Prediction of Bone Cell Probability Distribution in Weak ......To Bobin Xin, Cheuk Yu Lee, Haiyang Zhou and Shuang Zhang, thank you for your endless encouragement, concern, support,

i

Prediction of Bone Cell

Probability Distribution in

Weak Electromagnetic Fields

Song Chen, B.Sc, M.Eng (Hons)

This thesis is submitted to fulfil the requirements of The Australian National

University for the degree of

Doctor of Philosophy

September 2018

© Copyright by Song Chen 2018

All Rights Reserved

ii

Statement

The work presented within this thesis holds no material or information that has been

accepted for the award in any university for any degree. To the best of my knowledge,

this thesis does not contain any material written by another person except for the places

denoted by specific references. The content of this thesis is the product of research

work carried out at The Australian National University, since the starting of this

research program.

Supervisory Panel:

Professor Qinghua Qin, Research School of Engineering, The Australian National

University.

A/Professor Rachel W. Li, John Curtin School of Medical Research/ The Medical

School, The Australian National University.

Professor Paul N. Smith, The Medical School, The Australian National University.

Song Chen

September 2018

iii

Acknowledgements

My deepest appreciation and sincere gratitude for the guidance and support from

Professor Qinghua Qin. You have been great mentors throughout my journey in

completing this thesis. The enthusiasm that you have for the research is admirable, and

I am thankful for the knowledge you have imparted to me throughout the years.

I am extremely grateful to A/Professor Rachel W Li for the opportunity to undertake

such an exciting project. Your positivity, compassion and understanding have

provided me with the motivation to go through challenging times. You are like the

compass that has navigated me to the right direction of research.

I wish to thank Professor Paul N Smith, for your support and guidance throughout the

years, which has enabled me to successfully get through scientific research. Your

knowledge in the orthopaedic surgery and the bone field often inspired me to think

from a different perspective.

I am truly blessed to have the support and assistance from my colleagues. To Bobin

Xin, Cheuk Yu Lee, Haiyang Zhou and Shuang Zhang, thank you for your endless

encouragement, concern, support, assistance and above of all, for the fun times.

The completion of this thesis would not have been possible without the love, faith and

support from my family. I am grateful to my parents for the unconditional support and

sacrifices that you both have made to give me the best in life. To my dearest Haoning

Feng, you filled my heart with so much love and I feel blessed to have you in my life.

iv

Abstract

Electromagnetic field (EMF) effects on the cell membrane level, general and specific

gene expression and signal pathways of bone cells have been examined in numerous

studies. These studies were conducted on bioprocesses such as cell proliferation, cell

cycle regulation, cell differentiation, and metabolism. Genotoxic effects and apoptosis

were observed during in vitro experiments. However, several observations after EMF

exposure have been irreproducible and contradictory with other studies. Especially,

statistic insignificance of EMF effects in bioprocesses occurred when comparing the

exposure group with the control. Corresponding to the inconsistent observations, the

types of EMF apparatus in vitro are various in parameters. In this thesis, the biological

effects of EMFs on osteoblasts and osteoclasts were examined in the exposure of the

static magnetic field (SMF) and pulsed electromagnetic field (PEMF). At the

preliminary study, the nonlinear dose-response relationship was observed between the

intensity of EMF exposure and cell proliferation of osteoblasts. A hypothesis was

proposed for seeking the interpretation of nonlinearity by the principle of interference.

Several frameworks were formulated for building a theoretical structure of the

hypothesis. The verification of the hypothesis was rooted in the experimental design.

Two gradients of SMFs were applied to osteoblastic Saos-2 cells, and the biological

data represented the interference of cell probability. In the PEMF experiment, the

interference was also found in the cell proliferation of osteoblasts when they were

affected by the PEMFs. The similar results were reproduced at the co-culture of

osteoblasts and osteoclasts in SMFs and PEMFs, which resulted in a unique

entanglement of cells.

v

Abbreviations

ALP Alkaline phosphatase

BMU Bone multicellular unit

Cat K Cathepsin K

DMSO Dimethyl Sulphoxide

ELF Extremely low frequency

EMF Electromagnetic field

FBS Foetal bovine serum

FDM Finite difference method

FDTD Finite-difference-time-domain

FEM Finite element method

FSC Forward scatter

IGFs Insulin-like growth factors

M-CSF Macrophage colony-stimulating factor

mL Millilitre

mM Millimolar

mT Milli-Tesla

PBS Phosphate buffered saline

vi

PEMF Pulsed electromagnetic field

PTH Parathyroid hormone

RANK Receptor activator of NF-κB

RANKL RANK ligand

RT-PCR Reverse transcription polymerase chain reaction

SEM Scanning electron microscope

SEMs Standard errors of the mean

SMADs Sma and Mad-related proteins

SMF static magnetic field

SSC Side scatter

TRAP Tartrate-resistant acid phosphatase

v/v Volume/volume

w/v Weight/volume

Wnts Wingless-type protein

α-MEM Α-Modification of Eagle’s Medium

µL Microlitre

µm Micrometre

µM Micromolar

vii

Table of Contents

Acknowledgements ..................................................................................................... iii Abstract ....................................................................................................................... iv Abbreviations ............................................................................................................... v Table of Contents ....................................................................................................... vii

List of Figures .............................................................................................................. x List of Tables.............................................................................................................. xii Chapter 1 Biological Effects of Electromagnetic Fields .............................................. 1

1.1 Characteristics of electromagnetic fields ..................................................... 2 1.1.1 Frequency ................................................................................................. 2

1.1.2 Amplitude ................................................................................................. 3

1.1.2.1 Amplitude of electric field ................................................................ 3 1.1.2.2 Amplitude of magnetic field ............................................................. 3

1.1.2.3 Maxwell’s equations......................................................................... 4 1.2 Types of EMF applications in vitro and in vivo ........................................... 5

1.2.1 SMF .......................................................................................................... 5 1.2.1.1 Weak SMF ........................................................................................ 6

1.2.1.2 Moderate SMF ................................................................................. 6 1.2.1.3 Strong and ultra-strong SMF ........................................................... 8

1.2.2 PEMF ....................................................................................................... 9 1.2.3 Animal studies ........................................................................................ 11

1.3 Why effects of EMF in biology are irreproducible and contradictory ....... 11

1.3.1 Mechanisms of EMF .............................................................................. 12

Chapter 2 Bone Remodelling and Applied Mathematical Modelling ........................ 16 2.1 Bone modelling and remodelling ............................................................... 17

2.1.1 Osteoblast ............................................................................................... 18 2.1.2 Osteoclast ............................................................................................... 19 2.1.3 Gene network between osteoblasts and osteoclasts ............................... 20

2.2 Mathematical models for simulating cell growth and movement .............. 22 Chapter 3 Research Question and Hypothesis ........................................................... 29

3.1 Research question....................................................................................... 30 3.2 Hypothesis ........................................................................ 31

Chapter 4 Materials and Biological Methods ............................................................ 36 4.1 Materials ..................................................................................................... 37

4.1.1 Chemical reagents .................................................................................. 37 4.1.2 Commercial kits and molecular products ............................................... 38

4.1.3 Other products and consumables ........................................................... 39 4.1.4 Cytokines ............................................................................................... 39 4.1.5 Solutions ................................................................................................. 39 4.1.6 Equipment .............................................................................................. 41

4.1.7 Software ................................................................................................. 42

4.1.8 RT2 profiler PCR array........................................................................... 42 4.2 PEMF apparatus design ............................................................................. 46 4.3 Culture and in vitro studies of Sao-2 cell line ............................................ 46

4.3.1 Saos-2 cell line ....................................................................................... 47 4.3.2 Initiation of culture process .................................................................... 47

4.3.3 Subculture Saos-2 .................................................................................. 47 4.3.4 Count cell densities of Saos-2 cells ........................................................ 48

4.3.5 Cryopreservation and retrieval of cultured cells .................................... 48

viii

4.4 Isolation, culture and in vitro studies of primary cells ............................... 49 4.4.1 Isolation and culture of osteoblasts from trabecular bone ..................... 49 4.4.2 Harvest of monolayer cultures ............................................................... 49 4.4.3 In Vitro osteoblasts cell mineral formation assay .................................. 50

4.5 Osteoclasts induction and culture .............................................................. 51

4.5.1 Preparation of solution ........................................................................... 51 4.5.2 PBMC isolation ...................................................................................... 51 4.5.3 PBMC cell culture .................................................................................. 51 4.5.4 TRAP staining ........................................................................................ 52 4.5.5 Flow cytometer and analysis .................................................................. 53

4.6 Western blotting ......................................................................................... 53 4.6.1 Protein extract and assay ........................................................................ 53

4.6.2 Blotting and detection ............................................................................ 54

4.7 RNA extraction and quantification ............................................................ 55 Chapter 5 The Response of Cell Probability to External Stimulus ............................ 61

5.1 Introduction ................................................................................................ 62 5.2 Preliminaries ........................................................................ 62

5.2.1 Distribution of cell density ..................................................................... 62

5.2.2 Distribution of cell density under external stimulus .............................. 68

5.3 Cell probability ......................................................................................... 69

5.4 Joint cell probability ................................................................................... 77 5.5 Particular correlation between osteoblasts and osteoclasts ........................ 82

Chapter 6 Numerical Model: Predication of Osteoblast Cell Population under

Influence of Surface Grain Size ................................................................................. 90

6.1 Introduction ................................................................................................ 91 6.2 The relationship between osteoblast proliferation and substrate surface

roughness ............................................................................................................... 91 6.3 Governing equations for osteoblast adhesion on the material surface ....... 93

6.4 Numerical solution methods for the double-layered model ....................... 95 6.5 Numerical simulation on experimental data ............................................... 99 6.6 Comparison with experimental data and prediction ................................. 101

6.7 Parametric analysis................................................................................... 103 Chapter 7 Weak SMF Effects on Osteoblastic Cell Proliferation ............................ 118

7.1 Introduction .............................................................................................. 119 7.2 Experimental design ................................................................................. 119

7.3 Experimental results ................................................................................. 120 7.3.1 Effect of the SMF on the orientation of cultured Saos-2 cells ............. 120

7.3.2 Effect of SMF intensity on the proliferation of cultured Saos-2 cells . 121 7.3.3 Effect of SMF intensity on the cell viability of cultured Saos-2 cells . 121 7.3.4 The sensitivity of cultured Saos-2 cell proliferation to SMF intensity 121

7.4 A numerical model of the dose-dependent effect of SMF on osteoblastic

cell proliferation ...................................................................... 122

7.4.1 A stochastic model for cell probability in SMF ................................... 122 7.4.2 Numerical results ................................................................................. 124 7.4.3 Interference in SMF ............................................................................. 125

7.5 Discussion ................................................................................................ 125 7.5.1 Signal-to-noise ..................................................................................... 126

7.5.2 Physical mechanisms ........................................................................... 127 Chapter 8 PEMF Effects on Cell Proliferation of Osteoblasts ................................. 143

8.1 Introduction .............................................................................................. 144

ix

8.2 Experimental designs ............................................................................... 144 8.3 Experimental results ................................................................................. 145

8.3.1 Effect of PEMF intensity on the proliferation of cultured Saos-2 cells145 8.3.2 Effect of PEMF frequency on the proliferation of cultured Saos-2 cells ...

.............................................................................................................. 145

8.3.3 The sensitivity of cultured Saos-2 cell proliferation to PEMF ............ 146 8.3.4 Effect of PEMF on the orientation and mineralisation of human

osteoblasts ........................................................................................................ 147 8.3.5 Real-time RT-PCR of osteogenesis-related genes ............................... 147

8.4 Numerical solution of PEMF effects on osteoblastic cell proliferation .. 148

8.4.1 Model derivation for PEMF ................................................................. 148 8.4.2 Interference in PEMF ........................................................................... 149

8.5 Discussion ................................................................................................ 150

Chapter 9 PEMF and SMF Effects on Co-culture of Osteoblasts and Osteoclasts .. 167 9.1 Introduction .............................................................................................. 168 9.2 Prediction ................................................................................................. 169

9.2.1 The numerical interaction between osteoblasts and osteoclasts .......... 169 9.2.2 Numerical results ................................................................................. 172

9.3 Validation in experimental images........................................................... 172

9.3.1 TRAP staining ...................................................................................... 173 9.3.2 Fluorescence-based staining and flow cytometry analysis .................. 173 9.3.3 Scanning electron microscope (SEM) images ..................................... 174

Chapter 10 Conclusions ........................................................................................... 184 References ................................................................................................................ 187

Appendix .................................................................................................................. 202

x

List of Figures

Figure 2.1 BMP-SMAD and TGF-β/Activin pathway.. ............................................ 26 Figure 2.2 Map of osteo-gene network ..................................................................... 28 Figure 3.1 Double-slit model for osteoblasts ............................................................ 35 Figure 4.1 Self-made EMF device in controlling the frequency and signal type ...... 57

Figure 4.2 Coil tube ................................................................................................... 58 Figure 4.3 Picture of main body ................................................................................ 59 Figure 4.4 Support for each tube ............................................................................... 60 Figure 5.1 The structure factor .................................................................................. 89 Figure 6.1 Sketch of the double-layered model for osteoblast adhesion on the

material surface ................................................................................................ 107

Figure 6.2 Two examples of model fitting by experiment data .............................. 108 Figure 6.3 Function determination for molecular and cellular sensitivity in grain size

.......................................................................................................................... 110 Figure 6.4 Numerical models for Alumina and Titania .......................................... 111 Figure 6.5 Comparison of numerical results and experimental data.. ..................... 112 Figure 6.6 Simulation of molecular sensitivity ....................................................... 114

Figure 6.7 Simulation of cellular sensitivity ........................................................... 116 Figure 6.8 The impact of size factor ....................................................................... 117

Figure 7.1 Representation of the device used to generate the SMFs ...................... 130 Figure 7.2 Cell morphology of osteoblastic Saos-2 in SMF and control ................ 131

Figure 7.3 Effect of the SMFs on the proliferation of osteoblastic Saos-2 ............. 134

Figure 7.4 The effect of SMF exposure on cell viability of osteoblastic Saos-2 .... 135

Figure 7.5 Effect of the SMFs on the cell proliferation of osteoblastic Saos-2 in

combined G1+G2 ............................................................................................. 136

Figure 7.6 The sketch of the mathematical model for the movement of osteoblastic

cells in vitro under SMFs ................................................................................. 138 Figure 7.7 The predicted pattern of osteoblastic Saos-2 cell proliferation in weak

SMFs ................................................................................................................ 140 Figure 7.8 Parametric analysis for SMF coefficient ............................................... 142

Figure 8.1 Representation of the device used to generate the PEMFs .................... 153 Figure 8.2 Effect of the PEMF intensity on the proliferation of cultured Saos-2 cells

.......................................................................................................................... 154 Figure 8.3 Effect of the PEMF frequency on the proliferation of cultured Saos-2

cells .................................................................................................................. 155 Figure 8.4 Data distribution of PEMF groups along the control ............................. 156

Figure 8.5 A comparison between cultured Saos-2 cell proliferation to PEMF

generated by a solenoid and that generated by Helmholtz coils ...................... 157 Figure 8.6 PEMF effect on the orientation of human osteoblasts ........................... 158 Figure 8.7 Effect of PEMF and SMF on osteogenesis of human osteoblasts assessed

by Alizarin red-stained calcified nodules ......................................................... 159

Figure 8.8 Western blot analysis of IGF-1 protein at a loading of 10 and 30 µg ... 160 Figure 8.9 Waveform of frequency ......................................................................... 161 Figure 8.10 The interference of cell probability in PEMF ...................................... 162 Figure 8.11 Implicit functions of coefficient 1 and 2 .............................................. 164 Figure 9.1 Imaginary grid of osteoblasts and osteoclasts........................................ 175

Figure 9.2 Prediction of the relative position between osteoblasts ......................... 176 Figure 9.3 TRAP staining on osteoclasts performed in co-culture of human

osteoblasts and osteoclast in control, SMF and PEMF exposure .................... 179

xi

Figure 9.4 The fluorescence-based staining in control, SMF and PEMF exposure.

Compared with the control, the SMF and PEMF exposure prevented the

forming of mature osteoclasts .......................................................................... 181 Figure 9.5 Scanning electron microscope (SEM) images of osteoblasts and

osteoclasts co-culture in control, SMF and PEMF exposure ........................... 183

xii

List of Tables

Table 1-1 Animal studies .......................................................................................... 15 Table 8-1 The effects of SMF exposure on osteogenesis-related genes of human

osteoblasts ........................................................................................................ 165

Table 8-2 The effects of PEMF exposure on osteogenesis-related genes of human

osteoblasts ........................................................................................................ 165 Table 8-3 The frequency functions.......................................................................... 166

1

Chapter 1

Biological Effects of Electromagnetic Fields

2

1.1 Characteristics of electromagnetic fields

Bioelectrical phenomena play a vital role in bioprocesses by the separation, transport

and storage of electrical charge. The participation of electron transfer is essential for

electrochemical communication between molecules [1]. Electromagnetic fields

(EMFs) can influence the mobility of electron by induced forces in the structure of

matter, which might be responsible for biological effects ranging from increased

enzyme reaction rates to transcript levels of specific genes [2]. The electromagnetic

spectrums of EMFs interact with biological systems by various wavelengths in term

of frequency and amplitude.

1.1.1 Frequency

In unit of Hertz (Hz), frequency in an electromagnetic spectrum is categorized from

extremely low frequency (ELF) (0-103 Hz), radiofrequency (RF) (103-108 Hz),

microwaves (~109 Hz), infrared (~1012 Hz), visible light, ultraviolet, X-rays to gamma

rays [3]. Electromagnetic spectrum with a high order of frequencies is classified as

ionising radiation, while the radiation insufficient to break molecular bonds is

classified as non-ionizing radiation. Gamma rays, X- rays are examples of ionising

radiation. Radiations from microwaves, RF and ELF-EMFs are examples of non-

ionizing radiation. Therefore, non-ionizing radiation ranges from 0 to approximately

1011 Hz, while ionising radiation is considered above 1011 Hz [4]. The energy of

electromagnetic radiation G , is associated with frequency v by

G hv= , (1.1.1)

where h represents Planck’s constant.

3

1.1.2 Amplitude

The amplitude of an EMF wave comprises of electric and magnetic components. The

electric and magnetic components occur an EMF spontaneously by motions of charged

objects. The EMF weakens with increasing distance from the charged objects [5].

Generally, an EMF produces an electric field by the potential difference. The magnetic

field is created due to the electric current flowing in a conductor. Higher potential

difference correlates to the stronger electric field. Higher electric current value

correlates to the stronger magnetic field.

1.1.2.1 Amplitude of electric field

An electric field can be represented mathematically as a vector field E. The vector

field has a value defined at each point of space and time and is in a function of space

and time coordinates. The magnitude of electric field E is correlated with the vector of

electrical potential Φ ,

= −E Φ , (1.1.2)

with a solution for a uniform field in which the electric field is constant at each point,

Ed

= − , (1.1.3)

where d is the distance between potential difference, and the unit of the electric field

is measured in V/m.

1.1.2.2 Amplitude of magnetic field

A magnetic field can also be represented as a vector field B or H, which describes the

magnetic influence of electric currents and magnetised materials. The magnetic flux

density B is measured in units of Gauss (G) or Tesla (T) (1T= 10,000G), and the field

intensity H is measured in amperes per meter. In a vacuum, B and H are the same. In

4

a magnetised material, magnetic flux density B is proportional to field intensity H and

related by

=B H , (1.1.4)

where is the permeability of the magnetised material. In EMF, the magnetic field

generated by a steady current I is described by

0

2

ˆd

4

IB

r

=

l r , (1.1.5)

where dl is the vector line element with the same direction of the current I, 0 is the

magnetic constant and r is the distance between the location of dl and the location of

the measured magnetic field in the direction r .

1.1.2.3 Maxwell’s equations

If the electric field E is non-zero and constant in time, it is an electrostatic field.

Similarly, if the magnetic field B is non-zero and constant in time, it is a magneto-

static field. When either the electric or magnetic field is time-dependent, both fields

are considered as a coupled EMF governed by Maxwell’s equations. Coupled EMF

wave consists of electrical field E and magnetic field B vibrating in phase and

perpendicular to the direction of propagation. The Maxwell’s equations introduce the

electric and magnetic field with a time and location dependence,

0

02

0

0

1t

c t

+ =

− =

=•

•=

BE E

EB

B J

(1.1.6)

5

in which sources are represented by electric charge density and current density J.

The universal constants are the permittivity of free space 0 and the permeability of

free space 0 .

1.2 Types of EMF applications in vitro and in vivo

Numerous studies have examined EMF effects on the cell membrane level, general

and specific gene expression, and signal transduction pathways. These studies have

been conducted on bioprocesses such as cell proliferation, cell cycle regulation, cell

differentiation, and metabolism. However, several observations after EMF exposure

have been irreproducible and contradictory in other studies. Especially, statistic

insignificance of EMF effects in bioprocesses occurred in the comparison between the

exposure group and the control. Besides, the types of EMF apparatus in vitro and in

vivo are utilised in various parameters. In the following paragraphs, the biological

effects of EMFs are examined respectively in the term of the static magnetic field

(SMF) and pulsed electromagnetic field (PEMF).

1.2.1 SMF

SMF generated from the geomagnetic field is closely related to living and evolution

for organisms on the earth [6]. SMF can be produced by either a permanent magnet or

direct current (DC) electricity in EMF which has constant magnetic flux density over

the time interval. Man-made SMF is not associated with induced electric currents

except during activation and deactivation [7]. According to the magnetic flux density,

SMF is classified as weak (<1 mT), moderate (1mT to 1T), strong (1T to 5T), and

ultra-strong (>5T) [8]. The biological effects of SMFs have been extensively examined

in several biological systems although with inconsistent results.

6

1.2.1.1 Weak SMF

A weak SMF exposure of 20 μT for 30 minutes altered Ca2+ transport compared to

zero magnetic field exposure in cell-free conditions, measured by 660 nm absorbance

for calmodulin-dependent cyclic nucleotide phosphodiesterase activity [9]. Weak SMF

of 120 μT increased the proliferation of human umbilical vein cells by 40% throughout

2 days. Buemi, et al. [10] examined the effects of 0.5 mT SMF on the balance between

cell proliferation and death in renal cells and cortical astrocyte cultures from rats. After

2, 4 and 6 days of exposure to the SMF, they observed a gradual decrease in apoptosis

and proliferation, while a gradual increase in cells with a necrotic morphology

compared to the control group. Sonnier, et al. [11] measured transmembrane Na+ and

K+ currents of the action potential in SH-SY5Y neuroblastoma cells exposed to SMFs

of 0.1 and 0.5 mT. Application of the magnetic fields did not result in detectable

changes in any of the parameters of the action potential, suggesting that the studied

SMFs did not affect the cellular mechanisms responsible for generating the action

potential.

1.2.1.2 Moderate SMF

Exposure to a 120 mT SMF resulted in a reduction in the peak calcium current

amplitude and a shift in the current-voltage relationship in cultured GH3 cells [12].

When exposed to a 125 mT SMF, a less than 5% reduction in peak current in voltage-

activated Na+ channels was measured in GH3 [13]. Human osteosarcoma cell line

MG-63 became stellar shapes and formed multiple layers after exposed to SMF of 400

mT for 24 hours. The differentiation of MG-63 cells at 1-3 days was promoted with

increased expression of Alkaline phosphatase (ALP) [14]. Chiu, et al. [15] reported

that the proliferation of MG-63 was inhibited by SMF with average magnetic flux 100,

250 and 400 mT after 24 hours. The significant difference in cell population was only

7

observed between 400 mT SMF exposed group and control, with the reduction of cell

membrane fluidity. Cunha, et al. [16] analysed MG-63 after exposure to a 320 mT

SMF either continuously or by 1 hour at any 24 hours interval, demonstrating that SMF

significantly reduced the cell proliferation in both treatment manners after 7 days by

MTT assay. They also showed the absence of SMF effects on cell membrane integrity,

morphology and cytoskeleton organisation after 7 days. Feng, et al. [17] observed the

morphology change of MG-63 on a poly-L-lactide substrate when exposed to a 400

mT SMF for 24 hours. The effects of SMF 500 mT and 1 T on cells from human blood

were investigated by examining their influence on the frequency of gross lesions, sister

chromatid exchanges and on the proportion of amodal cells. Neither treatment had a

significant effect on any of the parameters measured [18]. Pacini, et al. [19] examined

morphological changes caused by exposure to a 200 mT SMF on human neuronal cell

culture (FNC-B4). The results showed dramatic changes in morphology in which

vortexes of cells were formed and exposed branched neurites featuring synaptic

buttons. Endothelin-1 release from FNC-B4 cells was also dramatically reduced after

5 minutes of exposure. They also reported that human skin fibroblast cell morphology

was modified with a concomitant decrease in the expression of some sugar residues of

glycoconjugates after 1-hour exposure to a 200 mT SMF [20]. Teodori, et al. [21]

investigated the exposure of HL-60 cells to SMF of 6 mT with or without DNA

topoisomerase I inhibitor, camptothecin for 5 hours. The SMF alone did not produce

any apoptogenic or necrogenic effect in HL-60. In combination with camptothecin,

SMF did not affect overall cell viability but accelerated the rate of cell transition from

apoptosis to secondary necrosis after induction of apoptosis by camptothecin.

8

1.2.1.3 Strong and ultra-strong SMF

Strong and ultra-strong SMFs are of sufficient intensity to alter the preferred

orientation of a variety of diamagnetic anisotropic organic molecules. Matrix proteins

such as fibrin fibres are orientated to the SMF of 8 T [22]. Kotani, et al. [23] reported

that cultured mouse osteoblastic MC3T3-E1 cells were transformed to rod-like shapes

and oriented in the direction parallel to the magnetic field 8 T after 60 hours exposure.

The SMF exposure did not affect cell proliferation of MC3T3-E1, but up-regulated

cell differentiation and matrix synthesis which was determined by ALP and alizarin

red staining. Wiskirchen, et al. [24] reported that population doublings and cumulative

population doublings of human fetal lung fibroblasts had an insignificant difference in

statistics between exposed and control cell groups after 9 hours exposure to an SMF

of 1.5 T in three weeks. Clonogenic activity, DNA synthesis, cell cycle, and

proliferation kinetics were not altered by an SMF exposure of 1.5 T. The SMF

exposure of 7 T after 64 hours produced a reduction in viable cell numbers in

melanoma, ovarian carcinoma and lymphoma cell lines. Prolonged exposure to the

SMF of 7 T slowed the growth of human cancer cells in vitro. Alterations in the cell

growth cycle and gross fragmentation of DNA were excluded as possible contributory

factors [25]. Aldinucci, et al. [26] investigated whether SMF at a flux density of 4.75

T, generated by an NMR apparatus (NMRF), could promote movements of Ca2+, cell

proliferation and the production of proinflammatory cytokines in human peripheral

blood mononuclear cells (PBMC) after exposure to the field for 1 hour. The same study

was also performed after the activation of cells with 5 mg/mL phytohemagglutinin

(PHA). The results demonstrated that the SMF had neither proliferative nor

proinflammatory effects on normal and PHA activated PBMC. The concentration of

interleukin‐1β (IL-1β), interleukin‐2 (IL-2), interleukin‐6 (IL-6), interferon and

9

tumour necrosis factor α (TNFα) remained unvaried in the exposed cells. Exposure to

SMF with various magnetic flux densities of less than 1.6 T had no significant effect

on either active or passive Rb+ influxes, the morphology of HeLa cells [27]. Gradient

magnetic fields of 6 T affected the convection of floating cell aggregations in the cell

culture flask, and reversibly changed the direction of conventional flow [28]. Hirose,

et al. [29] reported that human glioblastoma A172 cells embedded in collagen gels

were oriented perpendicular to the direction of the static magnetic field at 10 T. A172

cells cultured in the absence of collagen did not exhibit any specific orientation pattern

after 7 days exposure to the static magnetic field. Eguchi, et al. [30] observed that

cultured Schwann cells from dissected sciatic nerves of neonatal rats were oriented

parallel to the magnetic field at 8 T after the exposure for 60 hours.

1.2.2 PEMF

There is ongoing interest in the application of PEMF radiation as an alternative non-

invasive therapy for curing bone disease. PEMF is referred to as time-varying EMF

consisting of specific or arbitrary waveforms with pulse modulated frequency [31].

Significant numbers of peer-reviewed publications have demonstrated that PEMF

radiation facilities the process of wound repairs. For instance, low-frequency

sinusoidal waveforms from 10 Hz up to 500 Hz have been shown to enhance healing

when used as adjunctive therapy [32]. In PEMF, magnetic flux density changes at one

or more frequencies. A PEMF, produced by alternating current (AC) electricity, is

significantly more dynamic than SMF and can induce an electric charge in tissues

which creates a cascade of physiologic effects [33]. In past decades, efforts have been

made to elucidate the exact SMF’s effect on osteoblasts, but this topic is challenged

by the fact that the in vitro effects of SMF highly depends on cell type [34], magnetic

field intensity and modes of application [8]. For example, PEMF exposure induced

10

cell proliferation of MG-63 with 2.3 mT and 75 Hz [35]. PEMFs determine signal

transduction using the intracellular release of Ca2+ leading to an increase in cytosolic

Ca2+ and an increase in activated cytoskeletal calmodulin [36]. PEMFs induce a dose-

dependent increase in cartilage differentiation [37], and upregulation of mRNA

expression of extracellular matrix molecules, proteoglycan, and Type II collagen

[38]. The acceleration of chondrogenic differentiation is associated with the

increased expression of TGF-β1 mRNA and protein, suggesting the stimulation of

TGF-β1 may be a mechanism through which PEMFs affect complex tissue

behaviours such as cell differentiation, and through which the effects of PEMFs may

be amplified [39]. PEMFs also are postulated to affect membrane level by

influencing signal transduction of several hormones or growth factors such as

parathyroid hormone, IGF-2, producing the amplification of their transmembrane

receptors [40].

Extremely low-frequency electromagnetic fields (ELF-EMFs), with tissue gradients in

the range from 10-7 to 10-1 V/cm, are involved in essential physiological functions in

mammals. Numerous studies have addressed the effects of ELF-EMFs involving a

wide spectrum of calcium-dependent processes, such as cell membrane functions in

bone growth [41] and regulation of intercellular communication [42]. Noriega-Luna,

et al. [43] investigated the effect of a magnetic flux density of 0.65 mT and frequency

of 4 Hz on the proliferation of MG-63, observing a slight increase in cell number and

subtle difference in osteoblast morphology after 48 hours treatment. Zhou, et al. [44]

demonstrated that the effects of 50 Hz sinusoidal PEMF of different magnetic flux

density, from 0.9 to 4.8 mT with an interval of 0.3 mT, inhabited osteoblast

proliferation and promoted their differentiation and mineralisation with peak activities

at 1.8 and 3.6 mT. De Mattei, et al. [45] introduced an experiment showing a

11

correlation between PEMF exposure time and cell proliferation increase in human

osteosarcoma cell line MG-63 and human normal osteoblast cells. The results

indicated that a short PEMF exposure of 30 minutes could stimulate the cell

proliferation in MG-63 and the normal human osteoblasts in 6 to 9 hours. However,

an increase in the length of exposure time resulted in an insignificant difference in cell

proliferation.

