the role of cancellous bone architecture in misalignment ...€¦ · matthew b. l. bennison a...

The role of cancellous bone architecture in misalignment

and side effect errors

by

Matthew B. L. Bennison

A thesis submitted in partial fulfillment

of the requirements for the degree of

Master of Applied Science (MASc)

in Natural Resources Engineering

The Faculty of Graduate Studies

Laurentian University

Sudbury, Ontario, Canada

©Matthew B. L. Bennison, 2020

ii

THESIS DEFENCE COMMITTEE/COMITÉ DE SOUTENANCE DE THÈSE

Laurentian Université/Université Laurentienne

Faculty of Graduate Studies/Faculté des études supérieures

Title of Thesis

Titre de la thèse The role of cancellous bone architecture in misalignment and side effect errors

Name of Candidate

Nom du candidat Bennison, Matthew

Degree

Diplôme Master of Science

Department/Program Date of Defence

Département/Programme Engineering Date de la soutenance December 11, 2019

APPROVED/APPROUVÉ

Thesis Examiners/Examinateurs de thèse:

Dr. Brent Lievers (Supervisor/directeur de thèse)

Dr. Keith Pilkey

(Co-supervisor/Co-directeur de thèse)

Dr, Shailendra Sharan

(Committee member/Membre du comité)

Dr. Krishna Challagulla

(Committee member/Membre du comité)

Approved for the Faculty of Graduate Studies

Approuvé pour la Faculté des études supérieures Dr. David Lesbarrères

Monsieur David Lesbarrères

Dr. Heidi Ploeg Dean, Faculty of Graduate Studies

(External Examiner/Examinatrice externe) Doyen, Faculté des études supérieures

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Abstract

Cancellous bone is often found at the ends of long bones and similar load-bearing surfaces. Its

complex architecture allows high stiffness and strength, while also minimizing the mass and

metabolic needs of the bone. This architecture confounds attempts to measure cancellous

bone’s mechanical properties to produce a model for predicting its response to various loading

scenarios (e.g., falls). Two of these experimental challenges, specimen misalignment and “side

effects”, are known to be significant; however, the role of architecture on the magnitude of

these artefacts is unknown. The current study used finite element method (FEM) modelling

of bovine cancellous bone to examine this issue in more detail. Misalignment is strongly

dependent on bone volume fraction (BV/TV) and degree of anisotropy (DA). Side-effects are

affected by trabecular spacing (Tb.Sp), as well as BV/TV and DA. These findings will result

in more accurate testing, and hence more accurate modeling, of cancellous bone behaviour.

Keywords: finite element method (FEM); cancellous bone; experimental artefacts; mis-

alignment; specimen size; degree of anisotropy (DA); bone volume fraction (BV/TV); tra-

becular spacing (Tb.Sp)

iii

Acknowledgements

First, I would like to thank my co-supervisors Dr. Brent Lievers and Dr. Keith Pilkey. Your

knowledge of the experimental testing of cancellous bone has been incredibly useful to me

during my studies and has allowed me to research and better understand this interesting and

complex field of biomechanical testing. Thank you for answering my questions and guiding

my wording to ensure that my reasoning was clear and concise, as I sometimes struggle in

the regards to the latter. Lastly, thank you for helping me to develop my critical thought

process which will, hopefully, guide me to analyse all possible points of view moving forward,

and aid in a lifelong love of learning.

To my friends Colin Roos and Maxime Hogue, thank you for attentively listening to

my intrigue-fueled rants and complaints about my research. Without your ears, I certainly

would have either driven myself insane or driven others insane. Rest assured that my rants

will now be more focused on the topic at hand in our conversations. To my parents thank

you for your interest and support throughout my degree. I am lucky to have parents who

support me through all my interests and harbor a multi-faceted learning process. I hope you

enjoy this thesis and we can speak at great lengths about the time I spent researching it.

Lastly, I would like to thank Alex for your constant support. Mostly, I would like to

thank you for putting up with my constant hours at the computer, telling you that I only

needed to schedule a few more models and organize some data. I know that my obsessive

focus on my work could sometimes be challenging to put up with, but I hope it was made

up for with the many hours we shared speaking and learning together about biomechanics.

iv

I am very blessed to have you in my life and look forward to many more years of learning

with you.

v

Contents

Abstract iii

Acknowledgements v

List of Figures ix

List of Tables xii

Nomenclature xiii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Objectives and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Literature review 5

2.1 Cancellous bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Role of cancellous bone in whole bones . . . . . . . . . . . . . . . . . 7

2.1.2 Mechanical properties of cancellous bone . . . . . . . . . . . . . . . . 7

2.1.3 Cancellous bone structure . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.4 Cancellous bone remodeling process . . . . . . . . . . . . . . . . . . . 10

2.2 Architectural characterization . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Measures of architecture . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Effects of osteoporosis and ageing on cancellous bone architecture . . 15

vi

CONTENTS

2.2.3 How architecture varies . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Experimental testing of cancellous bone . . . . . . . . . . . . . . . . . . . . 17

2.3.1 End effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2 Misalignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Side effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.4 FEM modelling in cancellous bone . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.1 Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.2 Element geometry and model accuracy . . . . . . . . . . . . . . . . . 24

2.4.3 Element resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.4 Model material properties . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Literature review summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Misalignment error in cancellous bone 29

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 Materials & Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Specimen preparation & scanning . . . . . . . . . . . . . . . . . . . . 31

3.2.2 Conventional models . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.3 Eroded/dilated models . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.4 Morphological measures . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 Conventional models . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.2 Eroded/dilated results . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4 Models of “side effects” in cancellous bone 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

vii

CONTENTS

4.2 Materials & Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Model creation method . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Empirical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.2 Theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3.3 Correction factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3.4 Minimum specimen sizes . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Discussion & conclusions 65

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Comparison to previous work . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.4 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.5 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.6 Contributions & conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

References 72

A FE model convergence study 84

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.2 Materials & Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

A.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

A.4 Discussions & Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

viii

List of Figures

2.1 A display of the various structural levels of bone composition . . . . . . . . . 6

2.2 A femoral section from a lamb specimen . . . . . . . . . . . . . . . . . . . . 8

2.3 a) Cancellous bone b) Plate-like structure c) rod-like structure . . . . . . . . 9

2.4 An image depicting the cells involved in the bone remodeling process (http:

//ns.umich.edu/Releases/2005/Feb05/img/bone.jpg) . . . . . . . . . . . 10

2.5 Illustration of the 3D methods for measuring trabecular thickness (Tb.Th)

and trabecular spacing (Tb.Sp) . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 A comparison between cancellous bone structure of a third decade human

tibia specimen, and a ninth decade specimen. Significant decrease in bone

volume, and increase in degree of anisotropy was observed . . . . . . . . . . 16

2.7 Illustration of the theoretical model . . . . . . . . . . . . . . . . . . . . . . . 21

2.8 Hexhedral and tetrahedral meshing of cancellous bone . . . . . . . . . . . . 25

3.1 Five 5 mm sub-cubes, taken from the centers of the larger 12.86 mm cubes,

to illustrate architectural differences. These cubes have been aligned so that

the principal mechanical axis (PMA) is in the vertical direction. From left to

right, top to bottom, they are the sixth lumbar vertebrae (BLV6), greater

trochanter (BTR), sacral wing (BSW), humeral head (BHH), and lateral

femoral condyle (BLFC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

ix

LIST OF FIGURES

3.2 A 2D schematic illustrating the 3D process used to develop the ‘conventional’

FEM models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 A 2D schematic illustrating the 3D process used to create the eroded/dilated

FEM models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 a) The conventional model 5° misalignment fit. Each line represents a BV/TV

group within this study, while error bars represent standard deviation. b) The

conventional 20° misalignment fit which follows the same conventions . . . . 40

3.5 a) The eroded/dilated model 5° misalignment fit. Each line represents a

BV/TV group, while each group data point is represented by a different sym-

bol. b) The eroded/dilated model 20° misalignment fit which follows the same

conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1 Idealized model of side artefacts . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 A simplified 2D representation of the 3D model creation process used herein.

Note: Images not to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 The method used to crop smaller diameter cylinders from the aligned cylinder

region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 a) Model diameter vs apparent elastic modulus. Curves were fit individually

to each specimen. b) The previous curves, normalized by their individual E0

values as calculated in Eq. (4.1) . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 a) Model diameter normalized by trabecular spacing vs apparent elastic mod-

ulus. Separate curves are applied for each specimen’s trend as shown in

Eq. (4.2). b) Fig. 4.5a curves, normalized by their separate E0 values as

calculated in their individual fits . . . . . . . . . . . . . . . . . . . . . . . . 55

4.6 a) Specimen size fit according to BV/TV and DA values applied to apparent

elastic modulus values (Eq. (4.3)) b) Specimen size normalized by Tb.Sp vs

apparent elastic modulus by BV/TV and DA (Eq. (4.11)). . . . . . . . . . . 56

x

LIST OF FIGURES

4.7 a) The Theoretical model fit Eq. (4.5)applied to the 5 specimen apparent

elastic modulus values and b) the Fig. 4.7 fit normalised by E0 . . . . . . . . 57

4.8 a) The Eq. (4.6) fit, with lines representing values for each specimen b) The

Eq. (4.6) model for all specimens. Solid line shows our criteria based off of

our specimen most prone to side effects (BSW), while dashed indicates least

susceptible specimen (BLFC) to display a range of fits possible. . . . . . . . 58

A.1 The effect of resolution on BSW model structure. Resolutions are as follows:

A) 15µm, B) 30µm, C) 60µm, D) 90µm, . . . . . . . . . . . . . . . . . . . . 86

A.2 Results of specimen apparent modulus values dependence on element resolu-

tion. Both anatomical sites show similar trends at different magnitudes. . . . 87

A.3 Error of apparent modulus measurements based on element resolution. Error

is calculated as the percent difference from the 15 µm element resolution

model measurement. Dashed lines represent ± 2% error. . . . . . . . . . . . 88

A.4 Error of apparent modulus measurements compared to 30µm resolution model.

Once elements reach the 4 element across Tb.Th threshold, error is < 5%.

Dashed lines represent ± 2% error. . . . . . . . . . . . . . . . . . . . . . . . 88

xi

List of Tables

2.1 Apparent elastic modulus results from various studies implementing various

types of testing of both cortical and cancellous bone . . . . . . . . . . . . . . 9

2.2 A comparison between architectural measurements of a human and bovine

femoral trochanter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Model material property values . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Morphological parameters of PMA aligned cylinders at 15µm resolution . . 35

3.2 The combinations of dilations (D) and erosions (E) using 6- or 26-element

structuring elements used for each model. Some models were not eroded or

dilated (NED) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.1 Morphological parameters of thresholded and pruned 15 µm cylinder images,

representing a 5.6× 5.6× 10mm3 region . . . . . . . . . . . . . . . . . . . . 49

4.2 Correction factor ranges representing the minimum (BLFC), and the maxi-

mum correction factors (BSW) for both models as calculated using Eq. (4.7)

and Eq. (4.8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3 Specimen minimum required diameters at 5% and 1% error in asymptotic

apparent elastic modulus value . . . . . . . . . . . . . . . . . . . . . . . . . 59

A.1 The morphological parameters of PMA aligned cylinders at 15µm resolution

(reproduced from Tab. 3.1) highlighting the samples used for the convergence

study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

xii

Nomenclature

ϵ Strain

µCT Micro-computed tomography

σ Stress

BHH-C2 Bovine humeral head specimen

BLFC-C3 Bovine lateral femoral condyle specimen

BLV6-C3 Bovine lumbar vertebra specimen

BSW-C1 Bovine sacral wing specimen

BTR-C1 Bovine greater femoral trochanter specimen

BV/TV Bone volume fraction

Conn.D Connectivity density

DA Degree of anisotropy

D Diameter

E Apparent elastic modulus

E0 Asymptotic apparent elastic modulus

MIL Mean intercept length

xiii

NOMENCLATURE

MTD Mean trabecular direction

PMA Primary mechanical axis

Tb.Sp Trabecular spacing

Tb.Th Trabecular thickness

xiv

Chapter 1

Introduction

1.1 Motivation

Bone fractures in the elderly are a growing problem in Canada and around the world. The

decrease in bone quality caused by ageing, as well as diseases such as osteoporosis, are causing

populations to become more susceptible to fall-related fractures, with this trend expected

only to continue [1, 2]. Falls such as these have been shown to have related costs of $650

million in Canada in 1995 [1, 2], with more recent estimates being over $50 billion in the

United States [2]. These costs are in addition to the incalculable social costs. For these

reasons, more must be done to fully understand the mechanics behind these fractures, to

allow researchers and health care professionals to develop ways to minimize their effects.

To understand the mechanics involved in fractures, we first must understand the be-

haviour of bone both at a whole level, as well as the unique behavior of its base components,

cortical and cancellous bone. The complex structure of cancellous bone, as well as the dif-

ficulties related to mechanically testing samples in a way representative of in-vivo loading,

have been a particular focus of research. This complex structure is often quantified using

architectural measures such as apparent density and degree of anisotropy, in order to allow

1

CHAPTER 1. INTRODUCTION

researchers to compare mechanical testing results obtained in different anatomic sites, or

between different individuals.

Mathematical relationships between architecture and mechanical properties are well es-

tablished in the literature. Multiple studies have demonstrated that density and anisotropy

are important predictors [3, 4]; measures at the rod and plate level, such as trabecular

thickness and spacing, have also been shown to have a role, although to a lesser extent [5].

Studies such as these have allowed researchers to compare the behavior of skeletal regions

with different architectures, such as the femur and vertebra, allowing evaluation of cancel-

lous mechanical properties at a whole level, minimizing error and discrepancies due to the

variability between different structures of cancellous bone. Though these relationships have

been shown and used to analyse these architectural effects, many experimental methods

still rely on standards which are based on studies of single regions, ignoring the effects that

varying architectures have, possibly introducing error into studies and obscuring possible

relationships. By determining the role architecture plays in various standard experimental

procedures, previous studies can be evaluated and compared to each other more accurately,

while future studies can strive to introduce as little error as possible and further determine

relationships between region-dependent variance and mechanical loading response.

