Relationships of microstructure and composition with material and macro level viscoelastic properties of trabecular bone

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uef.fi

PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND Dissertations in Forestry and Natural Sciences

ISBN 978-952-61-2534-3 ISSN 1798-5668

Dissertations in Forestry and Natural Sciences

DISSERTATIONS | XIAOWEI OJANEN | RELATIONSHIPS OF MICROSTRUCTURE AND COMPOSITION... | No 271

XIAOWEI OJANEN

RELATIONSHIPS OF MICROSTRUCTURE AND COMPOSITION WITH MATERIAL AND MACRO LEVEL VISCOELASTIC PROPERTIES OF TRABECULAR BONE

PUBLICATIONS OF

THE UNIVERSITY OF EASTERN FINLAND

Trabecular bone is a metabolically active tissue with a highly hierarchical structure.

The viscoelasticity in trabecular bone allows dissipation of energy providing resistance to fractures under dynamic loading. Since our bones

are loaded dynamically daily during motion or standing, it is important to understand bone viscoelasticity at different hierarchical levels (i.e.

material and macro levels) and possible changes it may have during aging or disease. In this thesis, the relationships of material and macro level viscoelasticity of trabecular bone with tissue composition and microstructure are characterized.

XIAOWEI OJANEN

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RELATIONSHIPS OF MICROSTRUCTURE AND

COMPOSITION WITH MATERIAL AND

MACRO LEVEL VISCOELASTIC

PROPERTIES OF TRABECULAR BONE

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Xiaowei Ojanen

RELATIONSHIPS OF MICROSTRUCTURE AND

COMPOSITION WITH MATERIAL AND MACRO LEVEL VISCOELASTIC PROPERTIES OF TRABECULAR BONE

Publications of the University of Eastern Finland Dissertations in Forestry and Natural Sciences

No 271

University of Eastern Finland Kuopio

2017

Academic dissertation

To be presented by permission of the Faculty of Science and Forestry for public examination in the Auditorium SN201 in Snellmania Building

in the University of Eastern Finland, Kuopio, on June 19^th 2017, at 12 o’clock noon

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Grano Oy Jyväskylä, 2017

Editors: Pertti Pasanen, Matti Vornanen, Jukka Tuomela, Matti Tedre

Distribution: University of Eastern Finland / Sales of publications www.uef.fi/kirjasto

ISBN: 978-952-61-2534-3 (print) ISBN: 978-952-61-2535-0 (PDF)

ISSNL: 1798-5668 ISSN: 1798-5668 ISSN: 1798-5676 (PDF)

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Author’s address: University of Eastern Finland Department of Applied Physics P.O. Box 1627

70211 KUOPIO, FINLAND email: xiaowei.ojanen@uef.fi

Supervisors: Professor Juha Töyräs University of Eastern Finland Department of Applied Physics P.O. Box 1627

70211 KUOPIO, FINLAND email: juha.toyras@uef.fi

Dean Jukka Jurvelin

University of Eastern Finland Faculty of Science and Forestry P.O. Box 1627

70211 KUOPIO, FINLAND email: jukka.jurvelin@uef.fi

Associate Professor Hanna Isaksson Lund University

Department of Biomedical Engineering P.O. Box 118

221 00 LUND, SWEDEN

email: hanna.isaksson@bme.lth.se Senior Researcher Markus Malo University of Eastern Finland Department of Applied Physics P.O. Box 1627

70211 KUOPIO, FINLAND email: markus.malo@uef.fi

Reviewers: Professor Nancy Pleshko Temple University

Department of Bioengineering 19122 PHILADELPHIA, PA, USA email: npleshjko@temple.edu

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Associate Professor Jeffry Nyman Vanderbilt University

Department of Medicine and Division of Clinical Pharmacology

37209 NASHVILLE, TN, USA email: jeffry.s.nyman@vanderbilt.edu

Opponent: Professor Harrie Weinans

University Medical Center Utrecht Department of Orthopedics P.O. Box 85500

3508 GA UTRECHT, NETHERLANDS email: hweinans@umcutrecht.nl

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7 Ojanen, Xiaowei

Relationships of Microstructure and Composition with Material and Macro Level Viscoelastic Properties of Trabecular Bone

Kuopio: University of Eastern Finland, 2017 Publications of the University of Eastern Finland Dissertations in Forestry and Natural Sciences 2017; 271 ISBN: 978-952-61-2534-3 (print)

ISSNL: 1798-5668 ISSN: 1798-5668

ISBN: 978-952-61-2535-0 (PDF) ISSN: 1798-5676 (PDF)

ABSTRACT

Trabecular bone is a metabolically active tissue with a highly hierarchical structure.

With age, changes in the rate and balance of the remodeling process lead to loss of bone mass, making trabeculae thinner and increasing non-enzymatic collagen crosslinks. These changes make the bone brittle and more susceptible to fractures.

Human bones are loaded dynamically on a daily basis through normal physical activities, such as walking. Since it is the viscoelasticity in bone that ensures dissipation of energy providing resistance to fractures under dynamic loading, it is important to understand the viscoelastic properties and their changes with age or disease at different hierarchical levels (i.e. material (tissue) and macro (structural) levels).

In this thesis, viscoelastic properties of trabecular bone samples extracted from male cadavers were examined at the material and macro levels, with respect to age- related changes in composition, microstructure and the amount of collagen crosslinks. The material and macro level viscoelastic properties of samples were measured using nanoindentation and unconfined compression, respectively.

Furthermore, sample-specific hyper-viscoelastic finite element (FE) models using two termed Prony series for expressing viscoelasticity were developed to simulate experimental mechanical measurements. In addition, the spatial distribution of bone tissue elasticity and the effects of sample preparation and embedding process on the bone acoustic impedance were investigated using scanning acoustic microscope (SAM). Composition and collagen crosslinking of the samples were quantified using Raman micro-spectroscopy and high performance liquid chromatographic analysis, respectively.

At the material level, specific relationships between tissue composition and viscoelastic properties were revealed. The stiffness of the bone tissue increased with increasing crystallinity whereas viscoelasticity decreased with the increasing mature crosslink content. However, with the increase in the collagen content, the short-term

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8 load response was simultaneously prolonged and stiffened. At the macro level, similar associations between measured viscoelastic parameters and tissue composition or collagen crosslinks were not observed. However, relationships with microstructure were detected. The FE-analyses indicated that some of the viscoelastic material parameter values (equilibrium modulus, the second shear relaxation modulus and the short-term relaxation time) were different (p < 0.01) between nanoindentation creep and macro level stress-relaxation. The acoustic impedance values measured for hydrated samples were significantly lower than those measured for embedded samples (p < 0.001). However, the values from hydrated and embedded samples were linearly correlated (r = 0.854, p < 0.05).

The results revealed the high spatial variation of structural, compositional, mechanical and acoustic properties of the trabecular bone at the material level. This necessitates careful sample preparation, e.g., polishing of bone surface for SAM, but also emphasizes the need for a larger number of spatial measurements in order to obtain representation of the whole sample. In this study, the acoustic impedance measurement of hydrated trabecular bone was successful and provided material values that are in accordence with the nanoindentation testing. The present results demonstrated how the material level viscoelastic properties are dependent on tissue composition. Viscoelasticity tends to follow the age of the tissue, as indicated by crystallinity, rather than the age of the person. Furthermore, the viscoelastic properties of trabecular bone, as measured at the material level, may not fully predict the viscoelasticity at the macro level.