1.2.3 Animal studies

Few clinical studies concern EMF therapy, but animal studies have been carried out to

determine the effectiveness of EMF on bone growth. These studies indicate that EMF

can contribute to bone formation and healing process in various manners of magnetic

flux density, frequency and exposure duration. Representative animal studies in recent

years are summarised in Table 1 and generally show some positive effects.

1.3 Why effects of EMF in biology are irreproducible and contradictory

Cells and tissues are affected by the magnetic field of EMF, and not all EMF exposures

lead to an altercation at the cellular level. Some effects are only noted at discrete

frequencies and amplitude of the magnetic field. Others depend on the strength,

orientation and duration of the exposed field. The effects of EMFs on biological media

have been studied by many researchers using a variety of in vitro exposure systems

[46, 47]. Magnetic field therapy conducted by EMF is considered beneficial for

different diseases, especially those involving bones [48]. EMF stimulation has been

investigated as a therapy for wound healing following results that EMFs can promote

healing by potentially increasing collagen synthesis, angiogenesis, and bacteriostatic

[49]. Commercial EMF stimulators have been used to promote bone healing, with the

setting of EMF parameters varying significantly. Laboratory-based exposure systems

for studying EMF effects on biological samples utilise Helmholtz coils to generate

12

uniform electromagnetic exposures, but there are discrepancies between the set of

magnetic flux density and frequency. Hence, researchers argued that the controversial

effects of EMF on biological objects were due to experiments not carried out in well-

defined conditions [50]. Extensive studies have been done to categorise the parameters

of EMF for optimal laboratory settings [51]. EMF signal induces electric and magnetic

signal to initiate a cascade of biochemical reactions [52]. The EMF stimulation is

characterised by factors such as magnetic field intensity, waveform, dose-response

pattern [53], exposure duration [54], localisation of stimulation and spatial orientation

of the exposure system [55]. Other factors can influence the response to magnetic field

exposure by frequency and modulation, field uniformity, a combination of coil system

and precise placement, field intensity and polarisation, noise, vibration and

temperature of conducting materials, voltage carrying wires and metal equipment,

construction materials and the experiment schedule [56]. Therefore, EMF parameters

are proposed to be considered prior to evaluating the effect of magnetic field on a

biological sample in aspect of (i) type of magnetic fields, (ii) magnetic flux density,

(iii) frequency, (iv) exposure duration, (v) pulse shape, (vi) spatial gradient (dB/dx)

and temporary gradient (dB/dt) [57].

1.3.1 Mechanisms of EMF on biological systems

The mechanisms by which EMF affects biological systems are not fully elucidated. A

quantitative approach was presented in understanding the interaction between electric

fields and biological systems [58]. Biological samples react to external electrical

stimulation through a complex series of specific and non-specific responses [59]. The

specific response is determined by the physical nature of the stimulation, while the

non-specific response depends upon the intrinsic features of the organism system.

Electrical signals are the basis of information transportation in the nervous system

13

[60]. These electrical signals in the form of minute electric currents flow around and

within the cells and are of critical importance for their normal functioning and can

accelerate normal cellular function such as endocytosis [61]. EMF perturb these

currents and charges and positively influence the process of cellular functioning. It has

been suggested that EMFs may trigger specific, measurable cellular responses such as

DNA synthesis, transcription, and protein synthesis by altering or augmenting pre-

existing endogenous electrical fields [62]. This mechanism adds further evidence to

the fact that external non-invasive electric stimulation is a potent tool in augmentation

of cells, tissue and organs. A well-established phenomenon in physics, ion cyclotron

resonance (ICR) and ion parametric resonance (IPR) models were proposed, which

state that ions resonate when exposed to a specific combination of alternating and static

magnetic fields [63]. Hall effect may provide the electrical basis of EMF in the bone

lacuna canalicular system, which attaches the positive ions at the negative interface of

the bone matrix [64]. When EMF is applied, the moving charged particles encounter a

Lorentz fore perpendicular to their direction. Cations accumulate at the downward

surface, and anions go upward to forming a hall voltage [65].

The magnetic control operates in biology by strong and weak interactions. The strong

magnetic interaction, such as magnetic resonance imaging, requires a strong and

durable magnetic field. It can be manifested in processes involving particles and

membranes. If these particles and membranes have anisotropy of magnetic

susceptibility, the energy of the strong magnetic interaction will result in re-orientation

of particles and deformation of membranes. Consequently, the properties and the

chemical reactivity of particles and membranes change with the re-orientation and

deformation [66]. Evidence suggested that cell membrane is in response to transducing

EMF [67] where EMF can interact with moving charges by Lorentz forces [68].

14

However, an argument exists against Lorentz forces in cells since dielectric media and

plasma lack moving charges [69]. Buchachenko and Kuznetsov [70] emphasised the

molecular radical pair paradigm as a reliable basis for understanding and deliberately

using biochemical magnetic effects in medicine. Biochemical processes are

accompanied by generation or participation of ion-radicals in pairs. Radical pair

mechanism implies that two radicals are produced simultaneously with paralleled or

anti-paralleled electron spins. Chemical reactions are spin selective which are allowed

only for those spin states of reactants with identical total spins. The spin states of the

pair, singlet and triplet are different in chemical reactivity but identical in structure. In

triplet state, the reactions are forbidden. Weak magnetic interaction might provide a

manner to overcome the spin prohibition of processes in biochemistry. Magnetic

interactions induce singlet-triplet spin conversion and switch over the reaction

between spin-allowed and spin-prohibited channels, controlling the reaction pathways

and chemical reactivity. Radical pair mechanism has been shown functionally on the

molecular level in biochemical reactions [71]. Three types of magnetic interactions are

considered for catalysing chemical and biochemical reactions, namely Zeeman

interaction, Fermi interaction and microwaves. Magnetic catalysis is controllable and

switchable by using magnetic isotopes or paramagnetic ions [72]. The magnetic

interaction effects on the enzymatic ATP synthesis were detected for the creatine

kinase in vitro [73]. The magnetic control of enzymatic DNA synthesis was observed

in time-varying magnetic fields [74].

15

Table 1-1 Animal studies

Animal Aim Magnetic flux

density/Frequency

Exposure

duration

Results Reference

10-week-old

Wistar

female rats

Recovery of

osteoporosis

Gradient SMF

180 mT

3 weeks Significantly increased the BMD values of the

osteoporotic lumbar vertebrae in ovariectomized rats

without significantly influencing the E2 levels.

[75]

12-week-old

Sprague Dawley

male rats

Type 1 diabetes

mellitus (T1DM)

Helmholtz coils

4 mT

16 weeks Significantly prevented the deterioration of bone

architectural deterioration and strength reduction,

promotion of bone formation and weak modulation of

bone resorption.

[76]

12-week-old

Wistar albino

male rats

Chronic exposure

to bone

MRI

1.5 T

8 weeks Induced low-frequency fields within the tissues which

could exceed the exposure limits necessary to

deteriorate bone microstructure and vitamin D

metabolism.

[77]

20-week‐old

Sprague Dawley

male rats

Hyperthyroidism‐

induced

osteoporosis

Helmholtz coils

15 Hz/1 mT

12 weeks Significantly inhibited bone loss and microarchitecture

deterioration in hyperthyroidism rats, which might

occur due to reduced THR expression.

[78]

4-week-old Wistar

female rats

Effects of

exposure duration

Solenoid

50 Hz/1.8 mT

8 weeks Comparing the effects of 0.5, 1.0, 1.5, 2.0, 2.5, and

3.0 hours/day, 1.5 hours/day was the optimal exposure

duration to increase the peak bone mass of young rats.

[79]

12-week-old

Wistar albino

male rats

Bone fracture

healing

Helmholtz coils

50 Hz/1.5 ± 0.2 mT

4 weeks The EMF had a positive but modest effect on bone

fracture healing.

[80]

12-week-old

Sprague Dawley

male/female rats

Bone loss Helmholtz coils

8 Hz/3.8 mT

12 weeks The EMF prevented the diabetes-induced bone loss

and reversed the deterioration of bone

microarchitecture by restoring Runx2 expression

through regulation of Wnt/β-catenin signalling.

[81]

https://www-sciencedirect-com.virtual.anu.edu.au/topics/medicine-and-dentistry/type-1-diabetes

https://www-sciencedirect-com.virtual.anu.edu.au/topics/medicine-and-dentistry/type-1-diabetes

16

Chapter 2

Bone Remodelling and Applied

Mathematical Modelling

17

2.1 Bone modelling and remodelling

Bone is a continuously updated tissue and constituted mainly by BMSCs, osteoblast,

osteocytes and osteoclasts. The dynamic balance between bone formation and

resorption, such as bone modelling and remodelling, has a pivotal role to play in the

normal bone metabolism, bone integrity and appropriate bone strength [82]. Bone

modelling and remodelling are both involved in osteogenesis and skeletal growth.

Bone modelling is characterised by the process of bone formation and resorption with

a net increase in bone mass. The mechanism involves activation-formation and

activation-resorption, which primarily occurs in childhood [83]. Activation is signalled

by local tissue strain and involves the recruitment of progenitor cells that differentiate

into mature osteoblasts or osteoclasts. Once the appropriate cell population is

activated, the processes of formation and resorption happen until sufficient bone mass

is altered for normalising local strains. In contrast, bone remodelling is defined as

renewing and maintaining bone in which a coupled process occurs between the

catabolic effects of bone-resorbing osteoclasts and anabolic effects of bone-forming

osteoblasts [84]. During a bone remodelling process, bone mass is removed at sites

where the mechanical loads are low, while bone is formed where mechanical stimuli

are transmitted repeatedly.

Bone, therefore, can maintain itself, depending on the external mechanical and

physiological stimuli from the systemic environment [85]. The process of bone

remodelling is executed by a temporary anatomic structure incorporating a cohort of

cells known as the bone multicellular unit (BMU). The BMU mainly consists of

osteoblasts, osteoclasts and osteocytes. Bone-lining cells, osteomacs and vascular

endothelial cells have also been reported to be associated with the BMU. Bone is a

highly vascularized organ, and networks of blood capillaries have been observed at the

18

site of the BMU [86]. Vascularisation and angiogenesis are prerequisites for bone

formation and remodelling. They serve multiple purposed bone cell progenitors via

blood capillaries to establish the modelling and remodelling site. The blood supply

also allows the BMU to become accessible to immune cell infiltration and interaction

with bone cells.

Bone remodelling involves the balance between osteoclast and osteoblast activity,

which is regulated by numerous signalling pathways [87]. Deregulation of signalling

pathways in bone cells may lead to bone diseases such as osteoporosis and

osteoarthritis. In the big picture, osteoblasts facilitate bone formation by laying down

a matrix which subsequently is mineralised, and produce receptor activator of nuclear

factor - B ligand (RANKL) initiating osteoclastogenesis [88]. Osteoclasts, activated

by proinflammatory cytokines and RANKL/macrophage colony stimulating factor

(M-CSF), initiate bone resorption by releasing catalytic enzymes like Cathepsin K (Cat

K) and matrix metalloproteases (MMPs) in a resorptive pit formed on the bone surface

[89]. Protection against bone damage can be achieved by (i) inhibiting RANKL

production by increasing the production of osteoprotegerin (OPG), (ii) suppressing the

proinflammatory cytokines, (iii) inhibiting production of Cat K and MMPs, and (iv)

inhibiting osteoclast formation. The cellular coupling between osteoblasts and

osteoclasts is complex and is massively coordinated by several regulatory systems to

keep both remodelling and resorption processes synchronised. The accentuation of one

or the other process eventually leads to bone fragility and clinical diseases of the

skeleton, such as osteoporosis, arthritis and osteolysis [90].

2.1.1 Osteoblast

Osteoblasts are the primary cells responsible for bone formation. They originate from

mesenchymal stem cells, which have the potential to differentiate into other

19

musculoskeletal tissues such as cartilage, fat, muscle, ligament and tendon [91]. The

commitment of these stem cells to the osteoblast lineage is highly driven by the growth

factors wingless-type proteins (Wnts) and bone morphogenetic proteins (BMPs). Once

skeletal stem cells are committed to the osteoblast lineage, the proliferation of

osteoblast precursor cells begins. These cells produce type I collagen for the basic

building block of bone, osteocalcin and alkaline phosphatase for mineral deposition.

Osteoblast precursor cells mature into cuboidal osteoblasts. Besides their role in bone

formation, the osteoblasts are involved in the recruitment and maintenance of

osteoclasts by expressing M-CSF, RANKL and OPG [92]. Mature osteoblasts may

undergo apoptosis or coalesce into the heterogeneous population bone-lining cells, or

eventually become encased and trapped within the matrix to become osteocytes. The

proportion of osteoblasts following each possibility varies in all mammals and is not

conserved among different types of bone. The age of the mammal may also influence

the number of osteoblasts that transform into osteocytes [93].

2.1.2 Osteoclast

Osteoclasts are motile macrophage-like and multinucleated cells, which are formed by

the fusion of myeloid hematopoietic precursors. Osteoclast precursors are formed in

the bone marrow and circulate in the blood. The differentiation of osteoclast precursors

into mature osteoclasts requires factors like M-CSF and RANKL [94]. The coupling

of receptor-ligand complexes between osteoclast precursors and mature osteoblasts are

indispensable for osteoclastogenesis [95]. A variety of cytokines also regulate the

mechanisms of osteoclast differentiation and fusion in normal and pathological states.

Parathyroid hormone (PTH) stimulates bone marrow stromal cells and mature

osteoblasts to produce RANKL and enhances osteoclast formation [96]. Tumour

necrosis factor (TNF) activates the autocrine and paracrine mechanism in osteoclast

20

precursors, controlling their formation and the activity of bone resorption [97]. The

osteoclasts serve as bone-resorbing cells by eroding bone, enabling the tissue to be

remodelled during growth and in response to stresses. Bone resorption is an essential

event during bone growth, tooth eruption, fracture healing and the maintenance of

blood calcium level. Osteoclasts are derived from hematopoietic stem cells [98] and

are created by the differentiation of monocyte/macrophage precursor cells at or near

the bone surface [99]. Hence, osteoclasts share a common pathway with that of

macrophage and dendritic cells [100].

2.1.3 Gene network between osteoblasts and osteoclasts

Critical to maintaining the strength of bone, the process of bone remodelling must be

tightly regulated [83]. Thus, bone mass and structure are ultimately the consequence

of interactions of multiple pathways in the network that are modulated by hormonal,

cytokine and immune systems. This control is exquisitely sensitive to external stimuli

such as EMF [101, 102]. In vitro studies clearly show that a variety of molecules in

bone metabolism are affected by EMF application, including BMP-2, TGF-β and IGF-

II [103, 104] (Figure 2.1). EMFs resulted in the activation of the extracellular signal-

regulated kinase (ERK), mitogen-activated protein kinase (MAPK) and prostaglandin

synthesis, which may also lead to stimulatory effects on bone [105, 106].

TFG-ß/Nodal/Activin signals are transduced through type I and type II receptors for

each member to R-SMAD proteins, such as SMAD 2 and SMAD 3, while BMP/GDF

signals are transduced through type I and type II receptors for each member to R-

SMAD proteins. Phosphorylated R-SMADs associated with SMAD 4 are translocated

to the nucleus to activate transcription of target genes [107]. BMP/GDF family genes

within the human genome have been extensively studied [108]. However, the

transcriptional regulation of BMP/GDF family members by the canonical Wnt

21

signalling pathway remains unclear. Wnts constitute a family of proteins important in

cell differentiation, notably playing a critical role in OB cell differentiation and bone

formation [109]. Upon binding of Wnt to Frizzled receptors and the low-density

lipoprotein receptor-related protein (LRP) coreceptors-5 and -6, the activity of GSK-

3β is inhibited, leading to the stabilisation of β-catenin and its translocation to the

nucleus [110]. There, it associates with T cell factor (TCF) 4 or lymphoid enhancer

binding factor (LEF) 1 to regulate gene transcription [111]. Sclerostin (SOST)

produced by bone cells, has recently emerged as an essential modulator of anabolic

signalling pathways in bone, particularly PTH stimulation and mechanical loading

[112, 113]. These facts make Wnt a suitable target to derive a bone anabolic response

[114]. On the other hand, Wnt pathway components, including Wnts, Fzds, Lrps, and

Tcf family members, are also expressed in osteoclast lineage cells [115]. Thus, Wnt/β-

catenin signalling appears to reduce bone resorption. This area requires further

investigation to resolve the scope of Wnt influences on bone metabolism. The

landmark discoveries of the three molecules RANK, RANKL and OPG have moved

bone research into a new era [116, 117]. In bone, the expression of RANKL allows the

maturation, differentiation and activation of OCs by binding to its receptor, RANK,

present on the surface of pre-osteoclasts. On the other side of the coin, OPG exerts a

protective effect on bone acting as a decoy receptor for RANK–RANKL binding. The

balance between RANKL and OPG determines bone resorption [118, 119]. Activated

T cells express RANKL and support osteoclast formation and activation by cytokines,

including IL-1, IL-6, TNF-α and IL-17. In contrast, several other T cell-derived

interleukins and cytokines, IFN-γ, IFN-β, IL-4, IL-10, IL-13, GM-CSF, osteoclast

inhibitory lectin and secreted Frizzled-related proteins (sFRPs) potently inhibit

osteoclast formation [120, 121]. Complex interactions have been visualised by

22

conceptual block diagram using data and text mining in Figure 2.2. Established

knowledge from feedback control of dynamic systems in engineering, allows us to

define four primary functions for known regulations: induce, inhibit, relate, and

complex. These primary functions enable construction of a network with systematic

loops.

2.2 Mathematical models for simulating cell growth and movement

Efforts for modelling mechanism of EMF on bone remodelling have focused on

mechanical functions or biological functions based on signalling pathways [122, 123].

Mathematical models of bone remodelling have been established on the

RANK/RANKL/OPG pathway under the influence of PTH, mechanical force, and

EMF at the cellular level [124-126]. These models can be extended to different

waveforms of EMF and other external perturbation [127, 128].

Bone cell differentiation and proliferation are important factors during bone

remodelling, and clinical PEMF devices have been shown to affect differentiation and

proliferation of bone cells in vitro [129, 130]. It has been proposed that gap junctions

which are specialised intercellular junctions are considered as mediators of the PEMF-

related cellular responses [129, 131-133]. Nevertheless, the underlying mechanism at

the cellular level that regulates bone remodelling under PEMF remains poorly

understood because of the inconsistent or even contradictory results from experiments.

For example, cell proliferation, as assayed by cell number and H-thymidine

incorporation, has been reported to increase [134], decrease [135], and remain

unaffected [136] by PEMF exposure. Similarly, the production of alkaline phosphatase

has been reported to either increase [137] or decrease [132] following PEMF

exposure.

23

In order to remove the limitations to generalisation concerning causes and effects of

bone remodelling under PEMF, mathematical models provide a dynamic, quantitative

and systematic description of the relationships among interacting components of the

biological system [138]. Kroll [139] and Rattanakul, et al. [140] each proposed a

mathematical model accounting for the differential activity of PTH administration on

bone accumulation. Komarova, et al. [141] presented a theoretical model of autocrine

and paracrine interactions among osteoblasts and osteoclasts. Komarova [142] also

developed a mathematical model that describes the actions of PTH at a single site of

bone remodelling, where osteoblasts and osteoclasts are regulated by local autocrine

and paracrine factors. Potter, et al. [143] proposed a mathematical model for the PTH

receptor (PTH1R) kinetics, focusing on the receptor’s response to PTH dosing to

discern bone formation responses from bone resorption. Lemaire, et al. [144]

incorporated detailed biological information and a RANK-RANKL-OPG pathway into

the remodelling cycle of a model that included the catabolic effect of PTH on the bone,

but the anabolic effect of PTH was not described. Based on the Lemaire’s model,

Wang, et al. [145] developed a mathematical model that could simulate the anabolic

behaviour of bone affected by intermittent administration of PTH, as well as a

theoretical model and its parametric study of the control mechanisms of bone

remodelling under the mechanical stimulus. Pivonka, et al. [146], [147] extended the

bone-cell population model based on the Lemaire’s model to explore the model

structure of cell-cell interactions theoretically and then investigated the role of the

RANK-RANKL-OPG system in bone remodelling.

The numerical models of bone remodelling are built under external loads and

examined by bone density. Finite element method (FEM) [148-154] is employed by

such models to simulate the relationship between bone density change and mechanical

24

stimulus [155]. FEM is successful in analysing the macroscopic level of bone structure,

especially in studying microcracks [156] and reconstruction of bone models from CT

images [157, 158]. Such problems can refer to similar studies from material research

[159-165]. Although the FEM may put forward the quantitative prediction of cell

behaviours, it is not clear exactly how mechanical loading affect the activities of

osteoblasts and osteoclasts in each cell cycle. Besides, the parametric simulation of

cell signalling introduces unknowns into the model and raises a question of how

signalling pathways interact with each other instead of the answer for the coupling

mechanism between activities of osteoblasts and osteoclasts at the cellular level. It

might be feasible to describe each component mathematically on a small scale of

signalling (Figure 2.1), while the mission is impossible for a signalling network

(Figure 2.2). Further, Spatial and temporal heterogeneity of molecules or proteins

require a comprehensive mathematical description instead of connecting signalling

pathways with simple positive and negative circuit feedbacks.

The multi-scale methodology of bone remodelling simulation is created to find

solutions for unknowns from experiments. One method of organising the multi-scale

bone remodelling simulation is to integrate signalling-based bone cell population

model into a micromechanical representation of cortical bone [166]. Theoretically, the

bone cell population can affect the mechanical properties of bone and the unknowns

for the model of bone cell population could be found when the mechanical properties

of bone are calculated in the experiment. Bone cell development proceeds in three

main periods: proliferation, extracellular matrix maturation, and mineralisation [167].

Unfortunately, according to the model built by Owen, et al. [168], the bone cell

population cannot fully link with mechanical properties in these periods. Another

method of multi-scale bone remodelling simulation is based on the fact that electric

25

and electromagnetic fields gain significance in the therapy of bone fracture healing

and bone disease [169].

Mathematical modelling is a powerful tool for testing and analysing various

hypotheses in complex systems, yet it is hard to simulate the biological system

accurately. The challenges are to elucidate the interactions of a biological system

mathematically in both cellular and molecular level, and to verify whether the

prediction derived from the formulated biological system match the experimental

results.

26

Figure 2.1 BMP-SMAD and TGF-β/Activin pathway. The intracellular signalling

networks follow a specific pathway from one gene to another. Models for intracellular

signalling networks are written from the original physiological map and woven into a

web in which single elements in models can receive information from multiple inputs.

However, despite impressive accomplishments at small scale, current models fall apart

at the genome scale. The failures occurring at the large scale are attributed to the reason

that the pre-assumption of relatively homogeneous and static components hardly

reflects the heterogeneity of biological signalling network.

27

28

Figure 2.2 Map of osteo-gene network. The remodelling units of OCs and OBs and

related genes are divided into three levels: extracellular, intracellular and nuclear

levels. Known regulations are applied in the diagram to establish interactions among

genes, and unknown regulations are highlighted with a question mark. Osteo gene

network represents the importance of a gene or a protein in multiple and interacted

pathways. This network provides a possible frame for a systematic and quantitative

measure of organising the gene and protein expression of various samples in one

picture. It is possible to simulate and study how modulating specific molecules and

cellular functions affect critical physiological and pathological outcomes within an

integrated system context.

29

Chapter 3

Research Question and Hypothesis

30

3.1 Research question

Researchers have been confused about the effect of EMF on a biological system in

past decades. Chapter 1 has shown the efforts to elucidate one or several factors for

explaining the underlying mechanism of the EMF effect on a biological system.

Nevertheless, we realise that different dataset yield contradictions when articulating

them into one picture. In classical physics, data obtained by using various instruments

supplement each other and can be combined into a consistent picture where the

influence of the measuring procedure can easily be subtracted from its outcome. On

the contrary, the effect of EMF on cells illustrates that changing the experimental

arrangement is equivalent to changing the holistic phenomena. Consequently, we

could not predict the effect of EMF on the experiment results even though numerous

similar experiments have been carried out.

Such contradictions might be overcome by loosely articulating the information derived

from various experimental arrangements in complementarity. Bohr introduced

complementarity in 1927, “Evidence obtained under different experimental conditions

cannot be comprehended within a single picture but must be regarded as

complementary in the sense that only the totality of the phenomena exhausts the

possible information about the objects” [170]. The principle of complementarity was

applied (i) between incompatible variables, (ii) between causation and spatiotemporal

location, and (iii) between the continuous and discontinuous pictures. Following the

principle of complementarity, we could describe cell behaviours in a rough quantity

without giving precise laws, or we could give the precise laws in their abstract form

without the quantitative presentation. Interestingly, experimental results in Chapter 1

described the biological effects of EMF in a rough quantity, and cell signalling and

mathematical models in Chapter 2 illustrated the precise laws.

31

Different pieces of information are taken into being mutually exclusive, yet jointly

indispensable to characterise a specific object. We write research questions here about

the consistency of bone cells’ bio-data in weak EMF.

• In the same experimental setting except for magnetic intensity, whether the

relationship between osteoblasts population and magnetic intensity follows the

same pattern throughout the time.

• In the different types of EMFs but with a similar scale of magnetic intensity,

whether the relationship between osteoblasts population and magnetic intensity

obeys the same principle.

• Whether the principle derived from osteoblasts behaviour in weak EMF is

consistent in the co-culture of osteoblasts and osteoclast.

3.2 Hypothesis

The primary hypothesis assumes that EMFs can affect the movement of osteoblasts.

The mechanisms of EMFs on biological systems in Chapter 1 provide this hypothesis

with substantial evidence. However, it is an open question of how to describe the

movement of osteoblasts mathematically. We prefer the concept of “cell probability”

which means the chance that cells appear at a specified location by counting the

number of cells at a specified location in a particular time and then taking the ratio of

this number to the total number during that time.

The cell population is usually presented as a mean ± standard deviation when repeating

the same experiment of cell proliferation several times. Here we consider the cell

population as a normal distribution with the central of mean. This normal distribution

might not be established since the sample size is limited and each sample has the same

weight in occurrence. Instead, cell probability might be described. Cell probability is

an imaginary picture that indicates the tendency of cell movement.

32

We imagine the effects of EMF on cell probability as a slit, which is a derivative model

of the double-slit experiment (Figure 3.1). The culture environment consists of at least

two slits, with and without EMF. When osteoblasts are cultured in EMFs, the cell

probability has two possible patterns after passing through slits. Therefore, the

superposition of probability amplitudes could occur. We write the probability

amplitude as

Cell final at Cell initial at x s = . (3.2.1)

Following the expression in physics, the right part to the vertical line gives the initial

condition, and the one on the left indicates the final condition. The abbreviation of

(3.2.1) is

x s = . (3.2.2)

Such an amplitude might be a complex number. Each replicate in cell culture has the

same probability amplitude rather than the cell population. If a factor is amplified in

the culture environment such as hormones, this will create a new slit. We mark the two

slits as 1 and 2. The resultant amplitude is calculated by multiplying in succession the

amplitude for each of the successive events,

1 1 2 2x s x s x s= + , (3.2.3)

which equals to

1 2 = + . (3.2.4)

The cell probability P is given by

2

P = . (3.2.5)

Hence, cell probability could happen with or without interference. If there are mutually

exclusive, indistinguishable alternatives in an experiment, the interference happens,

33

2 2 2

1 2 1 2 1 22 cosP = + = + + , (3.2.6)

where is the phase difference between 1 and 2 . If an experiment is performed

with the capability of determining whether one or another alternative is taken, the

interference is lost,

2 2 2

1 2 1 2P = + = + . (3.2.7)

For instance, screening tests in biological experiments are taken without interference.

When a PEMF is specified in magnetic intensity and frequency, we calculate a much

more complicated problem. In physics, when an event can occur in several alternative

ways, the probability amplitude for the event is the sum of the probability amplitudes

for each way considered separately. We build two filters for PEMF, where the first

filter represents the magnetic intensity and the second filter represents the frequency.

If the first filter has i slits and the second filter have j slits, the complete amplitude

is written by

j i

x s x j j i i s= . (3.2.8)

When osteoclasts appear in the experiment, the disturbance happens between

osteoclasts and the slits. The amplitude of osteoblasts is described in the equation

(3.2.3). The amplitude 1 s occurs when osteoblasts go from s to slit 1 and multiply

the amplitude a when osteoblasts contact with osteoclasts at slit 1. Then the

amplitude that osteoblast go from s to x via slit 1 and contact osteoclasts is

1 1 1a x a s = . (3.2.9)

Similarly, the amplitude that osteoblasts go via slit 2 and contact osteoclasts is

2 2 2b x b s = . (3.2.10)

34

The probability of both amplitudes is

2

1 2P a b = + , (3.2.11)

where the time is involved in the amplitude and the corresponding probability.

Different results are detected if the measurement occurs at different places and times.

The function should satisfy a differential equation which could be a wave equation

analogous to the equation for electromagnetic waves. However, it must be noticed that

this function is not a real wave in space even when it satisfies the wave equation. Since

cells take a long time in movement compared with the particles with a given energy,

we may consider the cells equivalent to particles with a very long wavelength in the

interference. Therefore, we might observe the alteration of cell possibility and the

occurrence of interference when osteoblasts are cultured in EMFs.

35

Figure 3.1 Double-slit model for osteoblasts. Thinking that osteoblasts start to move

at S and stop at X, osteoblasts pass through the silts created by culture environment.

Cell probability has a pattern along the culture plate. Each replicate in the experiment

has the same cell probability rather than the cell population. The number of slits

depends on the variety of external stimulus such as EMFs. When osteoblasts pass slits,

the superposition of probability amplitudes could occur with or without interference.