Two experimental issues which have received limited architectural analysis are the effects

of specimen size on side artefacts and misalignment from the primary mechanical axis on

the underestimation of cancellous apparent elastic modulus. These effects occur when speci-

mens are cored from a whole bone misaligned from the strongest direction, due to cancellous

bone’s anisotropy. This causes mechanical results to misrepresent the maximum mechanical

properties. In addition, the natural boundary conditions of the bone are removed, causing

inaccurate test results due to side effect errors. Though various authors have shown that

underestimation of mechanical properties can occur below a certain size, minimum speci-

men dimension recommendations still rely heavily upon studies which analysed only a single

region and did not consider the effects of architecture on mechanical measures [6, 7]. Com-

2


pounding this issue is the fact that specimen misalignment may have a significant impact on

cancellous mechanical properties in uniaxial compression tests [8, 9]. This too has received

insufficient research. Analysis of the effect of cancellous architecture on these experimental

artefacts will allow researchers to both understand its role more fully with respect to me-

chanical behavior, as well as determine which architectural measures drive these effects. A

clearer understanding of these relationships will therefore enable researchers to more accu-

rately predict different regions’ responses to misalignment and side-effect errors.

1.2 Objectives and outline

This work seeks to quantify the effects of specimen architecture on two experimental artefacts

in cancellous bone testing to allow various researchers to calculate the magnitude of possible

error being introduced into their study and limit them as necessary. By determining the role

of cancellous bone architecture on misalignment and side-effect errors, researchers will be

capable of determining limitations in both current and previous study methodologies. These

limitations can then be reduced or corrected for to allow for more accurate experimental

testing. These effects will be determined using finite element method (FEM) modelling, and

will allow researchers to both actively, as well as retroactively, analyse studies of cancellous

bone to allow for more accurate cross-study comparison. In addition to this primary goal,

it is believed that the results of this study will allow for estimation of the decrease in

bone apparent elastic modulus in fall-like scenarios (off-axis loading), possibly aiding in

the development of a model to predict patient fracture risk.

This study will make contributions in two main areas under the broader scope of exper-

imental testing guidelines. First it will analyse what role misalignment from the primary

mechanical axis plays in overall elastic mechanical property measurements. This is impor-

tant as there are few studies which have examined this effect in depth and all that do have

significant limitations when looking to extrapolate their results to other anatomic sites. Sec-

3


ondly, it will compare two models of specimen “side effects” in order to understand what role

geometry plays in elastic mechanical property measurement. While there are many studies

that have considered this, they often have conflicting results, as well as significantly different

methods. Consequently, a standard method for accounting for side effects has not yet been

produced. Identifying such a standard would ease comparisons between different specimen

sizes and architectures in existing studies and could guide the selection of specimen sizes to

reduce their effects in future studies.

This thesis is organized using the manuscript style. The contents of the following chapters

are as follows:

• Chapter 2: a review of applicable literature as it pertains to the current study;

• Chapter 3: work done to determine the role architecture has in off-axis loading;

• Chapter 4: analysis of the predominant side effect models and their relation to archi-

tecture;

• Chapter 5: discussion and conclusions derived from the work presented; and,

• Chapter 6: references cited herein.

4

Chapter 2

Literature review

To understand the worked presented in this thesis, the applicable existing literature will first

be reviewed and summarized. This review will be broken into several sections to allow the

reader to comprehend the scope of the current study. First, cancellous bone will be reviewed

as to its role in the skeleton, how it is architecturally classified, and how the architecture

varies with disease, and age. Though intra-regional differences are important to consider in

experimental testing, the current study seeks only to quantify those due to inter-regional

differences. Following this, current experimental practices will be reported to give context

for the current study. Lastly, practices related to the finite element method such as imaging

and modeling approaches will be analysed to determine how to best to model cancellous

bone in the digital realm. By considering these regions of interest, readers, as well as the

author, should be able to accurately evaluate the results of the current studies critically.

2.1 Cancellous bone

Bone has a hierarchical structure that begins at the whole bone level and continues down

to the molecular level (Fig. 2.1). Though this is a study focused on how bone reacts to

loading at the more macroscopic level, some understanding of bone at smaller length scales

is useful to understand how the structure contributes to the mechanical properties. It should

5

CHAPTER 2. LITERATURE REVIEW

Figure 2.1: A display of the various structural levels of bone composition from [10]

be noted that the hierarchy of bone is a field of great complexity and the following is only a

simple breakdown of the review by Rho et al. [10] as it applies to the current study.

The skeletal system consists of various whole bones connected by joints, which gives our

body the underlying structure to allow movement. Whole bones consist of a shell of dense

tissue known as cortical (or compact) bone. Regions of more porous cancellous bone –also

known as trabecular or spongy bone– are found at the ends of long bones and in the bodies of

vertebrae. In studies of bone mechanics, researchers often study either cortical, or cancellous

bone, or how the two interact. Though the two are similar, given the complex structure of

cancellous bone, it is often studied with more depth.

Cortical and cancellous bone are both composed of three main components: collagen,

mineral, and water. Of these components, collagen gives the tensile strength, mineral gives

the compressive strength, and water keeps collagen hydrated, minimizing brittle behav-

ior [11]. Though these components are important, for the current study it is unnecessary to

6


review them in depth. Instead, more focus will be given to cancellous bone, its underlying

structure, and the role it plays in load bearing.

2.1.1 Role of cancellous bone in whole bones

The role of cancellous bone is of great importance for whole bone loading. In whole bones,

cancellous bone is typically found at the terminal sections of long bones; the areas where

forces are transferred between bones [11]. These areas tend to be much larger in shape and

surface area, which (along with joint cartilage and synovial fluid) lowers the stress on the

bone, during relative movement [11]. Cancellous bone is optimal for these areas due to its

highly porous structure, which allows for large volumes and surface areas, while having lower

mass when compared to cortical bone. Therefore, we find cancellous bone in areas such the

bodies of lumbar vertebrae or at the proximal end of the femur (Fig. 2.2). In that image,

cancellous bone exists only in the load bearing regions (the femoral head, femoral neck, and

trochanter) and tapers off around the metaphysis. This gradient of several types of bone is

useful for obtaining optimal mechanical properties. The cancellous bone limits the weight

while the thin cortical shell gives the cancellous bone a flat surface, allowing a more even

distribution of force across the trabecular network [11]. By working in tandem with cortical

bone, cancellous bone plays an important role in whole bone load-bearing properties.

2.1.2 Mechanical properties of cancellous bone

Cancellous bone is similar to cortical bone in a one notable way. The two types of bone

share similar proportions of their constituents (collagen, mineral, and water), with cancel-

lous bone being slightly less mineralized (90–95% that of cortical bone) [11]. Though these

similarities exist, cancellous bone still has significantly lower stiffness and strength when

compared to similar cortical bone specimens [11]. In addition to these differences, studies

of cancellous bone mechanical properties have found higher variability of the results, while

results from cortical testing have remained generally consistent (Tab. 2.1). These differences

7


Figure 2.2: A femoral section from a lamb specimen

arise from the greater geometric (architectural) variability associated with the trabecular

network. Therefore, more effort needs to be made to develop accurate and repeatable meth-

ods to determine the mechanical properties of cancellous bone as a material while accounting

for the architecture. Lastly, considering cancellous bone as a structure, due to its porosity

it is significantly weaker than a volume of cortical bone of the same size. Though this is

obvious, the differences between the two types of bone previously mentioned play a signifi-

cant role in mechanical test results, indicating that the differences in mechanical properties

of these two types of bones cannot be ignored.

8


Table 2.1: Apparent elastic modulus results from various studies implementing various types oftesting of both cortical and cancellous bone

Type of testing Apparent Elastic Modulus [GPa]Cortical Cancellous

Uniaxial tension 19.9 [12] 18.0 [12]Nanoindentation 20 [13] 13.4 [14]Uniaxial compression 19.09 [15] 6.6 [16]Nanoindentation 24.7 [17] 20 [17]

Figure 2.3: a) Cancellous bone [18] b) Plate-like structure [19] c) rod-like structure [19]

2.1.3 Cancellous bone structure

Perhaps the most notable characteristic of cancellous bone is its highly complex structure

(Fig. 2.3a). The cancellous architecture, as it is known in the medical literature, consists of a

3D network of rods and plates (Fig. 2.3b & Fig. 2.3c). These rods and plates, often referred

to as trabeculae, can be seen in all regions of cancellous bone; however, the ratio of the two

is often dependent on the region and its load-bearing requirements. In regions such as the

femur, many more plates are seen as more multi-directional loading occurs, when compared

to a very rod-like specimen from the vertebra. Though these structures seem to shoot out

in random directions, their overall structure is maintained in such a way to have a primary

loading direction, along which its apparent elastic modulus and strength are at a maximum.

9


Due to this anisotropic nature, as well as the complex rod and plate structure, it has become

necessary to develop architectural measures to fully analyse the complex geometry. The

definitions of some common measures follow in Sect. 2.2

2.1.4 Cancellous bone remodeling process

The source of the architectural variability seen in cancellous bone is the process of bone

remodeling. It is a topic of great complexity and one that is heavily studied in bone research

[11, 20–23]. The general process, as illustrated in Fig. 2.4, begins when a volume of old bone

tissue is removed by cells known as osteoclasts [11]. Other cells (osteoblasts), then attach to

a bone surface and lay down a collagen layer that will mineralize over time [11].

Figure 2.4: An image depicting the cells involved in the bone remodeling process (http://ns.umich.edu/Releases/2005/Feb05/img/bone.jpg)

This remodeling process is necessary for several reasons: to remove damaged and no

longer effective bone tissue, to reclaim resources from regions which have been overdevel-

oped based on current metabolic needs, and lastly, and of most importance to the current

study, to develop a structure which is more effective given current loading needs [11]. It is

for these reasons that those prone to bone density loss, such as menopausal women and/or

astronauts, are encouraged to exercise in ways which load their bones more evenly, encour-

aging remodeling [11]. This remodeling process can also be the cause of fractures related

to disease and ageing effects [24]. When a person becomes more sedentary, or affected by

10

http://ns.umich.edu/Releases/2005/Feb05/img/bone.jpg

http://ns.umich.edu/Releases/2005/Feb05/img/bone.jpg


a disease which disrupts the balance of remodeling (e.g., osteoporosis), their body removes

tissue which is damaged from wear, or no longer used (such as the trabeculae perpendicu-

lar to the primary loading direction), to conserve resources and energy [11]. This process

then causes the person to be more prone to fractures as their bone’s ability to bear loads

misaligned from the primary loading direction decreases [8, 9]. It is for these reasons that

research needs to be performed to determine the load bearing properties of cancellous bone in

such loading scenarios, to allow researchers to develop techniques to limit these remodeling

effects on those prone to them.

2.2 Architectural characterization

Given the importance of architecture to the mechanical properties, various methods and

measures have been proposed to characterize the geometry of cancellous bone. Originally,

measurements were taken from 2D images which represented a slice of the material [25].

These measurements analysed the trabeculae specifically, and included parameters like tra-

becular thickness, spacing, and number. These measures followed an assumption called the

parallel-plate method, which assumed an idealized structure. With the adoption of �CT scan-

ning, these measures have been converted to direct, 3D computed measurements to ensure

high accuracy. In addition to these measurements, measures of bone density and anisotropy

have been produced to most accurately analyse the complex structure, easing statistical

analysis of its role in mechanical properties. The definitions and techniques of measurement

follow.

2.2.1 Measures of architecture

Bone Volume Fraction

Bone volume fraction (BV/TV) is defined as the proportion of a given volume that is occupied

by bone. It is calculated by dividing the volume of bone tissue (BV) by the total volume

11


(TV) of the region analysed:

Bone Volume Fraction =BVTV (2.1)

Traditionally, these values were calculated using stereological point counting techniques on

2-D histological sections. This measure is now calculated in 3-D by counting the number of

voxels that represent bone in some region of interest and dividing it by the total number of

voxels within that region [26]. BV/TV has been shown to be very highly correlated with

the apparent density of specimens and is often chosen due to the simplicity of measurement

given the proper equipment is available. In a similar way to density, it has been shown

to significantly affect mechanical properties, with strength and apparent elastic modulus

holding an exponential relationship with it. These relationships take the form:

E = a (BV/TV)b or σ = a (BV/TV)b (2.2)

where a is a scalar coefficient, and the exponent b ≈ 2 [3, 27]. Though these findings are

widely spread, such an analysis is not fully indicative of the relationship between cancel-

lous architecture and mechanical properties. Due to this, several other measures are often

reported and analysed to fully understand this relationship.

Trabecular Thickness

Trabecular thickness (Tb.Th) is defined as the mean thickness of trabeculae within a speci-

men. Originally, Tb.Th was calculated using the parallel-plate model assumption and stere-

ological imaging. This method assumed an idealized structure, allowing 3D measures such

as Tb.Th to be estimated from 2D images. Though it was useful when high-resolution CT

scanning was not available, it has been shown to underestimate direct trabeculae-based mea-

surements [29, 30]. This measure is now computationally calculated as the diameter of the

sphere which can fit within a trabecula (Fig. 2.5) and is determined by the mean of a thick-

12


Figure 2.5: Illustration of the 3D methods for measuring trabecular thickness (Tb.Th) and tra-becular spacing (Tb.Sp), taken from [28]

ness map [31, 32]. Trabecular thickness is generally consistent within a specimen and has

been shown to stay fairly consistent even when bone volume fraction decreases. Rather than

strength being lost due to trabeculae thinning, the number of trabeculae in a region tends

to decrease. Though trabecular thickness is a useful measurement for some architectural

effects, it has been shown to not have a significant effect on mechanical properties when

analysed on its own [5].

Trabecular Spacing

In a similar way to trabecular thickness, trabecular spacing is the measure of the mean

spacing between trabeculae (Fig. 2.5). Though it too was originally measured using the

parallel plate model, it is now measured as the diameter of the largest sphere which can

fit between specimen and is calculated through a spacing map [31, 32]. As with trabecular

thickness, trabecular spacing is only slightly related to mechanical properties, and so is often

13


used to quantify architectural effects which may not be made apparent with simply measures

of BV/TV [5, 33].