National Library of Medicine Classification: QT 34.5, QT 36, QU 55.3, WE 200

Library of Congress Subject Headings: Bone; Materials Science; Biomechanics;

Viscoelasticity; Elasticity; Compressibility; Microstructure; Acoustic impedance;

Collagen; Crosslinking (Polymerization); Stress-relaxation (Physics); Finite element method; Numerical analysis

Yleinen suomalainen asiasanasto: luu; materiaalitutkimus; biomekaniikka; joustavuus;

puristus; puristuskokeet; rakenne; koostumus; akustiikka; kollageenit;

elementtimenetelmä; numeerinen analyysi

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ACKNOWLEDGEMENTS

This study was carried out during the years 2012 – 2017 in the Department of Applied Physics of the University of Eastern Finland and Kuopio University Hospital.

Firstly, I would like to express my deepest gratitude to my supervisors Juha Töyräs, Jukka Jurvelin, Hanna Isaksson and Markus Malo for their guidance, wisdom and endless support. I am grateful to have been given the honor to be part of the distinguished Biophysics of Bone and Cartilage (BBC) group.

I would also like to thank my thesis reviewers, Professors Nancy Pleshko and Jeffry Nyman for their insights on improving my work. In addition, I would like to thank emeritus Ewen MacDonald for doing the linguistic review on such a tight schedule.

I am grateful to my co-authors for their significant contributions to the studies. I am honored to have worked with such outstanding professionals. I would also like to acknowledge the help given by the staff of SIB Labs and Department of Applied Physics in the studies with special thanks to Ritva Savolainen for helping with the bone embedding process and Jukka Laakkonen for making all the “gadgets” for sample preparation.

I am fortunate to have had wonderful colleagues in the BBC group who have become my dear friends over the years. I want to offer a special thanks to all my ex- office roommates, Jarkko Iivarinen, Chibuzor Eneh, Hans Linder, Tuomo Silvast and Satu Inkinen for all the help and support you have given me in my professional and personal life.

This study has been funded by the strategic funding of the University of Eastern Finland, Academy of Finland (128863, 286526), Sigrid Juselius Foundation, Orion- Farmos Research Foundation, Emil Aaltonen Foundation, Kuopio University Hospital (VTR Project 5041741, 5041758, PY210), Finnish Cultural Foundation, Finnish Cultural Foundation North Savo Regional Fund and Doctoral Programme in Science, Technology and Computing (SCITECO) of University of Eastern Finland.

Finally, I would like to thank my family and loved ones. To my mother Sulin and father Yafei, who believed in me and who have always been there supporting me, a simple thank you is not enough to express the gratitude and love I have for you. This also extends to Paavo and Yan, who have been there for me as parents. My beloved Fredrick, you have seen me at my worst and still you are by my side and continue to love me. Although it sounds a cliché, you have shown me the meaning of true love.

Moumou, Joujou and Hattara, you will always be mommy’s lilla älsklingar.

Kuopio, 16^th May 2017 Xiaowei Ojanen

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LIST OF ABBREVIATIONS

2D two dimensional

3D three dimensional

AGE advanced glycation endproduct ANOVA analysis of variance

BMD bone mineral density BV/TV bone volume fraction

Ca calcium

CH² methylene side chains CV coefficient of variation DA degree of anisotropy DPD deoxypyridinoline

DXA dual energy X-ray absorptiometry FDTD finite-difference time-domain FE finite element

FEM finite element method FWHM full width at half maximum

HA hydroxyapatite

HP hydroxylysylpyridinoline

HPLC high performance liquid chromatography ISO International Organization for Standardization LP lysylpyridinoline

MSE mean square error

OH hydroxide

PBS phosphate buffered saline PDE partial difference equation

PEN pentosidine

PMMA polymethyl methacrylate

PO phosphate

PYD pyridinoline

SAM scanning acoustic microscopy SD standard deviation

SiC silicon carbide SMI structure model index Tb.N trabecular number Tb.Sp trabecular separation Tb.Th trabecular thickness TMD tissue mineral density

UV ultraviolet

µCT micro-computed tomography

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LIST OF SYMBOLS

A surface area or projected area A0 surface area at start

a radius of contact of a conical indenter C, c^ijkl stiffness tensor

d space dimension

dP/dh experimental stiffness of nanoindentation E Young’s modulus or elastic modulus E (t-t´ ) relaxation function

E* complex modulus

E’ storage modulus

E’’ loss modulus

E0 instantaneous modulus

Ec instantaneous elastic modulus of creep Eeq equilibrium elastic modulus

Ei elastic modulus of indenter or dimensionless relaxation modulus of i^th term

En normalized tensile-compressive modulus

Er reduced modulus

E^s elastic modulus of indented material

Esr instantaneous elastic modulus of stress-relaxation

∆E dissipated energy

F ^force

𝑓𝑓_𝑖𝑖 source value for force

G shear modulus

gi i^th normalized shear modulus term of Prony series H indentation hardness

h^a depth without tip contact

hc depth of contact at maximum load h^e elastic displacement during unloading

h^max depth from sample surface to the depth at maximum load h^r depth of residual impression

J(t-t’ ) creep compliance function

K bulk modulus

l length

l0 length at start

N number of terms in Prony series

n number of samples

P pressure or applied load

p p-value

𝑃𝑃_{𝑚𝑚𝑚𝑚𝑚𝑚} maximum applied load

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12 R reflection coefficient

r spherical indenter tip radius or correlation coefficient R^a surface roughness

t ^time

tan δ loss tangent

Tij stress tensor components

V ^volume

V0 volume at start

ΔV change in volume

𝑣𝑣𝑖𝑖 vector components of partial velocity field

W ^toughness

x space variable

x0 transverse length at start Δx change in transverse length y0 axial length at start

Δy change in axial length

Z acoustic impedance

Z1 acoustic impedance of a known coupling fluid Z2 acoustic impedance of object

α effective cone angle

δ phase shift

ε strain

ε strain tensor

ε0 strain at start εx transverse strain

εs shear strain

εy axial strain

η ^viscosity

θ ^angle

θij source value for tensor

λex wavelength of fluorescent exitation λem wavelength of fluorescent emission

ρ mass density

ρ0 mass density at start Δρ change in mass density

σ stress

σ stress tensor

σ0 stress at start

σs shear stress

τ characteristic time constant or nanoindentation creep time constant τi i^th viscoelastic relaxation time constant of Prony series

υ Poisson’s ratio

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ω angular frequency

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LIST OF ORIGINAL PUBLICATIONS

This thesis is based on data presented in the following articles, referred to with the Roman numerals I-III.

I Ojanen X, Isaksson H, Töyräs J, Turunen MJ, Malo MKH, Halvari A, Jurvelin JS (2015). Relationships between tissue composition and viscoelastic properties in human trabecular bone. J. Biomech 48, 269-275.

II Ojanen X, Töyräs J, Inkinen AI, Malo MKH, Isaksson H, Jurvelin JS (2016).