36

Chapter 4

Materials and Biological Methods

37

4.1 Materials

4.1.1 Chemical reagents

Chemical reagents involved in experiments conducted in this thesis were of analytical

grade unless indicated otherwise and were purchased from the following

manufacturers:

Description Manufacturer

1α,25-dihydroxyvitamin D3 Sigma-Aldrich, USA

4-(2-hydroxyethyl)-1-

piperazineethanesulfonic acid (HEPES)

ThermoFisher, Australia

α-MEM (α-modification of Eagle’s medium) Life Technologies, Australia

β-mercaptoethanol Sigma-Aldrich, USA

Alizarin red S Biochemicals Inc, USA

Bovine serum albumin Sigma-Aldrich, USA

Bromophenol blue Bio-Rad, USA

Chloroform ThermoFisher, Australia

Dexamethasone Sigma-Aldrich, USA

Dimethyl Sulphoxide (DMSO) ThermoFisher, Australia

D-MEM (Dulbecco’s-modification of Eagle’s

medium)

Life Technologies, Australia

Ethanol (100%) EMD Millipore Corporation, USA

FBS (Foetal Bovine Serum) Life Technologies, Australia

Hanks’ balanced salt solution Life Technologies, Australia

HEPES (4-(2-hydroxyethyl)-1-

piperazineethanesulfonic acid)

Sigma-Aldrich, USA

38

L-ascorbic acid phosphate Sigma-Aldrich, USA

L-Glutamine Sigma-Aldrich, USA

Liquid nitrogen (N2) Air Liquid, Australia

Penicillin Life Technologies, Australia

Skim Milk Powder Standard supermarket brand

Sodium chloride (NaCl) Bio-Rad, USA

Sodium dodecyl sulphate (SDS) Bio-Rad, USA

Sodium hydroxide (NaOH) Sigma-Aldrich, USA

Sodium phosphate dibasic (Na2HPO4) Sigma-Aldrich, USA

Sodium phosphate monobasic (NaH2PO4) Sigma-Aldrich, USA

Tween-20 Bio-Rad, USA

4.1.2 Commercial kits and molecular products


10 x Reaction Buffer, 1 mL of 200 mM Tris-

HCl, pH 8.3, 20 mM MgCl2

Sigma-Aldrich, USA

5 x First Strand Buffer Invitrogen, USA

Amplification Grade DNAse I ThermoFisher, Australia

BD Pharm LyseTM red cell lysis buffer (10 x) BD Bioscience, USA

DAPI (4’,6-diamidino-2-phenylindole),

dilactate

Biotium, USA

Fastlane Cell cDNA Kit QIAGEN Gmbh, Australia

RNEasy Mini kit QIAGEN Gmbh, Australia

TRIzol® Invitrogen, USA

39

4.1.3 Other products and consumables


0.2 mL PCR Tubes ThermoFisher, Australia

50 mL Centrifuge Tubes ThermoFisher, Australia

BD 1 mL Syringe BD Bioscience, USA

BD PrecisionGlideTM Needles 25 G BD Bioscience, USA

Coverslips-round 5 mM, 22×22 mm2 and

22×40 mm2

Knittle Glaser, Germany

Filter Paper Whatman International, UK

Glass Slides Knittle Glaser, Germany

Parafilm Laboratory Film Merck Millipore, Australia

Transfer pipettes Sigma-Aldrich, USA

4.1.4 Cytokines


Recombinant human macrophage-colony

stimulating Factor (M-CSF)

EMD Millipore Corporation, USA

Recombinant human soluble RANK ligand

(sRANKL)

EMD Millipore Corporation, USA

4.1.5 Solutions

Solution and buffers were prepared using Mili-Q distilled water (ddH2O). HCl and

NaOH were used to adjust the pH unless stated otherwise.

40

Solution Composition and preparation

Flow cytometry wash buffer PBS containing 1% FBS and 0.1% (w/v) sodium

azide, stores at 40C.

Phosphate buffered saline

(PBS)

10 x PBS stock solution in ddH2O: 70 mM

Na2HPO4, 30 mM NaH2PO4 and 1.3 M NaCl. 1 x

PBS:1 in 10 dilutions of 10 x PBS stock solution

and calibrated to pH 7.4.

SDS-PAGE running buffer 10x stock solution in ddH2O: 25 Trizma base, 1.92

M glycine and 1% (v/v) SDS. 1 x working solution:

1in 10 dilutions of the 10x stock solution in ddH2O.

SDS-PAGE sample loading

buffer

4x stock solution dissolved in ddH2O:240 mM Tris-

HCl (pH 6.8), 8% (w/v) SDS, 40% (v/v) glycerol,

0.04% (w/v) bromophenol blue and 5% (v/v) β-

mercaptoethanol. Stored at 4 0C.

SDS-PAGE separating gel

buffer

1.5 M Tris-HCl (pH 8.8) and 0.4% (w/v) SDS

prepared in ddH2O.

SDS-PAGE stacking gel

buffer

1 M Tris-HCl (pH 6.8) and 0.4% (w/v) SDS

dissolved in ddH2O.

Sodium deoxycholate

solution

10% (w/v) solution dissolved in ddH2O.

Tartrate-resistant acid

phosphatase (TRAP) stain

5 mg Naphthol AS-MX dissolved in 250 µL 2-

ethoxyethanol with the addition of 40 mg Fast Red-

Violet LB salt dissolved in TRAP stain solution A.

Aliquots are stored at -20 0C, thawed at 37 0C and

filtered before use.

41

TRAP stain solution A 100 mM sodium acetate trihydrate, 50 mM sodium

tartrate dihydrate and 0.22% (v/v) glacial acetic

acid. Adjusted to pH of 5.0.

Western blot transfer buffer 25 mM Trizma base, 192 mM glycine and 10%

(v/v) methanol in ddH2O.

4.1.6 Equipment


4300 SE/N Schottky Field Emission Electron

Microscopy

Hitachi, Japan

96-well Thermal Cycler Applied Biosystems, USA

Automatic CO2 Incubator Thermo Scientific, USA

Digital Teslameter with the Probe 3B Scientific, Germany

Eclipse TE2000s Fluorescent Microscope Nikon Instruments Inc., USA

Function Generator 3B Scientific, Germany

ImageQuant LAS 4000 GE Healthcare, USA

MiniSpin® microcentrifuge Eppendorf AG, Germany

MilliQ Water System Millipore Corporation, USA

Nanodrop® ND-1000 UV-Vis

Spectrophotometer V3.3

Thermo Scientific, USA

Nikon CoolPix S4 Digitial Camera Nikon Corp., Japan

Olympus CK30 Microscope Olympus Optical Co Ltd, Japan

Real-Time qPCR System Life Technologies, USA

42

4.1.7 Software


ANSYS Ansys, Inc. USA

FACSDiva BD Biosciences, USA

Microsoft Research Image Composite Editor Microsoft, USA

MATLAB MathWorks, USA

FlowJo FlowJo LLC, USA

Prism GraphPad Software, USA

Origin OriginLab, USA

4.1.8 RT2 profiler PCR array

The PCR array catalogue is PAHS-026z.

Symbol Description

ACVR1 Activin A receptor, type I

AHSG Alpha-2-HS-glycoprotein

ALPL Alkaline phosphatase, liver/bone/kidney

ANXA5 Annexin A5

BGLAP Bone gamma-carboxyglutamate (gla) protein

BGN Biglycan

BMP1 Bone morphogenetic protein 1







43

BMPR1A Bone morphogenetic protein receptor, type IA

BMPR1B Bone morphogenetic protein receptor, type IB

BMPR2 Bone morphogenetic protein receptor, type II

CALCR Calcitonin receptor

CD36 CD36 molecule (thrombospondin receptor)

CDH11 Cadherin 11, type 2, OB-cadherin (osteoblast)

CHRD Chordin

COL10A1 Collagen, type X, alpha 1

COL14A1 Collagen, type XIV, alpha 1

COL15A1 Collagen, type XV, alpha 1

COL1A1 Collagen, type I, alpha 1

COL1A2 Collagen, type I, alpha 2

COL2A1 Collagen, type II, alpha 1

COL3A1 Collagen, type III, alpha 1

COL5A1 Collagen, type V, alpha 1

COMP Cartilage oligomeric matrix protein

CSF1 Colony stimulating factor 1 (macrophage)

CSF2 Colony stimulating factor 2 (granulocyte-macrophage)

CSF3 Colony stimulating factor 3 (granulocyte)

CTSK Cathepsin K

DLX5 Distal-less homeobox 5

EGF Epidermal growth factor

EGFR Epidermal growth factor receptor

FGF1 Fibroblast growth factor 1 (acidic)

44

FGF2 Fibroblast growth factor 2 (basic)

FGFR1 Fibroblast growth factor receptor 1

FGFR2 Fibroblast growth factor receptor 2

FLT1

Fms-related tyrosine kinase 1 (vascular endothelial growth

factor/vascular permeability factor receptor)

FN1 Fibronectin 1

GDF10 Growth differentiation factor 10

GLI1 GLI family zinc finger 1

ICAM1 Intercellular adhesion molecule 1

IGF1 Insulin-like growth factor 1 (somatomedin C)

IGF1R Insulin-like growth factor 1 receptor

IGF2 Insulin-like growth factor 2 (somatomedin A)

IHH Indian hedgehog

ITGA1 Integrin, alpha 1

ITGA2 Integrin, alpha 2 (CD49B, alpha 2 subunits of VLA-2 receptor)

ITGA3

Integrin, alpha 3 (antigen CD49C, alpha 3 subunits of VLA-3

receptor)

ITGAM Integrin, alpha M (complement component 3 receptor 3 subunit)

ITGB1

Integrin, beta 1 (fibronectin receptor, beta polypeptide, antigen

CD29 includes MDF2, MSK12)

MMP10 Matrix metallopeptidase 10 (stromelysin 2)

MMP2

Matrix metallopeptidase 2 (gelatinase A, 72kDa gelatinase, 72kDa

type IV collagenase)

MMP8 Matrix metallopeptidase 8 (neutrophil collagenase)

45

MMP9

Matrix metallopeptidase 9 (gelatinase B, 92kDa gelatinase, 92kDa

type IV collagenase)

NFKB1 Nuclear factor of kappa light polypeptide gene enhancer in B-cells 1

NOG Noggin

PDGFA Platelet-derived growth factor alpha polypeptide

PHEX Phosphate regulating endopeptidase homolog, X-linked

RUNX2 Runt-related transcription factor 2

SERPINH1

Serpin peptidase inhibitor, clade H (heat shock protein 47), member

1, (collagen binding protein 1)

SMAD1 SMAD family member 1





SOX9 SRY (sex determining region Y)-box 9

SP7 Sp7 transcription factor

SPP1 Secreted phosphoprotein 1

TGFB1 Transforming growth factor, beta 1



TGFBR1 Transforming growth factor, beta receptor 1

TGFBR2 Transforming growth factor, beta receptor II (70/80kDa)

TNF Tumour necrosis factor

TNFSF11 Tumour necrosis factor (ligand) superfamily, member 11

46

TWIST1 Twist homolog 1 (Drosophila)

VCAM1 Vascular cell adhesion molecule 1

VDR Vitamin D (1,25- dihydroxy vitamin D3) receptor

VEGFA Vascular endothelial growth factor A

VEGFB Vascular endothelial growth factor B

ACTB Actin, beta

B2M Beta-2-microglobulin

GAPDH Glyceraldehyde-3-phosphate dehydrogenase

HPRT1 Hypoxanthine phosphoribosyltransferase 1

RPLP0 Ribosomal protein, large, P0

HGDC Human Genomic DNA Contamination

RTC Reverse Transcription Control

PPC Positive PCR Control

4.2 PEMF apparatus design

Self-made EMF device was used in controlling the frequency and signal type (

Figure 4.1). The EMF Apparatus consists of two tubes (Figure 4.2), one protective

shield, one platform (Figure 4.3) and two supports (Figure 4.4), generating PEMF

signal and magnetic field strength by Helmholtz coils or solenoid of around 300 turns

copper wires each. This apparatus also includes a digital oscilloscope and teslameter

for measurement.

4.3 Culture and in vitro studies of Sao-2 cell line

Cells were cultured in plates or flasks inside an incubator with 5% CO2, 95% air at 37

0C. Cultured cells were handled using aseptic techniques in UV sterilised biological

safety cabinets (Class II).

47

4.3.1 Saos-2 cell line

Saos-2 (sarcoma osteogenic) was purchased from Australian agent (Sigma Aldrich,

catalogue number: 89050205), which was a non-transformed cell line derived from

human primary osteogenic sarcoma of an 11-year old female Caucasian. The Saos-2

cell line was from the ECACC collection, used as a permanent line of human

osteoblast-like cells and a source of bone-related molecules.

4.3.2 Initiation of the culture process

The medium of 5 mL was pre-equilibrated within a 25 cm2 culture flask for 2 hours in

an incubator. Saos-2 cells were removed from frozen storage and thawed in a 37 0C

water bath with agitation. The cryovial was removed when the ice melted and then

rinsed with 70% ethanol. The cells were resuspended and transferred to a 25 cm2 flask

with 5 mL equilibrated medium, then incubated 30 minutes for the settlement. The

cells were rinsed with 5ml of warmed fresh medium to remove the cryoprotectant. The

cells were cultured in fresh media and monitored daily for use. Saos-2 cells could grow

in a stationary flask using medium supplemented with 10% FBS, 2 mM L-glutamine

and 0.01% kanamycin. The cells were kept in a 2 mm medium layer in a flask with the

caps opening a quarter turn to balance the aeration and nutrition.

4.3.3 Subculture Saos-2

For a 25 cm2 culture flask, cells were subcultured when reaching 70-80% confluent.

The cells were washed with 10 mL warm and sterile PBS after removing the culture

medium. Trypsin solution of 2 mL was added to cells. The reaction lasted for 6 minutes

in the incubator. Growth medium of 5 mL was added to stop the reaction, and detached

cells were transferred to a sterile 15 mL centrifuge tube. The cell suspension was

centrifuged for 6 minutes at 300 x g. The cell pellet was resuspended in complete α-

48

MEM of 2 mL supplemented with 10 ng/mL MCSF and cultured in culture flasks or

plates.

4.3.4 Count cell densities of Saos-2 cells

The cells of 100 µL were transferred to a well in a 96-well plate with 10 µL sterile

0.4% trypan blue. The mixture was ensured homogenous and transferred 10 µL to the

haemocytometer. The dead cell was counted in the colour of blue and divided by the

total number of cells for the proportion of dead or lysed cells. The viability was one

minus the proportion of dead cells.

4.3.5 Cryopreservation and retrieval of cultured cells

The cells were frozen when exceeding 80% in confluence. The cells were gently

dislodged and centrifuged for 6 minutes at 300 x g. The cell pellet was resuspended to

5-6 x106 cells/mL and added 10% DMSO. Plastic cryovials were loaded with a cell

suspension of 1 mL. The vials were placed on ice tub for 15-30 minutes to allow

equilibration of the cells with the DMSO-containing medium. The vials were

transferred to the tray of the cell freezer and adjusted the height of the ray according

to the manufacturer’s instructions. The freezing unit was inserted in the neck of the

liquid nitrogen refrigerator for approximately 3 hours. Transfer frozen vials were

transferred to a storage canister in liquid nitrogen.

In order to survive freezing and thawing, the cells were treated with a cryoprotective

agent. DMSO or glycerol was added to resuspension medium 10% acted to

permeabilize the plasma membrane. Cryoprotectants depressed the freezing point so

that ice crystals began to form at about -5 ºC. If at -5 ºC to -15 ºC, the cell was

sufficiently dehydrated accompanied by osmotic shrinkage and concentration of ice

crystals formed in the surrounding medium but not in the cell interior. In practice, the

warming rate was more challenging to control than the cooling rate. General practice

49

was to immerse frozen vials in a warm water bath (37-40 0C) so that the cells thawed

in about 1.5 minutes.

4.4 Isolation, culture and in vitro studies of human osteoblasts

4.4.1 Isolation and culture of osteoblasts from human trabecular bone

Pieces of excised human bone specimens were placed in sterile saline or PBS at time

of collection, and the samples were transferred in a foam box without ice. Completed

α-MEM (α-MEM supplemented with 10% FBS, 2 mM L-glutamine, 100 U/mL

Penicillin and 100 µg/mL Streptomycin) and filtered PBS were warmed in 37 0C water

bath. The centrifuge was set at 300 x g, 4 0C and 6 minutes. Aseptic technique was

used to cut the bone specimen into small pieces about 1-2 mm using a scalpel blade or

scissors. The bone chips were washed five times with sterile PBS and transferred into

a sterile Falcon tube. Contaminating red blood cells were removed using a vortex

mixer. 8-10 pieces of bone were placed into a T75 flask along with 20 mL completed

α-MEM. The flask was placed in an incubator with 5% CO2 at 37 0C. After 1 week,

the medium was removed, and fresh culture medium was added. The bone pieces were

cultured for another week without disturbing. After that, the culture medium was

changed every 2 days. Between 3-4 weeks the cells could be confluent about 1-2 x 106

cells/flask.

4.4.2 Harvest of monolayer cultures

Collagenase (1 mg/mL) of 1 mL was added to the flask for 2 hours at 37 0C. Cell

culture medium was removed using a vacuum, and the cells were immediately washed

with 10 mL warm and sterile PBS. 3-5 mL of TryExpress solution was added to cover

the cell surface for 6 minutes reaction in the incubator. α-MEM of 5 mL was added to

stop the reaction, and the cells were transferred to a 50 mL centrifuge tube. Cell

50

solution was centrifuged at 300 x g, 4 °C for 6 minutes, and the supernatant was

disposed. α-MEM of 3 mL was added to the flask for the cell suspension. A pipette

was used to break cell clusters by pressing the tip of the pipette against the bottom of

the flask. The cell suspension was passed through a needle of 10 mL syringe 5-6 times

with the needle bevel pressing against the tube wall to further separate the cells.

4.4.3 In Vitro osteoblasts cell mineral formation assay

Alizarin red-S (AR-S) is a dye-binding selectively to calcium salts to quantify calcium

mineral content by measuring the amount of AR-S bound to mineralised nodules in

the cultures. Cells were briefly washed with PBS 1 mL/well. Ice-cold ethanol (70%)

was added to the cultures and reacted at 4 ºC for a minimum of 1 hour. The cells were

washed with ddH2O and added 1 mL/35-mm well 40 mM AR-S, pH 4.2. The plate

was placed on the shaker at room temperature for 10 minutes. The cells were washed

with ddH2O for 5 times and with PBS once. The plate was then shaken at room

temperature for 15 minutes. AR-S solution of 40 mM was diluted in 1: 400 with CPC

buffer to make 100 µM AR-S solution, which was diluted in a series dilution of every

3-fold with CPC buffer to make AR-S standards. Eppendorf tubes were labelled, and

CPC buffer was added to each tube. AR-S solution (100 µM) of 200 µL was added to

the tube. CPC buffer was added to each well to extract AR-S for 15 minutes at room

temperature. CPC buffer of 90 µL was added to each well in a 96-well plate. 10 µL

the AR-S was added to each well (1:10 dilution) and mixed. The dilution factor can be

changed depending on estimating of mineral content such as colour. AR-S standard

solution was added to the wells in replicates. AR-S concentration was determined by

absorbance measurement at 562 nm. The AR-S concentration was calculated using

AR-S standard curve.

51

4.5 Induction and culture of human osteoclasts

4.5.1 Preparation of solution

Osteoclast cells can be differentiated from human peripheral blood mononuclear cells

(PBMCs) using differentiating culture medium. HANK’s solution was prewarmed in

a water bath at 37 0C and filtered using a filter cup (0.2 µM) before use. Completed

osteoclast culture medium was made by α-MEM supplemented with 1% L-Glutamine,

10% FCS, 1% PSN, 10-8 M Dexamethasone, 10-8 M Vitamin D3 and 25 ng/mL M-

CSF. DMSO was used to dissolve Vitamin D3. Human recombinant RNAKL was

prepared at 50 ng/mL.

4.5.2 PBMC isolation

The Buffy coat 15 mL in a falcon tube and 20 mL pre-warmed HANKs was added to

dilute the buffy coat. Ficoll-Paque of 7.5 mL was added to an empty 50 mL Falcon

tube, and diluted buffy coat of 10 mL was loaded over Ficoll solution using a syringe.

The ratio of Ficoll and diluted buffy coat was 3:4. After loading, all of the tubes were

placed on a centrifuge, spun at 500 x g, 25 C for 30 minutes. The mononuclear cells

(PBMCs) were collected at the interface using a sterile pipette directly inserted into

the PBMCs layer. This layer was like a cake in between orange colour and white

colour. The PBMCs were transferred to a fresh 50 mL falcon tube. Cells were washed

with 20 mL HANKs and centrifuged at 500 x g, 25 C for 5 minutes. The supernatant

was discarded and completed osteoclast cell culture medium was added to make 5-10

mL cell suspension, depending on how many cells isolated.

4.5.3 PBMC cell culture

The cell viability was checked using 20 µL, 0.2% trypan blue working solution plus

20 µL cell suspension. The cells suspension was diluted if the concentration was high.

52

The cells were counted under bright field microscopy. For example, seeding to 24-well

plate and the cell suspension was 0.5 mL/well, the volume of cell seeding suspension

was 12 mL. The cell seeding suspension was made by adding the above volume into

14 mL completed osteoclast cell culture medium. Cell seeding suspension of 0.5 mL

was added to each well of a 24-well plate. The cells were incubated at 37 0C, 5% CO2

and 95% moisture incubator. In day 4, the culture medium was changed. In day 7, the

culture medium was changed with adding human recombinant RANKL into the cell

culture to induce osteoclasts. The final concentration of RANKL in the culture media

was 50 ng/mL. In day 10, the osteoclasts could be observed under bright field

microscopy.

4.5.4 TRAP staining

When multinucleated osteoclasts were formed, the cells were gently washed once with

1 x PBS and fixed with 4% paraformaldehyde for 15 minutes at room temperature,

followed by three washes with 1 x PBS. The cells were subsequently stained with

filtered Tartrate-resistant acid phosphatase (TRAP) solutions. The TRAP solutions

were warmed up to 37 0C and filtered before staining. The slides were dewaxed and

hydrated. Excess liquid was removed from the slide without letting the section dry out.

Filtered pre-warmed TRAP solution was applied, and the slides were incubated at 37

0C. The slides were checked every 30 minutes to monitor the development of the stain.

After approximately 2 hours, with the observation of magenta-stained osteoclasts, the

slides were rinsed in 1 x PBS three times and stored at 4 0C until ready for imaging.

The osteoclasts were visualised using the Olympus microscope. Serial images were

taken at desired magnification covering the entire well.

53

4.5.5 Flow cytometer and analysis

The FACSCalibaur, FACSCanto II and LSR Fortessa TM SORP were used for flow

cytometric studies. FACSDiva software was used for application setup, data

acquisition and data compensation. The live and single mononuclear cells were gated

using Forward Scatter (FSC) and Side Scatter (SSC) plots. Events were recorded.

Acquired and compensated data was imported into FlowJo for analysis.

4.6 Western blotting

Western blotting allows the specific detection of proteins on membranes after

separation by SDS-PAGE. It combines the resolution of gel electrophoresis with the

specificity of immunochemical detection. This technique is beneficial for the

identification and semi-quantification of specific proteins in complex mixtures of

proteins.

4.6.1 Protein extract and assay

At the different time point of the treatment, cells were extracted by using RIPA buffer.

The cells were washed with PBS to remove residual media. RIPA buffer (1 x) of 400

μL was added for 10 cm2 (6-well-plate: 9cm2 needs 360 μL; 25cm2 needs 1 mL). The

cells were incubated on ice for 5 minutes. The cells were sonicated briefly and

centrifuged for 10 minutes at 14000 x g, 4 0C. The supernatant was transferred to a

new tube for use. Protein concentration was measured by using the Bio-Rad Protein

Assay. Dye reagent was prepared by diluting 1 Dye Reagent Concentrate with 4 parts

DDH2O. This diluted reagent may be used up to 2 weeks when kept at room

temperature. Five dilutions of a protein standard were prepared. The linear range of

this microtiter plate assay is 0.05 mg/mL to approximately 0.5 mg/mL, for instance,

0.05 mg/mL, 0.1 mg/mL, 0.2 mg/mL, 0.4 mg/mL and 0.5 mg/mL. Protein solutions

54

were assayed in duplicate or triplicate. The standard and sample solution of 10 μL each

were put into separate microliter plate wells. Diluted dye reagent of 200 μL was added

to each well. The sample and reagent were mixed thoroughly using a microplate mixer.

The mixture was incubated at room temperature for at least 5 minutes. Since

absorbance could increase over time, samples should incubate at room temperature for

no more than 1 hour. Measure absorbance was at 595 nm.

4.6.2 Blotting and detection

Proteins were loaded to each well of SDS-PAGE gels. The sample was heated for

denaturing electrophoresis for optimal results at 95 0C for 5 minutes. NuSep Tris-

Glycine SDS Running Buffer was used for the electrophoresis. Appropriate protein

molecular weight marker (10 µL Novex Sharp Pre-Stained Protein Standard) was

loaded. Electrophoresis was operated at room temperature based on gel type. PVDF

protein side was marked with a pencil and NuSep Transfer Buffer (1 x) of 1000 mL

was prepared. A PVDF membrane was soaked in methanol for 1 minute and placed in

distilled water. A PVDF membrane, four blotting pads, and two filter papers were

placed in a shallow tray filled with NuSep Transfer Buffer (1X) for 10 minutes. Gel

apparatus was disassembled. Stacking gel was cut off with a clean razor blade and

soaked in NuSep Transfer Buffer (1 x) for 10 minutes. Transfer apparatus gel cassettes

were open with the black panel lying flat on the bottom of the tray filled with NuSep

Transfer Buffer (1 x). The transfer sandwich was prepared on the black panel in the

tray filled with NuSep Transfer Buffer (1 x). The sandwich was covered with the

transparent panel, fasten with the latch, and inserted into the electrode module with the

black panel facing the cathode electrode panel. The gel from sandwich was removed

and rinsed with NuSep Transfer Buffer (1 x). PVDF membranes were blocked with

Blocking Buffer on a shaker for 2 hours at 37 0C or overnight at 4 0C. The membrane

55

was washed three times with Basic Buffer on the shaker for 10 minutes each time.

Primary antibody was diluted with Primary Antibody Dilution Buffer. The membrane

was incubated with the diluted primary antibody on a shaker for 1 hour at room

temperature. The membrane was then washed three times with Washing Buffer on the

shaker for 10 minutes each time. The secondary antibody was diluted with Primary

Antibody Buffer containing 5% dry milk. The membrane was incubated with the

diluted secondary antibody on a shaker for overnight at 4 0C. The membrane was then

washed three times with Washing Buffer on the shaker for 10 minutes each time.

The detection reagents were taken from the storage at 4 0C and equilibrated to room

temperature before opening. Detection solutions were mixed to the final volume of 0.1

mL/cm2. Protein side of the washed membrane was placed upon a transparency film.

The mixed detection reagent was added on the membrane for a reaction of 3 minutes

at room temperature. The reagents covered the entire surface of the membrane, held

by surface tension on to the surface of the membrane. Excess detection agent was

drained off by holding the membrane gently in forceps and touching the edge against

a tissue. The membrane ‘protein-side’ was placed facing the camera. Blots were

exposed to GE Healthcare ImageQuant LAS 4000 at least three different exposures.

4.7 RNA extraction and quantification

Total RNA was extracted from the cultured cells using TRIzol reagent. All media were

aspirated off, and the flask was washed twice with PBS. TRIzol was added directly to

cells (2 mL/25cm2, 4 mL/75cm2), and the homogenised cells were incubated in TRIzol

for 10 minutes at room temperature. The cell TRIzol extract was aspirated 10 times

through a 23 G needle using 5 mL syringe. Chloroform of 0.2 mL was added into 1

mL cell TRIzol extract. The mixture was vigorously shaken by hand for 15 seconds

and kept at room temperature for 3 minutes. The mixture was centrifuged at 12000 x

56

g, 4 °C for 15 minutes. Ethanol with DEPC H2O (70%) was prepared. The upper

aqueous phase after the centrifuge was transferred, containing the RNA, into a

microcentrifuge tube. An equal volume of 70% ethanol was added to mix thoroughly.

Visible precipitates could form after the addition of ethanol, resuspend precipitates by

vigorously shaking. Aliquots of the sample of 700 µL were transferred to an RNAeasy

mini spin column placed in a 2 mL collection tube. The sample was centrifuged at

≥8000 x g at room temperature for 15 seconds. The flow was discarded through after

each centrifugation. Buffer RW1 (commercial kit) of 350 µL was added to the column

with centrifuging at >8,000 x g for 15 seconds. DNaseI stock solution of 10 µL was

added to Buffer RDD of 70 µL. The DNase I solution of 80 µL was added directly

onto the RNeasy silica gel membrane and incubated at room temperature for 15

minutes. Buffer RW1 of 350 µL was pipetted into the column and centrifuged at

>8,000 x g for 15 seconds. The column was placed into a new collection tube, and

Buffer RPE of 500 µL was added to the column followed by centrifuging at >8,000 x

g for 15 seconds. Another 500 µL Buffer RPE was added to the column and centrifuged

at >8,000 x g for 2 minutes to dry the membrane. A new collection tube was placed in

a microfuge tube, and the old collection tube was discarded. To elute, the column was

placed into a new 1.5 mL RNase-free collection tube, and 50 µL RNase-free water was

added to the spin column membrane. The tube stood for 1 minute at room temperature

and was centrifuged at >8,000 x g for 1 minute. Another 50 µL RNase-free water was

added to the spin column membrane, and the tube was centrifuged as above if the

expected RNA yield is >30 µg. The RNA quality was measured by Nanodrop.

57

Figure 4.1 Self-made EMF device in controlling the frequency and signal type. The

EMF Apparatus consists of two tubes, one protective shield, one platform and two

supports, generating PEMF signal and magnetic field strength by Helmholtz coils or

solenoid of around 300 turns copper wires. This apparatus also includes a digital

oscilloscope and tesla-meter for measurement.

58

(A)

(B)

Figure 4.2 Coil tube. There are two tubes for this apparatus. Coils spool the cylinder

surface in a single layer. Longitude dimensions of the tube: 1 cm for each flange depth;

18cm for main tube depth. Thus, the total length of one coil tube is 20 cm. Transverse

dimensions of coil tube: D1 (internal) = 22.5 cm, D2 (external) = 25 cm; Flange

diameter: D3 = 28 cm.

59

(A)

(B)

Figure 4.3 Picture of the main body. The main body contains two parts: a protective

shield and a platform. Two tubes will be connected outside the shield. Protective shield

length is 40 cm as well as Platform length. Dimensions of transverse protective shield

are: D22 = 20 cm, D23 = 22.5 cm; H25 = 3.5 cm; V21 = 2.5 cm, V20 = V24 = 3 cm.

Dimensions of transverse platform are: H3 = 18 cm; V1 = 2 cm.