Connectivity Density

Connectivity density (Conn.D) replaces a previous 2D measure of trabecular number, which

was defined as the number of trabeculae per unit length as determined using the parallel

plate model. Connectivity density is similar but is rather expressed in terms of a volume

(unit3). The Euler characteristic of the volume analysed is calculated, and then is adjusted

by checking intersections between bone voxels and stack edges to define the contribution of

the structure’s connectivity to this characteristic [34, 35]. Connectivity density has been

found to have a limited relationship with elastic properties [36], but is a useful measure to

analyse bone quality decreases due to aging and disease effects.

Degree of Anisotropy

Degree of anisotropy (DA) is a measurement which defines the structure’s primary loading

direction dependence. It ranges from 0 to 1 and the larger the value, the more anisotropic

the specimen is. Analysing previous studies of human bone, DA values generally range from

0.2–0.6 depending on region [37–39]. It is calculated using the mean intercept length (MIL)

values in the 3 prominent directions, which are determined through the ellipsoid developed

during the MIL analysis. To determine the MIL in a given orientation, a linear grid is

projected across the cancellous bone structure in a specific orientation and the number of

intersections between the two structures is counted. The length of these lines is then divided

by the number of intersections to give MIL in that orientation:

MIL (w) =L

I (w)(2.3)

14


where L is the length of the drawn line, and I(w) is the number of intersections at a given

angle w.

Of the three prominent MIL values calculated, the smallest is divided by the largest and

subtracted from 1 to give DA [28]:

DA = 1− MILsmallest

MILlargest

(2.4)

The nature of this equation is such that DA = 0 for an isotropic material and DA → 1 as it

becomes more anisotropic.

2.2.2 Effects of osteoporosis and ageing on cancellous bone archi-

tecture

Architecture is heavily affected by bone loss due to diseases such as osteoporosis, as well

as ageing (Fig. 2.6). Several studies have shown that these states generally result in lower

BV/TV and Conn.D, and higher DA and Tb.Sp [1, 40–42]. As noted above, Tb.Th is

mostly unaffected. Measures of BV/TV have been seen to drop by 27.5% relative to original

measurements due to age [43], and 17% due to osteoporosis [44]. In addition, DA has been

shown to increase by up to 12% with age [43], causing certain regions to be more prone to

directional loads, as well as failures such as buckling [45]. Lastly, Tb.Sp has been shown

to increase significantly with osteoporosis, while Tb.Th stays relatively constant (drop of

3.13%) [43]. These changes to the architecture with age and disease are all a consequence

of the bone remodelling process described earlier (Sect. 2.1.4). The loss of bone tissue

explains why BV/TV decreases, and the loss of struts (as opposed to uniform thinning

everywhere) explains the increases in DA and Tb.Sp. Due to these effects, understanding

the role architecture plays in mechanical loading is important to fully understand how these

changes can be detrimental to whole bone mechanics.

15


Figure 2.6: A comparison between cancellous bone structure of a third decade human tibia speci-men, and a ninth decade specimen. Significant decrease in bone volume, and increasein degree of anisotropy was observed [46]

2.2.3 How architecture varies

The last, and possibly most important, characteristic of cancellous bone architecture to re-

view is its variability. Architecture has been shown to vary significantly between regions

due to the type of loading supported in that part of the skeleton. Region-dependent ar-

chitectural variation is generally focused on BV/TV and DA, which have high variability

between sites [4, 47], but measures such as Tb.Sp and Conn.D have also been shown to vary

[5]. These architectural variations are significant, making it challenging to determine how

different structures from different regions compare in certain loading conditions.

Compounding these challenges is the variation within individual sites. Specimens from

the same region have been shown to have significantly different architectures within local

architecture minima [48, 49]. These regions are areas where architectural parameters are

significantly lower than the specimen average [50]. This effect can be seen in a study of

misalignment effects by Öhman et al. [9], where the tested human femoral head specimens

varied from 15% to 40% BV/TV, indicating that different specimens from the same region

can have significantly different architectures. To account for this, specimens must either be

16


Table 2.2: A comparison between architectural measurements of a human [53] and bovine femoraltrochanter (taken from current study)

Measure Human measurement Bovine measurementBV/TV 12% 23%

DA 0.39 0.72Tb.Th (mm) 0.143 0.22Tb.Sp (mm) 0.850 0.72

Conn.D (1/mm3) 3.35 4.00

large enough to minimize the effects of these minima, or a large enough sample size must be

used to minimize variability introduced by these effects [51].

Finally, architecture is significantly different between species. Multiple studies have

shown that bovine and human cancellous bone are significantly different in nearly all regions

of cancellous bone, due to the significant anatomical differences between species (Tab. 2.2)

[27, 52]. Nevertheless, there have been findings to show that cancellous bone behaves the

same between the two species, when architectural differences are accounted for [52]. Due

to this, bovine cancellous bone should be an acceptable comparator to human cancellous

bone response to various loading scenarios [52]. Though specific values derived may not be

directly comparable to those expected from human studies, the overall trends are likely to

remain consistent.

2.3 Experimental testing of cancellous bone

Many different experimental methods of testing cancellous bone have been introduced over

the years. When methods have been found to introduce experimental artefacts, other testing

protocols have been introduced to try to eliminate or reduce these errors. For example, for

many years the most common method of axial testing was simply to compress a cancellous

bone specimen between two platens and use the resulting stress-strain curve to calculate

the mechanical properties. Though this method is fairly simple, research over the years has

17


determined that it underestimates the true properties due to increased compliance of the

severed trabeculae at the loading surfaces [54]. Similarly, it is now well understood that

specimen hydration must be controlled during testing because drying increases the stiffness

and reduces the ductility of bone [55]. Such standard practices are important for ensuring

accurate test results; unfortunately, some artefacts remain under-studied.

To attempt to further increase our knowledge of methodological effects on cancellous

bone mechanical compression results, specifically apparent elastic modulus measurements,

three experimental artefacts will be reviewed. This will be done to summarize what previous

works have concluded, as well as what further work needs to be performed to limit the effects

of biases and error on study results.

2.3.1 End effects

Originally, specimens of cancellous bone were compressed between two platens with mineral

oil applied at the specimen-platen surface to minimize frictional effects. From the stress-

strain curve produced by this test, specimen apparent elastic modulus could be determined.

This method was later found to be producing biased results. The increased compliance of

the specimen ends, resulting from the severed trabeculae when the specimen was cut, lead to

an overestimation of the strain in the specimen based on platen displacement. As a result,

the apparent elastic modulus was underestimated [54]. To combat this, the combination

of endcaps secured with PMMA, and an extensometer are used to determine strain in an

effective region of specimens, eliminating biases caused by end compliance. This method

was first introduced by Keaveny et al. [56] and is now widely used in the field.

2.3.2 Misalignment

Misalignment is an artefact that hasn’t been heavily analysed but is known to be a cause of

apparent elastic modulus underestimation. Although not considering misalignment per se, a

number of early studies quantified the orthogonal mechanical properties of cancellous bone

18


cubes (e.g. E1, E2 and E3) [57–59]. The results confirmed that cancellous bone specimens

are anisotropic in terms of mechanical properties as well as structure.

Subsequently, two studies have considered this effect in more depth, in an attempt to

quantify the error introduced due to misalignment of specimens from their primary mechan-

ical axis. The first was by Turner and Cowin [8] and consisted of a numerical analysis of

results from an experimental tri-axial study. Although the expected error due to misalign-

ment for a specific region of human cancellous bone was determined, the resulting values are

based off of theoretical assumptions and were obtained through a purely numerical analysis.

Furthermore, since the data used was obtained from platen compression of cubic samples, the

mechanical results used in this analysis are likely biased by end effects. Since end artefacts

may have different magnitudes in each of the three directions, it is unclear how this might

affect the analysis of Turner and Cowin [8].

The second study was experimental, and was performed by Öhman et al. [9] on human

femoral heads. Though this study experimentally determined error due to misalignment,

and attempted to control for BV/TV effects, their alignment protocol is of concern. Two

orthogonal 2D X-rays were analyzed and the main trabecular direction was estimated by

the researchers. Cylindrical specimens were then cored along this axis and compared to the

mean intercept length predicted alignment. It is possible that the specimens of Öhman et al.

[9] were more misaligned than predicted, which would mean that the true errors would be

larger than those reported.

Though these two studies have found that 24–40% error is introduced even at relatively

low misalignment angles when compared to the previous studies (e.g. 20°), values vary and

only two anatomical sites have been analysed, with the possible sources of error previously

mentioned drawing these values into question [8, 9]. Therefore, more work needs to be done

to determine if regions with structures more prone to misalignment than previously studied,

such as highly anisotropic specimens, are more heavily affected by misalignment error.

19


2.3.3 Side effects

The last experimental effect of interest for the current study is that of unconstrained trabec-

ulae along the periphery of specimens. Often termed “side effects”, cutting specimens from

regions of bone for compression testing causes the outer layer of trabeculae to be removed

from their natural boundary conditions. This effect is comparable to the previously men-

tioned end effects (Sect. 2.3.1) but is more challenging to solve as the specimen’s periphery

would need to be equally constrained while still being allowed to deform.

This effect has been shown to introduce variable amounts of error depending on the type

of experimental testing performed, as well as the size of specimens analysed. There have been

several analyses and methods developed to attempt to limit these errors such as specimen

size requirements, and experimental apparatus, though results have been variable.

One study of interest was performed by Lievers et al. [60] which attempted to describe

the effect of specimen size on apparent elastic modulus, in two anatomical sites of bovine

cancellous bone, using a empirical model of the form:

E = E0 (1− exp (aD)) (2.5)

where E0 is the asymptotic apparent elastic modulus, D is specimen diameter, and a is a

constant. This model assumes an exponential decrease in apparent elastic modulus mea-

surement error occurs as specimen size increases. Though this study was able to capture

the trend in apparent elastic modulus with specimen size in the two anatomical sites of

bone, it did not consider the effects of architectural differences between the sites and did not

implement an alignment protocol possibly introducing error into the results.

Another study has shown that variability due to these “side effects” can be attributed

to a disconnected periphery. Ün et al. [61] developed a theoretical model that assumed a

damaged region around the periphery of the specimen of some thickness, t, that would have

reduced properties and an inner core of material that would be unaffected (Fig. 2.7). They

20


Figure 2.7: Illustration of the theoretical model of Ün et al. [61]

assumed that the value of t would be a function of Tb.Sp. A correction factor, α, defined as

the ratio between the true (asymptotic) and measured moduli, was given as:

α =Etrue

Emeasured

=

(1

1− 2β

)2

(2.6)

where:

β =t

D=

aTb.Sp + b

D(2.7)

for specimens of diameter D. The value of t is assumed to be a linear function of Tb.Sp, and

a, b are the fitting parameters. They then fit their model to FEM results for 6 mm and 8 mm

diameter cylinders of human vertebral bone. The moduli from the 8 mm cylinder prediction

were assumed to represent Etrue. Though this model determined factors to attempt to correct

for the “side effects”, the FEM study considered only a single anatomical site and did not

consider other possible architectural effects.

21


Due to the uncertainty involved in both these studies, the most commonly followed

practices are to have a specimen of large enough size to attempt to minimize the effect of the

unconstrained trabeculae. Though this method is generally followed, different anatomical

sites have different upper limits and constraints on possible specimen size. Therefore, an

analysis determining the site’s architecture and its role in side effects should be performed

to evaluate if these sites require specimens with an architecture-based size constraint.

2.4 FEM modelling in cancellous bone

Though experimental testing is always preferable since it includes all the variability that

comes with a biological material, it can be nearly impossible to control certain variables

without incredibly high cost in terms of time as well as funding. Therefore, certain mod-

elling methods have been established to attempt to analyse cancellous bone through pairing

high-resolution µCT scanning with finite element method (FEM) modelling. This method

allows researchers to take a high-resolution (e.g., 15 µm3/voxel) 3D image and convert it

into a FEM model. Using FEM simulation software, the cancellous structure, converted into

elements, can be analysed in ways that can be impossible in experimental studies. Destruc-

tive sample preparation and testing mean that an experimental sample can only be tested

once; however, modeling techniques allows the same structure to be studied repeatedly under

different loading scenarios.

Though this method will introduce some level of error due to the steps and assumptions

required to create a model, studies have shown that using proper methods, FEM modelling

can replicate physical measurements with 95% accuracy [62]. To ensure that the current

study’s results are reliable and accurate, previous FEM modeling studies of cancellous bone

were reviewed and summarized below.

22


2.4.1 Imaging

Histomorphometry has long been used to produce simplified, idealized FEM models of can-

cellous bone, as well as determine complex architecture properties as explained previously.

This method has been replaced by X-ray micro-computed tomography (µCT) which allows

much higher accuracy of architectural analysis, as well as more accurate and complex FEM

modelling. This method, in simplified terms, uses an X-ray source whose photons are di-

rected towards a specimen for a given period of time. By using a scintillator to detect photons

which have passed through the sample, a 2D gray-scale image can be created. The grayscale

intensity corresponds to the attenuation of photons, where lighter regions have higher density

and darker regions are less dense. Multiple 2D images, taken at different angles around the

sample, can then be reconstructed to form a single 3D image which represents the specimen’s

structure. To attempt to control for different sources of error within this process, specimens

are encapsulated in epoxy to ensure a consistent layer of material for the photons to pass

through. This is expected to minimize any small effects of beam-hardening, but correction

factors can be applied in the image analysis scripts if the effects persist. Using images which

are produced with this method, highly accurate models of the specimen’s structure can be

produced from the resulting images.

Image resolution & architectural measurements

The images which are produced through the aforementioned imaging process go through sev-

eral steps of processing to be converted into FEM models. One common step is a resolution

downsample which is often used to reduce the size of an image by increasing the size of the

voxels. Various algorithms can be used to downsample, although the simplest is to take a set

of voxels (e.g, 2× 2× 2) and replace it with a single, larger voxel whose value is the average

of the eight original ones. Reducing the resolution before converting to a mesh results in

FEM models which do not require highly unreasonable computational resources.