Differences in acoustic impedance of fresh and embedded human trabecular bone samples – Scanning acoustic microscopy and numerical evaluation.

J. Acoust. Soc. Am. 140, 1931-1936.

III Ojanen X, Tanska PK, Malo MKH, Isaksson H, Väänänen SP, Koistinen A, Grassi L, Magnusson SP, Ribel-Madsen SM, Korhonen RK, Jurvelin J, Töyräs J (2017). Material viscoelasticity is related to tissue composition but may not fully predict the macroscale viscoelasticity in human trabecular bone. J.

Biomech (submitted).

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AUTHOR’S CONTRIBUTION

The publications in this thesis are original research papers representing a comprehensive study on human trabecular bone at different length scales. In all studies, the contribution of the co-authors has been significant.

I) The author took part in the design of the study, sample preparation and carried out the measurements and data analysis of hydrated human trabecular bone using nanoindentation and Raman micro-spectroscopy measurements. The author was the primary writer of the manuscript.

II) The author took part in the design of the study and carried out the sample preparation. Measurements of acoustic impedance of hydrated and embedded human trabecular bone using scanning acoustic microscope (SAM) and data analysis were done by the author. The author was the primary writer of the manuscript.

III) The author took part in the design of the study and carried out the sample preparation and data analysis. Imaging of the human trabecular sample microstructure using micro-computed tomography and bone surface acoustic impedance using SAM in addition to unconfined mechanical compression testing was done by the author. The author was the primary writer of the manuscript.

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1 INTRODUCTION

Since they have to provide adequate support and protection, the bones in the skeleton need to be strong and tough enough to absorb the mechanical loads exerted on the body but still light enough to allow movements by muscle contraction (Warwick and Williams, 1973). Bone, as a material, is essentially a network of mineralized collagen matrix (Buckwalter et al., 1996). This mixture of inorganic and organic material determines the strength and ductility that is specific to bone. In addition, bone has a hierarchical structure, which is molded by its regenerative properties (modeling and remodeling). This allows the bone structures to orientate and shape themselves in a manner that can optimally bear loading and resist fractures (Allen and Burr, 2014; Julius, 1986; Rho et al., 1998).

Bone is often assumed to be a fully elastic material when assessed in computational simulations, e.g, using finite element method (FEM) (Evans et al., 2012; Hambli, 2013; Müller and Rüegsegger, 1995). The elastic properties, e.g., the elastic modulus, have been extensively studied at both the material (tissue) (Hoffler et al., 2000; Zysset et al., 1999) and macro (structural) levels (Ding et al., 1997; Keller, 1994; Kopperdahl and Keaveny, 1998; Morgan and Keaveny, 2001). For example, due to human trabecular bone’s porous structure (Hambli, 2013; Rho et al., 2002), its elastic modulus at the macro level (apparent modulus) is in the order of 10⁶ Pa (Keaveny et al., 2001), while values at the material level are in the order of 10⁹ Pa (Crowin, 2001; Thurner, 2009). However, bone contains water (Crowin, 2001) and thus exhibits time-dependent viscoelastic behavior at both length scales (Isaksson et al., 2010b; Linde et al., 1988). It is important to understand the viscoelastic properties of bone since the tissue is constantly under dynamic loading during our daily living (walking, sports, etc.). Stress related damages to bone are repaired by a lifelong process of bone remodeling (Burr et al., 1989; Burr et al., 1985; Seeman, 2009).

However, the remodeling rate and the balance between bone removal and bone formation changes with age (Chaitou et al., 2010; Seeman, 2008).

After humans reach skeletal maturity, i.e. around the time they reach their early twenties, bone mass starts to decline after a short period of stability (Raisz and Seeman, 2001). This decline accelerates in women after menopause. Bone mass is lost due to the imbalance between bone removal and bone formation in the remodeling cycle. Since remodeling takes place at bone surfaces, this metabolic imbalance is usually first seen in trabecular bone, which has a higher surface-to-volume ratio than cortical bone (Seeman, 2009). Due to the imbalance in remodeling, trabecular thinning renders the bone more fragile and less fracture resistant (Mosekilde, 2000;

Seeman, 2009). Trabecular structures and possible morphological changes can be visualized using micro-computed tomography (µCT). µCT also offers a means to quantify volumetric bone mineral density (BMD) and volumetric tissue mineral

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20 density (TMD) (Barbe et al., 2014; Bouxsein et al., 2010; Chen et al., 2013; Ding et al., 1999).

Bone composition becomes altered during aging (Boskey and Coleman, 2010;

Paschalis et al., 2016) e.g., age related changes in collagen matrix reduce the toughness of bone (Wang et al., 2002). Changes in bone biochemical composition can be measured destructively by high performance liquid chromatography (HPLC) (Bank et al., 2000; Saito et al., 1997) or non-destructively by Raman micro- spectroscopy (Raghavan et al., 2012; Turunen et al., 2011). Raman micro- spectroscopy, requiring no complicated sample preparation protocols, provides compositional information about both the inorganic and organic phases (Morris and Mandair, 2011).

Mechanical measurements on elastic and viscoelastic properties of trabecular bone, however, are much dependent on the conditions in which they are measured.

For example, dehydration is known to increase the stiffness of bone both at the material and macro levels (Hengsberger et al., 2002; Nyman et al., 2006). Moreover, dehydration reduces the viscosity of bone under dynamic loading (Yamashita et al., 2001). Thus, it is always preferable to measure bone under physiological conditions, e.g., hydrated in physiological solution at 37 °C. However, the preparation of hydrated bone samples for analysis may be challenging (Bumrerraj and Katz, 2001).

This is especially true in high resolution measurement techniques, such as scanning acoustic microscopy (SAM) and nanoindentation. These techniques require a virtually mirror-like flat surface to determine tissue properties free from artefacts induced by imperfect polishing (Bumrerraj and Katz, 2001; ISO 14577-2:2015, 2015).

In order to achieve this property, bone samples are traditionally dehydrated and embedded in plastic for SAM or nanoindentation (Eckardt and Hein, 2001;

Rodriguez-Florez et al., 2013; Rupin et al., 2009). Possibly due to these challenges in bone measurements, the viscoelastic nature of human trabecular bone has not been fully explored.

It is unclear how the viscoelastic properties of trabecular bone at the material and macro levels are related to tissue composition and microstructure. Furthermore, age related changes in viscoelastic properties of human trabecular bone are not well characterized. In addition, some methodological aspects are still unresolved, e.g., how sample preparation (dehydration, embedding, polishing) affects the SAM measurements of trabecular bone.

In this thesis, systematic studies on the human trabecular bone were conducted to help to clarify these open issues. Age related changes in human trabecular bone composition and collagen crosslinks were examined using Raman micro- spectroscopy and biochemical analysis, respectively. Furthermore, the structure and density of trabecular bone were characterized using µCT. Variation in the acoustic impedance and mechanical properties of bone samples at material level were determined using SAM and nanoindentation, respectively. Macro level mechanical properties of the same samples were analyzed using a bi-axial servohydraulic

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21 material testing device. Furthermore, numerical simulations were conducted to examine the effect of dehydration and surface roughness of bone samples on SAM measured acoustic impedance and to predict the viscoelastic behavior of trabecular bone in the macro level based on viscoelastic properties derived from the material level.