60

(A)

(B)

Figure 4.4 Support for each tube. Dimensions following X axis are: tube fix = 18 cm;

flange fix = 1 cm; edge fix = 1 cm. Thus, the total depth is 22 cm. Transverse

dimensions for support are: H5 = 29 cm; V2 = 20 cm; D3 = 26.5 cm. Diameter for the

groove is D3 = 28 cm.

61

Chapter 5

The Response of Cell Probability to

External Stimulus

62

5.1 Introduction

The hypothesis in Chapter 3 describes the qualitative response of bone cells to EMFs.

The EMFs as an external stimulus alter the original environment of cell culture. In this

chapter, we attempt to establish several frameworks for quantitatively interpreting the

evolution of cell probability under a general situation of external stimulus. The

frameworks involve two major components, (i) an external stimulus and (ii) the

cellular response to the stimulus. The response is in two steps, (i) the detection of the

stimulus and (ii) the transduction of the stimulus into the controls of cellular interaction.

Ideally, each cell is distinguished from their direction of motion, reorientation

frequency and step length between reorientations in response to the stimulus. However,

the coupling between cell density and the local stimulus intensity might prevent the

process of distinguishing. For example, when cells aggregate or diffuse in response to

a stimulus at one time-step, the distributions of local cell density and stimuli are likely

to be altered at the following time-step.

Therefore, the spatiotemporal distribution of cell density is studied in preparation for

derivations of cell probability. The assumptions are made to correlate the local cellular

mobility with cell density. The cellular mobility is partially dependent on (i) available

cellular space, (ii) cellular adhesion, involving the attachments between neighbouring

cells via extracellular receptors, mediated cellular recognition and positions, and (iii)

the external stimulus which interact with cells by mediating a series of processes such

as cell migration and cellular signalling.

5.2 Preliminaries

5.2.1 Distribution of cell density

A cellular system is necessarily a balance sheet, based on components in the system

and the processes by which the components are determined. The components include

63

contents of proteins, nucleic acid, carbohydrate and lipid. The processes include such

as individual metabolites, related signalling pathways and cell differentiation. These

descriptions are considered equivalent to the exchanges of matter and energy between

the cells and their environment. If such exchanges are quantitatively represented by

chemical equations for metabolism and biosynthesis, they construct a conceptual

input-output matrix [171].

Following the concept of the input-output matrix in biology, we consider a cellular

system S with n components 1 2, nC C C , and each component has an

independent value id , then the value vector is

1 2{ , , }nd d d=d . (5.2.1)

When every component is presented in the same system S linked with a relationship

1 1 2 2i i i in n iC a C a C a C d= + + + + , (5.2.2)

where the coefficient ija , 1,2...j n= forms a n n matrix,

11 1

1

n

n nn

a a

a a

=

A . (5.2.3)

If the components have values under a specific condition

1 2{ , , }nc c c=c , (5.2.4)

these values satisfy the relation

= +c A c d . (5.2.5)

The equation (5.2.5) has a solution

1( )−= −c I A d , (5.2.6)

64

with the identity matrix I , in which 0, ; 1, ij ijI i j I i j= = = . Consider a set of cell

densities extracted from the same system S ,

1 2{ , , }mp p p=p , (5.2.7)

with m data series, m N , for example, time series 1 2, , mt t t . Suppose each cell

density is formed by n biochemical components 1 2, nC C C , then the same

coefficient matrix A exists in a relation which is analogous to (5.2.6),

1( )−=C I - A D , (5.2.8)

with

11 1 11 1

1 1

,

m m

n nm n nm

c c d d

c c d d

= =

C D . (5.2.9)

Hence C is a matrix for cell densities p , which could represent the history of cell

proliferation. Equations from (5.2.7) to (5.2.9) show that a solution of m cell

proliferation needs 1( )−I - A of dimension n n and D of dimension n m . The

equations (5.2.8) and (5.2.9) also indicate a linear relationship between matrix C and

matrix D , which is contrary to the nonlinearity of cell proliferation. To incorporate a

nonlinearity in the equations (5.2.8) and (5.2.9), at least one variable nmd in the

matrix D is a polynomial of degree which is not one. The similar methods of

modelling nonlinear biological systems have been described in the literature [172,

173].

However, this method has inherent defects. Taking a series of biosynthetic reactions

as an example, it is hard to write the degrees of components unless every component

is well-defined in related to enzymes. It is relatively more difficult to tell the difference

65

between normal and abnormal conditions of macromolecules. Even if a matrix D is

well-established after performing m independent experiments in the same system, the

degrees of components might be different when incorporating the 1m+ th data.

An invariant relationship 1( )−I - A is proposed to avoid the method of defects for the

same system S , which requires a unified approach to constructing the matrix D . This

approach should be able to build a sufficiently large and relevant matrix involving all

components during cell proliferation. If such a matrix was directly composed of

components in biosynthetic reactions, it could be lack of relevance since these

components have their chemical structures and biological properties. The cell

proliferation is highly dependent on cell migration that involves molecular

components such as adhesion receptors, cytoskeletal linking proteins, and extracellular

matrix ligands. At a specific time-step, cell migration is relevant to the distribution of

cell density at the position ix . Therefore, we expand (5.2.7) with n positions

1 1 1 2 1

2 1 2 2 2

1 2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

n

n

m m m n

p x p x p x

p x p x p x

p x p x p x

=

p , (5.2.10)

which is a matrix in the dimension of m n . The equation (5.2.8) multiples with

(5.2.10) forming a new n n matrix in (5.2.11) and this new matrix *

p represents the

distribution of the cell population after the interactions of components in the cellular

system

66

* *

11 11 1 1 1 1 1

*

* *

1 1 1

11 1 1 1 1

1

1 1

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

( )

( ) ( )

mn n

n n n n nm m m n

m n

n nm m m n

c cp x p x p x p x

p x p x c c p x p x

d d p x p x

I - A

d d p x p x

−

= =

=

p

(5.2.11)

with a relationship similar to (5.2.2),

*

1 1 2 2( ) ( ) ( ) ( )n n n n n n nm m np x c p x c p x c p x= + + + . (5.2.12)

Since cell population cannot be infinite, the limit of * ( )n np x must exist. Define the

requirement condition

lim 1nmm

m

c→

= , (5.2.13)

which indicates that the accuracy of * ( )n np x is determined by the amount of

experimental data. Based on this assumption, we can determine whether the accuracy

of the prediction results can be further improved and the affected components in the

following sections.

When m n= , (5.2.12) has a particular solution like formula (5.2.2),

0

1 1 2 2( ) ( ) ( ) ( )n n n n n n nn n n nnp x c p x c p x c p x p= + + + + . (5.2.14)

Hence，

1( )−= 0p I -C p , (5.2.15)

with

0 0

11 1

0 0

1

n

n nn

p p

p p

=

0p . (5.2.16)

67

This particular solution (5.2.15) shows that the cell density distribution p is linear

with the initial cell density distribution 0

p .

We further consider the conditions for the existence of this solution. Consider a time

series 1 2, , nt t t and corresponding cell density 1 2, , np p p . If it satisfies the

relation (5.2.15), the cell growth during time series is linear, which is different from

the log growth of cell proliferation. A linear growth can be approximately realised in

a short time interval between t and t t+ . When the relation 1( )−I -C is invariant in

this time interval, the distribution of cell density should be the same regardless of how

many times experiments are performed. When this time interval becomes longer, the

matrix D is affected by the experimental observation. Even if the relation 1( )−I - A

is invariant, the distribution of cell density p is altered with the number and duration

of experiments. Consequently, if there is an initial distribution of cell density 0

p ,

there must be a corresponding interval ct , satisfying

1

1

( ) ,

( ) ,

c

c

t t

t t

−

−

=

=

0

0

p I - C p

p I - A Dp (5.2.17)

where t indicates the time duration of the experiment. Equation (5.2.17) introduces

the matrix of cellular function to the distribution of cell density. When ct t , the

initial distribution of cell density p directly determines the final distribution of cell

density 0

p . When ct t , the final distribution of cell density p is determined by

the initial distribution of cell density 0

p and the matrix of cellular function D

together.

68

5.2.2 Distribution of cell density under external stimulus

According to the equation (5.2.17), both the matrix D and the initial distribution of

cell density 0

p can be affected by an arbitrary external stimulus. In this section, we

firstly assume a coupling effect of the external stimulus on the matrix D and initial

distribution 0

p . The coupling effect results in an evolution of cell distribution

changed with time under the external stimulus, which is different from the traditional

distribution of particles under a force. The difference might be observed when

inputting the data of in vitro experiment into a particle distribution model under an

external stimulus. The model from Keller and Segel [174] for the chemotactic

movement of particles is employed to develop a continuum-based model of cellular

movement,

2

1

2

2

{ ( , ) } ( , )

( , )

p

F

D ft

D ft

= − +

= +

pp p p F p p F

FF p F

(5.2.18)

where p represents the distribution of cell density ( , )p x t and F represents the

distribution of an arbitrary external stimulus ( , )F x t . PD and FD are referred to as

diffusion coefficients of cell and stimulus distribution respectively. The stimulus

affects the cell density in a negative-cross term which is modified by the sensitivity

( , ) p F . Cellular and external kinetics are given by 1f and 2f respectively.

This model has been shown to demonstrate pattern forming properties with cells

accumulating into high-density aggregations. Therefore, this model results in blow-up

after finite time steps without specific limitation.

Blow-up implies the formation of cell aggregates into infinite density, which is

unrealistic from a biological standpoint. The additional conditions can be set for the

69

sensitivity ( , ) p F with (i) (0, ) 0 F and (ii) existence of a 0ip for

( , ) 0i ip F = . The simplest solution is

0( , ) (1 )p F p = − , (5.2.19)

which was rigorously proved that the above condition could lead to repulsion at high

densities and a limit upon the aggregation size [175]. Equation (5.2.19) indicates the

density-dependence of cell motility, which may arise via a series of processes, such as

(i) the dependence between the cell migration and the space availability within cellular

environment, (ii) cellular adhesion involving the contact and attachment between

neighbouring cells via extracellular receptors, and (iii) the secretion of external

diffusible substances which allows cells to intercommunicate.

5.3 Cell probability

In the absence of any condition concerning the external stimulus, it is assumed that all

positions of the cells in the asymmetric cluster are equally probable. On this basis, it

is possible to determine the prior probability distribution of a structure factor. If an

experiment is performed and a certain magnetic intensity is observed, we may obtain

a posterior probability distribution of the structure factor. Our fundamental postulate

is that the posterior distribution is primarily determined by the values of observed

intensities when the number of observed intensities is sufficiently large, and this

distribution is relatively insensitive to the assumed prior distribution form. For this

purpose, we need to assume even less; the structure factor depends only on the values

of a sufficiently large number of observed magnetic intensities. This assumption is

plausible since the observed experimental results are not highly arbitrary. It is,

therefore, reasonable to suppose that, provided the number of available intensities is

greater than the minimum number required to determine the structure in the strict

70

algebraic sense, any structure factor of posterior distribution will agree with its real

value. The higher the number of observed intensities, the higher will be the statistical

significance, and the more reliable the final answer.

The probability distribution P(A) of a structure factor is an even function of A given

that the cells in the asymmetric cluster are assumed to occupy all positions with equal

probability. Thus, the structure factor can be positive as well as negative. However,

once a set of magnetic intensities is known, the cells in a cluster no longer occupy all

positions with equal probability. If the cells in the asymmetric unit are assumed to

move randomly throughout the unit cluster subject to the constraints imposed by the

magnetic intensities, the resulting probability distribution of a structure factor is no

longer an even function. The probability that the structure factors have a particular

sign is different from one-half. The purpose is to derive these probabilities on the basis

that specific magnetic intensities are specified and to derive a procedure for phase

determination. The probability distribution of a structure factor is derived for a cluster

of cells (Figure 5.1). The structure factor for the cluster is given by

1

( , , , )N

j j jh jhj

F f x y z h=

= , (5.3.1)

where jh

f is the scattering factor, N is the number of cells in the unit and

( , , , )j j jx y z h is a trigonometric function of h ( , , )h k l= . The probability that

( , , , )j j j jx y z h = lies between c and c+dc is denoted by ( )p c dc , where

( )p c is an even function within c. By this means, the coordinates of the jth cell cannot

exceed the region within c. Therefore, a cluster existing at c is given by

1 1( ) ( ) ( )N NR

Q c p p d d = , (5.3.2)

71

where the region R in the -space of N dimensions consists of points 1( , , )N

and the corresponding points ( , , )j j jx y z , 1,2, ,j N= yield to the equation

(5.3.1). Due to the region R within c, we introduce a discontinuous function

1( , , )NT to simplify the region with

1 1 1( ) ( ) ( ) ( , , )N N NQ c p p T d d

− −= , (5.3.3)

where

1

1

( , , ) 0

( , , ) 1

N k

N k

T if F c

T if F c

=

= . (5.3.4)

Moreover, kF inherits from (5.3.1),

-1

1

, , 1,2, , .k

k j j k k k k

j

F f F F f k N =

= = + = (5.3.5)

We establish the function 1( , , )NT to satisfy (5.3.4),

10

sin(( ) )1 1( , , )

2

kN

F c xT dx

x

−= − . (5.3.6)

Then

1 10

1 10

sin(( ) )1 1( ) ( ) ( ) ( )

2

1 1( ) ( ) sin(( ) )

2

kj N N

N k N

F c xQ p p dx d d

x

dxp p F c x d d

x

− −

− −

−= −

= − −

(5.3.7)

and

72

-1

-1 -1

( )sin(( ) )

( )sin(( ) )

( )sin(( ) )cos( ) ( )cos(( ) )sin( ) .

k k k

k k k k k

k k k k k k k k k k

p F c x d

p F f c x d

p F c x f x d p F c x f x d

−

−

− −

−

= + −

= − + −

(5.3.8)

When 1k = , we define

0 0F = , (5.3.9)

moreover, rewrite(5.3.8) at 1k = ,

1 1 1

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

( )sin(( ) )

( )sin( )cos( ) ( )cos( )sin( )

sin( ) ( )cos( ) cos( ) ( )sin( ) .

p F c x d

p cx f x d p cx f x d

cx p f x d cx p f x d

−

− −

− −

−

= − +

= − +

(5.3.10)

Since ( )p a is an even function, we wish that its integral is also an even function.

Then we approximate (5.3.10) at c by

1 1 1 1( )sin(( ) ) sin( ) ( )cos( )p F c x d cx p c f xc dc

− −− = − . (5.3.11)

Then

-1( )sin(( ) ) sin(( ) ) ( )k k k k kp F c x d F c x q f x

−− = − , (5.3.12)

where

( ) ( )cos( )k kq f x p c f xc dc

−= . (5.3.13)

The (5.3.12) and (5.3.13) are an approximation to (5.3.8), thereby simplifying

(5.3.7) by

73

0

1

1 1 sin( )( )

2

N

k

k

cxQ q f x dx

x

=

= + . (5.3.14)

According to (5.3.14), we find the probability that a cluster lies between arbitrary

area A and A+dA,

0

1

( ) 1( ) cos( ) ( )

N

k

k

Q AP A Ax q f x dx

dA

=

= = . (5.3.15)

To solve 1

( )N

k

k

q f x dx=

, we use

11

log ( ) log( ( ))N N

k k

kk

q f x dx q f x dx==

= , (5.3.16)

and

11

( ) exp( log( ( )))N N

k k

kk

q f x q f x==

= , (5.3.17)

and find the Maclaurin series of ( )kq f x from (5.3.13)

2

2

1

( 1) ( )( ) 1

(2 )!

n n

kk n

n

f xq f x m

n

=

−= + , (5.3.18)

where

2

2 ( )n

nm c p c dc

−= . (5.3.19)

n is a positive integer. Therefore, we rewrite (5.3.16) in Maclaurin series

74

11

2

2

1 1

2 2 22

2 2 2

1 1 1 1

2

2

1

( ) exp( log( ( )))

( 1) ( )exp( log(1 ))

(2 )!

( 1) ( ) ( 1) ( ) ( 1) ( )1 1exp( (1 ( ) ))

(2 )! 2 (2 )! 3 (2 )!

( 1)exp(

(2 )!

N N

k k

kk

n nNk

n

k n

n n n n n nNk k k

n n n

k n n n

n n

n

n

q f x q f x

f xm

n

f x f x f xm m m

n n n

x

n

==

= =

= = = =

=

=

−= +

− − −= − + −

−=

2 222 2

2 2 2

1 1

( 1) ( 1)1 1(1 ( ) )) ,

2 (2 )! 3 (2 )!

n n n n

n nn n n

n n

x xm m m

n n

= =

− −− + −

(5.3.20)

where

2

2

1

Nn

n k

k

f=

= . (5.3.21)

When 1n = the first term of (5.3.20) is

2 2 2 422 2 2

2 2 2

2 2 4 3 62 32 2 2

2 2 2

(12 4 12

.2 8 24

x x xm m m

x x xm m m

− + + +

= − − − −

(5.3.22)

When 2n = the second term is

4 2 8 3 122 34 4 4

4 4 4 .4! 2 4! 3 4!

x x xm m m

− + −

(5.3.23)

Substituting (5.3.22) and (5.3.23) into (5.3.20), and expanding Taylor series except

for the term 2x

22 2 42 2

2 2 4 4

1

22 2 42 2

2 2 4 4

1( ) exp( )exp( (3 ) )

2 4!

1exp( ) (1 (3 ) ) .

2 4!

N

k

k

m xq f x m m x

m xm m x

=

= − − −

= − − − +

(5.3.24)

75

Substituting (5.3.24) into (5.3.15), we find the cluster probability P(A),

2

2 2 42 22 2 4 4

0

2 2 2 4 42 2 42 2

2 2 4 40

1 1( ) cos( )exp( )(1 (3 ) )

2 4!

1 1exp( )(1 )(1 (3 ) ) .

2 2! 4! 4!

m xP A Ax m m x dx

m x A x A xm m x dx

= − − − +

= − − + − − − +

(5.3.25)

Using the form of Gaussian integral,

2 2

20exp( )

2

x adx

a

− = , (5.3.26)

2 2 1 2 22

2 10

(2 -1)!exp( ) ( ) (2 -1)!

2 2 2

nn n

n

x a n a ax dx n

a

+

+− = = , (5.3.27)

with 2

2 2

2a

m= to solve (5.3.25)

4 2 22

2 2 4 4

2 2

2 2 2 2 2 2

31( ) (1 )

2 2 4

A m mAP A

m m m

− += − + − . (5.3.28)

The form of (5.3.28) implies

2 22 2 4

2 2 4 4

2 2 2 2

2 2 2 2 2 2 2 2 2 2

31( ) exp( )(1 (1 ) ).

2 2 8 2 8

m mA A AP A

m m m m m

−= − − + − + −

(5.3.29)

Under the condition,

2 2( )n

n = (5.3.30)

the approximation of (5.3.29) is given by

76

22 2

2 4

2

2 2 2 2 2 2 2

22

2 4

2

2 2 2 2 2 2 2

31( ) exp( )(1 exp( ))

2 2 8 2

31 1exp( ) .

2 2 8 2

m mA AP A

m m m m

m mA

m m m m

−= − −

−= − −

(5.3.31)

From (5.3.31), we notice that P(A) has a very similar form to the probability density

of the normal distribution with a variance 2 2m which is independent of the

coordinate system. The boundary conditions of 2 2m as well as (5.3.31) depend on

the particular position of clusters (5.3.19) and scattering coefficient of cells (5.3.21),

2 2

2 2

1

( ) , N

k

k

m c p c dc f

−=

= = . (5.3.32)

The prior distribution may be found regardless that the cluster contains cells in

particular positions or general positions.

The phase problem is to determine the phases of structure factors h

F in (5.3.1) and it

may get a practical solution from a joint probability distribution of the structure factors.

Although the magnitude of h

F is independent of the choice of origin, the cluster

structure alone is not sufficient to determine all phases because the definition (5.3.1)

implies an initial origin. Both the functional form j and the values of the cell

coordinates ( , , )j j jx y z depend on which centre is chosen as the origin. However, for a

fixed functional form of structure factor, it is possible to find certain linear

combinations of the phases whose values depend on the structure alone and are

independent of the choice of origin within a range permitted by the chosen form of

structure factor. This property leads to structure semi-invariants which depend only on

the choice of the unit cluster, and their values depend on the chosen form of structure

factor. If a sufficient number of magnetic intensities can determine the cluster

77

structure, these intensities determine the values of structure semi-invariants rather than

the phases of cells. Therefore the phase problem can be started with the problem of

determining the values of structure semin-variants for each fixed form of structure

factor, once a sufficient number of intensities has been tested. The phases may then be

obtained from the values of the structure semi-invariants by specifying origin. As

introduced in the previous section, the probability theory provides a solution for the

identification of the structure sem-invariants. On the other hand, it should be

emphasised that the understanding of the form of structure semi-invariants is an

invaluable aid in interpreting the results from the probability theory. In the following

section, we devote to discuss this subject for a cluster with cells in general positions.

In general the function in (5.3.1) depends on the choice of origin which ideally is a

centre of symmetry. Two origins are equivalent if the functional forms of are

identical. In other words, two origins are equivalent if they are geometrically related

in the same way to all the symmetry elements. It is noticed that the value of referred

to two similar origins may be equal or contrary to each other. The concept of equivalent

origins leads to equivalent structure factors. If two structure factors are equivalent,

their structure semi-invariants have the same value. As previously mentioned, the

values of the structure invariants are only determined by the structure while the values

of the phased also depend on the choice of origin. It will be seen that the origin may

be chosen by selecting the form of the structure factor and then specifying the signs of

structure factors.

5.4 Joint cell probability

For the probability distribution of a structure factor when certain magnitudes or phases

are specified is readily derivable from the joint distribution. We define the joint

78

probability 1 1( , , )j jm j jmp d d for the interval between jn and jn jnd + , where

n 1,2, ,m= and m is a positive integer. Hence, a similar definition to (5.3.1) is

written

( , , , )njn j j jx y z h = . (5.4.1)

Let 1 1 1( , , )m mP A A dA dA be the joint probability for the interval nA and

n nA dA+ ,

1 m

1 1 m

1 m

Q(A , ,A )P (A , ,A )=

A A

m

, (5.4.2)

the probability 1( , , )mQ A A is estimated following the procedure from (5.3.3) to

(5.3.6) ,

1 1 1 1

1 1

( , , ) ( ( , , ) ) ( , , )N m

m j jm j jm n mn

j n

Q A A p d d T

− −= =

= ,

(5.4.3)

where

10

0

sin(( ) )1 1( , , )

2

exp( ( )) exp( ( ))1 1 ,

2 2

kn n nn mn n

n

n kn n n kn nn

n

F AT d

i F A i F Ad

i

−= −

− − − −= −

(5.4.4)

and

1

1

( , , ) 0

( , , ) 1 .

n mn kn n

n mn kn n

T if F A

T if F A

=

= (5.4.5)

79

We continue the procedure from (5.3.7) to (5.3.14), and find that 1( , , )mQ A A

can be written in two parts

1 1 2 1( , , ) ( , , )m mQ Q A A Q A A= + . (5.4.6)

It indicates

11 1 1 1 1 1

0 01

1

exp( )1 1

( , , ) ( , , ),2 (2 )

m

n n Nn

m m j jn mmm mj

n

n

i A

Q A A d d q f f

i

=

=

=

−

= −

(5.4.7)

where

1 1 1 1 1

1

( , , ) ( , , ) exp( )m

j jm m j jm jn jn n j jm

n

q f f p i f d d

− −=

= ,

(5.4.8)

12 1 1 2 1 1

0 01

1

exp( )1 1

( , , ) ( , , ),2 (2 )

m

n n Nn

m m j jn mmm mj

n

n

i A

Q A A d d q f f

i

=

=

=

−

= −

(5.4.9)

where

2 1 1 1 1

1

( , , ) ( , , ) exp( )m

j jm m j jm jn jn n j jm

n

q f f p i f d d

− −=

= − .

(5.4.10)

Therefore,

1 1 1 1 1 10 0

1 1

1( , , ) (exp( ) ( , , ) ,

(2 )

Nm

m n n j jm m mmn j

P A A i A q f f d d

= =

= −

80

(5.4.11)

and

2 1 2 1 1 10 0

1 1

1( , , ) (exp( ) ( , , ) .

(2 )

Nm

m n n j jm m mmn j

P A A i A q f f d d

= =

=

.

(5.4.12)

Equations (5.4.11) and (5.4.12) are the starting point for deriving the probability

distribution by the structure factors. As in the derivation of (5.4.11) and (5.4.12) the

cells in the asymmetric unit are assumed to move randomly throughout the cluster

except that they are restricted by known magnetic magnitudes or a specified set of

structure factors. Then we follow the procedures described from (5.3.18) to (5.3.19)

and obtain

1

1 1 1 1 1

0

( )

( , , ) ( , , )!

mk

jn jn n

nj jm m j jm j jm

k

i f

q f f p d dk

=

− −=

=

,

(5.4.13)

12 1 1 1 1

0

( )

( , , ) ( , , )!

mk

jn jn n

nj jm m j jm j jm

k

i f

q f f p d dk

=

− −=

−

=

.

(5.4.14)

By defining a mixed moment

1

1 1 1 1( , , ) m

m j jm j jm j jmm p d d

− −= , (5.4.15)

we obtain a simple form of (5.4.13) and (5.4.14)

1

11 1 1

0

( )

( , , )!m

mk

jn n

nj jm m

k

i f

q f f mk

=

=

=

, (5.4.16)

81

1

12 1 1

0

( )

( , , )!m

mk

jn n

nj jm m

k

i f

q f f mk

=

=

−

=

, (5.4.17)

where m N and the interdependence of the vectors is related to the value of

moments. For k is even, the joint probability distribution is likely to vanish. For k is

odd, the joint probability distribution is enhanced. We consider a joint probability of

two interdependent events ( 2m= ) and (5.4.16) expresses as

1 1 1 2 2

2 2 2 2

1 1 10 2 2 01 1 1 20 2 2 02 1 2 1 2 11

( , )

11 ( ) ( 2 )

2

j j

j j j j j j

q f f

i f m f m f m f m f f m

=

+ + − + + −

(5.4.18)

Similarly, (5.4.17) expresses as

2 1 1 2 2

2 2 2 2

1 1 10 2 2 01 1 1 20 2 2 02 1 2 1 2 11

( , )

11 ( ) ( 2 )

2

j j

j j j j j j

q f f

i f m f m f m f m f f m

= − + + + + +

(5.4.19)

Then, we can solve (5.4.11) and (5.4.12) in the scenario ( 2m= ) by following the

procedure from (5.3.20) to (5.3.24)

2 2 2 2

1 1 1 2 2 20 1 1 02 2 2

1 11

10 1 1 01 2 2 11 1 2 1 2

1 1 1 1

1 1( , ) exp( )

2 2

(1 ),

N N N

j j j j

j jj

N N N N

j j j j

j j j j

q f f m f m f

im f im f m f f

= ==

= = = =

= − −

+ + − −

(5.4.20)

and

2 2 2 2

2 1 1 2 2 20 1 1 02 2 2

1 11

10 1 1 01 2 2 11 1 2 1 2

1 1 1 1

1 1( , ) exp( )

2 2

(1 ).

N N N

j j j j

j jj

N N N N

j j j j

j j j j

q f f m f m f

im f im f m f f

= ==

= = = =

= +

− − + +

(5.4.21)

82

Substituting (5.4.20) and (5.4.21) into (5.4.11) and (5.4.12), we obtain

2 2

1 21 1 2

2 2 2 2

20 02 1 2 20 1 02 2

1 1 1 1

1 1( , ) exp( ) (1 )

22 2

N N N N

j j j j

j j j j

A AP A A

m m f f m f m f

= = = =

= − − +

(5.4.22)

and

2 2

1 22 1 2

2 2 2 2

20 02 1 2 20 1 02 2

1 1 1 1

1( , ) exp( ) (1 )

22 2

N N N N

j j j j

j j j j

A AiP A A

m m f f m f m f

= = = =

= + −

.

(5.4.23)

Therefore, the probability distribution 1 2( , )P A A contains the distribution of the

real part (5.4.22) and an imaginary part (5.4.23).

5.5 Correlation between osteoblasts and osteoclasts

The section specifically addresses the correlation between osteoblasts and osteoclasts.

These two types of cells are relevant and interacted in BMU which has been mentioned

in Chapter 2. Equations (5.4.22) and (5.4.23) propose a structure for two relevant types

of cells in a joint probability distribution. The structure illustrates the orthogonal

relationship between the joint probability distribution. We assume that the interaction

between osteoblasts and osteoclasts also satisfy an orthogonal relationship.

Set cell probability of osteoblasts with a vector ( , )r tOB and osteoclasts with ( , )r tOC

, where r represents space and t represents time. ( , )r tOB and ( , )r tOC satisfy the

orthogonal form in the following,

( )OBt

− = −

OCOB P , (5.5.1)

83

2

2

1( )OC

t t t

= +

OC OB OB , (5.5.2)

where , ,OB OC indicate transition constants for osteoblasts and osteoclasts

respectively. P denotes the factor of apoptosis. Submitting (5.5.1) into (5.5.2),

2

2

1[ ( )] 0OC OB

OB t t

− + + =

P OB OBOB , (5.5.3)

with the boundary conditions

0n =OB , (5.5.4)

0n =OB . (5.5.5)

For an isotropic condition, apoptosis is proportional to cell population

OB OB =P OB . (5.5.6)

Substituting (5.5.6) into (5.5.3),

2

2

1[ (1 ) ] 0OC OB OB

t t

− + + =

OB OBOB . (5.5.7)

The solution is obtained by the extreme value of the functional, which is a one-field

principle in term of OB ,

( ) ( )2

2

1 2( ) (1 ) 2 d ,

2OC OB OB

VV

t t

= − • + • + •

OB OBOB OB OB OB OB

(5.5.8)

and

( ) ( ) ( ) ( ) 1

(1 ) (1 ) d2

1 1 d .

OC OB OB OC OB OBV

V

V

V

= − • + − •

+ • + • + • + •

OB OB OB OB

OB OB E OB OB OB OB OB

(5.5.9)

84

The first variation on the time difference is set to zero for an arbitrary instance t ,

0, 0 = =OB OB . (5.5.10)

It can be simplified as

( ) ( ) ( ) ( ) 1

(1 ) (1 ) d2

1 d .

OC OB OB OC OB OBV

V

V

V

= − • + − •

+ • + •

OB OB OB OB

OB OB OB OB

(5.5.11)

Consider the first term on the right-hand side, which can be rewritten as

( ) ( ) ( ) ( )

( ) ( )

1(1 ) (1 ) d

2

(1 ) d .