23


This downsample is preferred to the alternative of simply performing imaging at a lower

resolution as it allows analysis at the highest image resolution, if necessary, to understand

the cancellous architecture. Some studies have shown that architectural measurements are

significantly affected by image resolution [63, 64]. Given these findings, it will be necessary

to consider the role of resolution in image architectural measurements within the studies

herein.

2.4.2 Element geometry and model accuracy

When converting µCT images to FEM models, an element geometry must be chosen. Gener-

ally, there are two options widely used within the study of cancellous bone: hexahedral and

tetrahedral elements (Fig. 2.8). While most models use exclusively either one or the other,

some researchers have chosen to use a combination of the two [65]. Though this has been

shown to perform slightly better than single-geometry models in some cases, it also requires

significantly more computational resources to create the meshes, as well as complex crite-

ria calculations to ensure the correct geometries are being chosen for the proper elements.

Therefore, researchers often use a single geometry to both limit resources required for model

production, as well as ease comparison between studies; different meshing algorithms will

produce different simulation results to some degree.

To combat these possible differences, hexahedral elements are often used in cancellous

bone studies due to their ease of implementation [16, 66, 67]. By converting single voxels

directly to hexahedral elements, researchers can be sure their methods are highly similar

as the process is quite simple. In addition, several studies analyzing element geometry

accuracy have found that hexahedral elements perform similarly to tetrahedral elements at

high resolutions (small voxels), with larger error compared to tetrahedrals only beginning

when voxels reached 168µm and higher [16, 66, 67]. Since the models used within this study

will be significantly higher resolution than this threshold (30-60 µm), hexahedral elements

will be used for their ease of implementation.

24


Figure 2.8: Hexhedral and tetrahedral meshing of cancellous bone Ulrich et al. [66]

2.4.3 Element resolution

Element resolution also has significant impacts on model accuracy, computational resources,

and solution time. If the resolution is too coarse, it will result in measurements that are

inaccurate as thinner trabeculae will be mistaken for marrow space resulting in lost con-

nections [66]. In a similar way, if the resolution is too high, the computational time for a

single model can become impractically long, and limit the total number of models which

25


can be calculated within the scope of a study. The relationship between resolution and

model size is exponential; as discussed above, a two-fold reduction in resolution leads to

an eight-fold decrease in the number of elements. Taking these facts into consideration, a

compromise between maximized model resolution accuracy and minimized computing time

must be achieved.

Several studies have compared the accuracy of FEM models of cancellous bone at differ-

ent resolutions, and the general consensus is that resolutions from 20–60µm produce similar

results [16, 26, 38]. Though this range is useful, it is notable that it is fairly large. This is

likely due to the various sites and species analysed having different responses to image down-

sampling. Therefore, the resolution recommended by Guldberg et al. [68], and confirmed by

Niebur et al. [38] will be used. This standard indicates that having 4–6 elements across

the thickness of the trabeculae will allow for accurate results. By ensuring this minimum

criterion is met, we can ensure model results are reliable with little error introduced due to

element resolution.

2.4.4 Model material properties

When using FEM software, elements must be assigned material properties. Though gen-

eral properties for cancellous bone tissue are known, those properties can vary throughout

the cancellous network due to heterogeneous mineral distribution and anisotropic crystal

properties. Given that the current study will compare specimens to themselves, it has been

determined acceptable to apply constant isotropic, linear elastic material properties across

the model elements (as outlined in Tab. 2.3). This method has been implemented in other

studies [12, 61].

Ignoring the variable nature of cancellous bone tissue and local tissue anisotropy [69],

by assigning constant isotropic material properties, allows for complex effects of trabecular

architecture to be studied in isolation. Furthermore, studies which have included material

heterogeneity have found its effects much smaller than the effects of architecture [70].

26


Table 2.3: Model material property values

Property ValueElastic Modulus 10000 N/mm2

Poisson’s ratio 0.3Density 1.9 g/mm3

2.4.5 Boundary conditions

Boundary conditions describe the loading and displacements constraints imposed on the

elements and nodes of the FEM model. These constraints are meant to represent an ide-

alized version of the experimental conditions being simulated. In the study of cancellous

bone, current experimental methods tend to allow movement in directions which are not

representative of in-vivo, such as horizontal movement of a specimen’s outer diameter, and

related side-artifact issues. To combat this, some methods have been produced to represent

kinematic boundary conditions [71].

Though these boundary conditions represent a more accurate analysis of cancellous bone

loading in-vivo within whole bones, the current study is focused on analyzing errors related

to architectural effects in current experimental techniques. To perform this task, simple end

constraints, as in previous studies [61], will be implemented to properly represent the loading

of cancellous bone specimens embedded in end caps similar to those utilized in experimental

studies [57, 72]. By having these boundary conditions, results should be as comparable

to current experimental practices as possible, allowing analysis of architecture to be better

understood.

2.5 Literature review summary

In closing, the information reviewed in this section will be used to accurately study the

topics in the following chapters. Previously developed and validated methods involving

27


specimen end effects, FEM methods, and cancellous bone mechanical properties will be

used to investigate questions which have received limited attention, such as the role of

architecture in misalignment and so-called “side effect” artefacts. By using these methods to

further these topics of interest, it is believed that results will accurately represent the trends

both in bovine cancellous bone, as well as possible likely trends in human cancellous bone

research. By determining these trends, the end goal of developing a model to predict ageing

and osteoporotic related fractures will become more achievable and will produce value for

the field of study and society.

28

Chapter 3

Misalignment error in cancellous bone

depends on bone volume fraction and

degree of anisotropy

3.1 Introduction

Cancellous bone is found in regions of the skeleton that require high load-bearing capabilities,

such as the ends of long bones, the hip, and vertebrae. Since these are the same sites where

age- and disease-related fractures are more likely to occur [46, 73], it is important to be

able to accurately quantify their mechanical behavior. Unfortunately, cancellous bone is

highly heterogeneous in terms of both its structure and its mechanical properties [48, 74].

The process of bone remodeling constantly optimizes the local architecture to minimize

mass and support loading requirements [46]. As a result, different skeletal anatomical sites

have distinctive bone volume fractions and anisotropy [48, 75]; age and diseases such as

osteoporosis can further increase this variability [41, 43]. In order to accurately predict the

response of cancellous bone to various loading scenarios, the mechanical properties must be

measured accurately and related to its architecture.

29

CHAPTER 3. MISALIGNMENT ERROR IN CANCELLOUS BONE

Experimental measurement of the mechanical properties requires great care to avoid in-

troducing bias into the results. The very act of preparing a test sample requires that some

trabeculae be severed which removes some of the natural constraint, increases compliance,

and leads to an underestimation of the apparent elastic modulus or strength [76]. To compen-

sate, certain testing protocols have been developed to better mimic the boundary conditions

experienced in vivo [56]. Further experimental artefacts can be introduced by variables such

as specimen size [7, 77] and hydration [55].

The architectural and mechanical anisotropy of cancellous bone is a further confound-

ing factor that must be controlled. Any misalignment from the principal mechanical axis

(PMA) will lead to an underestimation of the mechanical properties [8, 9]. Various align-

ment methods have been proposed for identifying the PMA [9, 78], but each assumes it will

approximately align with either the material fabric as determined using the mean intercept

length (MIL) or the main trabecular direction (MTD). This assumption is not universally

true, with differences between the PMA and MIL of up to 18° [79], and differences between

the PMA and MTD of up to 21° [9, 78]. However, even if it were possible to know the PMA

with 10% certainty, perfect sample alignment is difficult to achieve experimentally and some

underestimation of properties can always be expected [9, 78]. In an attempt to analyse the

magnitude of the error being introduced by misalignment, Turner and Cowin [8] performed

a numerical study on human cancellous bone from the tibia and found that a 5° and 20°

offset resulted in a < 5% and 20-30% decrease in apparent elastic modulus, respectively.

More recently, an experimental analysis on human femoral heads by Öhman et al. reported

a 40% error at 20°.

One limitation of these two studies is that they did not consider the role of architecture

on misalignment effects, which makes it difficult to extend these results to other species

and sites. For example, bone volume fraction (BV/TV) has been shown repeatedly to have

significant effects on mechanical properties [3, 80, 81]. A dependence of misalignment arte-

facts on BV/TV may reasonably be expected and may explain the differences in magnitude

30


observed by Turner and Cowin [8] and Öhman et al. [9]. In addition, off-axis errors may be

expected to be zero in a truly isotropic material, but to increase with increasing architectural

anisotropy [3, 82]. Since it seems unlikely that the same magnitude of misalignment error

can be expected in all cases, a better understanding of the architectural dependence of these

artefacts is needed to accurately assess the mechanical behavior of cancellous bone.

The goal of this study is to use finite element method (FEM) models of cancellous bone

to quantify the dependence of apparent elastic modulus on off-axis loading and architecture.

Samples were taken from five anatomical sites of the bovine skeleton and scanned in a micro-

CT scanner. FEM models were then created to determine the PMA within 1°. Multiple

misaligned models, at both 5° and 20° off-axis, were simulated to determine the effects of

misalignment and then related to bone volume fraction (BV/TV) and degree of anisotropy

(DA). Morphological image erosion and dilation steps were used to create a second series of

models with matched BV/TV in order to control for that variable. A better understanding

of the dependence of misalignment error on DA and BV/TV will ensure that experimental

testing procedures achieve repeatable and reliable results, particularly in highly anisotropic

or osteoporotic bone.

3.2 Materials & Methods

3.2.1 Specimen preparation & scanning

Roughly cubic specimens of nominal dimensions of 13-18 mm³ were cut from five sites in

the bovine skeleton: lateral femoral condyle (BLFC), sacral wing (BSW), greater trochanter

(BTR), sixth lumbar vertebrae (BLV6), and humeral head (BHH) (Fig. 3.1). All bovine

material was obtained from a slaughterhouse post-mortem. After being cut to size, each

specimen was demarrowed using a combination method of a dental water flossing system

(InterPlak; Conair, East Windsor, NJ) and boiling. Specimens were subsequently immersed

in slow-set epoxy (WEST SYSTEM 105 Resin; Gougeon Brothers, Bay City, MI), spun in

31


Figure 3.1: Five 5 mm sub-cubes, taken from the centers of the larger 12.86 mm cubes, to illus-trate architectural differences. These cubes have been aligned so that the principalmechanical axis (PMA) is in the vertical direction. From left to right, top to bottom,they are the sixth lumbar vertebrae (BLV6), greater trochanter (BTR), sacral wing(BSW), humeral head (BHH), and lateral femoral condyle (BLFC).

a centrifuge to ensure complete penetration into the inter-trabecular spaces, and allowed to

set for 24 hours.

The embedded specimens were scanned using a MicroXCT 400 (Xradia Inc., Pleasanton,

CA) at a resolution of 15 µm. From the center of each scan, a cube of bone was cropped

with a side length of 12.86 mm. This size was selected as it is sufficient to extract an 8 mm

diameter by 10 mm long cylinder in any possible orientation [83].

32


3.2.2 Conventional models

Two different groups of FEM models were evaluated to determine the relationship of can-

cellous architecture to misalignment error. The first of these two groups was entitled the

‘conventional’ models and were created directly from the scans of the five bovine specimens.

Model creation

The steps used to create the conventional FEM models are illustrated in Fig. 3.2. A rotation

matrix, R, was used to re-orient the 12.86 mm cubes, using tri-linear interpolation, to the

desired orientation. Only a single rotation from the original cube was ever performed in order

to avoid the potential for compounding errors from multiple interpolations. A downsample

was then performed by voxel averaging to coarsen the image from 15 to 30µm [38, 66] and

a cylinder (8 mm diameter by 10 mm long) was cropped from the center. The image was

thresholded using Otsu’s method [84] and bone voxels were converted to isotropic hexahedral

elements [66]. A cleaning step was then performed to remove unconnected and cantilever-

connected elements from the model.

Figure 3.2: A 2D schematic illustrating the 3D process used to develop the ‘conventional’ FEMmodels

All models were run using LS-DYNA Implicit version R10.0.0 (Livermore Software Tech-

nology Corporation) on a supercomputing cluster (SHARCNET). Each mesh consisted of

up to 6.5 million hexahedral elements. Models were compressed to 0.1% strain. From the

topmost layer of elements on the cylinder, total nodal force was extracted, allowing stress to

33


be calculated using total apparent cross-sectional area. From these known values, apparent

elastic modulus was calculated using E = σ/ε.

Primary mechanical axis (PMA) alignment

In order to study the effect of misalignment, one must first find the orientation along which

the apparent elastic modulus is at its maximum, referred to herein as the principal mechanical

axis (PMA). An initial estimate of the PMA was calculated using the method of Simmons

and Hipp [85]. This approach yields what may be termed the fabric direction using the mean-

intercept length (MIL), which is based on architecture and not the mechanical properties.

It has been shown to be misaligned from the PMA within 11° (95% CI) [78, 79], or up to

18.9° in another study [79]. Therefore, an iterative search was performed to determine the

PMA. Eight FE models were created, at a specified angular offset, θsearch, at eight positions

equally spaced around the current estimate of the PMA. Simulations were then performed

to predict the moduli at each of these locations. If one of the eight models was found to

have a higher apparent elastic modulus than the current estimate of the PMA, then that

orientation was reclassified as the PMA and another eight models were created. Otherwise,

θsearch was decreased, and a new search was performed. The search was performed using the

MIL as the initial estimate of the PMA and θsearch having decreasing values of 5, 4, 3, 2 and

1°. A similar search to identify the secondary and tertiary axes was not performed as the

subsequent analyses would account for differences in those directions. Using the method of

Wang et al., the MIL was found to be misaligned by 1.4-45.9° (mean=16.04°, median=11.54°)

when compared to the PMA across the five specimens under study.

