unit twist, 0. That is


C T (3.15)


Further analysis of elements of circular cross-section leads to

the result that


C = G Jeff (3.16)


where G is the shear modulus of the material and Jeff is the
eff
effective polar moment of inertia. For circular cross-sections

Jeff reduces to J. The noncircular cross-section requires a more










complex derivation to obtain the geometrical determination of the

torsional stiffness. Prandtl's "membrane analogy," as discussed

by den Hartog (1952) and Crandall (1956) allows for the determination

of an effective polar moment of inertia; thus the torsional stiffness

of noncircular cross-sections may still be expressed as a product of

a material and geometric property. Thus, for noncircular sections,


T
S= C = G J (3.17)
0 eff

The effective polar moment of inertia is still a measure of the

area's distribution. However, in this case its value cannot be

obtained by the simple integration of differential area elements

as shown for the polar moment of inertia since it also depends on

the convolutions of the sections periphery. The development of the

equations for determination of Jeff are beyond the scope of this

work,however, and the interested reader is referred to the above

references for further discussion. The SCADS computer program

contains the algorithm for mathematically modeling the "membrane

analogy," and thus calculating Jeff and these results have been

included for comparison with the polar moment of inertia, J, as

defined in the previous section.


Results of the Geometric Properties of the Femur

Raw data for the gross cortical area (A), area moments of

inertia (Ixx and I ), principal moments of inertia (I and I ),

principal angle (0), polar moments of inertia (J), and effective

moment of inertia (Jeff), were extracted from the SCADS computer









runs for the ten femur pairs. The left-to-right variation of these

data, except for the principal angles to be discussed later, were

first evaluated at sequential points along the bone length (from

20% to 80% of the total bone length in 10% increments). As dis-

cussed by Miller and Piotrowski (1974), the side-to-side difference,

d of any property is defined as

P P
d = (3.18)
p PL + PR


where PL and PR are the values of the property for the left and

right femurs respectively. The differences computed at each

incremental position were averaged to obtain the mean value of the

side-to-side difference, dp, and its standard deviation, S.D.,

for each pair of femurs. Table 3.1 summarizes these results. The

accumulated values, obtained by pooling the results from all ten

femur pairs, are also shown. The standard deviations have been

corrected for small sample sizes using the Bessel's Correction

(Moroney, 1951).

Figures 3.5-3.18 have been constructed to indicate the

overall values obtained for the geometrical properties of the right

femurs and the left femurs. The shaded band indicates the total

range over which the raw data falls. A line has been drawn to

indicate the calculated mean value of the property from 20% to

80% of the total bone length. Also included are vertical bars

which represent a range of plus and minus one standard deviation

from the mean.









Table 3.1. Values for the mean side-to-side difference, d ,
and standard deviations (in parentheses) for
the paired femurs

Bone dA d dl d d d
Bone # dA xx Vy Tn dJ Jeff

1 .005 -.079 .006 .014 -.005 .014 .003
(.022) (.058) (.033) (.034) (.034) (.032) (.041)

2 -.018 .035 -.059 -.007 -.034 -.016 -.0134
(.048) (.129) (.046) (.059) (.057) (.053) (.055)

3 -.006 -.036 -.038 -.004 -.002 -.018 -.009
(.019) (.155) (.035) (.033) (.015) (.023) (.0191)

4' .005 .054 -.004 -.010 .061 .018 .032
(.020) (.037) (.036) (.030) (.026) (.021) (.021)

5 .003 .043 -.024 .032 .010 .025 .026
(.030) (.042) (.043) (.037) (.060) (.032) (.041)

6 -.001 -.064 -.030 -.055 -.045 -.046 -.032
(.037) (.032) (.034) (.031). (.036) (.026) (.049)

7 .056 .024 .041 .020 .024 .034 .058
(.024) (.081) (.067) (.052) (.099) (.048) (.034)

8 -.002 .019 -.020 .012 -.024 .006 .002
(.016) (.043) (.055) (.040) (.057) (.032) (.043)

9 -.007 .002 .025 -.005 .018 .005 .006
(.020) (.080) (.049) (.074) (.063) (.064) (.052)

10 -.022 .012 .006 .010 .011 .014 .011
(.037) (.054) (.028) (.043) (.028) (.045) (.053)

Pooled
mean .001 .001 -.010 .001 .002 .004 .0035
Data
S.D. (.030) (.085) (.046) (.024) (.057) (.040) (.047)

t .11 .04 .69 .13 .11 .32 .24

P >.9 >.9 >.5 >.9 >.9 >.7 >.8


n = 10 for all cases











































20 30 40 50 60 70 80
PERCENTAGE OF TOTAL LENGTH


Figure 3.5.