OC OB OB OC OB OBV

OC OB OBV

V

V

− • + − •

= − •

OB OB OB OB

OB OB

(5.5.12)

To transfer the curl operator away fromOB , we invoke the differential vector

calculus identity

( )• = • − •a b b a a b , (5.5.13)

( )• = • −• a b b a a b . (5.5.14)

Let

( )(1 ) , OC OB OB − → →OB a OB b , (5.5.15)

Hence,

( ) ( )

( ) ( )

(1 )

(1 ) (1 ) ,

OC OB OB

OC OB OB OC OB OB

− •

= • − −• −

OB OB

OB OB OB OB

(5.5.16)

85

and

( ) ( )

( ) ( )

(1 ) d

(1 ) d (1 ) d .

OC OB OBV

OC OB OB OC OB OBV V

V

V V

− •

= • − − • −

OB OB

OB OB OB OB

(5.5.17)

Applying the divergence theorem

d dV S

V S• = • f n f , (5.5.18)

we obtain

( ) ( )

( ) ( )

(1 ) d

(1 ) d (1 ) d .

OC OB OBV

OC OB OBV

OC OB OBS

V

V S

− •

= • − − • −

OB OB

OB OB n OB OB

(5.5.19)

Using the vector identity

( ) ( ) ( )• = • = • a b c b c a c a b , (5.5.20)

we further have

( ) ( )

( ) ( )

(1 ) d

(1 ) d (1 ) d .

OC OB OBV

OC OB OBV

OC OB OBS

V

V S

− •

= • − − • −

OB OB

OB OB OB n OB

(5.5.21)

Consequently, the first variation equation of the function is rewritten as

( )

( )

(1 ) d

1d(1 ) d ,

OC OB OBV

VOC OB OB

S

V

VS

= • −

− • + •

• − +

OB OB

OB OB OB OBOB n OB

(5.5.22)

86

Rearranging,

( )

( )

( )

( )

( )

1(1 ) d

(1 ) d

1(1 ) d

(1 ) d

(1 ) d

OB

I

OC OB OBV

OC OB OBS

OC OB OBV

OC OB OBS

OC OB OBS

OC O

V

S

V

S

S

= • − + +

− • −

= • − + +

− • −

− • −

− •

OB OB OB OB

OB n OB

OB OB OB OB

E n OB

E n OB

E n ( )(1 ) d .OC

B OBS

S− OB

(5.5.23)

The first integral on the right-hand side enforce the governing equation (5.5.23), the

second integral should vanish due to the prescribed osteoblasts field constraint over

the boundary OBS , the third one enforces the continuity conditions on the inter-

element edges IS when co-considered with those from the neighbourhood elements,

and the last integral enforces the boundary condition on osteoclasts field OCS . When

the stationary requirement 0 = is imposed for arbitrary variation of OB . For the

convenience of the derivation of the stiffness equation, the functional is rewritten in

the form of a vector operation,

( ) ( )2

T T T

2

1 2( ) (1 ) 2 d .

2OC OB OB

VV

t t

= − + +

OB OBOB OB OB OB OB

(5.5.24)

In the finite element formulation, after a robust continuum is discretised into a finite

number of elements, the osteoblasts field OB at an arbitrary point is interpolated

regarding nodal electric fields d

( , ) ( ) ( )t t=OB x N x d , (5.5.25)

87

where N denote the shape function matrix.

1

1

1

2 2

1 1

3

3

1

1

( ) ( )

( , ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

n

i i

i

n n

i i i i

i i

n

i i

i

n

j ij i

i

N d t

E

t E d t N d t

E

N d t

OB N d t

=

= =

=

=

= = =

=

x

OB x N x x

x

x

, (5.5.26)

where n is the total number of unknowns

1

2

3

( )

( ) ( )

( )

i

i i

i

N

N

N

=

x

N x x

x

. (5.5.27)

Subsequently, the curl definition of the osteoblasts field OB is given by

( ) ( )1

( ) ( )n

i i

i

d t=

= = OB N x N d , (5.5.28)

with

1 2( ) ( ) ( )n = N N x N x N x . (5.5.29)

We obtain

( ) ( )

( )

TT T T

T T

1 2( ) (1 ) d d

2

d

OC OB OBV V

V

V V

V

= − +

+

E d N N d d N N d

d N N d

,

(5.5.30)

which can be written in matrix form as

T T T1

2 = + +d Kd d Cd d Md , (5.5.31)

88

with

( ) ( )

( )

T

T

T

(1 ) d

2d

d .

OC OB OBV

V

V

V

V

V

= −

=

=

K N N

C N N

M N N

(5.5.32)

The stationary condition of the functional regarding unknown d yields

T

1

2

= + + =

Kd Cd Md 0

d . (5.5.33)

The solution of the stationary condition (5.5.33) should have the similar form of

(5.4.22) and (5.4.23), which includes real part and imaginary part. If the real part is

assigned to reflect the possibility distribution of osteoblasts, the imaginary part will

indicate the possibility distribution of osteoclasts.

89

Figure 5.1 The structure factor. The structure factor can be positive as well as negative.

However, once a set of magnetic intensities is known, the cells in a cluster no longer

occupy all positions with equal probability. If the cells in the asymmetric unit are

assumed to move randomly throughout the unit cluster subject to the constraints

imposed by the magnetic intensities, the resulting probability distribution of a structure

factor is no longer an even function. The probability that the structure factors have a

particular sign is different from one-half. The purpose is to derive these probabilities

on the basis that specific magnetic intensities are specified and to derive a procedure

for phase determination.

90

Chapter 6

Numerical Model:

Predication of Osteoblast Cell Population

under Influence of Surface Grain Size

91

6.1 Introduction

The frameworks developed in Chapter 5 are utilised to investigate cell population of

osteoblasts on material surfaces of nano-grain size. In this case study, a numerical

model is proposed for simulating osteoblast behaviours on two different surface-

engineered biomaterials and investigating the functional importance of specific grain

sizes at the nanophase. The proposed model consists of molecular and cellular motions

based on theoretical and experimental evidence and creates predictive simulations

from sparse experimental data. The comparison of numerical solutions and

experimental data reveals that the proposed model can explain the nonlinearity of the

cell population on material surfaces concerning nanophase grain size (0-100 nm). The

numerical results further provide insight into the optimisation of nanophase grain size

on the surface of the biomaterial. Establishing a mathematical relationship in the

process between material surface and osteoblast allows deducing the coupling effects

of surface roughness and osteoblast adhesion.

6.2 The relationship between osteoblast proliferation and substrate surface

roughness

Consequent to the demographic changes in society with an ageing population and the

corresponding increase in the incidence of the musculoskeletal diseases of ageing such

as arthritis and osteoporosis, there is a pressing clinical need to develop new

biomaterials for orthopaedic implants. In an ideal situation, these materials would be

biocompatible, mechanically robust and actively direct osteogenic progenitor cells,

such as osteoblasts. The interaction between a material substrate and the biological

system is complex [176]. The interplay between the substrate and the extracellular

matrix (ECM) determines the cell response.

The development of a bone-implant biomaterial depends on the interactions of

osteoblasts and material surfaces. The quality of cell and material interactions

92

influence cellular proliferation and differentiation [177]. In surface engineering of

biomaterials, the primary strategy for enhancing the interactions between the cell and

the material surface is to modulate the properties of the material surface, such as

surface charge and roughness, to a state in which ECM adhesion proteins can

appropriately function [178]. Experiments highlight the importance of optimising the

roughness of the material surface in vitro and in vivo. [179] investigated the effect of

surface roughness of hydroxyapatite (HA) on human bone marrow cells and found that

cell adhesion, proliferation and detachment strength were sensitive to the surface

roughness of HA. Intriguingly, osteoblast adhesion and proliferation appear to vary

nonlinearly with the roughness of the material surface. Webster, et al. [180] reported

that osteoblast proliferation was significantly higher on nanophase Alumina and

titanium. Their experiment [181] also showed that the ability of osteoblasts to adhere

to these materials was not proportional to the reduction in grain size. Also, Huang, et

al. [182] observed that significant enhancements of osteoblast adhesion, proliferation,

maturation and mineralisation were exhibited on the nano-grained surface (below 100

nm), but little improvement was found on the ultrafine-grained (100 nm-1000 nm)

surface compared to the conventional coarse-grained surface.

The nonlinearity between material surface roughness and osteoblast adhesion might

be caused by multiple factors, such as cell population motility, proliferation and

synthesis of ECM proteins of anchorage-dependent cells [183]. Biological recognition

mediates the interaction between cells and implantable materials by binding specific

receptors on the cell surface to ligands on the material surface. Such ligands could be

proteins spontaneously absorbed upon the material surface when materials contact

with body fluid in vivo or cell culture medium in vitro [184]. Uncertainties exist as to

not only which aspects of cell behaviours are the most important in the interaction

93

between materials with a surface roughness but also what constitutes the best surface.

For instance, osteoblasts can adhere and maintain active when attached to some

surfaces (e.g., the cell culture plate or surface modified alloys), but almost inactive

when attached to others (e.g., many artificial surfaces).

In vitro and in vivo experimentation about materials and osteoblast cellular responses

to the substrate are expensive and time-consuming. Thus far because of the inherent

complexity of the biological milieu, there is no alternative to these experiments and

strategies in attempting to predict the response to a biomaterial at the cellular level.

Therefore, creating mathematical models of osteoblast responses to implantable

biomaterials is becoming increasingly popular as a strategy in biomaterial and drug

development. As an alternative to experimental approaches, a robust mathematical

model will facilitate more efficient optimisation of the molecular interface between

material surface and osteoblasts.

6.3 Governing equations for osteoblast adhesion on the material surface

We propose a double-layered model for investigating osteoblast adhesion on a material

surface (Figure 6.1). This model concerns the biological process of osteoblast

adhesion on the corresponding material surface with two relatively independent steps.

The first step describes the movement of molecules on the material surface due to the

diffusion-controlled transportation of proteins. This process includes biological

recognition [184]. During the transportation, the molecules have random motions in a

potential field formed by the interactions between the material surface and the cells

attached to the surface. The motions of molecules form a “molecular layer” in the

proposed model, which consists of cell signalling, ligand binding and adsorbed

proteins on the material surface [185]. The second step describes a “cellular layer”

including cell adhesion and migration at the cellular level. Since osteoblasts are

94

anchorage-dependent cells and the process of cell migration is inextricably linked with

the process of cell adhesion [186], we speculate that the process of osteoblast migration

is highly dependent on the process of osteoblast adhesion at the “cellular layer”. We

also assume that the osteoblast adhesion is related to the “molecular layer” because the

extracellular matrix (ECM) and specific transmembrane receptor (e.g., integrins)

contribute critically to the cell adhesion [187].

We introduce a local coordinate to describe the behaviours of the model on a material

surface. In the horizontal direction (x-axis) of the molecular layer, the model describes

the molecular motion by its probability distribution. Similarly, the model describes the

process of cell migration by a cellular probability distribution. Then we separate the

cell adhesion into two variables in the vertical direction (y-axis). One variable

represents the interaction between the material surface and the molecular layer.

Another variable represents the interaction between the molecular layer and the

cellular layer. Finally, the model links the variables in the vertical direction with the

probability distribution of cells and molecules in the horizontal direction. The double-

layered model consequently studies the osteoblast adhesion on a material surface by

identifying the probability distribution of molecules and cells. Based on the

assumptions above, we apply continuum mathematical descriptions to find the

probability distribution of molecules and cells. The descriptions assume the motions

of molecules and cells in stochastic processes, which lead to the Fokker-Planck

equations of stochastic processes for the probability density of molecules mp and cells

cp .

95

( )

( )

m mm m m

c cc c c

p pD p

t x x

p pD p

t x x

= −

= −

, (6.3.1)

where m and c refer to the drift coefficients among molecules and cells, which

determine their motions under external perturbations. mD and cD refer to the

corresponding diffusion coefficients indicating the random walk. We then consider the

influence of the cell adhesion and adsorbed proteins on the horizontal motions of

molecules and cells,

( ( ) )

( )

m mm m m

c cc c c

p pD p

t x x x

p pD p

t x x

= −

= −

, (6.3.2)

where path modulator expresses external perturbation from the surface roughness

R . When culture cells on a substrate, the surface roughness prevents the dispersion

of molecules through the friction between the molecular layer and material surface and

subsequently affects the cellular motion. Therefore, the probability density of

molecules mp and cells cp are bounded by conditions such that

( , , ( ), )

( , , )

m m m

c m c c

p f R D

p f p D

. (6.3.3)

The functions of drift coefficients m and c in Equation (6.3.2) are unknown

without experimental data.

6.4 Numerical solution methods for the double-layered model

Using the Ornstein-Uhlenbeck process as a reference for defining the motion of the

molecular layer, we attempt to obtain a numerical solution of the model. The Ornstein-

96

Uhlenbeck process describes the velocity of diffusing particles under the influence of

friction, in which there is a random walk tendency to move towards a central location.

Here we assume that the molecular motions follow a standard process, thereby creating

a probability distribution of molecules. The probability density of molecules mp of

the Ornstein-Uhlenbeck process satisfies

2

0 2( ) [( ) ]m

m m

px x p p

t x x

= − +

, (6.4.1)

where ( ) is environmental sensitivity of the molecular layer and is the

diffusion coefficient of the molecular layer. Taking 0 0x = for simplicity, the initial

condition of the equation (6.4.1) is

2

2 ( ) 2 ( )

( ) ( )exp[ ]

2 (1 ) 2 (1 )m t t

xp

e e

− −

−=

− − . (6.4.2)

Alternatively, the molecular sensitivity can be expressed as

( )

= . (6.4.3)

We substitute equation (6.4.3) into the equation (6.4.2) and solve the stationary

solution,

2

( ) exp( )2 2

m

xp x

= − , (6.4.4)

and the molecular sensitivity is dependent on the material surface property R ,

( )mf R = . (6.4.5)

Based on the equation (6.4.5), the surface property may hinder molecular motions and

alter the molecular probability density. For the continuity, the initial cellular

97

distribution should inherit the characteristics of the final molecular distribution. Thus,

we assume the initial probability density of osteoblasts to be approximately equal to

the stationary probability density of molecules.

0( , ) ( )c mp x t p x= , (6.4.6)

where 0( )cp t is the initial solution for cellular probability density. Considering the

evolution of the cell probability density cp in equations (6.3.2) and (6.3.3), we find

that the complexity of osteoblast adhesion leads to inhomogeneous cell distribution in

the cellular layer. Using the approach of Stevens and Othmer [188], we aim to

generalise an inhomogeneous distribution for a continuous-time discrete-space

random walk along the horizontal direction. This method restricts the time evolution

to one-step time jumps and postulates the discrete quantity ( )ip t as an

approximation to cellular probability density cp at the position ix ih= , iZ as

well as time t beginning at 0i = , 0t = . Thus, ( )ip t evolves in a manner such that

1 1 1 1

( )[ ( ) ( )] [ ( ) ( )]i

i i i i i i i i

dp tp t p t p t p t

dt

+ − + −

− − + += − − − . (6.4.7)

Here i

− denotes the transitional probability that cells enter from 1i − to i , and i

+

denotes the transitional probability that cells leave from i to 1i+ (Figure 6.1). The

equation (6.4.7) is highly dependent on the form of transitional probabilities i

that

reflect the physical parameters of the cell type under investigation. In the context of

the movement of osteoblasts, the transitional probabilities i

are determined by the

ambient cellular probability density 1ip , the cellular sensitivity . The cellular

98

sensitivity works to prevent cells from leaving the present position. Then, a simple

form of i

may be constructed as

2

1 1(1 )(1 ) /i i ip p h

= − − . (6.4.8)

The cellular sensitivity inherits the correlation with surface property R as

( )cf R = . (6.4.9)

Substituting equation (6.4.8) into equation (6.4.7) gives

1 22

( ) 1[ ( ) ( ) (1 ( )) ( )] 0i

i i

dp tp t f t p t f t

dt h+ − − = , (6.4.10)

where

1 1 1 1 1( ) (1 ( ))(1 ( )) (1 ( ))(1 ( ))i i i if t p t p t p t p t + − − += − − + − − , (6.4.11)

and

2 1 2 1 2( ) ( )(1 ( )) ( )(1 ( ))i i i if t p t p t p t p t − − + += − + − . (6.4.12)

Equations (6.4.11) are unconditionally stable for 0 ( ) 1ip t . For matching the

experiment data, the probability density of cells is converted to the cell density per unit

area. We estimate the probability density of cells at the discrete position i and

integrate the probability density by a circular domain on the assumption that the

cellular probability density cp is equal at each position with the same radius,

00

( ) 2r

k cP t P xp dx= , (6.4.13)

where 0P is the initial cell density at 0i = , 0t = . x and t denote the spatial and

temporal variables respectively. The scale of the studied cell determines the number

of points. The cell density kP is the integration of cellular probability density cp .

99

6.5 Numerical simulation on experimental data

In this section, we used our model to simulate osteoblast adhesion on two different

material surfaces of nanophase grain size by inputting the published experimental

results [181]. Webster et al. provided evidence that osteoblast adhesion enhanced on

nanophase Alumina ( 2 3Al O ) and Titania ( 2TiO ) in vitro. The method of evaluating cell

adhesion was to lift osteoblasts enzymatically using trypsin and to count the number

of adherent cells in five random fields per substrate. The authors observed that the

number of adherent osteoblasts increased with time, and the cell density of adherent

osteoblasts varied on the surfaces of Alumina and Titania with different grain sizes.

From their observation, they concluded that an increase in osteoblast adhesion could

be accompanied by a decrease in Alumina and Titania grain size.

Since the osteoblast size was around 20 µm [189], 250 osteoblasts could be arrayed

along a radius of 0.5 cm. In 1 cm2, the size was set at 500 points. The model was

initialised with molecular sensitivity and cellular sensitivity by fitting the

experimental data of adherent osteoblasts on Alumina and Titania. As an example, the

model fitted the experimental results with 2.2 = and 0.26 = on Alumina of

grain size 77 nm (Figure 6.2 A), and with 0.6 = and 0.22 = on Titania of

grain size 32 nm (Figure 6.2 B). We then fitted the data for each grain size of Alumina

and Titania in the experiment. All and were plotted along grain sizes to

determine the functions of molecular and cellular sensitivities. Trendlines were added

to find the modified functions of molecular and cellular sensitivities by curve fitting.

For Alumina, the functions were cellular sensitivity (Figure 6.3 A),

0.24 0.11exp( 0.018 )Al R = + − , (6.5.1)

and molecular sensitivity (Figure 6.3 B),

100

2.27 0.90exp( 0.027 )Al R = − − . (6.5.2)

The functions for Titania were found for cellular sensitivity and molecular sensitivity.

For both Alumina and Titania, the molecular sensitivity increased with grain size

whereas the cellular sensitivity decreased with grain size. We checked the correlation

of exponential fitting for both materials and found that the sensitivity functions for

Titania had better consistency with the experimental data. The experimental data

varied around the sensitivity functions of Alumina, in particular for the cellular layer.

These results may imply that the cell adhesion on Titania might be stable whereas the

cellular motions of osteoblasts on Alumina are likely to be affected by the external

perturbation.

Based on the modified functions, we recalculated the model to show the cell adhesion

in the profile of adherent cell density with both time and grain size. We further draw

the profile with different colours indicating the various cell density. As can been seen,

although both the cell density of adherent osteoblasts on Alumina (Figure 6.4 A) and

Titania (Figure 6.4 B) increased with time, their paths were predicted to be in different

ways. The predicted adherent cell density on Alumina was observed that the adherent

cell density keeps at the same number in the beginning, but the duration varied from 1

to 2 hours in the range of grain size from 0 to 100 nm. The duration approximately

reached an average of 1.5 hours when the grain size was 35 nm. Then the predicted

adherent cell density increased quickly and reached 3200 cells/cm2 after 4 hours. This

process was fast when the grain size of Alumina under 40 nm.

In contrast to the Alumina, the predicted adherent cell density on Titania increased in

a lag phase but varied on the grain size. When the grain size of Titania was beyond 40

nm, the predicted adherent cell density increased in a linear approximation of time.

Between 10 nm to 40 nm, the predicted path showed a similar trend with that of

101

Alumina. Under 10 nm, the predicted adherent cell density appeared no longer to

increase. The predicted results inspired us to think about the existence of critical grain

size on Alumina and Titania in the following section.

6.6 Comparison with experimental data and prediction

In this section, we compare the numerical results from the proposed model with the

original experimental results [181]. The numerical results reflect the historical profile

of adherent cell density concerning time and grain size. Here we focus on discussing

the influence of grain size on the adherent cell density. The numerical result of

adherent cell density on Alumina matches Webster’s experimental data (Figure 6.5

A). The adherent cell density is significantly higher on Alumina with grain sizes in the

range 0–40 nm than on Alumina with grain sizes in the range 60–100 nm. There is an

exponential decay on Alumina with grain sizes 40–60 nm. The numerical result of

Titania also agrees with the experimental data (Figure 6.5 B). The adherent cell

density is significantly higher on Titania with grain sizes in the range 0–20 nm and

followed by a linear decay from 20 to 100 nm. Webster et al. observed in their

experiments that variation existed in the adherent cell density of adherent osteoblasts

per grain size and concluded that there might be a critical grain size of Alumina and

Titania in mediating osteoblast adhesion. A critical grain size here can be defined as a

point on the axis of grain size in which the slope of adherent cell density and grain size

dependence significantly changes. On Alumina, our model identifies two critical grain

sizes at 36 nm and 60 nm, an observation which agrees on the with the conclusion of

Webster et al. that there may be a critical grain size between 49 nm and 67 nm. Our

model predicts a critical grain size at 36 nm with cell density 3042 cells/cm2. This

potential interaction between grain size and adherent cell density has not previously

been identified.

102

In contrast to the findings for Alumina, our model predicts a different interaction

between the osteoblast and Titania. The Webster’s experimental data implied a range

of the critical grain size for Titania from 32 nm to 56 nm, whereas the model predicts

a continuous decline of adherent cell density from 30 nm to 100 nm. Further

experiments, particularly examining grain sizes in the range of 40-70 nm, may well

clarify this issue further. The model surprisingly indicates that adherent cell density on

Titania drops significantly with the grain size smaller than 10 nm. There is limited

published experimental data regarding culturing osteoblasts on a substrate surface

made of Titania with grain size under 10 nm. Park, et al. [190] demonstrated that the

one-dimensional surface nano-topography of 15 nm Titania nanotubes promoted

osteoblast formation. They evaluated the adhesion, spreading and growth of

osteoblasts on the surface of Titania nanotubes from 15 nm to 100 nm diameters. Cells

were found to adhere and proliferate best on 15 nm tubes, and the adhesion decreased

with an increase in the nanotube diameter. Although the topography of nanophase

ceramic is different from the nanotube, it has been speculated that an optimal spacing

between cell and substrate exists for osteoblast adhesion on Titania. The putative

optimal spacing may derive from focal contacts or adhesion plaques, which are

junction locations of about 10-15 nm between adherent cells and material surfaces

[191]. On the other hand, theoretically, smaller grain sizes provide a more available

surface area of higher energy to promote protein interactions such as vitronectin and

fibronectin for adhesion [192].

The optimal spacing may well represent a combination of the coupling of focal

contacts and topography between osteoblasts and material surfaces. The topography is

a function not only of the material surface itself but also of the adherent protein layer

intimate with that surface. The focal contacts at the protein layer can create an optimal

103

spacing where the osteoblast adhesion is the highest, and adhesion is reduced if the

spacing becomes either smaller or larger. The spacing is altered by factors such as the

thickness or the composition of adsorbed proteins, both of which are highly dependent

on the grain size of the material surface. Smaller grain sizes promote protein

interactions so that they form a thicker layer. This potential mechanism may explain

the trends seen in the model’s predicted curve of Alumina with optimal grain sizes in

the range 36–60 nm and the curve of Titania with grain sizes in the range 10–100 nm.

It is notable that these predicted curves of both Alumina and Titania show stable trends

with increasing grain sizes beyond the above predicted critical values. This finding

agrees with the previous experimental observation [182] as well as with the

considerable clinical experience of the practical use of both these biomaterials where

grain size is not homogeneous in the end-use product. Other factors apart from

nanophase grain size may also play a role in cell-substrate adhesion. The surface

topographies of Alumina and Titania differ, with the arrangement of Alumina grains

being more linear than that on the Titania surface. As cell-substrate adhesion includes

integrins that have nanoscale features, cells will respond to surfaces with nanoscale

characteristics of the pores, ridges and fibres of the basement membranes [193].

6.7 Parametric analysis

We used the model to simulate the adhesion of osteoblasts on engineered surfaces in

this case. However, it is wondering whether this model is capable of simulating the

growth of osteoblasts in the long term. The growth curves of osteoblasts exhibit three

phases: lag phase, exponential phase and stationary phase. The lag phase of osteoblast

proliferation in vitro suggests the cells are adapted to the culture conditions and some

of them probably died. After that, cell growth is exponential, and a tendency for a

stationary phase is observed after a certain culture time. When osteoblasts are cultured

104

under different conditions, these three phases reflects the temporal characteristics of

osteoblast growth. The functions of molecular and cellular sensitivity can modify the

temporal properties of osteoblast proliferation in addition to influencing both the

maximal and minimal threshold of the cell proliferation.

We begin by considering the molecular sensitvity on cell probability of osteoblasts

(Figure 6.6). The numerical simulation represents the temporal characteristics in

which cell probability of osteoblasts with various molecular sensitvities changes from

different lag phase to one stationary phase. The increase in the molecular sensitvity

leads to long-term of lag phase with low cell density, suggesting that cells are hard to

settle down and likely to die under a high diffusion environment. If we consider that

both the morphological aspects of osteoblasts on materials and molecular sensitvities

are affected exclusively by the surface roughness of materials, the above statement

implies that the spreading of the osteoblast cell and the formation of continuous cell

layer are better on smooth surfaces as compared to the rough ones. The period of

exponential phase is also extended under high molecular sensitvity. The period of

exponential phase continues 100-time steps when the molecular sensitvity equals to

0.5, and it is extended to 276-time steps when the molecular sensitvity equals 10.

Moreover, the amplified area shows that the stationary phase is made by periodical

oscillation. This periodical oscillation indicates the fact that a dynamic balance of cell

proliferation and apoptosis exists even osteoblasts reach the stationary phase.

We further examine the influence of cellular sensitivity on cell probability (Figure

6.7). Numerical results show that the cellular sensitivity [0,1] determines the

qualitative behaviour of solutions to this simulation. For each value of , the

simulation is identified by analysis of dynamic changes in cell density in response to

a gradual increase in time step. The simulation exhibits the opposite behaviour of mode

105

as plotted by a function of . The cell probability of lag phase remains the same

regardless of various values. The increase in values can shorten the duration of

the lag phase and consequently reduce the time required for the stationary phase.

Compared with 0.05 = , the duration of lag phase at 0.5 = is shortened dramatically

by time step difference 590t = and the time requirement is shortened by 930t = . A

higher value of results in higher cell density at the stationary phase. The simulation

exhibits that osteoblast proliferation is promoted by greater cellular sensitivity, which

is observed in the study of cell-substrate adhesion [194]. When [0,1] , curves of cell

probability have a similar pattern. When 1 , unstable solutions appear and an

unpredictable pattern in the section [1.5,2] is shown in Figure 6.7. Cell probability

distributes symmetrically on both the positive and negative sides. This result is not

realistic for cell distribution since the value of cell probability cannot be negative.

Nevertheless, it may work for the binding of proteins if the positive values express the

unbinding of proteins while the negative values indicate the binding of proteins. In

other words, this model could be extended to study objectives at different size level by

changing the constants.

The size factor n affects the path-dependence integration and thereby justifying the

dynamic model behaviour (Figure 6.8). To illustrate the effect of the increase in size

factor, simulations of size 1000n = and 10000n = performance to compare. The results

show that the path of exponential growth is modified with the alteration for size factor.

When the size factor increases, time steps for reaching steady state increase as well as

the cell density. The higher accumulation of cell density at steady state can be

explained from the definition of size factor. A more substantial size factor indicates

that more cells existing at the target area and thereby results in higher cell density

while the size factor is highly dependent on the size of the target cell.

106

(A)

(B)

1i− i 1i +

1 1( ) ( )i i i ip t p t+ −

− − −1 1( ) ( )i i i ip t p t+ −

+ + −

( ) /ip t t

(C)

107

Figure 6.1 Sketch of the double-layered model for osteoblast adhesion on the material

surface. (A) The graph represents two osteoblast cells (OBs) on an undefined material

surface. The biological processes are assigned to the model. It is noted that osteoblast

cells are in monolayer in vitro after settlement. Small circles (pink) represent

molecules involved in cell signalling and ligand binding. The material surface

properties affect the molecular motions at the molecular layer and further affect

cellular motions at the cellular layer. Osteoblast adhesion works as the transmission

during this progress, deciding how strongly the double-layered system is disturbed.

(B) Mathematical description of cellular probability density ( )ip t during t . It

indicates the difference between cells that stay at the transitional site from 1i − to i

and cells that stay at the transitional site from i to 1i+ . This mechanism achieves a

continuous-time discrete-space random walk at i for equation. (C) Discrete

distribution of cellular probability density on the interval [0, ]r . This interval is

separated into n subintervals of length rhn

= , with endpoints

, 0,1,...,ix ih i n= = . The probability density at a point ix is denoted by ip .

108

(A)

(B)

Figure 6.2 Two examples of model fitting by experimental data. (A) Model fitting of

experimental data for Alumina on 77 nm grain size. (B) Model fitting of experimental

data for Titania on 32 nm grain size. The time scale used 400 time-step to match 4

hours of experimental duration.

109

(A)

(B)

110

Figure 6.3 Function determination for molecular and cellular sensitivity in grain size.

(A) Cellular sensitivity to Alumina and Titania; (B) molecular sensitivity to Alumina

and Titania. The functions on Alumina were calculated for sensitivity (A) and

molecular sensitivity. (C) The functions on Titania were calculated for cellular

sensitivity (B) and molecular sensitivity (D). It was noted that these functions had

similar trends but very different value scales because each point came from

independent experimental data. The functions linked to every independent experiment

together by defining the molecular and cellular sensitivity. Hence the errors between

the experimental data and the model result rooted from these functions.

111

(A)

(B)

Figure 6.4 Numerical models for Alumina and Titania. (A) Profile of adherent cell

density on Alumina with the scale of time and nanophase grain sizes. (B) Profile of

adherent cell density on Titania with the scale of time and nanophase grain sizes. The

colour bar indicates the number of cell density (cells/cm2).

112

(A)

(B)

Figure 6.5 Comparison of numerical results and experimental data. (A) Critical points

for Alumina are marked at grain sizes 36 nm and 60 nm. (B) The critical point for

Titania is marked at grain size 10 nm and 5 nm. The enlarged part shows the decline

of adherent cell density on Titania below grain size 10nm.