5 & 20° offset models

Once the PMA had been identified, eight 30 µm resolution models, equally spaced around

the PMA, were tested at both 5° and 20° offset. These two angles were chosen following the

“aligned” and “misaligned” groups of Öhman et al. [9], as well as the range given by Turner

34


and Cowin [8]. Each cylindrical model was compressed to 0.1% strain and the resulting

nodal force data was used to calculate the apparent elastic modulus. The average error (∆

E) was calculated relative to the apparent elastic modulus along the PMA using:

∆E =1

n

n∑i=1

Ealigned − Eimisaligned

Ealigned

× 100% (3.1)

where Ealigned is the apparent elastic modulus along the PMA, Eimisaligned is the apparent

elastic modulus of the i-th misaligned model, and n = 8 is the number of misaligned models

within the group analysed.

Morphological measures

Morphological measures were extracted from the largest continuous rectangular volume

within the “aligned” cylinder of each specimen (representing 5.6 × 5.6 × 10mm³). Bone

volume fraction (BV/TV), trabecular thickness (Tb.Th), trabecular spacing (Tb.Sp), con-

nectivity density (Conn.D), and degree of anisotropy (DA) were measured in FIJI [86] using

the BoneJ plugin [87] after thresholding using Otsu’s method [84] and pruning of floating

elements. Following the BV/TV, DA, Tb.Th, and Tb.Sp measurements, the mapped purify

algorithm was run in BoneJ –which removes disconnected bone particles and fills internal

holes– to ensure accurate Conn.D results. The morphological parameters for each of these

cylinders are summarized in Tab. 3.1

Table 3.1: Morphological parameters of PMA aligned cylinders at 15µm resolution

Site BV/TV Tb.Th Tb.Sp Conn.D DA(mm) (mm) (mm−3)

BSW 0.160 0.171 0.783 4.226 0.638BTR 0.229 0.222 0.720 4.003 0.715BLV6 0.264 0.190 0.606 9.295 0.762BHH 0.271 0.161 0.506 13.85 0.518BLFC 0.344 0.189 0.491 8.542 0.687

35


It should be noted that the morphological measures were not taken on the 30 µm res-

olution images used to generate the FE meshes. Instead, the same operations shown in

Fig. 3.2 were performed, except for the downsampling step, so that measurement of the

equivalent region could be performed at 15 µm. This step was necessary because a strong

resolution-dependence to some architectural measures was observed. While BV/TV for these

five specimens only changed by about 2% across a range of resolutions (15, 30, 45, 60µm),

DA varied by up 10%. Similar resolution-dependence has been noted in other studies [63].

Therefore, architecture parameters are reported based on measures performed at 15 µm res-

olution in order to best represent the structure of the bone specimen.

Statistical analyses

Data was analysed in Matlab (The MathWorks, Inc.; Natick, MA). A Kolmogorov–Smirnov

(K-S) test was used to evaluate normality for ∆E at each site. A value of p < 0.05 was

treated as significant. A Kruskal-Wallis test, as well as a multiple comparison with Bon-

ferroni correction evaluated at α = 0.005, was then performed to determine if ∆E differed

significantly by specimen.

In addition to these tests, the following model equation was fit using the least squares

method:

∆E = a (DA)b (1− BV/TV)c (3.2)

where a, b, and c are constants. This model equation was chosen because it has the prop-

erties that it will be zero for an isotropic material (DA= 0) or for a fully compact material

(BV/TV= 1). The exponent terms were included based on the well-established relationship

that apparent density and anisotropy generally have with apparent elastic modulus [3, 82].

The fits were compared to each other using the adjusted coefficient of determination (R̄2)

values and were used to analyse the influence that different morphological parameters had

on misalignment effects.

36


3.2.3 Eroded/dilated models

Since previous studies have shown that BV/TV has significant effects on apparent elastic

modulus, a second set of models was developed to better isolate the effects of anisotropy on

off-axis loading by controlling for BV/TV.

Model creation

A series of Boolean erosion/dilation steps were used to adjust the BV/TV values of the

original five specimen scans. Two different structuring elements (SE) were used to achieve

the largest range of BV/TV values for each model. An isotropic 6-element SE, as well as a

26-element SE, were used in various combinations. The steps needed to adjust each structure

to within 1-2% of the target BV/TV were determined by trial and error and are summarized

in Tab. 3.2. In four cases, no erosion or dilation (NED) steps were needed to achieve the

desired BV/TV.

Table 3.2: The combinations of dilations (D) and erosions (E) using 6- or 26-element structuringelements used for each model. Some models were not eroded or dilated (NED)

Site 15% BV/TV 20% BV/TV 27% BV/TV 34% BV/TVBSW NED D-6 D-6-26 D-6-26-26BTR E-26 E-6 D-6 D-6-26BLV6 E-6-6 E-6 NED D-6BHH E-26 E-6 NED D-6BLFC E-6-26 E-6-6 E-6 NED

As can be seen in Fig. 3.3, the operations performed for the eroded/dilated models are

nearly identical to those performed on the conventional models. One of the main differences

is that the threshold of the 15 µm cube is applied as the first step so that the erosion

and dilation steps could be performed on the highest resolution image; Boolean operations

performed on a coarser, downsampled image did not allow for a useful range of BV/TVs to

be achieved. After binarization and the erosion/dilation steps, the images were rotated and

downsampled which resulted in voxel values that were no longer truly binary but varied in

37


the range [0,255]. A second thresholding step was performed to convert voxels at 128 and

above to bone. It should be noted that, due to the differences in model creation steps, the

eroded/dilated models will differ from the conventional models, even when no erosion or

dilation is performed. Therefore, these models will be evaluated as a separate set.

Figure 3.3: A 2D schematic illustrating the 3D process used to create the eroded/dilated FEMmodels

Primary axis alignment

As with the conventional models, the PMA was determined using the iterative search tech-

nique described above.

5° & 20° offset models

Eight, equally displaced models were tested for both 5° and 20° offset from the identified

PMA. Because of the large range of architectures created due to the erosion/dilation steps,

was performed to 30, 45, or 60 µm. The model resolution was selected to obtain 4–6 elements

across the mean Tb.Th [68]. The average error was calculated relative to the apparent elastic

modulus along the PMA using Eq. (3.1).

3.2.4 Morphological measures

FIJI and BoneJ were used to obtain BV/TV, Tb.Th, Tb.Sp, DA, and Conn.D as described

previously. All morphological measures were obtained from the 15 µm aligned cylinder from

each specimen within each group to reduce resolution-dependent errors.

38


Statistical Analyses

Data were analysed in Matlab (MATLAB and Statistics Toolbox Release 2015b, The Math-

Works, Inc., Natick, Massachusetts, United States) as described for the conventional models.

A Kolmogorov–Smirnov (K-S) test was used to evaluate normality and a Kruskal-Wallis test

was then performed to evaluate if ∆E varied significantly by specimen (α = 0.05), fol-

lowed by pair-wise comparisons with Bonferroni corrections (α = 0.005). Results were fit

to Eq. (3.2) to determine the dependence of ∆E on BV/TV and DA for the eroded/dilated

models.

3.3 Results

3.3.1 Conventional models

The K-S test indicated that the data for the five conventional specimens were non-parametric.

Results of a Kruskal-Wallis analysis determined that reduction in apparent elastic modulus

(∆E) for both the 5° (p = 0.0091) and the 20° models (p < 0.0001) varied significantly

between specimens. Multiple comparisons performed on the 5° results with Bonferroni cor-

rection determined no specimens significantly varied from each other; however, significant

differences were found at 20° misalignment. The BTR was found to be significantly different

from the BLFC and BHH (p = 0.0045 and p = 0.0004, respectively), and the BLV6 specimen

was found to be significantly different from the BHH (p = 0.0049).

Eq. (3.2) was fit to both the 5° and 20° misalignment groups for the conventional data.

The data, curves, and fit equations are displayed in Fig. 3.4, along with the coefficients of

determination (R2) and adjusted coefficient of determination (R̄2). BV/TV groups repre-

senting the five specimens evaluated are displayed using lines, effectively presenting 3D data

in 2D.

39


(a) (b)

Figure 3.4: a) The conventional model 5° misalignment fit. Each line represents a BV/TV groupwithin this study, while error bars represent standard deviation. b) The conventional20° misalignment fit which follows the same conventions

3.3.2 Eroded/dilated results

Eroded/dilated results were determined to be non-parametric through a K-S test. Kruskal-

Wallis tests were performed on each BV/TV group to determine if specimens differed when

BV/TV was kept relatively constant. Results indicated that models within all BV/TV

groups significantly differed (p < 0.05). Bonferroni multiple comparisons were then per-

formed. At 5° misaligned, only two of the four BV/TV groups had significant differences

between specimens. At 15% BV/TV, the BLV6 was significantly different from the BSW

(p=0.0039), while at 34% BV/TV the BLV6 was significantly different from the BHH

(p = 0.0049). At 20° misaligned, the BTR and BLV6 specimens were both found to be

significantly different from the BHH in all BV/TV groups except 15%. At 15% BV/TV

the BTR was significantly different from the BLFC (p = 0.0045), while the BLV6 was sig-

nificantly different from the BHH. These results support the hypothesis that DA holds a

significant role in misalignment error.

40


As with the conventional models, Eq. (3.2) was fit to the data for 5° and 20° misalignment.

The data and selected constant BV/TV curves are shown in Fig. 3.5, along with the fit

equations and raw (R2) and adjusted (R̄2) coefficients of determination.

(a) (b)

Figure 3.5: a) The eroded/dilated model 5° misalignment fit. Each line represents a BV/TVgroup, while each group data point is represented by a different symbol. b) Theeroded/dilated model 20° misalignment fit which follows the same conventions

3.4 Discussion

Finite element method (FEM) modeling of cancellous bone was performed to determine how

architectural parameters such as bone volume fraction (BV/TV) and degree of anisotropy

(DA) affect the magnitude of misalignment error. The results from the conventional mod-

els at 20° off-axis indicate that misalignment errors (ΔE) varied significantly by anatomic

site. These differences were also observed in the eroded/dilated models that controlled for

BV/TV. Architecture explained a large portion of these differences, and misalignment error

was observed to increase with increasing DA and decreasing BV/TV. These results suggest

that off-axis errors are not constant and must be considered based on the architecture being

tested.

41


The regression equations fit to the FEM predictions in Fig. 3.4 can be used to compare

the current results to previous studies. In the mathematical analysis performed by Turner

and Cowin [8], error for the human proximal tibia was estimated to be < 5% at 5° and 28%

at 20° misalignment. Architectural parameters were not reported in that work; however,

other sources suggest the human proximal tibia has a BV/TV of approximately 30% [24]

and a DA of 0.84-0.88 [46]. Using these values, the equations in Fig. 3.4 predict an error of

20–25%, which compares well with the values of Turner and Cowin [8].

Our results do not agree as well with the findings reported by Öhman et al. for the human

femoral head. Using BV/TV = 26.5% [9] and DA = 0.37 [88], the regression equation in

Fig. 3.4 predicts an error of < 5% at 20° which is drastically lower than their value of

40%. This disagreement may be due to discrepancies in the DA values for the architectures

tested versus estimates of DA in the present study, extrapolation beyond the range of DA

(0.463–0.754), differences in human versus bovine architecture, or simply due to comparison

of experimental versus FEM model results. Since Öhman et al. [9] also estimated the PMA

based on the MTD, the differences in their “aligned” and “misaligned” values may not be as

defined in the current study. These effects need to be considered in future studies.

The dependence of misalignment error on BV/TV and DA are notable for several reasons.

Various studies [78, 79] have shown that, even with great care, perfect alignment of the

applied loading direction is not possible even if the principal mechanical axis (PMA) is

perfectly known. Current alignment practices, which estimate the PMA based on the MIL

or MTD, have been shown to be themselves misaligned by up to 18° and 21°, respectively.

Therefore, some amount of misalignment is expected resulting in an underestimation of the

true properties. The current findings indicate that different sites and architectures respond

significantly differently to the same misalignment; hence, the architecture being tested must

be considered to accurately estimate error due to off-axis loading.

These errors have the potential to confound not only site-to-site comparisons of mechan-

ical properties, but also comparisons made between healthy and diseased groups within the

42


same site. Osteoporosis has been shown to both decrease BV/TV [3, 89] as well as increase

anisotropy [90]. Both of these changes are expected to increase the magnitude of the off-

axis error based on Eq. (3.2). Therefore, even in studies where misalignment is somewhat

kept consistent between control and osteoporotic specimens, there will be larger artefactual

apparent elastic modulus reductions in the osteoporotic specimen. It is recommended that

future studies consider this effect with great scrutiny.

The current results also have implications for off-axis loading in whole bones. In a FEM

study on the effects of off-axis loading in falls, Troy and Grabiner [91] found that including

lateral components of loading in their simulations caused a 47% decrease in load required

to fracture; similar results have been observed in other studies [92, 93]. The current study’s

results –that off-axis decreases in mechanical properties are greatest at low BV/TV and high

DA– may be incorporated into such models to better predict the extent to which age- and

disease-related decreases in cancellous bone quality affect the strength of a whole bone in a

falling scenario. Though the results presented herein must not be extrapolated beyond their

bounds, they may be a useful starting point for future analyses of fall effects in vivo and the

role cancellous bone architecture plays in them.

Moving forward, further work is needed to understand the relationship of cancellous

bone architecture to orientation effects. An experimental analysis verifying the architectural

dependence demonstrated in the current work is paramount, particularly in human bone.

Lastly, a protocol should be developed to allow for alignment to the PMA (rather than just

MIL or MTD) to within 5°. Results from the current and previous studies suggest that

alignment to within 5° of the PMA should introduce reduction in mechanical properties of

< 5%. Improvement to specimen alignment protocols will reduce noise in experimental data

and allow for more accurate evaluation of cancellous bone response to loading.