Gross cortical bone area as a function
of position for the left femurs


0









6 -


o


PERCENTAGE OF TOTAL LENGTH


Figure 3.6.


Gross cortical bone area as a function
of position for the right femurs


t I I IL








































oT-
0


PERCENTAGE OF TOTAL LENGTH


Figure 3.7.


Area moment of inertia, Ixx, as a function
of position for the left femurs


I I r IL














51


oI

o0


PERCENTAGE OF TOTAL LENGTH


Figure 3.8.


Area moment of inertia, Ixx, as a function
of position for the right femurs










6r


5


. E







LU
U-
U-
LU
-







2







01-O I I I i I ,
0 20 30 40 50 60 70 80

PERCENTAGE OF TOTAL LENGTH



Figure 3.9. Area moment of inertia, Iyy, as a function of
position for the left femurs











































I-- I


40 50

PERCENTAGE OF TOTAL LENGTH


Figure 3.10.


Area moment of inertia, lyy, as a function of
position for the right femurs


Io-
0


ir










































.-4
0


PERCENTAGE OF TOTAL LENGTH


Figure 3.11.


Principal moment of inertia, ln, as a function
of position for the left femurs


__ ^1 __ __











































0 30 n
0 20 30 an n~ a


PERCENTAGE OF TOTAL LENGTH


Figure 3.12.


Principal moment of inertia, I as a function
of position for the right femurs



















4







V) 3


LLu

Ur-
-J 2







1-







0 II I
0 20 30 40 50 60 70 80

PERCENTAGE OF TOTAL LENGTH

Figure 3.13. Principal moment of inertia, IE as a function
of position for the left femurs
























































0 20 30 40 50 60 70

PERCENTAGE OF TOTAL LENGTH


Figure 3.14.


Principal moment of inertia, IEE, as a function
of position for the right femurs


.



CA
cr,

w
LJ

I-

0
CU



wu
*JJ


































E C I I I I T I
o 20 30 40 50 60 70 8
PERCENTAGE OF TOTAL LENGTH


Figure 3.15.


Polar moment of inertia, J, as a function
of position for the left femurs


111111














`K>


Ij


0 20 30 40 50 60 70 8
PERCENTAGE OF TOTAL LENGTH


Figure 3.16. Polar moment of inertia, J, as a function
of position for the right femurs























.1. 1111111 jIl


0 ,I I I 6 7
0 20 30 40 50 60 70
PERCENTAGE OF TOTAL LENGTH


Figure 3.17.


Effective polar moment of inertia, Jeff, as a
function of position for the left femurs















8








LJ -6



4-





2





0 I I
0 20 30 40 50 60 70 80
PERCENTAGE OF TOTAL LENGTH


Figure 3.18. Effective polar moment of inertia, Jeff, as a co
function of position for the right femurs


10r










The values of the principal angles are more difficult to

present. These values appear to have fallen into three main

groupings. Representatives of these three groups are shown

graphically in Figures 3.19-3.21. Here, the values of principal

moments along the length of a pair of femurs are shown as scaled

orthogonal lengths in their actual principal angle orientation.

In the first two cases, there is a gradual angular displacement

to the principal axes, going proximal to distal. The principal

axes were not well defined in case III.


Discussion of the Geometric Properties

Right-left variability. Using the pooled means which were

calculated from the data presented in Table 3.1, the Student's

t Test (Moroney, 1951) was applied to compare the resulting dis-

tributions with the hypothesis that there is no right-left bias

in the geometric properties, i.e. d = 0. In this particular case,
p
the value of t is indicative of the ratio of the mean to the

standard deviation. The nonzero mean values seen in Table 3.1

are quite likely to occur by chance since they are smaller than

the data's inherent scatter, represented by the standard deviations.

The actual statistical results have been included in the table and

indicate that P, the probability of t being this large or larger

by chance, is greater than 0.5 in all cases. One can thus conclude,

with considerable certainty, that there was no right-left bias in

the geometric properties of the femurs investigated.





































Figure 3.19.