113

(A)

(B)

114

Figure 6.6 (A) Simulation of molecular sensitivity. For 0.5 = (red line), the cell

probability density N moves through the lag phase 1 1.12N = at a time step 1t = ,

the exponential phase at 145t = and the stationary phase at 245t = ,245 1.61N = . For

1 = (blue line), the lag phase begins at 1t = ,1 0.99N = , the exponential phase

begins at 305t = , and the stationary phase is reached at 509t = ,509 1.60N = . For

10 = (purple line), the lag phase begins at 1t = ,1 0.39N = , the exponential phase

begins at 755t = , and the stationary phase is reached at 1031t = ,1031 1.59N = . (B)

Oscillation occurs at 0.016=b for 0.5 = , while 0.011=b for 1 = and 10 = .

Calculations are performed using the following set of parameters: 0.1 = , 100n = .

115

(A)

(B)

116

Figure 6.7 (A) Simulation of cellular sensitivity. For 0.05 = (red line), the cell

probability density starts at the lag phase 1 0.99N = , the exponential phase at 645t =

and reaches the stationary phase at 1049t = ,975 1.58N = . For 0.1 = (blue line),

dynamic behaviours repeat the blue line in Figure 6.6. For 0.5 = (purple line), the

lag phase begins at 1t = ,1 0.99N = , exponential phase begins at 55t = , and the

stationary phase is reached at 119t = ,119 1.65N = . Note that the oscillatory change

0.03=b for 0.05 = , 0.011=b for 0.1 = and 0.008=b for 0.5 = . (B) When 1.5 = ,

the simulation appears random walk. Calculations are performed using the following

set of parameters: 1 = , 100n = .

117

(A)

(B)

Figure 6.8 The impact of size factor. The comparison is made between the size 1000n =

and 10000n = . The results show that the path of exponential growth is modified with

the alteration for size factor. When the size factor increases, reaching steady state

needs more time steps, at the same time, cell density can be matched directly since the

value of y-axis is dramatically amplified. A high size factor indicates that substantial

cells existing at the target area and thereby results in a high cell density.

118

Chapter 7

Weak SMF Effects on Osteoblastic Saos-2

Cell Proliferation

119

7.1 Introduction

According to the research findings in Chapter 1 and the hypothesis in Chapter 3, the

“controversial results” for the EMF effect on osteoblasts may originate from the

alteration of ambient factors. Here we attempted to restrain the parametric study of the

EMF effect on osteoblasts. An in vitro experiment was designed for observing

osteoblastic Saos-2 cell proliferation in two gradients SMFs. In this chapter,

mathematical modelling was highly involved in the investigation of SMF effect on the

proliferation of osteoblasts. Therefore, we must guarantee the statistical accuracy and

reproducibility in each experiment. Compared with osteoblasts isolated from human

bone, osteoblastic Saos-2 cell line provided more stable statistical results for

quantifying cell proliferation.

We examined whether the osteoblastic cell proliferation was sensitive to the SMF

intensity on the order of milli-tesla (mT). The experimental data were analysed by the

numerical model developed in Chapter 6, exploring the relevance between the SMF

intensity and osteoblastic cell proliferation. The gradient field of SMF can generate six

different magnetic flux densities at the same time, which provide us with an

opportunity for observing the interference of cell probability.

7.2 Experimental design

This study used a solenoid coil to generate SMFs. The coil (125 mm in diameter, 200

turns of copper wire with a width of 33 mm) was mounted horizontally on a shelf

inside the incubator. A function generator drove the coil with a signal generator (3B

Scientific, Germany). The cells were placed in 6-well plastic culture plates mounted

on a platform across the centre of the solenoid. The culture plates with 85 mm x125

mm surface area were used throughout the experiment. The test area was 85 mm x 250

mm where two 6-well plates were placed horizontally. A magnetic field sensor (3B

120

Scientific, Germany) was used to measure the magnetic intensity at the centre of each

well (Figure 7.1 A).

In this study, multiple magnetic flux densities were generated at an identical

circumstance to reducing the statistical deviation from performing the experiments

separately. The device in Figure 7.1 was calibrated for generating weak SMF (~1 mT).

Limited by the accuracy of the device (~0.05 mT) and background noise (~0.01 mT),

the minimum difference of two magnetic flux densities was set > 0.06 mT.

Consequently, weak SMFs were established with proximal magnetic flux density at 2

and 2.5 mT, and with two gradient fields spontaneously. The first gradient field G1

was measured at the centre of the culture well in 2, 1.1, 0.6, 0.48, 0.37 and 0.3 mT.

The second gradient field G2 was measured at the centre of the culture well in 2.5, 1.6,

1, 0.75, 0.55, and 0.42 mT (Figure 7.1 B). Both fields satisfied the minimum

difference of magnetic intensity.

The control was placed in a different incubator. The background was placed in the

same incubator but different layer which was perpendicular to the coil at

approximately 50 cm. The exposure was applied for 2 hours at any 24-hour interval,

envisaging a possible clinical setting application. 6000 cells per square centimetre

were plated into a 6-well and cultured for two and four days, in the absence or presence

of SMFs.

7.3 Experimental results

7.3.1 Effect of the SMF on the orientation of cultured Saos-2 cells

The Saos-2 cells appeared fibroblasts-like cells in morphology after three days of

seeding. After SMF exposure for 4 hours, part of the cells formed a whirlpool-like

cluster, in contrast to the control which did not have such formation of the cluster

(Figure 7.2). The formed cluster indicated the alteration of cell distribution in SMF.

121

7.3.2 Effect of SMF intensity on the proliferation of cultured Saos-2 cells

Two test groups for SMF exposure, G1 and G2, were investigated on the Saos-2

proliferation (Figure 7.3). G1 was exposed in the gradient magnetic field from 2 mT,

and G2 was exposed in the field from 2.5 mT. Two groups were set in the same

condition except for the SMF intensity. Compared with the control, cell density in G1

only showed a significant increase after SMF exposure of 0.6 mT for 4 hours while

cell density in G2 was not changed significantly. The background of 0.01 mT in G1

and G2 was indifferent than control.

7.3.3 Effect of SMF intensity on the cell viability of cultured Saos-2 cells

The SMF exposure resulted in the lower cell viability compared with the background

and the control (Figure 7.4) The effect was pronounced in the early stage.

7.3.4 The sensitivity of cultured Saos-2 cell proliferation to SMF intensity

The sensitivity of cell proliferation to SMF intensity was focused on the difference

between adjacent data. The Saos-2 cell proliferation was not sensitive to the SMF

intensity of background in comparison with the control (Figure 7.3). The statistical

analysis in G1 and G2 illustrated that the Saos-2 cell proliferation was more sensitive

to SMF intensities after accumulated exposure for 4 hours than that after the exposure

for 8 hours. In G1, the difference of cell density was observed among adjacent SMF

intensities after 4 hours (Figure 7.3 A) but the difference was reduced to three intervals

with the apparent distinction (P<0.01) between the SMF exposure of 0.6 and 1.1 mT

(Figure 7.3 B). The similar phenomenon is also found in G2 that the sensitive intervals

were reduced from three to two (Figure 7.3 C, D). After the accumulated exposure of

8 hours, the distinction was found between the exposure of 0.75 and 1 mT in G2

(Figure 7.3 D). If combining G1 and G2 into one picture, the picture of G1+G2

showed sensitive to the SMF intensities after the accumulated exposure of 4 hours.

122

The distinction only appeared between 0.55 and 0.6 mT after 8 hours. It was

recognised for the feasibility of combined G1+G2. The distinction between the results

after the accumulated exposure of 4 hours and 8 hours implies that the SMF intensity

has a stronger influence on the Saos-2 proliferation at an early stage.

7.4 A numerical model of the dose-dependent effect of SMF on osteoblastic

cell proliferation

7.4.1 A stochastic model for cell probability in SMF

The first assumption is that a gradient SMF can affect the cell probability of

osteoblasts. Critical parametric functions are expected to be extracted by fitting cell

data in vitro experiment with a theoretical cell probability in silico. These parametric

functions facilitated the understanding of how osteoblastic cell perceived SMF. Our

model comprised of the mathematical analysis of random walks [188] and

Chemotactic movement [195], describing the spatial-temporal evolution of cell

probability density function under the influence of random forces as described in

Chapter 5 [196]. The fitting created an S-shaped curve for cell proliferation. In this

study, SMF can affect osteoblastic cell proliferation through the alteration of cell

formation. We write the Fokker-Planck equation of this stochastic process in term of

probability density function p and magnetic flux B ,

( , ) [ ( ) ( , ) ( ) ( , )]B Bp x t D p x t p x tt x x

= −

, (7.4.1)

where the evolution of the probability density function is affected by diffusion D and

drift . To simulate the dynamics of cell probability in the equation (7.4.1), we

suppose that the conditional probability density ip is implemented via an approximate

discretisation of the culture area ( i N ),

123

1 1 1 1( ) ( )i i i i i i i i ip W p W p W p W p+ − + −

− − + + = − − − (7.4.2)

where

1 1

1(1 )(1 )i i i iW p p

h

= − − (7.4.3)

Where the diffusion D is incorporated into the conditional probability density ip ,

and the drift indicates the correlation for neighbour coupling. Therefore, we

establish a three-term recurrence relation in a culture area where its radius r is divided

into points 0 ... i with a step h . Each point i corresponds to a conditional cell

probability density ip . After this step, the algorithm guarantees the nearest neighbour

coupling, and the whole system of cell probability density can be arranged in a finite

chain. In an ideal circumstance, we can fit any cell population N with the initial cell

population 0N by integrating the cell probability density p with a spatial component

i and temporal component j ,

0 ( )ji

j i

N N f p= (7.4.4)

where N represented the observed cell population and 0N represented the initial cell

population, for example, cell number cultured at day 0. The equation (7.4.4) was

calculated by a circular integration. Hence, the cell population tN matched the cell

number in a culture well after culturing certain days. When designing the experiment

for this study, we selected three temporal points to check the cell number at day 0, 2

(4 hours) and 4 (8 hours）respectively. The algorithm was solved by MATLAB® with

three predicted function. These functions were interpolated into the equation (7.4.4),

124

0 3 2 1( ) ( ) ( ( ))B B ji B

j i

N N f f p f= . (7.4.5)

The exact formation of these functions was related to the experimental data.

7.4.2 Numerical results

The numerical results of the stochastic model for G1 and G2 represented that Saos-2

cell proliferation was not linear with SMF intensities, but an approximation of the sine

curve in term of SMF intensities (Figure 7.7). After 8-hour SMF exposure in four

days, a period was found by the peak-to-peak amplitude as 1 mT in G1 and 0.8 mT in

G2. The system was meta-stable when examining the results after 4-hour SMF

exposure in two days. The oscillation was evident in the intensities from 2 to 3 mT in

G1, and from 0 to 0.5 mT in G2. The oscillation was reduced by combining G1 and

G2 into G1+G2. However, the oscillation of G1+G2 remained the same as that of the

G1 from 2 to 3 mT. The oscillation is an essential part of the stochastic model referring

to the interference of cell probability. A sine function is written based on the

relationship between osteoblastic cell proliferation and SMF intensity,

( ) ( )sin( )cN t N A t kB= + , (7.4.6)

where ( )N t is the cell population at the time t , and cN is a constant that could be a

population of control. The amplitude ( )A t of the sine function changes with time and

might satisfy a wave function. Its property illustrates a harmonic wave function in

which each point on the axis of the wave has a constant amplitude at a defined time.

The wave oscillates in time but has a spatial dependence that is stationary. Substitute

time t with a spatial component i and temporal component j ,

( ) sin( )c ji i

j i

N t N a p kB= + . (7.4.7)

125

We can write a square matrix to satisfy the equation (5.2.17) by

11 1 1

1 1

sin( B ) sin( B )

=

sin( B ) sin( B )

n n

n nn n

p k p k

p k p k

p . (7.4.8)

7.4.3 SMF Interference

According to the equation (7.4.5), the description of the interference is related to the

parametric functions f . Three coefficients are analysed for the linearity (Figure 7.8

A). The coefficient 1 represents the spatial modification on cell probability density,

and the coefficient 2 represents the temporal modification. The coefficient 3

contributes to the interference if applicable. The linearity reduces from coefficient 1

to coefficient 3. The correlation between coefficient 1 and the SMF intensity is linear,

and the coefficient 2 is approximately linear with the SMF intensity. The coefficient 3

is nonlinear with the SMF intensity. The results indicate that the interference occurs

without the dependence on the cell probability density,

1: 1.25sin[ ( 0.47)],0.49

2 : 0.68sin[ ( 0.36)],0.41

1 2 : 0.79sin[ ( 0.47)].0.51

B

B

B

G f

G f

G G f

= −

= +

+ = −

(7.4.9)

The interference results in various cell probability distribution along the cell plate

(Figure 7.8 B). The experimental results only showed several possibilities. The

possible variety of the cell population is enormous.

7.5 Discussion

Data presented in this chapter demonstrate that the osteoblastic Saos-2 cell

proliferation is sensitive to the weak-intensity SMF. The numerical results of the dose-

126

dependent effect of SMF on osteoblastic cell proliferation satisfy a wave equation.

This wave equation changes with the SMF circumstance. The experiment of 2.5 mT

has an enhanced frequency and reduced amplitude compared with the experiment of 2

mT. Hence the difference of the SMFs intensities and the arrangement of the

experiment can both modify osteoblastic cell proliferation. The wave equation is also

the representation of the interference, which could be the reason for the observation of

various experimental results at the same SMF intensity.

7.5.1 Signal-to-noise

Although weak-intensity SMFs may stimulate the unique responsiveness of organisms

in vitro, experimental data for the biological effects of weak-intensity SMFs on cell

cultures are limited compared with the effects of moderate-intensity SMFs [7].

Notably, the weak-intensity SMFs rarely expose on osteoblastic cells since the

underlying signal-to-noise problem between identification of weak-intensity SMFs

bio-effects and standard population deviation of cell proliferation [197]. However, if

the signal-to-noise problem can be solved, the study of weak-intensity SMFs on

osteoblastic cells will serve two purposes. The first one is to understand the dose

pattern of osteoblastic growth in SMFs. Compared with moderate-intensity SMFs,

weak SMFs have fewer combinations of physical parameters and more consistent

biological reactions at the cellular level, just because of the narrow range of SMFs

intensities. The second purpose is to verify the role of weak-intensity SMFs in clinical

applications by answering whether the weak-intensity SMFs can enhance cell

proliferation. We realise that the signal-to-noise problem in weak-intensity SMFs bio-

effect is part of the reasons for the controversial results of the SMFs influence on

osteoblastic cells mentioned above, which is highly conditional on their physical

mechanisms. Therefore, we propose a computational model to simulate the physical

127

process of osteoblastic cell in weak SMFs as the framework for experimental design.

Different from previous models focusing on Ca2+ and ion channels [198], our model

emphasises the cellular movement and general interaction between cells and

molecules. Inspired by the window effect, we made the underlying assumption of this

article that any magnetic signal can be understood by cells through finite physiological

processes. This assumption led to two derivations for designing in vitro and in silico

model. (i) Optimal cell response can be observed in any range of magnetic flux density.

(ii) Cell data in SMF can be understood by parameters of physiological processes and

magnetic flux densities. The first assumption can be partially proved by the evidence

that the Ca2+ and ion transport system can be affected by weak-intensity SMFs (< 1mT)

[199]. The second assumption is made to guarantee that the biological system can

reach a steady state after a finite time.

7.5.2 Physical mechanisms

If the evidence mentioned above are considered reasonable, then the osteoblastic cell

must perceive SMF by more than one local mechanism. There were three types of

physical mechanisms of SMF interacting with biological processes, including

electrodynamic interactions, magneto-mechanical interactions and effects on

electronic spin states [200]. Insight into these mechanisms was provided by

examination of the interaction of SMF with moving charges, enzyme reaction rates

and signal-transduction pathways involved in response to SMF [1]. Although these

mechanisms might target different subtle cell functions, their influence (individually

or collectively) was suspected of being observed from cell distribution and

morphology at the end of cell development in vitro. This provides a chance to answer

the question of how osteoblastic cell perceives SMF in vitro by a quantitative

description of osteoblastic cell distribution and magnetic flux intensity. A quantitative

128

description was conceptually built by a hypothesis “window effect”, which indicated

that specific magnetic flux intensities could make biological systems achieve optimal

response [57]. An analogy of the window hypothesis is a quantitative resonant

interaction of magnetic field with ions in biological systems [201]. Different from the

resonant interaction, the window effect implied the optimal response of a biological

system caused not only by the resonance of ions but also by other physiological

processes, and thereby allows multiple optimal responses to a range of magnetic flux

densities. The existence of probability cell density causes the uncertain of position and

velocity. Therefore, we may infer that a strong SMF would generate a fluctuation with

high frequency and small amplitude in cell growth which means that a strong SMF

would not affect cell proliferation but reduce the deviation of cell number between the

exposed group and control group. A study investigated the effects of a strong SMF (8

T) exposure on cultured mouse osteoblastic MC3T3-E1 cells and found that cells were

orientated in the direction parallel to the magnetic field. The interesting fact was that

this strong SMF exposure did not affect cell proliferation regardless of 24-hour or 60-

hour exposure. Also, there was no significant difference between the growth curve for

the exposed and non-exposed groups in the following three days and six days [23].

The underlying mechanisms for the existence of fluctuations in reconstructed dose-

dependent patterns may root in the physical mechanisms of SMFs interacting with the

biological process of bone cells. We examined two primary proposed physical

mechanisms, electrodynamic interactions and magneto-mechanical interactions [202].

The electrodynamic interactions may result in Hall Effect which leads to redistribute

surface charges on cell membrane [203] and may affect bone cell activity by altering

streaming potential. It has been shown that streaming potential can affect bone cell

129

activity in vitro, but how altered streaming potential is perceived by bone cells remains

poorly understood [204].

On the other hand, magneto-mechanical effects are related to the uniformity of SMFs

and inherent magnetic properties of materials. In an inhomogeneous SMF, bone cells

may experience a force (Lorentz force perhaps) and orientate along the direction of the

field gradient. Mouse osteoblastic MC3T3-E1 cells were orientated in the direction

parallel to a strong SMF (8 T) after 60 hours exposure in vitro as well as the orientation

of bone formation in and around the BMP-2/collagen pellets implanted subcutaneously

in vivo [23]. Interestingly, the migration of osteoblastic cells under strong SMF follow

the direction of magnetic flux rather than encounter a Lorentz force in the case that the

motion direction is not parallel to the applied magnetic field.

The diffusion function in osteoblastic cells after the treatment of SMFs indicates that

SMFs can reduce the diffusion activity with increasing magnetic flux. This conclusion

explains the phenomenon that we observed in the experiment. Naturally, the magneto-

mechanical effects are related to the uniformity of SMFs and inherent magnetic

properties of materials. In inhomogeneous SMFs, materials experience a force and

orientate along the direction of the field gradient. If the magnetic gradient is large

enough, diamagnetic levitation can be achieved. In a uniform magnetic field, there is

no force, but magnetic torque, which makes materials with anisotropy of magnetic

susceptibility tend to rotate until they reach a stable orientation. It has been approved

that osteoblasts can be oriented along the magnetic direction under high SMFs [23].

Another possibility is that SMFs can influence the rates of specific chemical reactions

in chemistry and biology because on radical ion pair of the reaction intermediates. The

radical ion pair can be altered by SMFs of modest and weak intensity [205, 206].

130

(A)

(B)

Figure 7.1 Representation of the device used to generate the SMFs. (A) The device

consisted of a signal generator and platform. The platform included a solenoid and two

six-well plates in line. (B) The strength of SMF was measured by a tesla-meter in the

centre of each well, labelled by A1 to A6. Two SMF groups G1 and G2 were treated

on osteoblasts. Two replicates were tested in one experiment, each data point with n=4.

The platform was placed in an incubator of 5% CO2, 37 0C.

131

(A)

(B)

Figure 7.2 Cell morphology of osteoblastic Saos-2 in SMF and control. After SMF

exposure of 0.6 mT, part of the cells formed a whirlpool-like cluster, in contrast to the

control which did not have such formation of the cluster.

132

(A)

(B)

133

(C)

(D)

134

Figure 7.3 Effect of the SMFs on the proliferation of osteoblastic Saos-2. Cells of

G1(A, B) and G2(C, D) were harvested after SMF exposure for accumulated 4 hours

and 8 hours respectively. The number of cells per well in 6-well plate was counted.

Two-tailed t-tests are performed on each intensity versus control and means of them

are not significantly different. Statistical significance is indicated among different

intensities. Data are expressed as the mean (symbols) ± SEMs (error bars), *P<0.05

**P<0.01 ***P<0.001.

135

Figure 7.4 The effect of SMF exposure on cell viability of osteoblastic Saos-2. The

SMF exposure resulted in the lower cell viability compared with the background and

the control. The effect was pronounced in the early stage. Data are expressed as the

mean (symbols) ± SEMs (error bars), *P<0.05 **P<0.01 ***P<0.001.

(A)

(B)

136

(A)

(B)

Figure 7.5 Effect of the SMFs on the cell proliferation of osteoblastic Saos-2 in

combined G1+G2. The picture of G1+G2 was sensitive to the SMF intensities after

the accumulated exposure of 4 hours. The obvious distinction was only shown between

0.55 and 0.6 mT after 8 hours. It was recognised for the feasibility of combined

G1+G2. Data are expressed as the mean (symbols) ± SEMs (error bars), *P<0.05

**P<0.01 ***P<0.001.

137

138

Figure 7.6 The sketch of the mathematical model for the movement of osteoblastic

cells in vitro under SMFs. (A) The movement of the osteoblastic cell is separated into

cellular movement and molecular movement under SMFs for modelling. (B) A logic

concept is built for initialising the mathematical description of modelling, established

in the relationship that the magnetic density can affect the molecular motions and

subsequently affect cellular motions. Cellular motions may reversely influence the

property of SMFs. (C) A realistic scenario is established for enriching the details of

modelling. A logic description can only provide a null hypothesis to the influence of

SMFs on osteoblastic cells, and we further set a detailed scenario to calculate the

cellular probability density in a culture well, for instance, one well in a 6-well plate.

(D) The algorithm of cellular probability density is written in both spatial and temporal

scale. (E) The mathematical model can be solved by fitting the experimental data into

the algorithm.

139

(A)

(B)

140

(C)

(D)

Figure 7.7 The predicted pattern of osteoblastic Saos-2 cell proliferation in weak

SMFs. The prediction is consistent with the experimental data. The oscillation was

evident in the intensities from 2 to 3 mT in G1 (A), and from 0 to 0.5 mT in G2 (B).

The oscillation was reduced by combining G1 and G2 into G1+G2 (C). However, the

oscillation of G1+G2 remained the same as the G1 from 2 to 3 mT. The symmetric

interval was observed with the central of 0.7 mT (D).

141

(A)

(B)

142

Figure 7.8 Parametric analysis. (A) Three coefficients are analysed for the linearity.

The coefficient 1 represents the spatial modification on cell probability density, and

the coefficient 2 represents the temporal modification. The coefficient 3 contributes to

the interference if applicable. The linearity reduces from coefficient 1 to coefficient 3.

The correlation between coefficient 1 and the SMF intensity is linear, and the

coefficient 2 is approximately linear with the SMF intensity. The coefficient 3 is

nonlinear with the SMF intensity which mainly contributes to the interference. (B) The

interference results in various cell probability distribution along the cell plate.

143

Chapter 8

PEMF Effects on Cell Proliferation of

Human Osteoblasts

144

8.1 Introduction

Despite a long history of clinical use of PEMFs and some mechanistic studies, the

optimal parameters for usage remain poorly understood. PEMFs can modulate the

proliferation and differentiation of osteoblasts and alter the morphology and

intracellular calcium levels of osteoblasts. Optimal therapeutic effects of PEMFs

require a series of optimisation in the methodology of treatment. PEMFs obtain various

results at different frequencies and intensities, which also diverse on signal waveform,

signal duration, cell type and cell stage. The study in Chapter 7 indicated that weak

SMFs resulted in the interference of cell probability in osteoblastic Saos-2 cells. A

periodic function was written for the relationship between SMF intensities and cell

population at the time. In this chapter, we investigated the effects of PEMF on cell

proliferation of osteoblastic Saos-2 cells, and further on the formation, mineralisation

and gene expression of primary human osteoblasts (passage 2). The interference of

cell probability was found in term of PEMF frequency and intensity.

8.2 Experimental designs

This study used two cell types, osteoblastic Saos-2 cells and primary human

osteoblasts (passage 2). Saos-2 cells were tested for the effects of PEMF on cell

proliferation. Primary human osteoblasts (passage 2) were used for the observation of

morphology, mineralisation and gene expression. Two types of apparatus were set to

generate PEMFs, a solenoid and Helmholtz coils. The coils were mounted horizontally

on a shelf inside the incubator and driven by a signal generator with alternative current

(AC) (3B Scientific, Germany). The signal function was sinusoidal. The cells were

placed in 6-well plastic culture plates mounted on a platform across the centre of the

coils. The culture plates of 85 mm125 mm were used throughout the experiment. A

self-made magnetic field sensor with an oscilloscope was used to measure the

145

magnetic intensity at the centre of each well (Figure 8.1 A). The effects of PEMF

intensity and frequency were analysed on cultured Saos-2 cells. The exposure method

utilised the decline of magnetic density at the proximal to the solenoid. Six magnetic

flux densities were investigated at the centre of each well: 0 (control), 0.1, 0.15, 0.25,

0.53 and 0.69 mT. The range of frequency was investigated in 40, 80, 160, 320, 640

and 800 Hz (Figure 8.1 B). The exposure was turned on for 2 hours at any 24-hour

interval, envisaging a possible clinical setting application. The control was set in a

separate incubator. 6000 cells per square centimetre were plated into a 6-well and

cultured for five days, in the absence or presence of a PEMF. Cells were settled in day

0, followed by PEMF exposure in day 1-4 and harvested on day 5.

8.3 Experimental results

8.3.1 Effect of PEMF intensity on the proliferation of cultured Saos-2 cells

Compared to the control group, PEMF exposure significantly promoted the cell

proliferation in the combination treatment of 0.15 mT with 40 and 800 Hz, 0.25 mT

with 320 and 800 Hz, and 0.69 mT with 160 Hz (Figure 8.2). No significant difference

was found between the control and PEMF exposure with 80 and 640 Hz. At the same

frequency, the perturbation of the cell population was observed under different

intensities of PEMF which was like the effect of SMF exposure.

8.3.2 Effect of PEMF frequency on the proliferation of cultured Saos-2 cells

At the same intensity of PEMF, the various frequency induced fluctuations around the

control (Figure 8.3). Compared with the range of control, 80, 160 and 640 Hz showed

adverse effects on cell population at 0.1 mT. 40, 320 and 800 Hz indicated positive

impacts on cell population at 0.15 mT. At 0.25 mT, 40 and 800 Hz showed positive

impacts on cell population while 160 Hz had a negative impact. 160 and 640 Hz

146

indicated adverse effects on cell population at 0.53 mT. At 0.69 mT, 80 Hz presented

negative impact on cell population while 160 Hz presented positive effect.

8.3.3 The sensitivity of cultured Saos-2 cell proliferation to PEMF

The sensitivity of Saos-2 cell proliferation to PEMF was illustrated by Figure 8.4 in

the format that the variety of PEMF intensities were correlated with cell proliferation

along the PEMF frequencies. The data on cell proliferation in various PEMF intensities

at the same frequency were collected at the same time. The sensitivity of cell

proliferation to PEMF was evaluated by the continuity of plotted data at each

frequency.

The Saos-2 cell proliferation was not sensitive to PEMF at 40, 80 and 320 Hz.

Compared with the control, the cell proliferation was sensitive to the points of 0.69

mT combined with 160 and 640 Hz, and 0.15 mT combined with 640 Hz. The cell

proliferation was also sensitive to the intervals of 0.25 ~ 0.53 mT at 160 and 640 Hz,

and 0.15 ~ 0.25 mT at 800 Hz. The combination of 0.69 mT and 160 Hz might be

unique since it was a discrete point. The significant perturbation of cell proliferation

was observed at 640 Hz, while the consistency of cell proliferation was observed at

320 Hz. It indicated that the Saos-2 cell proliferation might react to specific PEMF

frequencies by strengthening or weakening the perturbation.

A comparison was made between the sensitivity of cultured Saos-2 cell proliferation

to PEMF generated by a solenoid and that generated by Helmholtz coils (Figure 8.5).

The solenoid created a field gradience of 0.25 mT when the central magnetic flux

density was 0.69 mT. The Helmholtz coils created a consistent field of 0.69 mT. Two

types of PEMFs were carried out in the same protocol and platform. The results

showed a good continuity in data of cell proliferation at 40, 80, 320 and 800 Hz. The

solenoid indicated a better promotion of cell proliferation at 160 and 640 Hz. Weak

147

PEMFs from solenoid and Helmholtz coils presented a similar effect on Saos-2 cell

proliferation. The solenoid could strengthen the perturbation of cell proliferation in

comparison with Helmholtz coils.

8.3.4 Effect of PEMF on the orientation and mineralisation of human osteoblasts

The human osteoblasts presented the rod-like shape in morphology (Figure 8.6).

PEMF exposure of 0.69 mT with 160 Hz for 8 hours induced a variety of the

orientation of osteoblasts. At the same scale, the control illustrated consistent

orientation. After 12-day culture, cells stained by Alizarin red-aligned along a

whirlpool-like circle in PEMF, while the control did not form this unique shape

(Figure 8.7). Interestingly, the SMF exposure also showed a similar orientation pattern

with the group in PEMF. Effect of PEMF and SMF on osteogenesis of human

osteoblasts were assessed by Alizarin red-stained calcified nodules formed. The red

nodule was observed in PEMF exposure of 0.69 mT with 160 Hz, while the control

and the SMF exposure showed the insignificant difference.

8.3.5 Real-time RT-PCR of osteogenesis-related genes

The effects of PEMF and SMF exposure on osteogenesis-related genes of human

osteoblasts were assessed by quantitative reverse transcription polymerase chain

reaction (RT-PCR) with RNA isolation from osteoblasts after 8 hours treatment of

SMF (Table 8-1) and PEMF (Table 8-2). Compared with the control, both PEMF and

SMF exposure at 0.69 mT significantly increased IGF-1 and decreased BMP-1

expression after 8 hours of treatment. However, PEMF significantly increased PHEX

while SMF decreased PHEX expression in comparison with the control. The protein

level of IGF-1 indicated a slight increase by SMF and PEMF compared with the

control, which is not compatible with the significance of RT-PCR result.