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2 STRUCTURE AND COMPOSITION OF BONE

2.1 BONE AS AN ORGAN

Bone is an organ that consists of osseous tissue, epithelial tissue (e.g. blood vessels), other connective tissues (e.g. blood and fat) and nervous tissue (Karaplis, 2008).

Human bones can be categorized according to their shape and size into four main groups: short (e.g. metatarsal), long (e.g. femur), flat (e.g. rib) and irregular bone (e.g.

vertebra) (Warwick and Williams, 1973). The bones make up the human skeleton.

Bone has many functions, one of which is to form protective cavities for the soft organs such as the brain (protected by the skull) and the heart (protected by the rib cage). It can also serve as an attachment site for skeletal muscles where the rigid bone is used as a lever to produce motion (Warwick and Williams, 1973). The characteristic locomotion of the animal dictates the shape of the skeleton and ultimately the shape of the body. Moreover, bone serves as a reservoir for minerals (Sapir-Koren and Livshits, 2011). Most of the body’s calcium, phosphate and sodium are stored in the skeleton from where these elements are released into the bloodstream to maintain the mineral homeostasis. Thirdly, bone takes part in blood cell formation and energy storage (Buckwalter et al., 1996; Huang and Terstappen, 1992). Red and white blood cells including platelets are mainly formed in red (hematopoietic) bone marrow.

Lipids are stored in the adipose cells of the yellow (stromal) bone marrow.

2.2 HIERARCHICAL STRUCTURE OF BONE

Bone is highly hierarchical in structure (Figure 2.1) (Rho et al., 1998). At the macro level, bone can have a spongy or a more compact structure. At the micro level (10 - 500 µm), individual components of the macro bone structure can be detected, such as the Harvesian system or a single trabecula. Furthermore, the compositional units of bone such as collagen fibrils and apatite crystals, are visible in the nano scale (< 1 µm).

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24 Figure 2.1. The human skeleton and the hierarchical structure of bone in different length scales.

10

0

m 10

-2

m 10

-3

m 10

-7

m 10

-9

m 10

-6

m Ma cro str uc tu re Sk ele to n Mi cro str uc tur e N ano str uc tur e

Long bone(femur) Irregular bone(vertebra) Flat bone(rib)

Short bone(metatarsal)Cortical bone Trabecular bone

Single trabecula

50-300 µm Collagen fiber

100 nm Mineralized collagen fibrilCollagen molecule300 nmHole Zone 35 nmOverlapzone32 nm HA crystals Tropocollagen triple helix

1.5 nm

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25 2.2.1 Macrostructure

At the macro level, bones can be differentiated into cortical (compact) and trabecular (spongy or cancellous) bones (Rho et al., 1998). Cortical bone is found at the outer edges surrounding the whole bone (cortex), while trabecular bone can either fill the entire inner cavity of the bone (e.g. vertebra) or only partially, such as in the epiphyses of long bones (Reznikov et al., 2014). About 80% of the adult bone mass is attributable to cortical bone and the rest is trabecular bone (McNamara, 2011).

Cortical bone consists mainly of tube-like Haversian systems (osteons) while trabecular bone has marrow filled cavities formed by plate- and rod-like trabeculae (McNamara, 2011; Rho et al., 1998). At this length scale, cortical bone has a higher mineral content and a lower porosity than trabecular bone (Reznikov et al., 2014).

However, the distinction between the two types of bone is not always clear when examining structure alone (Rho et al., 1998).

Bones can be further distinguished into woven and lamellar bones (Currey, 2003;

McNamara, 2011; Weiner et al., 1999). Woven bone is formed rapidly and has randomly oriented fine collagen fibers. On the contrary, lamellar bone is formed more slowly and has, as its name suggests, a more organized structure. With growth and aging, woven bone is replaced by lamellar bone, which possesses more optimal load bearing properties. The lamellae are organized in packets with cement lines separating new from old tissue.

2.2.2 Microstructure

The typical length scales within the microstructure and sub-microstructure of bone are around 10 - 500 µm and 1 – 10 µm, respectively (Rho et al., 1998). In cortical bone, an intricate network of canals and small cavities contribute to its porosity. The larger pores originate from cylindrical osteons, which are roughly 100 - 250 µm in diameter with a central cavity (Haversian canal) having a diameter of about 30 - 40 µm (Reznikov et al., 2014). The cavity is occupied by neurovascular vessels (McNamara, 2011; Reznikov et al., 2014). The osteons run virtually in parallel with each other along the long axis of the bone and are connected by Volkmann’s canals. In mature bone, the bone cells (osteocytes) reside in small cavities called lacunae which are connected to each other via canaliculi. The walls of the osteons are lamellar bone.

Lamellar bone is formed by several layers of lamellae that are 3 to 7 µm thick (Reznikov et al., 2014; Rho et al., 1998). Between the osteons, there are interstitial

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26 lamellae; these are remnants of old osteons originated during the bone remodeling process.

In trabecular bone, the individual trabeculae are interconnected, forming a spongy structure, where the pores are in the order of 1 mm (Keaveny et al., 2001).

The average trabecular thickness is between 50 - 300 µm but the trabecula can be up to 2000 µm in length (McNamara, 2011; Rho et al., 1998). The shape of the trabeculae depends on their anatomical location and applied loading (McNamara, 2011). The trabecular bone pores are filled with bone marrow (Buckwalter et al., 1996). The trabecular bone cells acquire nourishment from the surrounding bone marrow.

Trabecular bone also contains lamellar bone, but unlike the cylindrical osteons, the lamellae in the trabeculae form extended parallel arrays (Weiner et al., 1999).

2.2.3 Nanostructure

The nanostructures and sub-nanostructures of the bone have lengths ranging from a micron to few hundred nanometers (Rho et al., 1998). Bone essentially consists of mineralized collagen fibrils. The fibrils are mainly type I collagen and are about 100 nm in diameter. Bone forming cells i.e. osteoblasts, secrete 300 nm long and 1.5 nm thick triple helical collagen molecules that self-assemble into 300 nm long sheets of collagen molecules (Figure 2.1). The sheets are stacked on top of each other with a 35 nm gap between the ends of the sheets (hole zone). The sheets are also stacked behind each other in such a way that the edges of the sheets are arranged in parallel to one another, overlapping by 32 nm (overlap zone) (Fratzl et al., 2004). The gaps between the collagen sheets are filled with hydroxyapatite (HA) crystals (Ca⁵ [PO⁴]³OH) (Landis et al., 1993). Mature crystals are plate like in shape (~50 nm in length, ~25 nm in width and ~3 nm in thickness) and are discrete and discontinuous (Ziv and Weiner, 1994). They grow in a vertical crystallographic axis (c-axis) which is roughly parallel to the long axis of the collagen fibrils (Rho et al., 1998).

2.3 COMPOSITION OF BONE

Bone is a natural composite consisting of organic material (25% of total bone mass), minerals (65%) and water (10%). About 90% of the organic matrix consists of collagen and 10% of noncollagenous proteins (Buckwalter et al., 1996; Crowin, 2001). Collagen provides, ductility, tensile strength, structural integrity and shape to the bone (Burr, 2002a; Olszta et al., 2007). It is synthesized by bone forming cells called osteoblasts.