Principal angle orientation and relative magnitude
of the principal moments as a function of position
for a representative pair of femurs from group I
(the darker line indicates the larger moment
of inertia, I )








DISTAL


Left Right


7LF9


PROXIMAL




































Figure 3.20. Principal angle orientation and relative magnitude
of the principal moments as a function of position
for a representative pair of femurs from group II
(the darker line indicates the larger moment of
inertia, I )








DISTAL


Left


Right


3LF9


PROXIMAL




































Figure 3.21. Principal angle orientation and relative magnitude
of the principal moments as a function of position
for a representative pair of femurs from group III
(the darker line indicates the larger moment of
inertia, I )
rnn






DISTAL


+e


Left


Right


PROXIMAL


8LF9










Miller and Piotrowski (1974) showed that one can expect the

standard deviation of the side-to-side difference in the torsional

strength of paired animal bones to be about 9%. The results obtained

above for the standard deviation of the side-to-side difference in

the polar moment of inertia was of the order of 4%. This indicates

that, while the geometric variability has some effect on the observed

strength variations of whole bones, it does not account for all of

it. The strength of whole bones, however, depends on both geometric

and material properties. Reilly (1974) reported that one can expect

variability in ultimate torsional strength of bony material to be

approximately 4%. Thus, it would appear that the combination of

geometric and material variability adequately describes the scatter

found in the strength of whole bones.

Gross Cortical Area. The flared outward shape of the femur

might imply a similar cortical bone area distribution. This is not

the case, however, due to the thinning of the cortex which occurs

in the flared metaphysial regions. The cross-sectional area of

bone as shown graphically in Figures 3.5 and 3.6, tends to increase

only slightly from the level of the lesser trochanter to approxi-

mately 40% of the bone length. This is where the isthmus, or

narrowest section of the intramedullary canal, is found. The cortical

area then decreases monotonically to the knee. At the most distal

data point taken, 80% from proximal, the area is approximately

2.5 cm2. This resulting area, just proximal to the knee, is slightly

larger than the proximal tibia area, just distal to the knee, pre-

sented by Minns et al. (1975) and Piziali et al. (1976). Thus,










ignoring the discontinuity at the knee, the lower limb appears to

be a tapered column of bone mass. This shape reduces the amount

of bone appearing distally, and results in a minimization of the

bones' contribution to the mass moment of inertia of the limb about

the hip. Thus, compared to a limb with bone mass evenly distributed

along its length, the torques necessary to accelerate the extremity

during locomotion are reduced, with the result that the energy

necessary to accelerate the limb is also decreased and higher

efficiency can be maintained.

The slight increase in area seen in the proximal portion of

the femur is due to the presence of the gluteal tuberosity and the

beginning of the linea aspera. These areas allow for the attachment

of the muscle groups which, while providing for rotations about the

hip, also apply significant axial loading in the region due to their

poor mechanical advantage. These axial loads tend to reduce the

tensile stresses set up by bending of the femur due to the loading

of the femoral head, but also requires the need for additional bone

area to maintain suitable compressive stress levels.

Area moments of inertia. It is important to note at the

outset of this discussion of the area moments of inertia (Figures

3.7-3.10) and polar moment of inertia (Figures 3.15 and 3.16) that

there is a relationship between the two, as shown in equation (3.14).

These properties influence the stress levels caused by different

forms of loading, namely bending and torsion. These properties

also influence the rigidity of the system under these different

loading configurations.










The bending stresses, o, developed in the outer surface of

a beam with an applied bending moment, M, may be represented by


Mc
a = M (3.19)


where c is the distance from the centroid of the section to the

outer surface, and I is the area moment of inertia. The quantity

I/c is often referred to as the section modulus, Z, of the beam.

Figure 3.22 shows the results obtained from the calculation of the

section modulus for two representative femurs. Also included is

the result obtained by Koch (1917) in his detailed analysis of a

single femur. While the magnitudes are quite similar, the results

obtained here indicate that the section modulus decreases slightly

from proximal to distal. Koch (1917) and Rybicki et al. (1972)

also note that when a downward load is applied to the femoral head,

the bending moment in the x-plane increases rapidly from the femoral

head to the level of the lesser trochanter and then decreases

monotonically to the knee. Thus, the bending stresses, equal to

M/Z, decrease from the lesser trochanter to the knee, but more

gradually than the bending moment itself. This corresponds well

with the bending stress distribution reported by Rybicki in his

analytical modeling of the femur.

The increase in the area moment of inertia I in the
xx'
proximal femur, also results in an increase in the flexural rigidity

of the bone in this region. This is due to the direct proportionality

between the rigidity and the moment of inertia. The femoral condyles,

which are the bearing surfaces of the knee joint, allow muscular


































2 *



**.. data from Koch (1917)


1- Femur 3L


-- Femur 3R


Ia


SPROXIMAL


DISTAL


Figure 3.22. Section modulus, Z, as a function of length
along the femur for two sample femurs compared
to data derived from Koch (1917)


E



I I
NI










insertions, and provide areas for attachment of the ligaments which

stabilize the knee joint, are responsible for the increase in I
xx
in the last 15%-20% of length. This results in an increase in the

flexural rigidity of the femur in this region as well.