148

8.4 Numerical solution of PEMF effects on osteoblastic cell proliferation

8.4.1 Model derivation for PEMF

The parametric functions were extracted by fitting experimental data in both scenarios

of intensity and frequency, which indicated how osteoblastic cell perceived PEMF.

The derivation described the spatial-temporal evolution of cell probability distribution

under the PEMF effect. The fitting created an S-shaped curve for cell proliferation.

PEMF affected osteoblastic cell proliferation by the alteration of cell distribution, and

a Fokker-Planck equation of this stochastic process was written in term of the

probability density function p , magnetic flux B and frequency v ,

( , ) [ ( , ) ( , ) ( , ) ( , )]B Bp x t D p x t p x tt x x

= −

, (8.4.1)

where the evolution of the probability density function was affected by diffusion and

drift. The equation (8.4.1) was incorporated with transitions to simulate the effect of

PEMF on the dynamics of cell distribution. Suppose that the conditional probability

density ip ( i N ) is implemented via an approximate discretisation of the culture

area, and attributed to the magnetic intensity B

ip and frequency v

ip respectively,

1 1 1 1( ) ( )B B B B B

i i i i i i i i ip W p W p W p W p+ − + −

− − + + = − − − , (8.4.2)

and

1 1 1 1( ) ( )i i i i i i i i ip W p W p W p W p + − + −

− − + + = − − − , (8.4.3)

where the diffusion is incorporated into the conditional probability density, and the

drift shows the correlation for neighbour coupling as in an equation (7.4.3). Therefore,

we establish a three-term recurrence relation in a culture area where its radius is

divided into points 0 ... i with a step h . Each point i corresponds to a conditional cell

probability density. After this step, the algorithm guarantees the nearest neighbour

149

coupling, and the whole system of cell probability density can be arranged in a finite

chain. Any cell population N is built with the initial cell population 0N by

integrating the cell probability density p with a spatial component i and temporal

component j ,

0 3 2 1( , ) ( , ) ( ( , ))B B ji B

j i

N N f f p f = , (8.4.4)

where N represented the observed cell population and 0N represented the initial cell

population. Using the matrix for the expression of functions, column entries show

inputs to a process sector while row entries represent outputs from a given sector as

the equation (5.2.17),

1

1

( ) ,

( ) ,

c

c

t t

t t

−

−

=

=

0

0

N I - C N

N I - A DN.

This format shows how dependent each sector is on every other sector. The coefficient

matrix 1( )−I -C and 1( )−I - A are redefined by n square matrix. The actual construction

of the coefficient matrix may require addressing the affected processes in PEMF.

8.4.2 PEMF Interference

The numerical results showed that Saos-2 cell proliferation was not dose-dependent

on PEMF frequency but fitted a sine function in term of PEMF frequency (Figure 8.9).

After 8-hour PEMF exposure, the fitting on PEMF frequency was created in five

equations (Table 8-3). The fitting was stable by comparing the results from PEMF

exposure of different intensities. The data were combined into one picture,

[0.37cos( ) 0.15sin( )] [0.18cos( ) 0.45sin( )]141.3 141.3 0.13 0.13

B BN

= − − + .

(8.4.5)

150

The equation articulated the information derived from five experimental arrangements

in complementarity as the hypothesis in Chapter 3. The equation (8.4.5) established a

pattern of interference which met the whole data set with an adjusted R-square of 0.5

(Figure 8.10 A). The interference inherited the similar properties of the mathematical

description in Chapter 7 (Figure 8.10 B). The metastability was found in the cell

probability in PEMF.

Set

1 2, 141.3 0.13

B

= = , (8.4.6)

the equation (8.4.5) had a general form as

2 1 2 1 2 11 22 sin( )[sin( ) cos( )]

2 2 2N C C

+ − −= − + , (8.4.7)

where C1 and C2 referred to Coefficient 1 and 2. The difference between each dataset

and the standard equation (8.4.5) was expressed in implicit functions (Figure 8.11).

Coefficient 1 indicated numerical distinction near the combination of 0.2 mT and 800

Hz. Coefficient 2 showed a numerical distinction near the combination of 0.3 mT and

300 Hz. The incorporation of two implicit functions into the standard equation built a

general model for analysing the PEMF effect on osteoblastic cell proliferation.

8.5 Discussion

The PEMF interference (8.4.5) represented a consistent description of cell probability

with SMF interference (7.4.9). In comparison with the SMF interference, the PEMF

interference employed two more components for the frequency influence and thereby

using more dataset to create a coefficient matrix. Twelve datasets were used in SMF

while thirty datasets were used in PEMF. Interestingly, the number of datasets was

proportional to the number of gene expression alteration in Table 8-1 and Table 8-2.

151

Interaction with signalling pathways is a potential mechanism by which deficient

energy PEMF might produce metabolic responses. Hormone and neurotransmitter

receptors are specialised protein molecules involved in a variety of biochemical

processes to passing chemical signals from the outside of a cell across the plasma

membrane to the interior. Since weak PEMFs have too little energy to traverse the

membrane directly, it is possible that they may modify the existing signal transduction

process in cell membranes, providing both transduction and biochemical amplification

[207].

We postulate that the observed effects of PEMF on bone cell metabolism correlate to

PEMF coupling with signal transduction which contributes to the hormonal regulation

of osteoblast proliferation and differentiation. Compared with the SMF, the PEMF

coupling covered a wide range of signal pathways in Table 8-2. For example, PHEX

was upregulated after the PEMF exposure, which was reported to enhance osteogenic

differentiation, extracellular matrix deposition and mineralisation [208]. We also

observed the enhanced mineralisation in the PEMF group, yet the insignificant

difference between the SMF group and the control (Figure 8.7). Hence, the PEMF

coupling exhibited stronger influence on calcium-dependent transduction than the

SMF coupling.

The phenomenon of signal transduction is central to a wide range of cellular activities

triggered by ligand-gated binding of hormones, antigen molecules, growth factor and

other cell-surface agonists. Calcium ions appear to be essential in the first steps of

transduction coupling of exogenous physical signals at the cell membrane and the

ensuing steps of calcium-dependent signalling in intracellular enzyme systems. The

modulation of calcium signalling by PEMF is thereby suggested to be a plausible

candidate for the activation of biochemical reactions. PEMF coupling with cellular

152

targets may occur via highly cooperative steps. For example, calcium-dependent steps

in the target pathway may include, (i) initial detection of PEMF at specific binding

sites with resultant electrochemical changes, (ii) membrane-bound proteins signalling

to the cell interior, and (iii) PEMF coupling with the cytoskeleton and other subcellular

constituents. Consequently, the PEMF coupling builds a hierarchical system from top

to bottom, starting with the calcium signalling. The depth of the hierarchical system

may depend on the magnetic intensity and frequency. At the same level of magnetic

intensity, the PEMF coupling illustrates no difference than the SMF coupling with

signal transductions (Figure 8.8), which implies that frequency may not affect the

activation of biochemical reactions. The frequency of PEMF may provide an

opportunity to couple with a series of pathways such as IGFs (Table 8-2). This

mechanism gives the PEMF an ability to influence the immune system (Figure 2.2).

An impact on the immune system leads to more evident results in vivo than in vitro,

which explains that the PEMF has significant therapeutic results on bone healing for

animals (Table 1-1) rather than the increase on cell proliferation of cultured

osteoblasts.

153

(A)

(B)

Figure 8.1 Representation of the device used to generate the PEMFs. (A) The device

consisted of a signal generator and platform. The platform contained a solenoid or

Helmholtz coils. (B) The PEMF intensity was measured by a tesla-meter in the centre

of each well, labelled by A1 to A6. Two replicates were tested in one experiment and

each data point with n=4. The platform was placed in an incubator of 5% CO2, 37 0C.

154

Figure 8.2 Effect of the PEMF intensity on the proliferation of cultured Saos-2 cells.

In each experiment, osteoblasts were exposed to an EMF with a fixed frequency.

Exposures were applied on osteoblasts for 2 hours per day and continued for four days.

The data are represented by mean ± SEMs (n=4). PEMF groups were compared with

the control, *P<0.05 or ** P<0.01.

155

Figure 8.3 Effect of the PEMF frequency on the proliferation of cultured Saos-2 cells.

Perturbations were induced by different frequencies of PEMF exposures for 8 hours.

The control was shown by an interval of mean ± SEMs (n=24). Six frequencies 40, 80,

160, 320, 640 and 800 Hz were compared at each PEMF intensity respectively. The

curves indicated the frequency-dependent tendency. Results are presented as mean ±

SEMs (n=4).

156

Figure 8.4 Data distribution of PEMF groups along the control. The data on cell

proliferation in various PEMF intensities at the same frequency were collected at the

same time. The sensitivity of cell proliferation to PEMF was evaluated by the

continuity of plotted data at each frequency. The data are represented by mean ±SEMs.

Each point in the EMF group includes n=4, and the interval of control includes n=24.

157

Figure 8.5 A comparison between cultured Saos-2 cell proliferation to PEMF

generated by a solenoid and that generated by Helmholtz coils. The solenoid created a

gradient field with the central magnetic flux density of 0.69 mT. The Helmholtz coils

created a consistent field of 0.69 mT. Two types of PEMFs were carried out in the

same protocol and platform. The results showed a good continuity in data of cell

proliferation at 40, 80, 320 and 800 Hz. The data are represented by mean ± SEMs.

Each point in the EMF group includes n=4, and the interval of control includes n=24.

158

(A)

(B)

Figure 8.6 PEMF effect on the orientation of human osteoblasts. (A) The human

osteoblasts presented the rod-like shape in morphology. (B) PEMF exposure of 0.69

mT with 160 Hz for 8 hours induced a variety of the orientation of osteoblasts.

159

(A) Control (1 mm)

(B) Control (500 µm)

(C) PEMF 0.69 mT,160 Hz (1 mm)

(D) PEMF 0.69 mT,160 Hz (500 µm)

(E) SMF 0.69 mT (1 mm)

(F) SMF 0.69 mT (500 µm)

Figure 8.7 Effect of PEMF and SMF on osteogenesis of human osteoblasts assessed

by Alizarin red-stained calcified nodules. The red nodule was observed in PEMF

exposure of 0.69 mT with 160 Hz, while the control and the SMF exposure showed

the insignificant difference.

160

(A) IGF-1 10 µg

(B) IGF-1 30 µg

Figure 8.8 Western blot analysis of IGF-1 protein at a loading of 10 and 30 µg. The

protein level of IGF-1 indicated a slight increase by SMF and PEMF compared with

the control, which is not compatible with the significance of RT-PCR result.

161

Figure 8.9 Waveform of frequency. The numerical results showed that Saos-2 cell

proliferation was not dose-dependent on PEMF frequency but fitted a sine curve in

term of PEMF frequency. After 8-hour PEMF exposure, the fitting on PEMF

frequency was created in five wave equations. The fitting was stable by comparing the

results from PEMF exposure of different intensities.

162

(A)

(B)

Figure 8.10 The interference of cell probability in PEMF. (A) The graph articulated

the data derived from five experimental arrangements, which established a normalised

pattern of interference (B) on the whole dataset with an adjusted R-square of 0.5. The

interference inherited the mathematical description.

163

(A) Coefficient 1

(B) Coefficient 2

164

Figure 8.11 Implicit functions of coefficient 1 and 2. The metastability was found in

the cell probability in PEMF. The implicit functions of Coefficient 1 and 2 were

expressed in graphs. Coefficient 1 (A) indicated numerical distinction near the

combination of 0.2 mT and 800 Hz. Coefficient 2 (B) showed a numerical distinction

near the combination of 0.3 mT and 300 Hz. The incorporation of two implicit

functions into a normalised pattern created a general model for analysing the PEMF

effect on osteoblastic cell proliferation.

165

Table 8-1 The effects of SMF exposure on osteogenesis-related genes of human

osteoblasts. SMF treated cells with the setting of a magnetic flux density of 0.69 mT

for 8 hours in 5 days. Fold difference is 1.5.

Genes Under-Expressed in

Genes Over-Expressed in

Gene Symbol Fold Gene Symbol Fold

ALPL -2.0 BMP4 1.9

BMP1 -1.8 BMP6 1.6

CSF3 -2.0 IGF1 2.6

CTSK -2.1

MMP9 -2.0

PHEX -1.5

VCAM1 -1.7

Table 8-2 The effects of PEMF exposure on osteogenesis-related genes of human

osteoblasts. PEMF treated cells with the setting of the magnetic flux density of 0.69

mT and a frequency of 160 Hz for 8 hours in 5 days. Fold difference is 1.5.

Genes Under-Expressed in

Genes Over-Expressed in

Gene Symbol Fold Gene Symbol Fold

AHSG -1.7 BMPR1B 1.5

BMP1 -1.6 CSF2 2.7

COL2A1 -2.6 FLT1 2.2

COL5A1 -3.0 IGF1 1.9

DLX5 -1.6 MMP8 1.5

IGF2 -1.5 PHEX 1.5

ITGA3 -1.9 TGFB3 1.6

166

Table 8-3 The frequency functions.

Function: sin( )cc

xN y A f

w w

= + −

yc A xc w Adj.R-square

0.1 mT 2.45 0.43 -88.78 142.27 0.84

0.15 mT 2.90 -0.34 473.23 184.92 0.79

0.25 mT 2.77 0.69 215.49 135.61 0.98

0.53 mT 2.47 0.24 153.83 0.40 0.98

0.69 mT 2.78 0.55 179.55 215.99 0.0035

167

Chapter 9

PEMF and SMF Effects on Co-culture of

Human Osteoblasts and Osteoclasts

168

9.1 Introduction

Bone remodelling plays an essential role in the maintenance of the integrity of skeletal

structures spatially and temporally, which is a coordinated process involving bone cells

in BMU. Osteoblasts and osteoclasts work in a coupling following an activation-

resorption-formation sequence [138]. This coupling has been simulated quantitatively

in signalling-based models [144, 209]. Signalling-based models are developed to

interpret bone remodelling from physiological aspects [142]. Parathyroid hormone

(PTH) is considered to mediate the underlying mechanisms of bone remodelling, and

the temporal effect of PTH is investigated as the key to establish a nonlinear

remodelling process. Signalling interactions among osteoblasts and osteoclasts, such

as RANK, RANKL and OPG signalling pathways, are also incorporated into the

mathematical model for bone remodelling [146]. However, signalling-based models

inevitably introduce unknowns for data weights of signalling factors and raise a

question of how to balance signalling weights in a network.

In Chapter 3 and 5, we proposed a hypothesis and two frameworks for solving joint

cell possibility in EMFs. The effects of the hypothesis and frameworks aim to describe

the coupling between osteoblasts and osteoclasts mathematically by path integration

instead of signalling weight. The path integration deals with cell movement by

physical principles, while the signalling weight must face biochemical reactions.

Therefore, path integration could satisfy the simplicity of formulation. In this chapter,

we solved the framework proposed in Chapter 5 under a simplified circumstance. Even

so, a prediction was presented for showing the change of relative positions between

osteoblasts and osteoclasts in EMFs. Further, we examined the prediction by the

experiment of osteoblasts and osteoclasts co-culture in EMFs.

169

9.2 Prediction

The exploration of the physical structure of bone cells discussed here is not limited to

single cell type but expanded to the interactions of two cell types. To simplify the

analysis without loss of generality, we omit the influence of cell size and membrane

and study the relative positions between osteoblasts and osteoclasts. Under this

assumption, we consider the interaction between osteoblasts and osteoclasts in a two-

dimensional picture in which osteoblasts and osteoclasts are put separately into two

different layers (Figure 9.1). In each layer, an imaginary grid is built with black or

grey nodes. After osteoblasts and osteoclasts occupy the nodes of grids, their

generalised mechanical properties are automatically defined. Consequently, we denote

cell probability of osteoblasts by a vector ( , )r tOB and cell probability of osteoclasts

by ( , )r tOC , where r represents spatial parameter and t represents temporal

parameter. Ideally, ( , )r tOB and ( , )r tOC attach to each node of black and grey grid

respectively, which can be solved by finite difference methods (FDM).

9.2.1 The numerical interaction between osteoblasts and osteoclasts

Although cell probability ( , )r tOB and ( , )r tOC are expected to be inhomogeneous,

the cell probabilities have the same temporal coordinate. In other words, time might

be the only consistent coordinate for different cell types. The temporal derivative of

( , )r tOB is related to the spatial derivative of ( , )r tOC when they have interactions

with each other, and vice-versa. This property is expressed as follows:

( , ) ( , )

( , ) ( , )

OC

OB

r t r tt x

r t r tt x

+

−

OB OC

OC OB

. (9.2.1)

We assume that the relations of (9.2.1) have a general form of equations,

170

( , ) ( ) ( , ) ( , )a

r t r r t r tt r

= +

OB OC , (9.2.2)

and

( , ) ( ) ( , ) ( , )b

r t r r t r tt r

− = +

OC OB . (9.2.3)

Where and are coefficients only containing a spatial variable, ( , )r t and

( , )r t contain both spatial and temporal variables. The solution relies on the

transformation in the spatial coordinate. On a finite domain, when , (1,2]a b ,

equations in (9.2.3) are fractional diffusion equation. When 1a b= = , we can solve

equations with finite-difference-time-domain (FDTD) under the simplest assumption,

1, ( , ) 0, ( , ) 0r t r t = = = = . (9.2.4)

Then,

( , ) ( ) ( , )r t a r r tt r

=

OB OC , (9.2.5)

( , ) ( ) ( , )r t b r r tt x

− =

OC OB . (9.2.6)

Following the construction of FDTD, the temporal derivative of ( , )r tOB with second-

order accuracy is approximated，

1

( , / 2) ( ( , ) ( , ))r t t r t t r tt t

+ = + −

OB OB OB . (9.2.7)

At the same temporal point, the spatial derivative of ( , )r tOC is evaluated with the

same method as:

1

( , / 2) [ ( / 2, / 2) ( / 2, / 2)]r t t r r t t r r t tr r

+ = + + − − +

OC OC OC

(9.2.8)

171

( , ) ( , ) a( ) [ ( / 2, / 2) ( / 2, / 2)]t

r t t r t r r r t t r r t tx

+ = + + + − − +

OB OB OC OC

(9.2.9)

Note that the values of r and t exist on the discrete points in space and time. We

continue to update the second equation from time to time / 2t t+ at the spatial point

/ 2r r− with a similar method, and obtain:

( / 2, / 2) ( / 2, / 2) b( ) [ ( , ) ( , )]t

r r t t r r t t r r r t r tx

− + = − − + − −

OC OC OB OB

(9.2.10)

/r t is defined as the diffusion velocity, which is proportional to cell proliferation

during a finite time domain. Then a( ) /r t r is inversely proportional to cell

proliferation of osteoclasts during a finite time domain and b( ) /r t r is inversely

proportional to cell proliferation of osteoblasts during a finite time domain. In the

program of simulation, the finite time domain is realised by several time steps.

Therefore, more time steps are divided in a finite time domain, less cell proliferation

of bone cells would be at each time step, which directly affects the magnitude of

a( ) /r t r and b( ) /r t r . It is noted that their magnitude may not be constant for

each time step since the cell proliferation is nonlinear.

The equations can exist only if the initial ( , )r tOB and ( , )r tOC keep stable.

Considering the cell cycle of osteoblasts and osteoclasts, they can only survive during

a limited time. Hence the time step is limited. After cell death, the system needs to be

re-initialised. However, history affects the re-initialisation and leads to the

accumulation in a new time step.

172

9.2.2 Numerical results

The boundary of ( , )r tOB in the numerical simulation was set as 21/( t) (0,1] , and

positions of cells were expressed as lg( )r . The magnitude of lg( / )r t determined the

relative position of OB and OC . When lg( / )r t increased from 1 to 9, the spatial

interval between OB and OC was minimised and the interference of cell probability

occurred (Figure 9.2). The function lg( / )r t was evaluated as a factor of diffusion.

The factor of diffusion profoundly affected the interference between cell probability

of osteoblasts and osteoclasts. High diffusion resulted in substantial entanglement

among osteoblasts and osteoclasts. In contrast, low diffusion led to spatial separation

among osteoblasts and osteoclasts. When osteoblasts and osteoclasts are cultured in

EMFs, the diffusion may vary from the situations of on/off EMFs, which forms a co-

existence of high and low diffusion. This circumstance generates interference in a

wave pattern over the cell culture. Therefore, entanglement and separation between

cells are expected to exist at the same time.

9.3 Validation in experimental images

Co-culture of human osteoblasts and osteoclasts began with separately culturing

osteoblasts and osteoclasts with the protocols in Chapter 4. Co-culture was initiated on

Day 8 by seeding osteoblasts on the plate of RANKL induced PBMC. The medium of

co-culture used completed osteoclast cell culture medium with RANKL. The medium

was changed every three days. The EMFs exposure started from day 9, and the

experiment finished on day 14. The co-culture was exposed under SMF of 0.69 mT

and PEMF of 0.69 mT with 160 Hz.

173

9.3.1 TRAP staining

TRAP staining on osteoclasts was performed in co-culture of human osteoblasts

(passage 2) and human osteoclasts in control, SMF and PEMF exposure (Figure 9.3).

The intensity of SMF exposure was 0.69 mT, and the PEMF exposure was at 0.69 mT

with 160 Hz. The substantial entanglement of cells was observed in the group under

the SMF and PEMF exposure (Figure 9.3 E-L). The entanglement illustrated that

osteoblasts were more likely to contact with osteoclasts under the SMF (Figure 9.3 J)

and PEMF (Figure 9.3 F) in comparison with the control (Figure 9.3 B). The

entanglement with the PEMF exposure promoted the forming of mature osteoclasts in

Figure 9.3 G, while inhabited the forming of mature osteoclasts in Figure 9.3 E. The

entanglement with the SMF exposure showed similar results. Moreover, the SMF

exposure limited the size of mature osteoclasts in contrast to the control and PEMF.

9.3.2 Fluorescence-based staining and flow cytometry analysis

The fluorescence-based staining illustrated similar results with TRAP staining.

Compared with the control (Figure 9.4 A), the SMF (Figure 9.4 E) and PEMF

exposure (Figure 9.4 C) limited the size of mature osteoclasts. Additionally, the SMF

exposure inhabited the forming of mature osteoclasts in contrast to PEMF exposure.

The results of flow cytometry presented the promotion on cell proliferation of

osteoblasts under the PEMF (Figure 9.4 D) and SMF exposure (Figure 9.4 F). The

SMF exposure resulted in higher cell proliferation of osteoblasts in contrast to PEMF

exposure. The observation indicated that the reduction of forming mature osteoclasts

companied with the increase in cell proliferation of osteoblasts in co-culture under

EMFs, which is consistent with the theoretical interaction between osteoblasts and

osteoclasts in the equation (9.2.1). The interference of cell probability in EMFs could

contribute to the separation of cell groups in Figure 9.4 D, F.

174

9.3.3 Scanning electron microscope (SEM) images

The SEM images were taken to elucidating the possible cellular interaction in SMF

and PEMF exposure (Figure 9.5). The co-culture of osteoblasts and osteoclasts in

control distributed evenly (Figure 9.5 A, B). After the PEMF and SMF exposure, the

overlapping cells were observed (Figure 9.5 C-F). The overlap was shown densely

after the SMF exposure (Figure 9.5 E, F) which could be the reason for preventing the

forming of mature osteoclasts. The dense overlap of cells after the SMF exposure also

showed that magnetic intensity worked in a major role for the cellular contact among

bone cells. The interference of cell probability in EMFs may originate from the overlap

of cells which is equivalent to the superposition of cell probability in Chapter 3.

175

Figure 9.1 Imaginary grid of osteoblasts (OB) and osteoclasts (OC). Two types of

cells are assigned to black and grey nodes respectively. Any given node of the black

grid is surrounded by neighbours that belong to the grey grid. The spatial variables

change with time.

176

(A) Initial stage

(B) lg( / ) 3r t =

(C) lg( / ) 6r t =

(D) lg( / ) 9r t =

Figure 9.2 Prediction of the relative position between osteoblasts ( OB ) and

osteoclasts ( OC ). When lg( / )r t increases from 1 to 9, the intervals between

osteoblasts and osteoclasts tend to be minimised, which leads to the interference of

cell probability. The function lg( / )r t contains the factor of diffusion. The numerical

results illustrated that high diffusion resulted in a substantial interference of osteoblasts

and osteoclasts.

177

(A) Control 1; (100 µm)

(B) Control 1; (50 µm)

(C) Control 2; (100 µm)

(D) Control 3; (100 µm)

(E) PEMF 1;

0.69 mT, 160 Hz (100 µm)

(F) PEMF 1;

0.69 mT, 160 Hz (50 µm)

178

(G) PEMF 2;

0.69 mT, 160 Hz (100 µm)

(H) PEMF 3;

0.69 mT, 160 Hz (100 µm)

(I) SMF 1; 0.69 mT (100 µm)

(J) SMF 1; 0.69 mT (50 µm)

(K) SMF 2; 0.69 mT (100 µm)

(L) SMF 3; 0.69 mT (100 µm)

179

Figure 9.3 TRAP staining on osteoclasts performed in co-culture of human osteoblasts

and osteoclast in control, SMF and PEMF exposure. The SMF exposure was at the

intensity of 0.69 mT, and the PEMF exposure was at 0.69 mT with 160 Hz. The active

interference was observed in the group under SMF and PEMF exposure. Compared

with PEMF exposure, SMF exposure showed substantial entanglement.

180

(A) Control

(B) Control

(C) PEMF 0.69 mT, 160 Hz

(D) PEMF 0.69 mT, 160 Hz

(E) SMF 0.69 mT

(F) SMF 0.69 mT

181

Figure 9.4 The fluorescence-based staining in control, SMF and PEMF exposure.

Compared with the control, the SMF and PEMF exposure prevented the forming of

mature osteoclasts. SMF and PEMF exposure inhabited the forming of mature

osteoclasts while promoted cell proliferation of osteoblasts. This phenomenon is

consistent with the prediction of spatial interaction between osteoblasts and osteoclasts

182

(A) Control (2k x)

(B) Control (7k x)

(C) PEMF 0.69 mT, 160 Hz (2k x)

(D) PEMF 0.69 mT, 160 Hz (7k x)

(E) SMF 0.69 mT (2k x)

(F) SMF 0.69 mT (10k x)

183

Figure 9.5 Scanning electron microscope (SEM) images of osteoblasts and osteoclasts

co-culture in control, SMF and PEMF exposure. The co-culture of osteoblasts and

osteoclasts in control distributed evenly. In the PEMF and SMF exposure, the

overlapping cells were observed. The overlap was shown densely after the SMF

exposure which could be the reason for preventing the forming of mature osteoclasts.

184

Chapter 10

Conclusions

185

In conclusion, this thesis achieved the following objectives.

1. A weak PEMF and SMF system is designed and constructed. The exposure

system for studying the effects of weak PEMF and SMF on bone cells was

constructed with self-design parts and commercial devices. The exposure

chamber is switchable between the solenoid and Helmholtz coil systems. The

magnetic flux can be either a gradient field or a uniform field. The structural

design of the exposure system allows experiments to be conducted at a proper

temperature on a maximum of four 6-well plates at the same time.

2. A mathematical modelling is built for articulating different dataset into one

picture. A general model was established for analysing the experimental data

in PEMF and SMF exposure. This model can be incorporated with the various

framework and able to calculate the numerical results in the various scenario

of the experimental environment.

3. Experimental effects of weak PEMF and SMF on osteoblasts and osteoclasts

are evaluated at various parameters, including (i) type of magnetic fields, (ii)

magnetic flux density, (iii) frequency, (iv) exposure duration, and (v) spatial

gradient (dB/dx). For SMF, two gradient fields of SMFs were applied to

cultured Saos-2 cells. The effects of cell proliferation were examined after the

exposure of 4 hours and 8 hours respectively. The first gradient field contained

intensities at the centre of the culture well with 2, 1.1, 0.6, 0.48, 0.37 and 0.3

mT. The second gradient field contained intensities at the centre of the culture

well with 2.5, 1.6, 1, 0.75, 0.55, and 0.42 mT. For PEMF, the effects of

sinusoidal PEMF intensity and frequency were analysed on cultured Saos-2

cells. Six magnetic flux densities were investigated at the centre of each well:

186

control, 0.1, 0.15, 0.25, 0.53 and 0.69 mT. The range of frequency was

investigated in 40, 80, 160, 320, 640 and 800 Hz.

4. The interference term is derived from the mathematical description of cell

behaviour in EMF. Cell probability was proposed to be evolving with spatial

and temporal factors. Different results could be obtained when the experiment

was carried out in different places and measured at a different time. The

interference should satisfy a differential equation which could be a wave

equation analogous to the equation for electromagnetic waves. However, it

must be emphasised that this function is not a real wave in space even when it

satisfies the wave equation. The existence of interference assumed that the cells

were equivalent to particles with a very long wavelength. Hence the

interference can be observed in the experiment if the various experimental

arrangement were set up at the same time.

5. The cell proliferation of osteoblasts and co-culture of osteoblasts and

osteoclasts are affected by the mutual interference generated in the

environment of SMFs and PEMFs. The proof included the numerical analysis

of cultured Saos-2 cell proliferation, morphology of human osteoblasts and

imaging evidence in co-culture of osteoblasts and osteoclasts. Based on the

observation of interference term in cell proliferation, the optimised EMF

exposure for promoting the growth of bone cells in vitro might be obtained by

the combination of various PEMFs or adoption of gradient SMFs.

187

References

188

[1] P. Kovacic, R. Somanathan, Electromagnetic fields: mechanism, cell signaling,

other bioprocesses, toxicity, radicals, antioxidants and beneficial effects,

Journal of Receptors and Signal Transduction, 30(4) (2010) 214-226.

[2] R. Goodman, M. Blank, Insights into electromagnetic interaction mechanisms,

Journal of cellular physiology, 192(1) (2002) 16-22.

[3] K. Hug, M. Röösli, Therapeutic effects of whole‐body devices applying pulsed

electromagnetic fields (PEMF): A systematic literature review,

Bioelectromagnetics, 33(2) (2012) 95-105.

[4] R.H. Funk, T. Monsees, N. Ozkucur, Electromagnetic effects - From cell biology

to medicine, Progress in histochemistry and cytochemistry, 43(4) (2009) 177-

264.

[5] R.G. Martin, Electromagnetic field theory for physicists and

engineers:Fundamentals and Applications, 2006.

[6] K.J. Lohmann, Q&A: Animal behaviour: Magnetic-field perception, Nature,

464(7292) (2012) 1140.

[7] L. Dini, L. Abbro, Bioeffects of moderate-intensity static magnetic fields on cell

cultures, Micron, 36(3) (2005) 195-217.

[8] A.D. Rosen, Mechanism of action of moderate-intensity static magnetic fields on

biological systems, Cell Biochemistry and Biophysics, 39(2) (2003) 163-173.