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27 The structural basis of collagen consists of three polypeptide strands called α-chains, which are twisted together to form a triple helix coil (tropocollagen). There are many forms of collagen in biological tissues; these are commonly designated using Roman numerals (McNamara, 2011). The collagen mostly seen in bones is type I collagen (95%). About 10% of the organic matrix consists of noncollagenous proteins (Crowin, 2001). They have many functions, i.e., organizing the collagenous matrix, mediating cell attachment and regulating the rate of growth and stability of mineral crystals (Gehron, 1989; Heinegard and Oldberg, 1989; Tye et al., 2005).

The bone minerals exist mainly in the form of impure HA crystals with substitutions of ions such as carbonate and sodium (Olszta et al., 2007). They are deposited by osteoblasts between the spaces of collagen fibrils during the calcification of the organic matrix (Landis and Silver, 2009). Within 5 to 10 days of their initial deposition, the HAcrystals grow in size (secondary mineralization) and 70% of the collagen matrix is mineralized. It takes about three to five months to reach complete mineralization at any local site and years for the entire skeleton to reach this state (McNamara, 2011). The growth and orientation of the mineral crystals are limited by the spaces between the collagen fibrils and the structure of the underlying collagen matrix (Paschalis et al., 1996).

About 15 to 25% of the bone’s volume consists of water and it is found within different hierarchical levels of the bone (Granke et al., 2015). Most of the water in the bone resides in the vascular-lacunar-canalicular spaces such as Haversian and Volkmann’s canals as mobile free water (pore water). The free water moves under the influence of pressure gradients that are formed in the skeleton during movement.

At the nanoscale, water is loosely bound to the surface of collagen fibrils and between collagen and mineral phases. It has been speculated that the water bound to the surfaces of the mineral crystals helps to orientate the apatite crystals during matrix mineralization (Wang et al., 2013). Water, at the molecular level, is tightly bound to the triple helix of the collagen molecule (Granke et al., 2015). Hydrogen bonds or water bridges make connections within and between the tropocollagen chains. The term structural water referes to those water molecules found within the apatite crystals.

2.4 BONE REMODELING AND AGING

Bone is a living tissue that adapts to the changes in the mechanical environment and repairs microscale tissue damage through an active cycle of bone removal and formation (Burr, 2002b; Julius, 1986; Manolagas and Jilka, 1995). In bone modeling,

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28 bone is formed prior to or without bone removal (resorption) while in the remodeling process, the bone is formed after bone resorption (Seeman, 2009). Unlike modeling, which occurs mainly during growth, there is no change in the size or shape of individual bone during remodeling. Bone remodeling is a lifelong process. It has been estimated that after reaching maturity, most of the skeleton is replaced every decade due to remodeling (Office of the Surgeon General, 2004). With age, however, the delicate balance between bone resorption and formation becomes disrupted.

Bone volume is significantly lost and bones become more fragile (RW.ERROR - Unable to find reference:414; Seeman, 2008; Slemenda et al., 1996).

2.4.1 Remodeling cycle

Remodeling occurs in cycles of bone resorption and formation. It can be classified into targeted and nontargeted remodeling (Parfitt, 2002). Targeted remodeling is localized and is usually signaled by microdamage or osteocyte death (apoptosis) (Bentolila et al., 1998; Burr et al., 1985) whereas nontargeted remodeling is thought to be a process related to calcium homeostasis (Allen and Burr, 2014).

The remodeling cycle starts with a 10 day long activation phase. Then mononuclear precursors gather at the surface of the bone and differentiate into multinucleated cells called osteoclasts. Once the osteoclasts are fully matured, they bind to the bone matrix and begin to dissolve the minerals and fragment the underlining collagen. The resorption phase lasts about 21 days, after which the osteoclasts undergo apoptosis i.e. programmed cell death. (Allen and Burr, 2014;

Crockett et al., 2011)

Before bone formation starts, there is a brief (5 day) transition (reversal phase) where the resorption debris is cleared from the exposed bone surface and a thin layer of new bone matrix called the cement or reversal line is deposited. The cement separates the new matrix from the surrounding old matrix. The final phase of remodeling is bone formation by osteoblasts. (Allen and Burr, 2014; Crockett et al., 2011) After bone matrix formation, the osteoblasts either undergo apoptosis, develop into osteocytes or form a layer of inactive bone lining cells (Allen and Burr, 2014).

The bone lining cells can be reactivated back to osteoblasts.

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29 2.4.2 Bone and aging

The bones in the female body reach peak mass and size around the age of 15-20 years while in men that happens a few years later (Raisz and Seeman, 2001; van der Sluis et al., 2002). At this point, the modeling and remodeling rate has slowed down from the growth period (Seeman, 2009). Soon after this time point, around 30 years of age, a decrease in bone mass is observed due to reduced bone formation during remodeling (Riggs et al., 2008). Remodeling takes place at bone surfaces and since trabecular bone has a higher surface per unit bone volume than cortical bone, the initial bone loss is greater in trabecular than in cortical bone (Seeman, 2008). By the age of 50, about 40% of lifetime loss of trabecular bone has occurred in both men and women. Substantial cortical bone loss in women is seen already by the time they are in their midlife, while in men it is observed much later (after 75 years of age) (Riggs et al., 2008). In addition, women experience an increase in the remodeling rate after menopause. This leads to a greater bone loss in women than in men during midlife (Seeman, 2009). Bone loss results in trabecular thinning and loss of connectivity in trabecular bone as well as an increase in the porosity of cortical bone (Seeman, 2008).

The ultimate outcome of these changes is decreased bone strength and increased fracture risk (Ding et al., 1997; Szulc et al., 2006).

Age related changes are detected also in the composition of bone (Boskey and Coleman, 2010). The collagen fibers are interconnected via collagen crosslinks, adding complexity to the collagen network, which provides structural integrity and shape to the bone (Knott and Bailey, 1998; Olszta et al., 2007). The crosslinks can be distinguished into those that are formed enzymatically (lysine hydroxylase and lysyl oxidase-controlled) and those that are formed non-enzymatically through glycation or oxidation (Saito and Marumo, 2010). Crosslinks can be further distinguished into immature and mature crosslinks (Knott and Bailey, 1998). There are four crosslinking sites on a collagen molecule that have a specific amino acid sequence with which immature crosslinks can bond. Individual collagen molecules are connected to one another via immature crosslinks, forming a sheet of collagen fibril. Mature crosslinks, however, are non-reducible and link the collagen fibrils together. Enzymatic crosslinks are positively associated with bone strength (Saito and Marumo, 2010).

When there is a decrease in the concentration of enzymatic crosslinks, bone strength has been shown to decrease although the collagen density was unchanged (Oxlund et al., 1996). Mature non-enzymatic crosslinks are known as advanced glycation endproducts (AGEs). With age, there is an increase of non-enzymatic crosslinks in the collagen matrix which has adverse effects on the mechanical properties of the bone, e.g., decrease in bone strength and toughness (Wang et al., 2002).

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30 As mentioned earlier, the bone minerals exist mainly in the form of impure hydroxyapatite crystals. The impurities in the HA crystals are attributable to substitutions of other ions, such as carbonate and sodium (Olszta et al., 2007). The ion substitution rate changes with age, specifically carbonate (type-B) substitution increases with age (Yerramshetty et al., 2006) but decreases with increased rate of remodeling (Isaksson et al., 2010c).