The antero-posterior moment of inertia, I (Figures 3.9

and 3.10) is smaller in magnitude than I in the proximal femur.
xx
It also decreases in magnitude over the proximal 40% of'length,

and then remains almost constant until the femoral condyles are

reached. One reason for the initially larger relative values of

I is the large bending loads which appear in this plane due to
Yy
the femoral head loading, as well as forces generated by the large

gluteus maximus muscle used in extending the hip.

Principal moments and directions. The principal moments of

inertia, In and I were calculated and their principal angle,

relative to the medio-lateral direction, was determined. The

magnitudes of these principal moments show a curved profile

(Figures 3.11-3.14) similar to that for the values of I
xx
However, the orientation of the principal axes was quite remarkable.

Study of the principal axes of the ten femur pairs led to the obser-

vation that the femoral cross-sectional shape at the midshaft

appears to fall into three basic categories. A midshaft section

from the left femur of representatives of each of these groups has

been included in Figures 3.19-3.21 along with scaled orthogonal

lines representing the relative magnitudes of the principal moments.

(The heavier of the two lines indicates the larger value, I .)

Of the ten femur pairs examined, four pairs were found to be










in the first group, four pairs in the second group and two pairs

in the third group.

The first group, shown in Figure 3.19,has an elliptical

cross-sectional shape whose major axis, I runs antero-posterior

at the midshaft. There is definite right-left symmetry as seen in

this figure. At the proximal end of the femur, the major axis is

inclined at an angle which provides greatest rigidity in the

approximate direction of the femoral neck and head. As the midshaft

is approached, the linea aspera dominates the cross-section, result-

ing in a principal angle of approximately 90 (the antero-posterior

direction). In the distal portion, the major axis rotates and

comes to an almost medio-lateral orientation corresponding well

with the femoral condyles.

The second group, Figure 3.20, has an elliptical cross-section

at the midshaft as well, but is perhaps less pronounced than in the

previous group. However, the I axis begins in a more medio-lateral

position and remains as such throughout the length of the femur.

Thus the more rigid direction, corresponding to the I direction,

is now 90 from that of the first group. Here again, the right-left

symmetry is maintained.

The third group, Figure 3.21, has a profile in'the proximal

segment which is similar in magnitude and direction to the second

type described above. However, as the midshaft is approached, the

magnitudes of I and I are almost equal, in contrast to groups

I and II, and determination of the principal direction is uncertain

due to the relatively small size of the Mohr's circle in comparison










to the magnitudes of I and I Thus, the bone has no preferen-
xx yy
tially rigid direction until the femoral condyles are reached. The

only generalization which can be made here is that in all groups

the principal direction is medio-lateral at the femoral condyles.

An attempt to explain the above observations led to the

consultation of the anthropological literature. Comas (1960) makes

several observations relative to the point of femoral shape. The

linea aspera, in the femur of people of the more modern civilizations,

is more pronounced in males than females. The more robust skeleton

of the male has broad areas for muscular attachment and the linea

aspera may develop in such a way as to produce a large, rough, crest

shaped protrusion not unlike that seen in group I (Figure 3.19).

Antero-posterior flattening, as represented by the second group

presented above, is observed in the female skeleton.

This observation was reiterated by Maples (1977) who indicated

that the anthropological and forensic medicine literature suggests

that a pronounced linea aspera is indicative of the male and a

flattened diaphyseal shape of the femur may be indicative of the

female. Maples cautioned, however, that generalization is very

tenuous in this regard.

From a biomechanics point of view, the large linea aspera of

the male increases the rigidity of the system in the plane needed

to resist the large bending loads of the flexors and extensors

of the thigh and knee so that moderate stress levels are maintained.

One might also conjecture that the wider pelvis of the female

results in larger bending loads in the medio-lateral plane and a









principal axis oriented in that plane. Unfortunately, the sex of

the femoral samples studied here was unknown, and substantiation of

this hypothesis must remain for further study.

Polar moment of inertia. The polar moment of inertia, J, is

distributed in a manner similar to I (Figures 3.15 and 3.16), due
XX
to its dependence on I and I as discussed in the previous
xx yy
section. In the proximal femur, the relative increase in this

property appears to be due to the several internal and external

rotators of the hip which attach in this area and apply torsional

loading to the bone.