43


3.5 Conclusion

Some amount of misalignment between the primary mechanical axis (PMA) and the applied

loading axis is to be expected when conducting an experimental test on a cancellous bone

sample, which will result in an artificial decrease in the measured mechanical properties. The

findings of this study indicate that the magnitude of misalignment error is not constant but

varies as a function of cancellous architecture. While small levels of misalignment from the

primary mechanical axis (within 5°) were found to have a small effect on apparent elastic

modulus error (< 5%) across all specimens tested, larger misalignment (20°) had errors

ranging from 8-24% depending on bone volume fraction (BV/TV) and degree of anisotropy

(DA). These misalignment errors could confound to comparisons of the mechanical behavior

of cancellous bone with different architectures, either from different skeletal sites or due to

disease-related changes.

44

Chapter 4

Evaluating a theoretical and an

empirical model of “side effects” in

cancellous bone

4.1 Introduction

Cancellous bone is found in skeletal regions that require high load-bearing capabilities such

as the ends of long bones, the hip, and the vertebrae. Since these are the same sites where

age- and disease-related fracture commonly occur, it is important to be able to accurately

predict cancellous bone’s mechanical behavior in order to assess someone’s risk of fracture.

Prediction, in turn, requires reliable knowledge of the mechanical behavior of cancellous bone.

Unfortunately, the nature of cancellous bone makes it very difficult to test. Cancellous bone’s

three-dimensional network of plate- and rod-like trabeculae is characterized by significant

variability in both the architecture of the network itself and its underlying components of

collagen, water, and mineral content. In addition to this, the tissue is heterogeneous in

its mechanical properties due to varying mineral distribution and crystal structure. These

45

CHAPTER 4. MODELS OF “SIDE EFFECTS” IN CANCELLOUS BONE

sources of variability make accurate mechanical characterization difficult because there are

multiple ways in which to introduce testing errors.

Many experimental methods have been developed with the goal of maximizing the ac-

curacy of cancellous bone testing by minimizing various testing artefacts. For example,

the very act of preparing a test sample requires that trabeculae be severed, which removes

some of the natural constraint, which can lead to an underestimation of the apparent elastic

modulus or strength. To limit these negative effects, Keaveny et al. [54] developed testing

protocols to restore some of the constraint lost at the load-bearing ends of an axial test spec-

imen. Specimen hydration and alignment must also be controlled to limit the introduction

of artefacts that can raise or lower the measured properties [8, 9, 55, and Chap. 3]. Methods

such as these are critical to ensuring that the bone properties are measured as accurately as

possible.

Although the method of Keaveny et al. [54] is commonly used to minimize inaccuracies

due to so-called “end effects”, there is less consensus on how to deal with the related issue of

side artefacts resulting from unconstrained trabeculae around the periphery of the specimen.

Some authors have proposed a method of constrained mechanical testing [94], but it is unclear

what effect friction between the specimen and the constraining cylinder has on testing results.

Additionally, while this constrains the trabeculae from displacing radially, they can still travel

vertically since they lack the rigid connection to adjacent trabeculae they would have in vivo.

Another approach is to use test samples with larger dimensions to minimize the proportion

of unconstrained to constrained trabeculae. Various authors have recommended a “minimum

size” ranging from 6.5–10 mm side lengths in cubic specimens [7, 77], or diameters of 5.6–

7.5mm and lengths of 6.5–10mm for cylindrical specimens [6, 7, 60]. Unfortunately, the

anatomy of the skeletal sites of interest imposes physical upper limits on the possible sizes

of specimens; hence, some researchers have suggested using correction factors to account for

these size effects when larger specimens cannot be used. In addition to these limitations,

46


Figure 4.1: Idealized model of side artefacts (modified from Ün et al. [61])

it is also likely that a strong architectural dependence is responsible for the variability in

so-called “minimum” sizes, which must also be considered.

Two different approaches have been used to understand the relationship between appar-

ent elastic modulus and the “side effects” caused by unconstrained peripheral trabeculae,

although each has its own limitations. Ün et al. [61] proposed a theoretical model to ex-

plain the underestimation of apparent elastic modulus due to side artefact (Fig. 4.1). They

assumed an annular region around the periphery of a cylindrical specimen that would be

affected, for which they sought to account with a correction factor. Unfortunately, because

the model was validated only to a single site, it is unclear whether it is applicable to differ-

ent architectures. An empirical model, which assumes apparent elastic modulus reaches an

asymptotic value as specimen dimensions increase, has also been proposed [60]. Although

this has been validated on two sites, it considers diameter only and fails to incorporate a

47


dependence on cancellous architecture. A shortcoming shared by both models is that they

require the “true” apparent elastic modulus to be known a priori in order to be fit to exper-

imental data, which may not always be possible given skeletal limitations. Although both

of these models have shown promise, they require further modification and validation be-

fore they can describe the artefactual effects of specimen size on cancellous bone mechanical

properties

The goal of this study is to use finite element method (FEM) models of cancellous bone to

quantify the dependence of uniaxial apparent elastic modulus on specimen size and trabecular

architecture. Five micro-CT scans acquired for a previous study (Chap. 3), each from a

different site in the bovine skeleton, were used. The existing theoretical and empirical models

were generalized, so that they were based purely on architectural parameters, and then fit to

the FEM predictions to evaluate their performance. Identifying a general model for specimen

size effects based on architectural parameters will result in a better understanding of “side

effects”, allowing for correction factors to be calculated and minimum specimen sizes to be

recommended. These developments will improve the accuracy of cancellous bone mechanical

testing, which is critical for predicting fracture behaviour.

4.2 Materials & Methods

The FEM models were created from micro-computed tomography (micro-CT) scans of can-

cellous bone, acquired as part of a previous study (Chap. 3), from five sites in the bovine

skeleton: the lateral femoral condyle (BLFC), sacral wing (BSW), greater trochanter (BTR),

sixth lumbar vertebrae (BLV6), and humeral head (BHH). Each specimen was demarrowed

using a dental water flosser (InterPlak, Conair) combined with boiling, immersed in epoxy,

spun in a centrifuge to ensure complete penetration into the specimen, and then allowed to

cure for 24 hours. The embedded specimens were scanned using a MicroXCT 400 (Xradia

Inc.; Pleasanton, California) at a resolution of 15 µm. The specimens were large enough

48


Table 4.1: Morphological parameters of thresholded and pruned 15 µm cylinder images, repre-senting a 5.6× 5.6× 10mm3 region



that an uninterrupted 12.3× 12.3× 12.3mm3 cube could be cropped from the center of the

sample scan. With this size of cube, it was possible to extract an 8 mm diameter × 10 mm

length cylinder from any possible orientation; these constraints determined the upper limit

of specimen geometry used within this study.

Since morphological measures of cancellous bone are known to be sensitive to voxel size

[63, Chap. 3], all measurements were performed at the highest possible resolution (15 µm)

to minimize the potential for errors. Values for bone volume fraction (BV/TV), trabecular

thickness (Tb.Th), trabecular spacing (Tb.Sp), and degree of anisotropy (DA) were obtained

using FIJI [86] and the BoneJ plugin [87]. Prior to taking morphological measurements,

image segmentation was performed using Otsu’s thresholding method [84] and floating voxels

were pruned. Connectivity density (Conn.D) was also calculated after running the purify

algorithm in BoneJ –which removes disconnected bone particles and fills internal holes– to

avoid measurement error. The morphological parameters for each of these cylinders are

summarized in Tab. 4.1.

4.2.1 Model creation method

The steps used to create the FEM models are illustrated in Fig. 4.2. A rotation matrix,

R, was used to re-orient the original 12.3 mm cropped cube using tri-linear interpolation

to align with the principal mechanical axis (PMA). The orientation of the PMA had been

49


Figure 4.2: A simplified 2D representation of the 3D model creation process used herein. Note:Images not to scale

Figure 4.3: The method used to crop smaller diameter cylinders from the aligned cylinder region[95]

identified in Chap. 3 to within 1° using an iterative search based on FEM simulations of

uniaxial compression. A downsample step was then performed to decrease the resolution

from 15 to 30 µm via voxel averaging [38, 66] and a cylinder of the desired dimensions was

extracted from the center. The image was binarized using Otsu’s method [84] and bone

voxels were converted to isotropic hexahedral elements. A cleaning step was then performed

to remove unconnected and cantilever-connected elements from the model. All models were

run using LS-DYNA Implicit version R10.0.0 (Livermore Software Technology Corporation;

Livermore, CA).

As mentioned previously, the largest randomly oriented cylindrical model of uninter-

rupted bone that could be obtained from the 12.3 mm images was 8 mm in diameter and

10 mm in length. Therefore, only one model at 8 mm diameter could be created for each

skeletal site. For each diameter group less than 8 mm, nine models of equal distribution

50


were cropped from the original 8 mm diameter cylinder (Fig. 4.3). One of the nine smaller

cylinders was taken directly from the center of the specimen, while the remaining eight were

taken with their outer edge touching the outer edge of the original 8 mm diameter cylinder.

Each model was then subjected to a 0.1% compressive strain. The nodal forces at the top

surface were averaged over the apparent cross-sectional area to determine an apparent stress

(σ). The apparent elastic modulus was then calculated using E = σ/ε.

Empirical model

Based on previous work [60], the change in apparent elastic modulus (E) with specimen

diameter (D) has been shown to follow an equation of the form:

E = E0 [1− exp (−a0D)] (4.1)

where E0 and a0 are constants. Least-squares fitting routines in Matlab (The MathWorks,

Inc.; Natick, MA) were used to fit the apparent elastic modulus values for each site to

Eq. (4.1). In order to investigate whether all five sites display the same fundamental be-

havior, two types of normalization were investigated. First, the results of Eq. (4.1) were

normalized by the asymptotic value E0 to account for the differences in apparent elastic

modulus. Various methods for normalizing the specimen diameter were also investigated

using an equation of the form:

E

E 0=

[1− exp

(−a0

D

P

)](4.2)

where a0 is a fitting term and P is a morphological parameter such as Tb.Th, Tb.Sp,

Tb.Th+Tb.Sp, or 1/ 3√

Conn.D. While Eq. (4.2) is helpful for understanding the funda-

mental dependence of apparent elastic modulus on diameter, it is not useful for calculating

correction factors or selecting specimen sizes for experimental studies since it requires that

E0 be known in advance. Therefore, an additional fit was performed to find a single equation

51


that could be used for experimental planning based purely on architectural parameters:

E = E∗0

[1− exp

(−a0

D

P

)](4.3)

where E∗0 takes the form

E∗0 = a1 (BV/TV)a2 (1 + a3DA) (4.4)

and ai are constants. Since there are four fitting parameters, it is possible to arrive at

various sets of parameters based on the assumed starting values. Therefore, 100 iterations of

the fit were performed with randomized initial values within a suitable range. The fit with

the largest R2 was chosen to avoid local minima. Within the results for all these models,

comparisons of both R2 and R̄2 (adjusted R2) will be performed to attempt to control for

improvements in model accuracy due to the addition of parameters.

Theoretical model

The apparent elastic modulus versus diameter data were also fit to the theoretical model

proposed by Ün et al. [61] as given by:

E = E0

[1− 2 (b0Tb.Sp + b1)

D

]2(4.5)

where bi are constants. Similar to Eq. (4.3), this theoretical model was then generalized for

unknown asymptotic moduli in the form:

E = E∗0

[1− 2(b0Tb.Sp + b1)

D

]2(4.6)

where E∗0 is given in Eq. (4.4).

52


Correction factors

Following Ün et al. [61], correction factors (α) for experimental testing can be calculated

using:

α =E∗

0

E=

[1− exp

(−a0

D

P

)]−1

(4.7)

for the empirical model, Eq. (4.3), and

α =E∗

0

E=

[1− 2(b0Tb.Sp + b1)

D

]−2

(4.8)

for the theoretical model given in Eq. (4.6).

Minimum specimen sizes

These equations can also be used to calculate a minimum specimen size needed to achieve a

particular level of average error (δ). The minimum diameters, Dδ, are given by:

Dδ =P

a0ln(δ) (4.9)

and

Dδ =2(b0Tb.Sp + b1)

1−√1− δ

(4.10)

for the two models, Eq. (4.3) and Eq. (4.6), respectively.

4.3 Results

4.3.1 Empirical model

A series of finite element method models of varying diameters were created for five sites in

the bovine skeleton. The predicted moduli from the FEM models are shown in Fig. 4.4,

along with the individual fits using the empirical model given in Eq. (4.1) to relate moduli

53


with increasing diameter. The fits are given by:

BLFC : E = 2.095 [1− exp (−1.201D)] , R2 = 0.9037 R̄2 = 0.8877

BLV6 : E = 1.636 [1− exp (−1.250D)] , R2 = 0.7324 R̄2 = 0.6773

BTR : E = 1.232 [1− exp (−0.927D)] , R2 = 0.9188, R̄2 = 0.9053

BHH : E = 1.145 [1− exp (−0.832D)] , R2 = 0.9799, R̄2 = 0.9765

BSW : E = 0.6211 [1− exp (−0.483D)] , R2 = 0.9668, R̄2 = 0.9613

All specimens display a similar trend that is well described by the empirical equation. One

noteworthy feature are the higher levels of variance that occur in some specimens as diameter

decreases.

Figure 4.4: a) Model diameter vs apparent elastic modulus. Curves were fit individually to eachspecimen. b) The previous curves, normalized by their individual E0 values as calcu-lated in Eq. (4.1)

The FEM moduli were also fit to Eq. (4.2) to determine if a relationship between min-

imum specimen geometry and architecture could be identified. Specifically, diameter was

normalized by Tb.Sp, Tb.Th, Tb.Sp+Tb.Th, or 1/ 3√

Conn.D. Trabecular spacing (Tb.Sp)

gave the highest average coefficient of determination (R2) across all five specimens. The

54


results are plotted in Fig. 4.5 and the individual equations for each specimen are:

BLFC : E = 2.095

[1− exp

(−0.589

D

Tb.Sp

)], R2 = 0.9037, R̄2 = 0.8877

BLV6 : E = 1.636

[1− exp

(−0.758

D

Tb.Sp

)], R2 = 0.7324, R̄2 = 0.6773

BTR : E = 1.232

[1− exp

(−0.667

D

Tb.Sp

)], R2 = 0.9188, R̄2 = 0.9053

BHH : E = 1.145

[1− exp

(−0.421

D

Tb.Sp

)], R2 = 0.9799, R̄2 = 0.9765

BSW : E = 0.6211

[1− exp

(−0.378

D

Tb.Sp

)], R2 = 0.9668, R̄2 = 0.9613

Figure 4.5: a) Model diameter normalized by trabecular spacing vs apparent elastic modulus.Separate curves are applied for each specimen’s trend as shown in Eq. (4.2). b)Fig. 4.5a curves, normalized by their separate E0 values as calculated in their indi-vidual fits

In order to produce a more predictive equation –one that does not require prior knowledge

of the asymptotic apparent elastic modulus (E0)– the average moduli for all five specimens

were fit to Eq. (4.3). The result was:

E = E∗0

[1− exp

(−0.608

D

Tb.Sp

)], R2 = 0.98217, R̄2 = 0.98128 (4.11)

55


where:

E∗0 = 7.747× 10−3 (BV/TV)1.448 (1 + 1855.7DA) (4.12)

and is plotted in Fig. 4.6.