Due to the constant remodeling process, mineral crystal maturity may not be related to the age of the person. This gives rise to the concept of tissue age (Rey et al., 2009). Tissue age refers to the crystallinity of the minerals deposited before resorption of the tissue (Glimcher, 2006). The material level mechanical properties of bone can therefore be independent of the subject’s age but not of his/her tissue age (Nyman et al., 2016).

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31

3 MECHANICAL PROPERTIES OF TRABECULAR BONE

3.1 ELASTIC AND VISCOELASTIC PROPERTIES

Bone strength, toughness, structure and density are typical characteristics that determine the mechanical resistance of bone to withstand impact fractures (Parkkari et al., 1999; Yeni et al., 1998). Fatigue fractures, due to repeated loading of the bone, can happen at loads that are below the ultimate strength limit (Fleck and Eifler, 2003).

Repeated loading over time causes micro cracks and cumulative damage in the bone (Pattin et al., 1996). Most of the fatigue damage in bone can be repaired through the modeling and remodeling processes (Parfitt, 2010). However, with age and in osteoporosis (bone loss caused by metabolic bone disease), the regenerative properties of bone decrease substantially and the probability of fatigue fractures increases (Keaveny et al., 2001; Seeman, 2009). Due to the mechanical role of bone, a closer examination of the mechanical properties of bone is necessary for understanding the factors that may lead to fatigue fractures.

3.1.1 Elasticity and stress-strain curve

Materials have an inherent ability to resist deformation exerted by outside forces. If the material’s deformation is fully restored after removing the external force, then it is considered to be elastic (Hooke’s law). Elasticity is described by the stress-strain curve, where stress is the force excreted on the material per unit area and strain is the relative deformation of the material (Figure 3.1). In specific loading geometries, the slope of stress-strain relation is referred to as the elastic modulus. When the stress- strain relation is linear, the material is considered to be linearly elastic. The elastic modulus can be specified to be Young’s modulus, bulk modulus and shear modulus according to the direction of the stress and strain (Table 3.1). The three moduli are related through Poisson’s ratio, which describes the relationship between transverse and axial strains. Trabecular bone has a Poisson’s ratio between 0.2 - 0.3 at the macro level (Dalstra et al., 1993; Reilly et al., 1974); at the material level, the Poisson’s ratio has been assumed to be 0.3 in most previous studies (Hoffler et al., 2000; Kim et al., 2013; Rho et al., 1997; Turner et al., 1999). An object can have different stress or strain responses in different directions. The resulting stiffness or elasticity tensor is described with a 6x6 matrix. For an isotropic material, the stiffness tensor is reduced to only bulk and shear moduli describing the material’s elastic properties.

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32 Table 3.1. Mathematical expressions of Poisson’s ratio and different types of elastic moduli.

Parameter Equation

Poisson’s ratio (-) 𝜐𝜐 =𝜀𝜀𝑚𝑚

𝜀𝜀𝑦𝑦=∆𝑥𝑥 𝑥𝑥⁄ 0

∆𝑦𝑦 𝑦𝑦⁄ 0

Young’s modulus (Pa) 𝐸𝐸 =𝜎𝜎

𝜀𝜀 =

𝐹𝐹 𝐴𝐴⁄ ₀ (𝑙𝑙 − 𝑙𝑙0) 𝑙𝑙⁄0

Bulk modulus (Pa) 𝐾𝐾 = ∆𝑃𝑃

∆𝑉𝑉 𝑉𝑉⁄ =0

∆𝑃𝑃

∆𝜌𝜌 𝜌𝜌⁄ =0

𝐸𝐸 3(1 − 2𝜐𝜐)

Shear modulus (Pa) 𝐺𝐺 =𝜎𝜎𝑠𝑠

𝜀𝜀_𝑠𝑠 = 𝐹𝐹 𝐴𝐴⁄ tan 𝜃𝜃 =

𝐹𝐹 𝐴𝐴⁄

∆𝑥𝑥 𝑙𝑙⁄ = 𝐸𝐸 2(1 + 𝜐𝜐) εx= transverse strain, εy = axial strain, x0= transverse length at start, y0 = axial length at start, Δx = change in transverse length, Δy = change in axial length, σ = stress, ε = strain, F = force, A0 = surface area at start, l = length, l0= lengthat start, ΔP = change in pressure, V = volume, V0 = volume at start, ΔV = change in volume, Δρ = change in mass density, ρ0 = mass density at start, σs = shear stress and εs = shear strain, A = surface area.

Trabecular bone at the macro level consists of plate and rod-like trabeculae and marrow filled pores. The Young’s modulus of human trabecular bone at the macro level (apparent modulus) varies greatly depending on the anatomical site (e.g.

vertebra 67 MPa, proximal tibia 445 MPa, proximal femur 441 MPa and calcaneus 68 MPa) (Keaveny et al., 2001). The values are much higher (2 GPa - 21 GPa) at the material level (Crowin, 2001). The discrepancy is thought to be due to the porous structure of bone at the macro level (Hambli, 2013; Rho et al., 2002).

When a high enough stress or strain is exerted on a material, it will not recover elastically and the deformation becomes permanent even after the removal of the stress. This type of deformation is considered as being plastic. On the stress-strain curve, the transition from elastic to plastic deformation is referred to as the yield point. In some materials like bone, the yield point from elastic to plastic deformation is not well defined (Callister, 2007). In this case, the initial deformation from the linear portion of the curve is referred to as the proportional limit and the yield point is commonly defined at the intercept between the stress-strain curve and a line parallel to the elastic slope with an offset in strain (usually at 0.2%) (Figure 3.1). The strength of a material is its ability to withstand external stress before total failure. In bone, this is usually defined as the maximum or ultimate stress on the stress-strain curve. If the

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33 material deforms extensively with plasticity before reaching ultimate stress, it is considered ductile, while a short plastic deformation indicates that the material is brittle. Two different materials can have the same strength but one can be brittle while the other is ductile. Usually it is preferred to have strength and ductility because the material can then absorb more energy before failure. The absorbed energy per unit volume before failure is called toughness (Figure 3.1).

3.1.2 Viscoelasticity

Viscosity is the resistance of liquid to flow. When liquid is compressed between two plates, viscosity is the ratio between the applied shear force to the change in shear flow (Newton’s law). (Callister, 2007; Findley et al., 1989) A viscoelastic material exhibits both viscous and elastic behavior when undergoing deformation. Elastic deformation is instantaneous while viscous deformation is time dependent. The applied loading rate determines whether the response is more elastic or viscous.

When the deformations are small, the viscoelastic response can be considered to be linear. A viscoelastic material dissipates energy during a loading and unloading cycle. This is seen as a hysteresis loop in the stress-strain curve and as a phase difference between stress and strain during dynamic loading (Figure 3.2). The more Figure 3.1. A stress-strain curve of a trabecular bone sample under compressive stress or strain. The elastic modulus is derived from the slope of the elastic region. The elastic and plastic regions are separated by the proportional limit (1). The yield point (2) is set at the 0.2%

offset strain in the plastic region of the stress-strain curve. Toughness is the energy needed to cause deformation per unit volume prior to fracture and it can be determined as the area under the curve until the point of maximum stress (3).