In the midshaft region, the values for J do not vary con-

siderably due to the uniform torsional loading which is distributed

along the length of the bone. As was discussed in the previous

section on area moments of inertia, the distal femur must also

maintain a fairly large area for support of the cartilagenous

bearing surface of the knee. It accomplishes this by using a

preponderance of cancellus bone which aids in distribution of the

joint loading force. In order to maintain the torsional strength

of the system, however, a thin layer of cortical bone is distributed

at the periphery of this area and the polar moment of inertia is

maintained with efficient use of material. This larger polar

moment of inertia also aids in reducing the loads in the collateral

ligaments necessary to maintain joint stability.

Effective polar moment of inertia. The effective polar

moment of inertia calculated for the noncircular bone cross-sections,

using the SCADS computer program, are shown in Figures 3.17 and 3.18.









While the profile of the distribution is similar to that found for

the polar moment of inertia given in the previous section, the

magnitudes of Jeff are approximately 10%-15% less. The greatest

differences appear in the distal portion of the bone. Thus, the

nonsymmetry of the bone cross-sections result in a decrease in the

actual torsional rigidity of the bone.


Discussion of the Observed Scatter in Properties

The magnitude of the observed scatter in the geometric cal-

culations, characterized by the standard deviations, ranged up to

20%-25% of the mean values. The somewhat larger standard deviations

present at the ends of the bones are due to the difficulty in

discerning the cortico-cancellus bone interface of the intra-

medullary canal. In light of the discussion of the inherent

differences in individuals presented above, this large range does

not seem unreasonable. The maximum error of the SCADS computer

program, for calculations on well defined geometries, has been

shown to be 5% or less (Piotrowski and Kellman, 1973).

In an attempt to quantify the remaining 20% of variation,

nondimensionalization of the observed properties using other

dimensions of the bones was carried out. It was hoped that a

general characteristic shape of each of the property distributions

along the bone length could be generated in this way. Using the

overall length, Lmajor, and the lesser trochanter to femoral condyle

length, Llesser, (Figure 3.1) as nondimensionalization parameters,

did not, however, reduce the magnitudes of the scatter. In fact,










as can be seen in Figures 3.23 and 3.24, there is no correlation

between these lengths and the average calculated bone area. There

is also no correlation between the average area and the midshaft

outer.diameter (Figure 3.25). One must conclude, therefore, that

there are significant differences in the bone properties of dif-

ferent individuals and that these differences are dependent on many

variables. This is consistent with the thought (Comas, 1960) that

it is build and/or body weight, sex, age and general physical con-

dition and not height alone, which determines the size and shape of

an individuals bones. Unfortunately, this information was not

available for consideration in this study.


Conclusion

Even with the substantial scatter present in the geometric

properties discussed in this chapter, the general property dis-

tribution profiles are valid. Several conclusions can be made

from the results presented.

1. No statistically significant right-left bias is evident
in the geometric properties presented.

2. The geometric properties calculated using the SCADS
computer program adjoins the tibial information pre-
sented by Minns et al. (1975) and Piziali et al. (1976).

3. The area distribution profiles do not mirror the out-
ward appearance of the femur. In fact, the entire
lower limb has a "tapered column" distribution of
bone area, with the greater area appearing in the
proximal femur. This distribution of bone aids in
efficient ambulation.

4. The flared shape of the femur is more pronounced in the
medio-lateral plane than in the antero-posterior plane,
and I and I the respective area moments of inertia,
mirroxthis ard appearance.
mirror this onward appearance.













0 AR vs. Lmajor r=0.21


* AL vs. Lmajor r=0.41



0 0


52.0


50.0


48.0


46.0


44.0


42.0


40.0


0


* 0
0o


L-^----- I ----- 1 --- I ------
2.5 3.0 3.5 4.0 4.5 5.

AVERAGE AREAS (cm2)


Figure 3.23.


Graphical representation of the overall femoral length versus
average cortical bone area with the respective correlation
coefficients, r, for the right and left femurs


0
0

















o AR vs. Lesser r=0.31


* AL vs. Llesser- r=0.50

0 O


44.0


42.0


40.0


38.0


36.0


AVERAGE


AREAS (cm2)


Figure 3.24.


Graphical representation of the lesser trochanter to femoral
condyle length, Llesser, versus average cortical bone area
with respective correlation coefficients, r, for the right
and left femurs


00
0
*


34


I I 1
r


-Ot
















0 AR vs. dR r=0.38


* AL vs. dL- r=0.42


S00


0
0


S0


AVERAGE AREAS (cm2)


Figure 3.25.


Graphical representation of the outer midshaft diameter
versus average cortical bone area with the respective
correlation coefficients, r, for the right and left femurs


3.2


3.0


2.8


2.6


I I 1


4










5. The principal directions were found to fall into three
major groups. There are those femurs which are most
rigid in the antero-posterior plane; those which are
most rigid in the medio-lateral plane; and those which
appear to have no preferential orientation. Anthro-
pological information available suggests that these
differences are sex-linked.