Figure 4.6: a) Specimen size fit according to BV/TV and DA values applied to apparent elasticmodulus values (Eq. (4.3)) b) Specimen size normalized by Tb.Sp vs apparent elasticmodulus by BV/TV and DA (Eq. (4.11)).

4.3.2 Theoretical model

The same FEM moduli were fit using the theoretical model of Ün et al. [61] in order to

compare it with the empirical results. Given that it was originally validated based on moduli

from a single site (human vertebrae), the model was first fit individually to the five sites in

the current study to ensure its robustness. The following equations were found and are

shown in Fig. 4.7:

BLFC : E = 2.221

[1− 2(4.424Tb.Sp − 2.089)

D

]2, R2 = 0.9768, R̄2 = 0.9675

BLV6 : E = 1.732

[1− 2(10.42Tb.Sp − 6.282)

D

]2, R2 = 0.8583, R̄2 = 0.8017

56


BTR : E = 1.333

[1− 2(0.7079Tb.Sp − 0.392)

D

]2, R2 = 0.9814, R̄2 = 0.974

BHH : E = 1.251

[1− 2(−560.7Tb.Sp + 286.1)

D

]2, R2 = 0.9966, R̄2 = 0.9953

BSW : E = 0.7015

[1− 2(0.2718Tb.Sp + 0.0464)

D

]2, R2 = 0.9832, R̄2 = 0.9765

Figure 4.7: a) The Theoretical model fit Eq. (4.5)applied to the 5 specimen apparent elasticmodulus values and b) the Fig. 4.7 fit normalised by E0

Note that the predicted values reverse direction and approach infinity for very small

values of D.

When the generalized form of Ün’s theoretical model, as given in Eq. (4.6), was fit

simultaneously to all the data, the following equation was determined (Fig. 4.8):

E = E∗0

[1− 2(0.092892Tb.Sp + 0.044417)

D

]2, R2 = 0.98477, R̄2 = 0.98374 (4.13)

where:

E∗0 = 8.2160× 10−3 (BV/TV)1.4770 (1 + 1945.3DA) (4.14)

57


4.3.3 Correction factors

Following the equations for both the empirical (Eq. (4.7)) and theoretical models (Eq. (4.8)),

correction factors were determined for both models (Tab. 4.2). Correction factors are calcu-

lated based off the assumption of a 6 mm diameter unaffected region. This assumption was

made to allow comparison between the current study’s results and those of Ün et al. [61]:

α =E∗

0

E=

[1− 2(0.092892Tb.Sp + 0.044417)

D

]−2

, R2 = 0.98477, R̄2 = 0.98374

α =E∗

0

E=

[1− exp

(−0.60788

D

Tb.Sp

)]−1

, R2 = 0.98217, R̄2 = 0.98128

Figure 4.8: a) The Eq. (4.6) fit, with lines representing values for each specimen b) The Eq. (4.6)model for all specimens. Solid line shows our criteria based off of our specimen mostprone to side effects (BSW), while dashed indicates least susceptible specimen (BLFC)to display a range of fits possible.

Table 4.2: Correction factor ranges representing the minimum (BLFC), and the maximum cor-rection factors (BSW) for both models as calculated using Eq. (4.7) and Eq. (4.8)

Model Minimum MaximumEmpirical 1.00 1.01

Theoretical 1.06 1.08

58


4.3.4 Minimum specimen sizes

The two models can also be used to estimate minimum specimen size requirements to meet

some level of average apparent elastic modulus accuracy. The minimum specimen diameters

required for all five anatomical sites, as calculated by Eqs. (4.9) & (4.10), are displayed

in Tab. 4.3. The empirical and theoretical models predict a range of minimum specimen

sizes, across all five specimens, from 7.1–9.2 mm and 35.9–46.6 mm at 5% and 1% error,

respectively. Though these values can be used as guidelines for minimum specimen size

requirements, the large differences between the two models indicate the need for further

research.

One convenient feature of the empirical model is that the minimum specimen size is a

function of D/Tb.Sp. Therefore, it is possible to suggest a rule-of-thumb that specimens

should be a minimum of 4.9 or 7.6 trabecular spacings, for 5% and 1% error respectively.

Table 4.3: Specimen minimum required diameters at 5% and 1% error in asymptotic apparentelastic modulus value

5% error 1% errorSpecimen Theoretical Empirical Theoretical Empirical

BLFC 7.1 2.4 35.8 3.7BLV6 8.0 3.0 46.5 4.6BTR 8.8 3.5 44.3 5.7BHH 7.2 2.5 36.5 3.9BSW 9.2 3.8 46.5 5.9

4.3.5 Discussion

FEM simulations were performed to evaluate both the empirical and theoretical models

when applied to cancellous bone from five sites in the bovine skeleton. It was observed

that apparent elastic modulus increases as specimen diameter increases, trending towards

an asymptotic value; however, the rate at which it approaches that asymptotic value varies

by anatomical site. In addition to this, apparent elastic modulus variance was observed

59


to increase as diameter decreases, with the magnitude of this effect also being specimen

dependent. When evaluating the empirical model predictions, it was observed that inclusion

of the specimen architecture, specifically Tb.Sp, resulted in better fits. This observation,

along with the arguments that Ün et al. used to support the inclusion of Tb.Sp in the

theoretical model, supports the important role of specimen architecture in “side effects”.

Lastly, both models fit the FEM simulation data well, with the theoretical model slightly

out performing the empirical model in 4 of the 5 regions analysed.

Given that both models performed similarly, the overall nature of each model must be

considered to determine if one is recommended over the other. The Ün model was developed

based on a theoretical argument about the nature of specimen “side effects”. Although it

accounts for the effect of specimen architecture on this phenomenon, the model predictions

become unstable as specimen size decreases (Fig. 4.8). It should be recognized that this

behavior is more of a mathematical quirk than a genuine limitation, however, since it only

occurs at diameters so small as to be impossible to manufacture and test. In addition to

this, the minimum specimen size predictions made by the theoretical model were generally

as high or higher than previous studies, with values at 99% the asymptotic value being

unreasonably large. Given that previous studies’ minimum size estimations are expected

to be over-estimations, due to expected errors caused by misalignment and end effects, the

theoretical model, though it produces a superior fit with data when compared to the empirical

model, appears to over-predict minimum specimen size requirements.

The empirical model, similarly to the theoretical, has issues at small specimen diam-

eters in that it predicts that apparent elastic modulus will decay smoothly to zero as D

approaches 0. In reality, there is some minimum diameter, which will itself be a function

of the architecture, below which no cylindrical specimen will be machinable. Though this

is of less concern than the upward trends possible in the theoretical model, it should be

noted. Lastly, minimum specimen size requirements of this model were notably lower than

the recommendations of 5.6–10 mm in previous studies [6, 7, 60, 77], as well as the theoret-

60


ical model. Therefore, it was deemed necessary to consider the errors possible in previous

studies’ minimum specimen size estimations, to allow analysis of the accuracy of the model’s

results, as well as compare its accuracy to the theoretical model’s.

Two recent studies will be considered to attempt to minimize error due to differences in

methodology. In an experimental study by Lievers et al. [60], they found a 9% apparent

elastic modulus reduction at 5.1 mm compared to 10.6 mm diameter specimens from the

bovine femoral condyle. They also observed a 17% decrease in apparent elastic modulus

in bovine vertebral cores when reducing the diameter from 10.6 to 6.6 mm. The current

empirical model’s predictions for those same changes are 0.1–6.8% and 0.8–7% for a femoral

condyle and vertebrae, respectively. The model’s underestimation of error when compared to

the experimental results is likely due to all the simplifying assumptions contained within the

FEM analysis. For example, the FEM model assumes homogeneous, isotropic elastic material

properties which are not representative of the complexity of the true tissue properties of

trabecular bone. Modeling also assumes perfect loading conditions and ignores how the

presence and development of microdamage might alter tissue behaviour. Finally, since it

is limited to one example from each anatomic site, it does not include the variability in

architecture from within a site that would be present in experimental testing. In addition to

these differences, Lievers et al.’s lack of alignment protocol could cause some small amount

of error and/or variation as no alignment protocol was followed and the region analysed is

likely prone to these errors (Chap. 3).

The second study of interest is one performed by Ün et al. [61]. Their findings indicate

an average error of 27% within their human vertebral specimens, with a maximum of 50% in

one specimen at 6 mm diameter. Focusing on their average error, the current models predict

a range of 0.7–5.4% and 7.5%–10.4% for the empirical and theoretical models, respectively.

Perhaps the most likely explanation for these differences is species. Though bovine cancellous

bone has been shown to have a very similar response to loading when compared to human

cancellous bone [52], differences are expected. In addition, Un et al.’s specimens had a large

61


range in BV/TV (0.05–0.20), while the current study examined a similar range but over

denser specimens (0.15-0.34). Given this variation, and the highly anisotropic structure of

vertebrae, it is possible misalignment introduced some small degree of error in the earlier

study. When comparing FEM modeling approaches, some differences are noted. Their initial

scanning resolution was 22 µm compared to the current study’s initial scanning resolution

of 15 µm. Un et al.’s models were analyzed by a custom-designed code using roller-type

constraints which allow the bottom-platen surface to expand laterally, hence, some amount of

strain in the load bearing direction would be transferred to the lateral direction, which would

not occur in the current study. Lastly, given their study only analysed a single anatomical

site, the extrapolation of their results may be limited. The current study’s wider range

of cancellous architectures may be prone to underestimating error in regions particularly

affected by these phenomena.

As mentioned previously, results from the current study’s correction factor fits corre-

sponded with previous results. Correction factors for a 6 mm diameter effective region were

calculated to be around 1 for both models presented herein, somewhat lower than the value

of 1.27 found by Ün et al. at this diameter. To achieve a correction factor of 1.27 our mod-

els require a 1.60–2.08 mm diameter for the theoretical model (depending on Tb.Sp) and

a 1.99 mm diameter model for vertebrae specimen using the empirical model (while no real

solution exists for the femoral specimen). This significant difference is expected to be caused

by Un et al’s assumption that values for Etrue were equivalent to their 8 mm diameter model

results. Fits determined with both the theoretical and empirical models indicated that this

assumption is not likely correct, with one specimen (BSW) having a ratio of E/E0 < 1 and

all specimens having a ratio of E/Etrue < 1. As a result, the current models indicate that

the “side effect” phenomenon is likely more extreme than noted in Un et al.’s study, due

to their assumption of an asymptote at 8 mm diameter. In addition to these findings, the

current authors suggest that correction factors be used with caution. Given the architectural

62


heterogeneity of specimens, these correction factors should be applied only to averages of

multiple specimens, as opposed to single specimens.

In terms of limitations, it must be recognized that this is a purely numerical study. The

advantage of such an approach is that it allows the same architecture to be tested repeatedly

in order to isolate the effects of specimen size. This is not possible with experimental testing

due to destructive specimen preparation. However, the effects of the simplified homogeneous,

isotropic, linear-elastic properties assumed as part of the model are not considered. These

should be studied in future work; however, it is expected that the architecture itself will

have a dominant role [3]. The fact that the cancellous bone sites are all from the bovine

skeleton is also a limitation, at least with respect to extrapolating the current results to

human sites. The validity of these model parameters, as applied to the human skeleton,

needs to be considered in more detail. Nevertheless, the five bovine sites studied have shown

the ability of both the theoretical and empirical model to perform well over a wide range of

architectures.

Moving forward, future work must be performed to determine the accuracy of the em-

pirical model, as well as to confirm the current study’s findings. To perform this analysis,

it is recommended that an experimental procedure involving proper alignment of specimens

is used to ensure only that side effect error is being analysed. The work should analyse

various regions of human cancellous bone which represent a wide range of architectural mea-

surements. By analyzing the two proposed models in such a study, results should be able

to predict minimum specimen sizes for human specimens, allowing researchers to efficiently

and effectively use donated cancellous bone specimens to further understand these effects

and for accurate analysis of cancellous bone mechanical properties. The authors believe that

following research is paramount for minimizing possible error due to specimen size in future

studies.

63


4.4 Conclusion

Error introduced by “side effects” must either be minimized or corrected for to ensure ac-

curate mechanical testing results for cancellous bone. The current findings indicate that a

empirical model may be of the most use in this regard. Furthermore, specimen architecture

has been shown to affect the magnitude of the “side effects” across a range of cancellous bone

sites, in particular Tb.Sp. Findings suggest that measures including Tb.Sp relate different

anatomical sites loading response better than measures of geometry alone. Due to this, it

may be possible to standardize a minimum specimen geometry across various anatomical sites

using a measure of or including Tb.Sp. The authors recommend this model be applied to the

experimental study of various regions of human cancellous bone to determine what role ar-

chitecture holds in minimum specimen size requirements, to allow determination of a criteria

for minimum specimen size based on architecture, which will minimize any error introduced

by “side-effects”. By validating and developing this model in such a study, researchers will

be able to minimize error, allowing easier and more accurate comparisons between studies

of cancellous bone, furthering the field to the final goal of an accurate mechanical model of

cancellous bone and its role in age- and disease-related fractures.