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34 viscous the material, the more energy it dissipates. Thus, a larger hysteresis loop area and a greater phase difference are evidence of a higher viscosity of the material.

Table 3.2. Constitutional models commonly used in modeling of bone creep. Elastic modulus and viscosity are denoted by E^andη, respectively.

Constitutive model Equation Burger

𝜀𝜀(𝑡𝑡) = 𝜎𝜎0�1 𝐸𝐸1+ 𝑡𝑡

𝜂𝜂1+ 1

𝐸𝐸2�1 − 𝑒𝑒^{−𝑡𝑡 𝜏𝜏}^��

Standard linear solid

𝜀𝜀(𝑡𝑡) = 𝜎𝜎0�1 𝐸𝐸0+ 1

𝐸𝐸1�1 − 𝑒𝑒^{−𝑡𝑡 𝜏𝜏}^��

Generalized Maxwell

𝜀𝜀(𝑡𝑡) = 𝜎𝜎₀�1 𝐸𝐸₀+ 1

𝐸𝐸₁�1 − 𝑒𝑒^{−𝑡𝑡 𝜏𝜏}^�¹� + 1

𝐸𝐸₂�1 − 𝑒𝑒^{−𝑡𝑡 𝜏𝜏}^�²��

ε = strain, t = time, σ0 = stress at start and τ = creep time constant

The viscoelastic nature of a material can be characterized by using transient or dynamic testing. In transient experiments, the material is deformed and the response is recorded as a function of time. The time dependent deformation of a viscoelastic material under constant load is called viscoelastic creep. Viscoelastic stress- Figure 3.2. Hysteresis curve (left) and phase shift between stress and strain (right) caused by viscosity.

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35 relaxation, on the other hand, is the time dependent decrease in load as the strain is held constant. Creep and stress-relaxation are dependent on the material’s temperature. Thus, mechanical tests should be performed at a constant temperature.

Creep and stress-relaxation tests are usually performed by applying an instantaneous stress or strain on the material, after which the stress or strain, respectively, are maintained for a certain period of time. A model is then fitted to the resulting stress or strain vs. time data. In the linear viscoelastic model, the Bolzmann superposition theory is applied; this states that the total creep (likewise for stress- relaxation) is made up of independent and additive increments of loads:

𝜀𝜀(𝑡𝑡) =𝜎𝜎(𝑡𝑡)

𝐸𝐸_𝑐𝑐 + � 𝐽𝐽(𝑡𝑡 − 𝑡𝑡^′)𝑑𝑑𝜎𝜎(𝑡𝑡^′) 𝑑𝑑𝑡𝑡^′ 𝑑𝑑𝑡𝑡^′

𝑡𝑡

0 , (3.1)

𝜎𝜎(𝑡𝑡) = 𝐸𝐸_{𝑠𝑠𝑠𝑠}𝜀𝜀(𝑡𝑡) + � 𝐸𝐸(𝑡𝑡 − 𝑡𝑡^′)𝑑𝑑𝜀𝜀(𝑡𝑡^′) 𝑑𝑑𝑡𝑡^′ 𝑑𝑑𝑡𝑡^′

𝑡𝑡

0 , (3.2)

where t is time, ε and σ are strain and stress, respectively, Ec and Esr are the instantaneous elastic moduli of creep and stress-relaxation, respectively, J(t-t’ ) is the creep compliance function and E(t-t’ ) is the relaxation function. (Crowin, 2001;

Vincent, 2012) Viscoelasticity can also be described with constitutive models using Hookean springs and Newtonian dashpots connected in series or in parallel (Vincent, 2012). These types of models are achieved using exponential response functions, where the exponential decay factor is called the characteristic time constant (τ^{) that} describes the rate of the creep (stress-relaxation) (Table 3.2) (Shepherd et al., 2011).

The creep and stress-relaxation expressions are related through Laplace transformation (Lakes, 2009). The exponential function can be expressed as the sum of exponential functions similarly to Fourier transformation. The resulting function (stress-relaxation) is called the Prony series

𝐸𝐸𝑛𝑛(𝑡𝑡) =𝐸𝐸_{𝑒𝑒𝑒𝑒}

𝐸𝐸0 = 1 − � 𝐸𝐸𝑖𝑖�1 − 𝑒𝑒^{−𝑡𝑡 𝜏𝜏}^�^𝑖𝑖�

𝑁𝑁 𝑖𝑖=1

, (3.3)

where Eⁿ is the normalized time dependent tensile-compressive elastic modulus, Eeq

is the equilibrium modulus, E0 is the instantaneous modulus, Ei and τi are the elastic component and relaxation time, respectively, and N is the number of terms in the Prony series.

In dynamic testing, materials are tested under a cyclic stress or strain. The stresses and strains in a dynamic test need to be small enough to avoid plastic deformation of the material. Linear loading and unloading cycle produces a hysteresis pattern of the stress-strain curve. The area of the hysteresis is the dissipated energy per unit volume of the material. When a fully elastic material is stressed or deformed

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36 sinusoidally at frequency ω, the stress and strain reach their peak at the same time point (in-phase), but for a fully viscous material stress and strain peaks show a 90°

phase shift (Vincent, 2012). When the material is viscoelastic, the relation between stress and strain becomes as follows:

𝜎𝜎0= 𝜀𝜀0𝐸𝐸^∗sin(𝜔𝜔𝑡𝑡 + 𝛿𝛿), (3.4)

where σ⁰ and ε⁰ are the stress and strain at the start, respectively, t is time, δ is the phase shift between stress and strain and E* is the complex modulus that is a combination between the elastic or storage modulus (E’ ) (real part) and the viscous or loss modulus (E’’ ) (imaginary part).

𝐸𝐸^∗= 𝐸𝐸^′+ 𝑖𝑖𝐸𝐸^′′. (3.5)

The storage and loss moduli can be further defined as

𝐸𝐸^′=𝜎𝜎0

𝜀𝜀₀cos 𝛿𝛿, (3.6)

𝐸𝐸^′′ =𝜎𝜎0

𝜀𝜀₀sin 𝛿𝛿. (3.7)

The dissipated energy is determined as

3.2 MATERIAL TESTING

Mechanical testing of elastic and viscoelastic properties of bone may be conducted using material testing systems that can bend, compress, extend, shear or twist the bone. The size of a bone specimen can vary from a whole bone to a single trabecula.

Typically, a material testing device consists of a load frame that is the supporting body of the machine. There is a movable cross head attached with a load cell between the frames. The material testing system has means to measure the material deformation which can be tracked internally (rotation of stepper motors) or through externally attachable devices like an extensometer. The sample is placed on fixtures that are shaped depending on the required test, e.g., metallic plates for compression testing. The system is usually controlled with computer software which records also the output signals.

∆𝐸𝐸 = 𝜋𝜋𝐸𝐸^′′(𝜀𝜀0)². (3.8)

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37 NANOINDENTATION

If the tested material is a thin film, then conventional material testing is unsuitable.