6. The calculated effective polar moment of inertia
using the "membrane analogy" indicates that the
noncircularity of the bone cross-section results
in torsional rigidities which are 10%-15% less than
is predicted by the symmetrical polar moment 'of
inertia calculations.

7. The geometric property profiles are distributed along
the bone length in a highly efficient use of the material
so that "strength to weight" optimization appears to be
maintained under complex loading configurations.

The results which have been obtained in this geometric study

of the femur are consistent with the small amount of information

available in the literature and the data presented in Figures

3.5-3.22 can be used in the subsequent structural modeling of this

bone.


The Intramedullary Nail


Materials and Methods

Determination of the geometric properties of five intra-

medullary nails was carried out. These included four nails, of

differing types (Figure 2.2), intended for implantation in a 13 mm

reamed intramedullary canal (13 mm Schneider, 13 mm Kuntscher,

13 mm x 11 mm Hansen-Street, and 12 mm Fluted). A 14 mm Fluted

nail, intended for subsequent use in an in vitro osteosynthesis


model, was also processed.










A transverse section, approximately 2 mm thick, was cut

from the midportion of commercially supplied intramedullary nails

of the above nominal sizes. The sections were then ground and

carefully deburred. In a manner similar to that described in the

previous section for femoral geometry determinations, these sections

were projected and an enlarged tracing of their cross-sectional

outline made. Coordinate points of the boundaries were produced

using the digitizing unit and were input to the SCADS computer

program for geometric property determination. The 14 mm Fluted

nail was processed with and without its flute-like protrusions to

appraise the significance of these gripping points on the overall

geometric properties of the device.


Results

Figure 3.26 shows the results obtained for the geometric

property determination of the four nail types considered. These

results include the gross area, area moments of inertia, polar

moment of inertia, and principal moments of inertia as defined at

the beginning of this chapter. Also included in the figure are the

axes orientations used for geometric property determination. The

principal angles were.zero in all cases due to the orientations

used, i.e. the nails were oriented in their principal directions

during processing. Table 3.2 shows the results obtained from

the geometric processing of the 14 mm Fluted intramedullary nail.










xx lyy Ill 122


L -e- -e-


0.251-


0.20)


0.15-


0.10-


4-e


0.031-


LEGEND

-- FLUTED (12mm)

- HANSEN-STREET (13mm)


-- SCHNEIDER

-9- KUNTSCHER
ORIENTATION
y

S ,X Fluted


-4


0.051


5~~x


(13mm)

(13 mm)


Hansen-Street


Schneider


Kuntscher


Figure 3.26. Calculated geometrical properties for several intramedullary nail shapes
(all nails are intended for use in a nominal 13 mm reamed medullary canal)


1.11


0.09F


0.95


0.7F


0.071


0.5-


0.05


0.3-


I


- .. _










Table 3.2. Geometric properties of
with and without flutes


the 14 mm Fluted type nail,


14 mm Nail


With Without
Flutes Flutes

Area (.cm2) 1.12 1.00

Ix (cm4) 0.19 0.16
xx
I (cm4) 0.19 0.16
yy
J (cm4) 0.38 0.32

Ill (cm4) 0.19 0.16

122 (cm4) 0.19 0.16

0 0.0 0.0









Discussion

The results obtained using the SCADS computer program are

approximate due to the difficulties inherent in processing the

intricate geometry of these cross-sections. Independent cal-

culations indicate that the results obtained are within 5% of the

exact values. The Fluted nail has the largest values for all the

geometric properties determined. This has a pronounced effect on

its relative bending and torsional rigidity. Its very high

cross-sectional symmetry also makes it insensitive to the orienta-

tion of the applied loading. The Schneider nail also has a high

degree of symmetry, while the Hansen-Street and Kuntscher nails

are sensitive to orientation due to large differences observed in

their orthogonal geometric properties.

The results obtained from the processing of the 14 mm

Fluted nail were quite interesting. The flute-like projections

of this nail, primarily intended to maintain torsional stability

of the bone fragments, have a significant effect on the area and

polar moment of inertia. These properties increase almost 20%

with the addition of the flutes to the cross-section while

increasing the area of the section by only 12%.

It is important to note that only one sample of each nail

type was processed. Measurement of the outer dimensions of a few

other samples of similar nails from different manufacturers indicate

that there is considerable difference in the cross-sections of nails

made by different companies. Changes in the wall thickness of a







74.