64

Chapter 5

Discussion & conclusions

5.1 Introduction

Experimental testing is invaluable to the assessment of age– and disease–related changes in

cancellous bone. One must minimize error, bias, and variation in studies of both diseased

and healthy bone in order to be able to properly and accurately determine these effects.

As explained in Chaps. 1 & 2, although standardized protocols exist for cancellous bone

experimental testing, the effects of alignment and specimen size need to be better understood

to ensure that results are both reliable and repeatable. These issues were then addressed

using FEM modeling in Chaps. 3 & 4, with individual discussions in each, following the

manuscript style. This final chapter presents a common discussion of the results of this

thesis, the contributions it makes to the literature, and its conclusions.

5.2 Findings

The results outlined in both Chaps. 3 & 4 have confirmed that cancellous architecture plays

a significant role in specimen alignment and specimen geometry artefacts. In both studies,

specimens with lower BV/TV and higher DA have larger, as well as more variable, levels of

error when either misaligned or undersized. In addition, substantial differences in the mag-

65

CHAPTER 5. DISCUSSION & CONCLUSIONS

nitude of the errors were found between specimens, indicating that different anatomical sites

require standards based upon their own unique architectural measures to evaluate possible

errors. One cannot simply assume that the magnitude of the errors obtained in the verte-

brae, for example, will be the same as those in the femoral condyle, even with equivalently

sized or aligned specimens. Lastly, DA was found to hold a significant role in this behavior,

along with BV/TV, indicating that previous studies promoting BV/TV or apparent density

as the primary indicator of architectural-related behaviors do not necessarily fully determine

the effects of misalignment and required specimen geometry. It is recommended that more

attention be given to DA in future studies.

5.3 Comparison to previous work

One notable difference when comparing this study to existing research is that architecture’s

role in determining the magnitude of misalignment and side-effect artefacts was not consid-

ered in previous studies [6, 8, 9, 61]. Therefore, findings such as those reported in the current

study may have been missed or overlooked. As with previous studies [3, 27], apparent elastic

modulus and BV/TV were found to have an exponential relationship in the models studied

herein. Though this role is an important one, the current study expands this relationship

through the inclusion of DA. Previous studies have found that anisotropy affects apparent

elastic modulus measurements [3]. While these findings are important to consider, most

used either a fabric anisotropy measurement, or a DA measurement ranging from 1 to ∞.

The form used within the current study (0 to 1) simplifies modelling, with complete isotropy

having no effect in misalignment loading. By developing such a model, one can analyse two

regions having the same BV/TV (such as BLV6 & BHH in the current study) and determine

what role the architectural anisotropy has in these experimental loading effects. This finding

further indicates the need to use a model which considers both BV/TV and DA as measures

of significant importance, rather than DA being considered a measurement secondary to

66


BV/TV. Other than this difference, all results reported herein followed the general trends

seen in past studies, though the magnitudes of these trends were lessened to a degree. This

is likely due to proper alignment to the PMA in the current study, as well as the fact that

the current study considers multiple models from the same specimen. Hence, it is likely that

the current study’s results include less error due to variation caused by these effects, which

in turn lessen the magnitude of the trends.

5.4 Limitations

When comparing the current results to previous studies, one must consider that the current

work is not without its own limitations. First, it must be recognized that every step of the

FEM model creation process introduces some level of error due to the approximations and

assumptions they require. For example, the voxel-to-hexahedral element meshing approach,

while common and convenient for such complicated cancellous models, produces a mesh

surface that is discontinuous. These errors, however, are minimized by using resolutions

that ensured a minimum number of elements across the thickness of a trabecula. Even for

the models at the highest resolutions within this study, some rounding errors are expected,

but should be ≤ 5% [62].

The assumption of a constant isotropic elastic material for each bone element likely in-

troduces error. Cancellous bone tissue has been shown to have variable mineral distribution

and, as a result, heterogeneous material properties [96]. Previous FEM modelling stud-

ies have found that introducing different populations of material properties led to a minor

overestimation of apparent elastic properties, when compared to homogeneous models, of

2.2% [70]. Moreover, this previous work only investigated heterogeneity by region gener-

ally, rather than attempt to replicate a specimen’s true property distribution. An accurate

method for measuring and replicating the material property distributions of specific samples

within an FE model does not currently exist. Therefore, although this assumption is an

67


over-simplification of reality, it was necessary. It is expected to cause only a small error, but

future work will be needed to quantify the effect of this assumption. The exact value of the

constant elastic modulus (10 GPa) is not critical, as most apparent elastic modulus values

for the specimens reported in this thesis are normalized in some way.

The study is limited to five different anatomical sites. Given the large number of models

that were required, particularly for measuring the effects of specimen misalignment, and

the super-computing time to perform the simulations, this number was deemed reasonable.

The five samples chosen also spanned a large range of BV/TV and DA values; however,

further sites would be beneficial to expand that range. A greater diversity of architectural

parameters would ensure that the results remain valid over a wider range of cancellous bone.

Adding to the previous point of selected material properties, the current study relies

solely on numerical simulations. This form of analysis is useful as it allows researchers to

examine relationships while controlling and/or eliminating outside effects on specimens (such

as hydration and variable loading surfaces). Though these analyses are useful for determining

relationships between cancellous architecture and apparent elastic modulus, experimental

studies are needed to confirm the nature and magnitude of these effects.

Lastly, though bovine cancellous bone is a useful material for such studies because it is

easily acquired, it is significantly different from human anatomy. Due to this, results based

on this animal model may not be directly extrapolated to studies of human cancellous bone,

which is of greater clinical interest due to the effects of age- and disease-related as previously

mentioned. When evaluating the findings of this study, both within as well as to other

applications, these limitations must be considered.

5.5 Future work

Future work must be performed to further explore the findings in this thesis. First, exper-

imental studies must be performed to confirm whether the trends and predictions obtained

68


from numerical modelling hold true in mechanical testing. This would require that both spec-

imen size requirements and specimen alignment be tested separately to ensure the individual

trends are correct.

Paramount to this field of research is an experimental method to align cancellous bone

specimens to within 5° degrees of the primary mechanical axis. The development of a

reliable and repeatable specimen alignment protocol would allow for the near elimination

of misalignment error, allowing minimization of noise, error, and biases in results. Perhaps

the most reliable way would be a computer numerical control (CNC) method, following the

form of a previous study performed by Wang et al. [78], as it would not be as susceptible

to the human error inherent in having an operator identify the main trabecular direction

[9]. Initial estimates of the PMA could be calculated through first the MIL method, with

confirmation or recalculation of the PMA using FEM models of lower resolution to limit

the computational requirements. This calculated alignment could then be related back to

the initial alignment of the scan, with Euler angles being calculated to determine necessary

transformations to align the specimen. A CNC method could then be used to align the

specimen to within 4° of this estimation, ensuring specimen alignment is within 5°. Finally,

after these practices, a sample could be cored aligned to this axis, allowing specimens to be

within 95% of the expected apparent elastic modulus value.

Upon the completion of this CNC alignment method, specimens from multiple anatomi-

cal sites should be analysed with varying geometries to evaluate the results of the calculated

minimum specimen size. By ensuring specimens are aligned, the relationship between speci-

men size and apparent mechanical properties can be determined with the minimum amount

of noise possible which should result in clearer trends. Future studies could then base their

required specimen geometries off of both a computational, as well as an experimental anal-

ysis. In addition to this, the specimen size requirements of various sites can be determined,

allowing researchers to optimize the use of human cancellous bone specimen donations. This

69


will allow for these valuable donations to be used to their maximum efficiency by permitting

more analyses to be performed with the same number of whole specimens.

Following these analyses, experimental studies should be performed on human cancellous

bone specimens using the developed methods. Multiple regions, specifically low BV/TV,

high anisotropy regions which are prone to ageing and osteoporotic effects, as well as regions

that are not heavily changed by these effects, should be analysed. By performing such

a study, cancellous architecture’s role in apparent elastic modulus measurements can be

more accurately determined for use in developing a model for clinical analysis. This analysis

should be considered the primary goal of these methodological developments to provide more

accurate and reliable data for patient care and treatment.

Lastly, the misalignment study’s findings may be of use to produce a model predicting

the mechanical strength decrease in cancellous bone during fall-like scenarios. This model

would be extremely useful to minimize the effects of fractures due to ageing and osteoporosis

in our increasingly elderly population.

5.6 Contributions & conclusions

While acknowledging these limitations, the current thesis makes a number of novel contri-

butions to the field of cancellous bone research:

1. The current study is the first to demonstrate the effects of cancellous bone architecture

on specimen misalignment artefacts. A mathematical model has been developed to

predict the magnitude of the error based on two architectural parameters, BV/TV

and DA.

2. An iterative modelling methodology is introduced whereby the primary mechanical

axis of cancellous bone can be determined to within some specified tolerance (e.g., 1°).

While such alignment accuracy is near impossible within experimental studies using

70


the current methodology, it is helpful to reduce misalignment artefacts in numerical

simulations.

3. This study has expanded two previous models of specimen side-effects, one phenomeno-

logical and one theoretical, to predict the experimental artefacts based on architecture

(BV/TV, DA, Tb.Sp) and specimen diameters alone. An evaluation of the two models

showed they both fit the data similarly, although the phenomenological model is con-

ceptually simpler and provides more realistic predictions of minimum specimen size

and correction factors.

Specimen architecture was shown to play a significant role in determining both minimum

specimen size and specimen misalignment effects, as well as its effect on apparent elastic

modulus measurements. Alignment error is increased in regions of lower bone volume and

higher anisotropy, while trabecular spacing was found to be a determinant of minimum

specimen geometry. These architectural-based measurements will allow for quantification of

error introduced by these effects. By following the guidelines produced in this work, it is

expected that error will be kept to an acceptable level to avoid introduction of significant

biases between different specimens and regions. Experimental methods must be produced

to meet these guidelines to allow for study of cancellous bone mechanical properties to be

as accurate as possible, allowing more reliable models to be produced, which could be used

to better understand the effects of osteoporosis and ageing on fall-like scenarios.

71

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Appendix A

FE model convergence study

A.1 Introduction

Several studies have shown that, below some threshold, increased element resolution has

little effect on the accuracy of hexahedral finite element method models of cancellous bone

[38, 68]. Though these results suggest that maintaining 4–6 elements per trabecular thickness

ensures convergence of the predicted moduli [68], it was deemed necessary to confirm these

findings with specimens from the current study. To perform this analysis, specimens with

different architecture from the current studies were modelled to confirm the role element

resolution holds in apparent elastic modulus convergence.

A.2 Materials & Methods

Two specimens were chosen for this study based on notable differences in their architectures:

the sacral wing (BSW) and humeral head (BHH). The morphological parameters highlighted

in Tab. A.1 demonstrate how these two selections achieve a wide range of BV/TV and Conn.D

values.

To conserve computational resources, smaller cylindrical specimens of 5 mm diameter

and 5 mm length were chosen for the convergence study. FE meshes were generated as in

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APPENDIX A. FE MODEL CONVERGENCE STUDY

Table A.1: The morphological parameters of PMA aligned cylinders at 15µm resolution (repro-duced from Tab. 3.1) highlighting the samples used for the convergence study



Chaps. 3 & 4 at resolutions of 15, 30, 45, 60, 75, and 90µm. Images of four meshes for BSW

are shown in Fig. A.1.

These meshes were then subjected to apparent compressive strains of 0.1% using LS-

DYNA. The predicted apparent moduli were then plotted with respect to resolution. The

percentage error was also calculated for each resolution, i, using:

Error (%) = Ei − E15µm

E15µm

× 100% (A.1)

where E15µm is the apparent modulus for the 15µm model. Because the scan resolution was

15µm, it will be considered the baseline (e.g. 0% error).

A.3 Results

The predicted apparent moduli for the two specimens are plotted as a function of element

size in Fig. A.2. As in previous studies [38, 68], a decrease in specimen apparent modulus

was seen as voxel size increased. More importantly, the predictions for both models converge

at higher resolutions (i.e., smaller voxel sizes).

85


Figure A.1: The effect of resolution on BSW model structure. Resolutions are as follows: A)15µm, B) 30µm, C) 60µm, D) 90µm,

86


Figure A.2: Results of specimen apparent modulus values dependence on element resolution.Both anatomical sites show similar trends at different magnitudes.

Plotting the error (relative to the 15µm results) of coarser models shows this converge

more clearly (Fig. A.3). The error is 2% or less at the proposed model resolution of 30µm.

Convergence appears to begin at 45µm resolution for BSW and 60µm for BHH.

Additionally, the findings of Guldberg et al. [68] were confirmed in our specimens; four

elements or more across Tb.Th in both our specimens resulted in < 2% error (Fig. A.4). A

small overestimation (1%) of the 15µm apparent modulus was observed in the BHH specimen

for the 30 and 45µm models. Given that this overestimation is small, and such trends did

not occur in the BSW models, it is concluded that the model had indeed converged.

A.4 Discussions & Conclusions

The FEM modelling work described in Chaps. 3 & 4 relied on one of two criteria to determine

the model resolution: 30µ voxels or 4–6 elements per trabecular thickness (Tb.Th). The

results of this convergence study confirm that the selected resolutions for the current studies

87


Figure A.3: Error of apparent modulus measurements based on element resolution. Error iscalculated as the percent difference from the 15 µm element resolution model mea-surement. Dashed lines represent ± 2% error.

Figure A.4: Error of apparent modulus measurements compared to 30µm resolution model. Onceelements reach the 4 element across Tb.Th threshold, error is < 5%. Dashed linesrepresent ± 2% error.

88


are adequate to ensure < 5% error in FEM models. In addition to this, the trends indi-

cate apparent elastic modulus measurement error is indeed tied to the number of elements

spanning Tb.Th of a specimen. Both these results are supported by previous work [38, 68].

Therefore, it was deemed appropriate to employ a 30µm resolution for the conventional

and ’side–effects’ specimens, and the 4–6 elements per Tb.Th rule for the eroded/dilated

specimens.

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the role of cancellous bone architecture in misalignment ...€¦ · matthew b. l. bennison a...

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