Indentation is a method of material testing where mechanical properties and hardness are determined by indenting the tested material with a probe (i.e., indenter) which has known material properties. Nanoindentation is a special kind of indentation technique where the indentation displacement is in the order of nanometers and the projected area of contact is indirectly assessed from the indentation depth. For bone, nanoindentation is used to test mineralized collagen and lamellar scale structures (Figure 3.3). The indenter is usually made of diamond which has a Poisson's ratio of 0.07 and an elastic modulus of 1140 GPa. For analytical purposes, the geometry of the indenter needs to be well defined. Table 3.3 lists some typical nanoindentation indenters and their geometries (Fisher-Cripps, 2004).

Table 3.3. Projected areas of various indenter tips. The semi-angle of pyramidal indenters is the angle between the face and the center of the pyramid.

Indenter tip Projected area Semi-angle θ Effective cone angle α

Sphere 𝐴𝐴 ≈ 𝜋𝜋2𝑟𝑟ℎ𝑐𝑐 N/A N/A

Cone 𝐴𝐴 = 𝜋𝜋ℎ_𝑐𝑐²tan²𝛼𝛼 α α

Berkovich 𝐴𝐴 = 3√3ℎ𝑐𝑐2tan²𝜃𝜃 ^65.27° ^70.3°

Vikers 𝐴𝐴 = 4ℎ_𝑐𝑐²tan²𝜃𝜃 68° 70.3°

Knoop 𝐴𝐴 = 2ℎ𝑐𝑐2tan 𝜃𝜃1tan 𝜃𝜃2 θ1 = 86.25°, θ2 = 65° 77.65°

Cube corner 𝐴𝐴 = 3√3ℎ_𝑐𝑐²tan²𝜃𝜃 ^35.26° ^42.28°

hc = depth of indenter tip in contact with specimen at maximum load, r = radius of spherical indenter tip.

Figure 3.3. Nanoindentation marks made by a Berkovich tip (circled) on a single trabecula.

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38 The measured elastic modulus (Er), also known as the reduced modulus, is a combination of the elastic modulus of the indenter (Ei) and the indented material (Es) (Fisher-Cripps, 2004):

1

𝐸𝐸_𝑠𝑠 =(1 − 𝜐𝜐𝑠𝑠2) 𝐸𝐸_𝑠𝑠

�1 − 𝜐𝜐_𝑖𝑖²�

𝐸𝐸_𝑖𝑖 , ^(3.9)

where υ is the Poisson’s ratio and the subscripts s and i refer to the indented specimen and the indenter, respectively. When using nanoindentation for testing elasticity, hardness and creep of bone, a fast ramp load is applied and held at a specific load (in the order of mN) for a period of time (in the order of seconds) and then the load is released at the same rate that it was applied. The resulting force-displacement curve consists of elastic-plastic loading with an elastic unloading (Figure 3.4). The reduced modulus of bone is calculated from the unloading curve as follows:

𝐸𝐸_𝑠𝑠 =𝑑𝑑𝑃𝑃 𝑑𝑑ℎ 1 2√𝜋𝜋

√𝐴𝐴, (3.10)

where dP/dh is the experimental stiffness derived from the elastic portion of the unloading curve with a linear fit and A is the projected contact area of the indenter with a known geometry. This analysis technique was developed by Warren C. Oliver and George M. Pharr and is known as the Oliver-Pharr method (Oliver and Pharr, 1992). Indentation hardness is defined as

𝐻𝐻 =𝑃𝑃_{𝑚𝑚𝑚𝑚𝑚𝑚}

𝐴𝐴 , (3.11)

where Pmax is the maximum applied load. A constitutive spring and dashpot model such as Burger’s model can be fitted to the creep data by taking the indenter’s geometry into account (effective cone angle) in the equations (Fisher-Cripps, 2004).

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39

3.3 INFLUENCE OF DEHYDRATION

According to the International Organization for Standardization (ISO 14577-2:2015, 2015), the surface roughness R^a (defined as the arithmetic average of absolute values of the roughness profile ordinates) of the tested material should be less than 5% of the maximum nanoindentation depth. This minimizes the error caused by the surface roughness on the indentation depth. In order to achieve such surface smoothness in

𝑑𝑑𝑃𝑃 𝑑𝑑ℎ

h

^r

h

^e

h

^c

h

^max

h

^r

h

^e

h

^c

h

^max

h

^a

h

^a

a

Figure 3.4. A) Force-displacement curve of trabecular bone matrix. First, the bone matrix is loaded elastic-plastically to maximum load. At the maximum load, the depth from sample surface (dashed-dot line, subfigure C) to the depth at maximum load is called the maximum displacement (hmax) and the distance between the indenter tip having contact with bone to the maximum depth is called the depth of contact (h^c) (solid gray line, subfigure C). The difference in distance between h^maxand h^c is the displacement without tip contact (h^a). B) Subsequently, the indenter tip is held at maximum load for a period of time for creep testing.

After creep testing, the load is released. From the elastic portion of the unloading curve, the experimental stiffness slope (dP/dh) is derived with a linear fit. The intercept between the linear fit and the displacement axis represents the depth of elastic displacement during unloading (h^e). C) This is also the depth of unloading with the tip in contact with the bone.

An indentation with a residual depth (h^r) and radius (a) is left in the bone after the indenter tip is unloaded. For a conical indenter tip α is the angle between the face and center of the pyramid.

A

B

C Elastic-plastic

loading

Fo rc e

Hold

Elastic unloading

Displacement

Time Creep

Displacement

Conical indenter tip

unloaded

loaded

Relationships of microstructure and composition with material and macro level viscoelastic properties of trabecular bone

uef.fi

Dissertations in Forestry and Natural Sciences

XIAOWEI OJANEN

RELATIONSHIPS OF MICROSTRUCTURE AND COMPOSITION WITH MATERIAL AND MACRO LEVEL VISCOELASTIC PROPERTIES OF TRABECULAR BONE

XIAOWEI OJANEN

RELATIONSHIPS OF MICROSTRUCTURE AND

COMPOSITION WITH MATERIAL AND

MACRO LEVEL VISCOELASTIC

PROPERTIES OF TRABECULAR BONE

Xiaowei Ojanen

RELATIONSHIPS OF MICROSTRUCTURE AND

COMPOSITION WITH MATERIAL AND MACRO LEVEL VISCOELASTIC PROPERTIES OF TRABECULAR BONE

ABSTRACT

ACKNOWLEDGEMENTS

LIST OF ABBREVIATIONS

LIST OF SYMBOLS

LIST OF ORIGINAL PUBLICATIONS

AUTHOR’S CONTRIBUTION

CONTENTS

1 INTRODUCTION

2 STRUCTURE AND COMPOSITION OF BONE

2.1 BONE AS AN ORGAN

2.2 HIERARCHICAL STRUCTURE OF BONE

10

m 10

m 10

m 10

m 10

m 10

m Ma cro str uc tu re Sk ele to n Mi cro str uc tur e N ano str uc tur e

2.3 COMPOSITION OF BONE

2.4 BONE REMODELING AND AGING

3 MECHANICAL PROPERTIES OF TRABECULAR BONE

3.1 ELASTIC AND VISCOELASTIC PROPERTIES

3.2 MATERIAL TESTING

3.3 INFLUENCE OF DEHYDRATION

h

h

h

h

h

h

h

h

h

h

Fo rc e

Displacement

Conical indenter tip