Kuntscher nail, for example, without altering the outside dimensions,

causes significant changes in the flexural rigidity of the device.

The preliminary results reported here are sufficient to allow for

verification of the osteosynthesis modeling to be carried out at

this time. However, a more detailed study is indicated so that

the effect of implant variability on geometric properties of the

devices can be accurately determined.
















CHAPTER IV

MODELING OF THE OSTEOSYNTHESIS


The Preliminary Idealized Model

To gain insight into the parameters affecting the deflections

occurring in an osteosynthesis under bending loading, an elementary

model was analyzed using engineering beam theory. This model, as

shown in Figure 4.1, consisted of an outer tube with a transverse

cut, representing the fractured bone, and a central rod, representing

the intramedullary nail, passing the full length of the system.

Four-point bending loading was assumed to be applied to the outer

tube with equal spacing between the loading points. Clearances

were assumed to be such that the rod and tube interacted only at

idealized point contacts, as shown in Figure 4.1, and that no

deformations occurred at these points. The space between the

fracture surfaces was assumed to be just wide enough so that these

surfaces did not contact during deflection. The moduli of elasticity

of the outer tube and inner rod, Eb and En, were assumed to be con-

stant throughout, and the components were restricted to linearly

elastic behavior. The cross-sectional geometry of these components

was also assumed to be constant along the system length. Thus, the

moments of inertia of the tube and rod, Ib and In, were unvarying

along the length of the model.







































Figure 4.1. Schematic representation of the preliminary idealized
osteosynthesis model with equally spaced four-point
external loading and point contact reactions between
the nail and bone fragments









Free Body Diagram of the Outer Tube

Due to the symmetry of this system, a free body diagram for

half of the outer tube is constructed as shown in Figure 4.2. With

an applied load P at each of the external loading points, and rod

reactions R1 and R2 as shown, vertical force equilibrium yields


P + R1 + R2 = 0 (4.1)


and


R1 = R2 = R (4.2)


Static moment equilibrium about the left end of the tube yields


PL RL
3 =- 0 (4.3)
3 2

so that

2
R = R =R P (4.4)


Free Body Diagram of the Inner Rod

As shown in Figure 4.3, a free body diagram of the inner

rod can be constructed with reaction loads as shown. Note the

idealized point contacts of the nail-bone interaction.


Deflection Equations

The flexural rigidities of the inner rod (N) and the outer

tube (B) are defined as


N = E I (4.5)
n n


B = Eb Ib


(4.6)












P
L/3 L/6





R1 R





P


Figure 4.2.


Figure 4.3.


Free body diagram of half of the
outer tube with external and point
contact reaction loads


Free body diagram of the inner rod
with idealized point contact reactions


IT6P


'12 % PIIY '12


/3









where En and Eb are the modulus of elasticity of the inner rod and

outer tube respectively, and In and Ib are the area moments of

inertia for the inner rod and outer tube respectively.

The reaction loads on the rod are applied to each end and the

midspan of the component (Figure 4.3). The midspan deflection, z,

for the three point loaded span in this configuration can be written

(Roark, 1954) as
pL3
z= (4.7)
36N

where P is the load applied to the outer tube, of length L, at its

four loading points.

Similarly, from the outer tube free body diagram (Figure 4.2),

an angular deflection of the fracture surface, relative to its

original centerline, can be defined as


PL2
0 PL2 (4.8)
b 64.8B

The outer tube has been constrained to contact the inner rod

only at its idealized contact points, and the centerline of the

inner rod is now inclined at an angle 0 which may be approximated,

for small deflections, as


S z (4.9)
n (L/2)

The fracture interfacial deflection angle, Ot, relative to the

vertical axis, can be obtained, as shown in Figure 4.4, by super-

imposing equations (4.8) and (4.9) as defined above. Due to the

symmetry of the system, the total angular mismatch of the system,













































Figure 4.4. Idealized angular mismatch representation
with the overall angulation, Ot, equal to
the difference in nail angulation, 0 ,
and bone angulation, 0b








T0 is twice 0t and if


pL2 pL2
t n b 18N 64.8B10)

then

PL2 B
T 2 x 2 (3.6 1) (4.11)
TP@L/3 64.8B N


The values obtained for the angular deflection of the fracture

interface are dependent on the applied load configuration. For

example, if the downward applied loads are assumed to be located

at positions L/4 in from each end of the system, instead of at L/3

as discussed above, the resulting angular deflection becomes

PL2 B
2 x 128B (2.66 1) (4.12)
P@L/4


While the form of the equations remain the same, their

coefficients take on new values. Nonetheless, several observations

can be made. The application of bending loads to an idealized

system such as the one shown indicates that an optimum flexural

rigidity ratio exists (N/B = 3.6 or 2.66 depending on the configura-

tion) which results in no angular mismatch of the fracture surfaces.

This implies that in an actual osteosynthesis, where the outer tube

represents the bone, and the rod represents the intramedullary nail,

some optimum property ratio could result in a minimum fracture

surface angulation.

While the many assumptions used to derive the results obtained

above were necessary to allow for the application of elementary








engineering beam equations, these assumptions are not necessarily

consistent with the configuration of an actual intramedullary

osteosynthesis as discussed in Chapter I. The geometry of the

bone, as discussed in Chapter III, is not uniform along its length.

The intimate contact between the nail and the bone, needed to

maintain torsional rigidity of the fragments, does not allow for

large enough clearances to assure only point contacts. 'Large

clearances would also allow for unacceptable relative transverse

motion at the fracture site under shearing loads. Also, as loads

are applied to the system, the reactions cause simultaneous, but

unequal, deflections to take place in the components. Interactions

at other than the original contact points will take place once the

mismatch exceeds the clearance, and, as the loads increase, local

deformations at the contact points will occur. These deformations

result in redistribution of the reaction forces along the nail-bone

interface, changing the loading configurations and the relative

deflections.

While the simplified model indicates that an optimum relation-

ship between the component properties may minimize deflections at

the fracture site, a more general model, which more closely

duplicates the actual osteosynthesis, is necessary before a

quantitative evaluation of this hypothesis can be obtained.

In light of the previous discussions, the material properties

of the nail and the bone, and the geometric properties of the nail,

may be considered to be arbitrary constants whose values depend on

the nail and bone under consideration. The model should, however,









account for the nonuniformity of the bone's geometric properties so

that the bone's flexural rigidity is properly represented as it

varies along its length. It should also provide intimate contact

between the nail and the bone, but should allow for longitudinal

sliding of the components. The nail-bone interface should be

assumed to develop an arbitrary interfacial loading profile which

is dependent on the relative deflections occurring between the nail

and the bone. Contact between the fracture surfaces will be assumed

to be nonexistent.

The previous work cited in proceeding chapters allows for

quantification of all the parameters discussed in the previous

paragraph except for the nail-bone interfacial reactions. In order

to obtain quantitative information on this interaction, an experi-

mental investigation was performed.


Intramedullary Interfacial Behavior


Methods and Materials

A matched pair of embalmed femurs was cleaned of soft tissue

and periosteum. At all times the bones were kept moist using a

saline solution drip. The bones were cast in plaster, and as soon

as the plaster had set, several 20 mm thick sections were cut from

the diaphysial and metaphysial regions of the bone. These sections

were then placed in saline solution until testing.

A Kuntschner nail (13 mm), Schneider nail (13 mm), or Fluted

nail (12 mm) was placed in the intramedullary canal of these

sections as shown in Figure 4.5. Each of these samples was placed






















.4


Figure 4.5.


Photograph showing the intramedullary nail in
place in the cast bone section for interfacial
stiffness determination










on a self-aligning compression fixture (Figure 4.6) for loading

of the nail-bone interface. Using an Instron universal testing

machine at a cross-head speed of 0.02 mm/min, the nails were pressed

against the endosteal surface, which contained either cancellus or

cortical bone. Load-deflection curves were obtained, along with a

measurement of the section thickness, for further calculations.

The results obtained from the load-deflection curves were

used to obtain the interfaciall stiffness" of the cancellus and

cortical bone of the femur in contact with different nail shapes.

This stiffness was defined as the slope of the load-deflection

curve divided by the specimen thickness. Thus, a parameter relating

force per deflection per unit length could be obtained for subsequent

use in the general modeling of the osteosynthesis.


Results

Representative curves for the load-deflection response of the

different nail types tested in cancellus and cortical bone are shown

in Figures 4.7 and 4.8 along with the testing configurations used.

Even with the small number of samples tested, the similarity

in the load-deflection response of the different nail types during

testing in cancellus bone led to the definition of three overall

interfacial stiffness regimes. The first interfacial stiffness range

was low in magnitude and appeared to be due to the initial crushing

of the cancellus bone. A transition region was observed as the

deflection increased. This was followed by a higher stiffness

level after significant deflection and packing had taken place.

These different regimes are shown as different line types in the






























INTRAMEDULLARY
NAIL


BONE
IN


SECTION
PLASTER


SELF-ALIGNING
SUPPORT


Figure 4.6.


Schematic drawing of the self-aligning
compression fixture for interfacial
stiffness determination




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