[9] A. Liboff, S. Cherng, K. Jenrow, A. Bull, Calmodulin‐dependent cyclic

nucleotide phosphodiesterase activity is altered by 20 μT magnetostatic

fields, Bioelectromagnetics, 24(1) (2003) 32-38.

[10] M. Buemi, D. Marino, G. Di Pasquale, F. Floccari, M. Senatore, C. Aloisi, F.

Grasso, G. Mondio, P. Perillo, N. Frisina, Cell proliferation/cell death balance

in renal cell cultures after exposure to a static magnetic field, Nephron, 87(3)

(2001) 269-273.

[11] H. Sonnier, O. Kolomytkin, A. Marino, Action potentials from human

neuroblastoma cells in magnetic fields, Neuroscience Letters, 337(3) (2003)

163-166.

[12] A.D. Rosen, Inhibition of calcium channel activation in GH3 cells by static

magnetic fields, Biochimica et Biophysica Acta (BBA)-Biomembranes,

1282(1) (1996) 149-155.

[13] A.D. Rosen, Effect of a 125 mT static magnetic field on the kinetics of voltage

activated Na+ channels in GH3 cells, Bioelectromagnetics, 24(7) (2003) 517-

523.

[14] H.M. Huang, S.Y. Lee, W.C. Yao, C.T. Lin, C.Y. Yeh, Static magnetic fields

up-regulate osteoblast maturity by affecting local differentiation factors,

Clinical Orthopaedics and Related Research, 447 (2006) 201-208.

[15] K.H. Chiu, K.L. Ou, S.Y. Lee, C.T. Lin, W.J. Chang, C.C. Chen, H.M. Huang,

Static magnetic fields promote osteoblast-like cells differentiation via

increasing the membrane rigidity, Annals of Biomedical Engineering, 35(11)

(2007) 1932-1939.

[16] C. Cunha, S. Panseri, M. Marcacci, A. Tampieri, Evaluation of the effects of a

moderate intensity static magnetic field application on human osteoblast-like

cells, American Journal of Biomedical Engineering, 2(6) (2012) 263-268.

[17] S.W. Feng, Y.J. Lo, W.J. Chang, C.T. Lin, S.Y. Lee, Y. Abiko, H.-M. Huang,

Static magnetic field exposure promotes differentiation of osteoblastic cells

grown on the surface of a poly-L-lactide substrate, Medical & Biological

Engineering & Computing, 48(8) (2010) 793-798.

189

[18] P. Cooke, P. Morris, The effects of NMR exposure on living organisms. II. A

genetic study of human lymphocytes, The British Journal of Radiology,

54(643) (1981) 622-625.

[19] S. Pacini, G.B. Vannelli, T. Barni, M. Ruggiero, I. Sardi, P. Pacini, M. Gulisano,

Effect of 0.2 T static magnetic field on human neurons: remodeling and

inhibition of signal transduction without genome instability, Neuroscience

Letters, 267(3) (1999) 185-188.

[20] S. Pacini, M. Gulisano, B. Peruzzi, E. Sgambati, G. Gheri, S.G. Bryk, S.

Vannucchi, G. Polli, M. Ruggiero, Effects of 0.2 T static magnetic field on

human skin fibroblasts, Cancer Detection and Prevention, 27(5) (2003) 327-

332.

[21] L. Teodori, J. Grabarek, P. Smolewski, L. Ghibelli, A. Bergamaschi, M. De

Nicola, Z. Darzynkiewicz, Exposure of cells to static magnetic field

accelerates loss of integrity of plasma membrane during apoptosis,

Cytometry, 49(3) (2002) 113-118.

[22] M. Iwasaka, S. Ueno, H. Tsuda, Effects of magnetic fields on fibrinolysis,

Journal of Applied Physics, 75(10) (1994) 7162-7164.

[23] H. Kotani, H. Kawaguchi, T. Shimoaka, M. Iwasaka, S. Ueno, H. Ozawa, K.

Nakamura, K. Hoshi, Strong static magnetic field stimulates bone formation

to a definite orientation in vitro and in vivo, Journal of Bone and Mineral

Research, 17(10) (2002) 1814-1821.

[24] J. Wiskirchen, E. Groenewaeller, R. Kehlbach, F. Heinzelmann, M. Wittau, H.

Rodemann, C. Claussen, S. Duda, Long‐term effects of repetitive exposure

to a static magnetic field (1.5 T) on proliferation of human fetal lung

fibroblasts, Magnetic Resonance in Medicine: An Official Journal of the

International Society for Magnetic Resonance in Medicine, 41(3) (1999) 464-

468.

[25] R.R. Raylman, A.C. Clavo, R.L. Wahl, Exposure to strong static magnetic field

slows the growth of human cancer cells in vitro, (1996).

[26] C. Aldinucci, J.B. Garcia, M. Palmi, G. Sgaragli, A. Benocci, A. Meini, F.

Pessina, C. Rossi, C. Bonechi, G.P. Pessina, The effect of strong static

magnetic field on lymphocytes, Bioelectromagnetics: Journal of the

Bioelectromagnetics Society, The Society for Physical Regulation in Biology

and Medicine, The European Bioelectromagnetics Association, 24(2) (2003)

109-117.

[27] H. Miyamoto, H. Yamaguchi, T. Ikehara, Y. Kinouchi, Effects of

electromagnetic fields on K+(Rb+) uptake by HeLa cells, in: Biological

effects of magnetic and electromagnetic fields, Springer, 1996, pp. 101-119.

[28] M. Iwasaka, K. Yamamoto, J. Ando, S. Ueno, Verification of magnetic field

gradient effects on medium convection and cell adhesion, Journal of Applied

Physics, 93(10) (2003) 6715-6717.

[29] H. Hirose, T. Nakahara, J. Miyakoshi, Orientation of human glioblastoma cells

embedded in type I collagen, caused by exposure to a 10 T static magnetic

field, Neuroscience Letters, 338(1) (2003) 88-90.

[30] Y. Eguchi, M. Ogiue-Ikeda, S. Ueno, Control of orientation of rat Schwann cells

using an 8-T static magnetic field, Neuroscience Letters, 351(2) (2003) 130-

132.

[31] Z. Zhao, J. Tao, J. Su, C. Ai, Y. Liu, J. Wang, The Design and Measurement of

Pulsed Magnetic Field, in: Measuring Technology and Mechatronics

190

Automation (ICMTMA), 2011 Third International Conference on, 2011, pp.

744-747.

[32] W. Latham, J.T.C. Lau, Bone Stimulation: A Review of Its Use as an Adjunct,

Techniques in Orthopaedics, 26(1) (2011) 14-21.

[33] L.Y. Sun, D.K. Hsieh, P.C. Lin, H.T. Chiu, T.W. Chiou, Pulsed electromagnetic

fields accelerate proliferation and osteogenic gene expression in human bone

marrow mesenchymal stem cells during osteogenic differentiation,


[34] B. Tenuzzo, A. Chionna, E. Panzarini, R. Lanubile, P. Tarantino, B.D. Jeso, M.

Dwikat, L. Dini, Biological effects of 6 mT static magnetic fields: a

comparative study in different cell types, Bioelectromagnetics, 27(7) (2006)

560-577.

[35] V. Sollazzo, G.C. Traina, M. DeMattei, A. Pellati, F. Pezzetti, A. Caruso,

Responses of human MG‐63 osteosarcoma cell line and human osteoblast‐like cells to pulsed electromagnetic fields, Bioelectromagnetics: Journal of

the Bioelectromagnetics Society, The Society for Physical Regulation in

Biology and Medicine, The European Bioelectromagnetics Association, 18(8)

(1997) 541-547.

[36] C.T. Brighton, W. Wang, R. Seldes, G. Zhang, S.R. Pollack, Signal transduction

in electrically stimulated bone cells, Journal of Bone and Joint Surgery-

American Volume, 83(10) (2001) 1514-1523.

[37] L. Massari, F. Benazzo, M. De Mattei, S. Setti, M. Fini, Effects of electrical

physical stimuli on articular cartilage, Journal of Bone and Joint Surgery-

American Volume, 89(suppl_3) (2007) 152-161.

[38] R.K. Aaron, D.M. Ciombor, G. Jolly, Stimulation of experimental endochondral

ossification by low‐energy pulsing electromagnetic fields, Journal of Bone

and Mineral Research, 4(2) (1989) 227-233.

[39] R.K. Aaron, D.M. Ciombor, H. Keeping, S. Wang, A. Capuano, C. Polk, Power

frequency fields promote cell differentiation coincident with an increase in

transforming growth factor‐β1 expression, Bioelectromagnetics, 20(7)

(1999) 453-458.

[40] R.J. Fitzsimmons, J.T. Ryaby, F.P. Magee, D.J. Baylink, IGF‐II receptor

number is increased in TE‐85 osteosarcoma cells by combined magnetic

fields, Journal of Bone and Mineral Research, 10(5) (1995) 812-819.

[41] R. Selvam, K. Ganesan, K.N. Raju, A.C. Gangadharan, B.M. Manohar, R.

Puvanakrishnan, Low frequency and low intensity pulsed electromagnetic

field exerts its anti-inflammatory effect through restoration of plasma

membrane calcium ATPase activity, Life sciences, 80(26) (2007) 2403-2410.

[42] M.L. Pall, Electromagnetic fields act via activation of voltage‐gated calcium

channels to produce beneficial or adverse effects, Journal of Cellular and

Molecular Medicine, 17(8) (2013) 958-965.

[43] B. Noriega-Luna, M. Sabanero, M. Sosa, M. Avila-Rodriguez, Influence of

pulsed magnetic fields on the morphology of bone cells in early stages of

growth, Micron, 42(6) (2011) 600-607.

[44] J. Zhou, L.G. Ming, B.F. Ge, J.Q. Wang, R.Q. Zhu, Z. Wei, H.P. Ma, C.J. Xian,

K.M. Chen, Effects of 50 Hz sinusoidal electromagnetic fields of different

intensities on proliferation, differentiation and mineralization potentials of rat

osteoblasts, Bone, 49(4) (2011) 753-761.

191

[45] M. De Mattei, A. Caruso, G.C. Traina, F. Pezzetti, T. Baroni, V. Sollazzo,

Correlation between pulsed electromagnetic fields exposure time and cell

proliferation increase in human osteosarcoma cell lines and human normal

osteoblast cells in vitro, Bioelectromagnetics: Journal of the

Bioelectromagnetics Society, The Society for Physical Regulation in Biology

and Medicine, The European Bioelectromagnetics Association, 20(3) (1999)

177-182.

[46] S. Ahmadian, S.R. Zarchi, B. Bolouri, Effects of extremely‐low‐frequency

pulsed electromagnetic fields on collagen synthesis in rat skin, Biotechnology

and Applied Biochemistry, 43(2) (2006) 71-75.

[47] C. Bauréus Koch, M. Sommarin, B. Persson, L. Salford, J. Eberhardt,

Interaction between weak low frequency magnetic fields and cell membranes,


[48] V. Akpolat, M.S. Celik, Y. Celik, N. Akdeniz, M.S. Ozerdem, Treatment of

osteoporosis by long-term magnetic field with extremely low frequency in

rats, Gynecological Endocrinology, 25(8) (2009) 524-529.

[49] N.M. Shupak, F.S. Prato, A.W. Thomas, Therapeutic uses of pulsed magnetic-

field exposure: a review, URSI Radio Science Bulletin, 76(4) (2003) 9-32.

[50] B. Strauch, C. Herman, R. Dabb, L.J. Ignarro, A.A. Pilla, Evidence-Based Use

of Pulsed Electromagnetic Field Therapy in Clinical Plastic Surgery,

Aesthetic Surgery Journal, 29(2) (2009) 135-143.

[51] A.R. Sul, S.N. Park, H. Suh, Effects of sinusoidal electromagnetic field on

structure and function of different kinds of cell lines, Yonsei Medical Journal,

47(6) (2006) 852-861.

[52] C.F. Martino, D. Belchenko, V. Ferguson, S. Nielsen Preiss, H.J. Qi, The effects

of pulsed electromagnetic fields on the cellular activity of SaOS‐2 cells,


[53] V. Grote, H. Lackner, C. Kelz, M. Trapp, F. Aichinger, H. Puff, M. Moser,

Short-term effects of pulsed electromagnetic fields after physical exercise are

dependent on autonomic tone before exposure, European Journal of Applied

Physiology, 101(4) (2007) 495-502.

[54] K. Chang, W.H.S. Chang, M.L. Wu, C. Shih, Effects of different intensities of

extremely low frequency pulsed electromagnetic fields on formation of

osteoclast‐like cells, Bioelectromagnetics, 24(6) (2003) 431-439.

[55] J.L. Kirschvink, Uniform magnetic fields and double‐wrapped coil systems:

improved techniques for the design of bioelectromagnetic experiments,


[56] W.R. Adey, Biological effects of electromagnetic fields, Journal of Cellular

Biochemistry, 51(4) (1993) 410-416.

[57] M.S. Markov, How living systems recognize applied electromagnetic fields, The

Environmentalist, 31(2) (2011) 89-96.

[58] K. R. Foster, Mechanisms of interaction of extremely low frequency electric

fields and biological systems, Radiation Protection Dosimetry, 106(4) (2003)

301-310.

[59] N. Salansky, A. Fedotchev, A. Bondar, Responses of the nervous system to low

frequency stimulation and EEG rhythms: clinical implications, Neuroscience

& Biobehavioral Reviews, 22(3) (1998) 395-409.

[60] A. De Loof, The electrical dimension of cells: the cell as a miniature

electrophoresis chamber, in: International review of cytology, Elsevier, 1986,

pp. 251-352.

192

[61] Y. Antov, A. Barbul, R. Korenstein, Electroendocytosis: stimulation of

adsorptive and fluid-phase uptake by pulsed low electric fields, Experimental

Cell Research, 297(2) (2004) 348-362.

[62] R. Goodman, A.S. Henderson, Some biological effects of electromagnetic

fields, Bioelectrochemistry and Bioenergetics, 15(1) (1986) 39-55.

[63] P. Volpe, Interactions of zero-frequency and oscillating magnetic fields with

biostructures and biosystems, Photochemical & Photobiological Sciences,

2(6) (2003) 637-648.

[64] D.H. Trock, Electromagnetic fields and magnets: investigational treatment for

musculoskeletal disorders, Rheumatic Disease Clinics of North America,

26(1) (2000) 51-62.

[65] R.C. Riddle, H.J. Donahue, From streaming‐potentials to shear stress: 25 years

of bone cell mechanotransduction, Journal of Orthopaedic Research, 27(2)

(2009) 143-149.

[66] C. Zhu, G. Bao, N. Wang, Cell mechanics: mechanical response, cell adhesion,

and molecular deformation, Annual Review of Biomedical Engineering, 2(1)

(2000) 189-226.

[67] J. Kirschvink, J. KDiaz‐Ricci, S. Kirschvink, Magnetite in human tissues: a

mechanism for the biological effects of weak ELF magnetic fields,

Bioelectromagnetics, 13(S1) (1992) 101-113.

[68] C. Polk, Physical mechanisms by which low-frequency magnetic fields can

affect the distribution of counterions on cylindrical biological cell surfaces,

Journal of Biological Physics, 14(1) (1986) 3-8.

[69] A. Buchachenko, Why magnetic and electromagnetic effects in biology are

irreproducible and contradictory?, Bioelectromagnetics, 37(1) (2016) 1-13.

[70] A. Buchachenko, D. Kuznetsov, Magnetic control of enzymatic

phosphorylation, Journal of Physical Chemistry and Biophysics, 4(2) (2014)

9.

[71] P. Hore, Are biochemical reactions affected by weak magnetic fields?,

Proceedings of the National Academy of Sciences, 109(5) (2012) 1357-1358.

[72] I.A. Shovkovy, Magnetic catalysis: a review, in: Strongly Interacting Matter in

Magnetic Fields, Springer, 2013, pp. 13-49.

[73] A.L. Buchachenko, D.A. Kuznetsov, Magnetic field affects enzymatic ATP

synthesis, Journal of the American Chemical Society, 130(39) (2008) 12868-

12869.

[74] A. Liboff, j.T. Williams, D. Strong, j.R. Wistar, Time-varying magnetic fields:

effect on DNA synthesis, Science, 223(4638) (1984) 818-820.

[75] S. Xu, H. Okano, N. Tomita, Y. Ikada, Recovery effects of a 180 mT static

magnetic field on bone mineral density of osteoporotic lumbar vertebrae in

ovariectomized rats, Evidence-Based Complementary and Alternative

Medicine, 2011 (2011).

[76] H. Zhang, L. Gan, X. Zhu, J. Wang, L. Han, P. Cheng, D. Jing, X. Zhang, Q.

Shan, Moderate-intensity 4 mT static magnetic fields prevent bone

architectural deterioration and strength reduction by stimulating bone

formation in streptozotocin-treated diabetic rats, Bone, 107 (2018) 36-44.

[77] H.R. Gungor, S. Akkaya, N. Ok, A. Yorukoglu, C. Yorukoglu, E. Kiter, E.O.

Oguz, N. Keskin, G.A. Mete, Chronic Exposure to Static Magnetic Fields

from Magnetic Resonance Imaging Devices Deserves Screening for

Osteoporosis and Vitamin D Levels: A Rat Model, International Journal of

Environmental Research and Public Health, 12(8) (2015) 8919-8932.

193

[78] C. Liu, Y. Zhang, T. Fu, Y. Liu, S. Wei, Y. Yang, D. Zhao, W. Zhao, M. Song,

X. Tang, H. Wu, Effects of electromagnetic fields on bone loss in

hyperthyroidism rat model, Bioelectromagnetics, 38(2) (2017) 137-150.

[79] B.Y. Zhu, Z.D. Yang, X.R. Chen, J. Zhou, Y.H. Gao, C.J. Xian, K.M. Chen,

Exposure Duration Is a Determinant of the Effect of Sinusoidal

Electromagnetic Fields on Peak Bone Mass of Young Rats, Calcified Tissue

International, 103(1) (2018) 95-106.

[80] Y. Atalay, N. Gunes, M.D. Guner, V. Akpolat, M.S. Celik, R. Guner,

Pentoxifylline and electromagnetic field improved bone fracture healing in

rats, Drug Design, Development and Therapy, 9 (2015) 5195.

[81] J. Zhou, X. Li, Y. Liao, W. Feng, C. Fu, X. Guo, Pulsed electromagnetic fields

inhibit bone loss in streptozotocin-induced diabetic rats, Endocrine, 49(1)

(2015) 258-266.

[82] J.C. Crockett, M.J. Rogers, F.P. Coxon, L.J. Hocking, M.H. Helfrich, Bone

remodelling at a glance, Journal of Cell Science, 124(7) (2011) 991-998.

[83] H.K. Datta, W.F. Ng, J.A. Walker, S.P. Tuck, S.S. Varanasi, The cell biology of

bone metabolism, Journal of Clinical Pathology, 61(5) (2008) 577-587.

[84] N.A. Sims, J.H. Gooi, Bone remodeling: Multiple cellular interactions required

for coupling of bone formation and resorption, Seminars in Cell &

Developmental Biology, 19(5) (2008) 444-451.

[85] E. Eriksen, Cellular mechanisms of bone remodeling, Reviews in Endocrine &

Metabolic Disorders, 11(4) (2010) 219-227.

[86] S. Khosla, The bone and beyond: a shift in calcium, Nature medicine, 17(4)

(2011) 430-431.

[87] K. Miyazono, S. Maeda, T. Imamura, BMP receptor signaling: Transcriptional

targets, regulation of signals, and signaling cross-talk, Cytokine &

Growth Factor Reviews, 16(3) (2005) 251-263.

[88] S.i. Harada, G.A. Rodan, Control of osteoblast function and regulation of bone

mass, Nature, 423(6937) (2003) 349.

[89] J. Crockett, D. Mellis, D. Scott, M. Helfrich, New knowledge on critical

osteoclast formation and activation pathways from study of rare genetic

diseases of osteoclasts: focus on the RANK/RANKL axis, Osteoporosis

International, 22(1) (2011) 1-20.

[90] T. Phan, J. Xu, M. Zheng, Interaction between osteoblast and osteoclast: impact

in bone disease, Histology and histopathology, 19(4) (2004) 1325-1344.

[91] P. Ducy, T. Schinke, G. Karsenty, The osteoblast: a sophisticated fibroblast

under central surveillance, Science, 289(5484) (2000) 1501-1504.

[92] P. Chatakun, R. Núñez-Toldrà, E.D. López, C. Gil-Recio, E. Martínez-Sarrà, F.

Hernández-Alfaro, E. Ferrés-Padró, L. Giner-Tarrida, M. Atari, The effect of

five proteins on stem cells used for osteoblast differentiation and

proliferation: a current review of the literature, Cellular and Molecular Life

Sciences, 71(1) (2014) 113-142.

[93] T. Nakashima, M. Hayashi, T. Fukunaga, K. Kurata, M. Oh-hora, J.Q. Feng,

L.F. Bonewald, T. Kodama, A. Wutz, E.F. Wagner, J.M. Penninger, H.

Takayanagi, Evidence for osteocyte regulation of bone homeostasis through

RANKL expression, Nature medicine, 17(10) (2011) 1231-1234.

[94] W.J. Boyle, W.S. Simonet, D.L. Lacey, Osteoclast differentiation and

activation, Nature, 423(6937) (2003) 337-342.

[95] D.J. Mellis, C. Itzstein, M.H. Helfrich, J.C. Crockett, The skeleton: a multi-

functional complex organ: the role of key signalling pathways in osteoclast

194

differentiation and in bone resorption, Journal of Endocrinology, 211(2)

(2011) 131-143.

[96] T. Wada, T. Nakashima, N. Hiroshi, J.M. Penninger, RANKL–RANK signaling

in osteoclastogenesis and bone disease, Trends in Molecular Medicine, 12(1)

(2006) 17-25.

[97] F. Shen, M.J. Ruddy, P. Plamondon, S.L. Gaffen, Cytokines link osteoblasts and

inflammation: microarray analysis of interleukin‐17‐and TNF‐α‐induced genes in bone cells, Journal of Leukocyte Biology, 77(3) (2005) 388-

399.

[98] Y. Tanio, H. Yamazaki, T. Kunisada, K. Miyake, S.-I. Hayashi, CD9 molecule

expressed on stromal cells is involved in osteoclastogenesis, Experimental

Hematology, 27(5) (1999) 853-859.

[99] W.J. Boyle, W.S. Simonet, D.L. Lacey, Osteoclast differentiation and

activation, Nature, 423(6937) (2003) 337.

[100] T. Nakashima, H. Takayanagi, The dynamic interplay between osteoclasts and

the immune system, Archives of Biochemistry and Biophysics, 473(2) (2008)

166-171.

[101] R.K. Aaron, S. Wang, D.M. Ciombor, Upregulation of basal TGFβ1 levels by

EMF coincident with chondrogenesis – implications for skeletal repair and

tissue engineering, Journal of Orthopaedic Research, 20(2) (2002) 233-240.

[102] T. Bodamyali, B. Bhatt, F.J. Hughes, V.R. Winrow, J.M. Kanczler, B. Simon,

J. Abbott, D.R. Blake, C.R. Stevens, Pulsed Electromagnetic Fields

Simultaneously Induce Osteogenesis and Upregulate Transcription of Bone

Morphogenetic Proteins 2 and 4 in Rat Osteoblastsin Vitro, Biochemical and

Biophysical Research Communications, 250(2) (1998) 458-461.

[103] H.H. Guerkov, C.H. Lohmann, Y. Liu, D.D. Dean, B.J. Simon, J.D. Heckman,

Z. Schwartz, B.D. Boyan, Pulsed Electromagnetic Fields Increase Growth

Factor Release by Nonunion Cells, Clinical Orthopaedics and Related

Research, 384 (2001) 265-279.

[104] J.H. Jansen, O.P. van der Jagt, B.J. Punt, J.A. Verhaar, J.P. van Leeuwen, H.

Weinans, H. Jahr, Stimulation of osteogenic differentiation in human

osteoprogenitor cells by pulsed electromagnetic fields: an in vitro study,

BMC musculoskeletal disorders, 11 (2010) 188.

[105] K. Nie, A. Henderson, MAP kinase activation in cells exposed to a 60 Hz

electromagnetic field, Journal of Cellular Biochemistry, 90(6) (2003) 1197-

1206.

[106] M. Schnoke, R.J. Midura, Pulsed electromagnetic fields rapidly modulate

intracellular signaling events in osteoblastic cells: comparison to parathyroid

hormone and insulin, Journal of Orthopaedic Research, 25(7) (2007) 933-

940.

[107] C.H. Heldin, K. Miyazono, P. ten Dijke, TGF-beta signalling from cell

membrane to nucleus through SMAD proteins, Nature, 390(6659) (1997)

465-471.

[108] A. von Bubnoff, K.W.Y. Cho, Intracellular BMP Signaling Regulation in

Vertebrates: Pathway or Network?, Developmental Biology, 239(1) (2001) 1-

14.

[109] J.J. Westendorf, R.A. Kahler, T.M. Schroeder, Wnt signaling in osteoblasts

and bone diseases, Gene, 341 (2004) 19-39.

[110] E. Canalis, Update in new anabolic therapies for osteoporosis, The Journal of

Clinical Endocrinology & Metabolism, 95(4) (2010) 1496-1504.

195

[111] T.L. McCarthy, M. Centrella, Novel links among Wnt and TGF-beta signaling

and Runx2, Molecular Endocrinology, 24(3) (2010) 587-597.

[112] D.G. Monroe, M.E. McGee-Lawrence, M.J. Oursler, J.J. Westendorf, Update

on Wnt signaling in bone cell biology and bone disease, Gene, 492(1) (2012)

1-18.

[113] M.K. Sutherland, J.C. Geoghegan, C. Yu, E. Turcott, J.E. Skonier, D.G.

Winkler, J.A. Latham, Sclerostin promotes the apoptosis of human

osteoblastic cells: a novel regulation of bone formation, Bone, 35(4) (2004)

828-835.

[114] S.L. Holmen, S.A. Robertson, C.R. Zylstra, B.O. Williams, Wnt-independent

activation of beta-catenin mediated by a Dkk1-Fz5 fusion protein,

Biochemical and Biophysical Research Communications, 328(2) (2005) 533-

539.

[115] Y.W. Qiang, Y. Chen, N. Brown, B. Hu, J. Epstein, B. Barlogie, J.D.

Shaughnessy, Jr., Characterization of Wnt/beta-catenin signalling in

osteoclasts in multiple myeloma, British Journal of Haematology, 148(5)

(2010) 726-738.

[116] B.R. Wong, R. Josien, S.Y. Lee, B. Sauter, H.L. Li, R.M. Steinman, Y. Choi,

TRANCE (tumor necrosis factor [TNF]-related activation-induced cytokine),

a new TNF family member predominantly expressed in T cells, is a dendritic

cell-specific survival factor, Journal of Experimental Medicine, 186(12)

(1997) 2075-2080.

[117] J. Xu, J.W. Tan, L. Huang, X.H. Gao, R. Laird, D. Liu, S. Wysocki, M.H.

Zheng, Cloning, sequencing, and functional characterization of the rat

homologue of receptor activator of NF-kappaB ligand, Journal of Bone and

Mineral Research, 15(11) (2000) 2178-2186.

[118] J. Caetano Lopes, H. Canhao, J.E. Fonseca, Osteoimmunology--the hidden

immune regulation of bone, Autoimmunity Reviews, 8(3) (2009) 250-255.

[119] E. Ang, Q. Liu, M. Qi, H.G. Liu, X. Yang, H. Chen, M.H. Zheng, J. Xu,

Mangiferin attenuates osteoclastogenesis, bone resorption, and RANKL-

induced activation of NF-kappaB and ERK, Journal of Cellular Biochemistry,

112(1) (2011) 89-97.

[120] M.T. Gillespie, Impact of cytokines and T lymphocytes upon osteoclast

differentiation and function, Arthritis Research & Therapy, 9(2) (2007) 103.

[121] J. Xu, H.F. Wu, E.S. Ang, K. Yip, M. Woloszyn, M.H. Zheng, R.X. Tan, NF-

kappaB modulators in osteolytic bone diseases, Cytokine & Growth Factor

Reviews, 20(1) (2009) 7-17.

[122] B.P. Ayati, C.M. Edwards, G.F. Webb, J.P. Wikswo, A mathematical model of

bone remodeling dynamics for normal bone cell populations and myeloma

bone disease, Biology Direct, 5 (2010) 28.

[123] E.F. Eriksen, Cellular mechanisms of bone remodeling, Rev Endocr Metab

Disord, 11(4) (2010) 219-227.

[124] Y. Wang, Q.H. Qin, A theoretical study of bone remodelling under PEMF at

cellular level, Computer methods in biomechanics and biomedical

engineering, 15(8) (2012) 885-897.

[125] X.Q. He, C. Qu, Q.H. Qin, A theoretical model for surface bone remodeling

under electromagnetic loads, Arch Appl Mech, 78(3) (2008) 163-175.

[126] C.Y. Qu, Q.H. Qin, Y.L. Kang, A hypothetical mechanism of bone remodeling

and modeling under electromagnetic loads, Biomaterials, 27(21) (2006)

4050-4057.

196

[127] J.C. Crockett, D.J. Mellis, D.I. Scott, M.H. Helfrich, New knowledge on

critical osteoclast formation and activation pathways from study of rare

genetic diseases of osteoclasts: focus on the RANK/RANKL axis,

Osteoporosis International, 22(1) (2011) 1-20.

[128] N.J. Pavlos, T.S. Cheng, A. Qin, P.Y. Ng, H.T. Feng, E.S. Ang, A. Carrello,

C.H. Sung, R. Jahn, M.H. Zheng, J. Xu, Tctex-1, a novel interaction partner

of Rab3D, is required for osteoclastic bone resorption, Molecular and

Cellular Biology, 31(7) (2011) 1551-1564.

[129] C.H. Lohmann, Z. Schwartz, Y. Liu, Z. Li, B.J. Simon, V.L. Sylvia, D.D.

Dean, L.F. Bonewald, H.J. Donahue, B.D. Boyan, Pulsed electromagnetic

fields affect phenotype and connexin 43 protein expression in MLO-Y4

osteocyte-like cells and ROS 17/2.8 osteoblast-like cells, Journal of

Orthopaedic Research, 21(2) (2003) 326-334.

[130] K. Chang, W.H.S. Chang, M.L. Wu, C. Shih, Effects of different intensities of

extremely low frequency pulsed electromagnetic fields on formation of

osteoclast-like cells, Bioelectromagnetics, 24(6) (2003) 431-439.

[131] K.J. McLeod, C.T. Rubin, The effect of low-frequency electrical fields on

osteogenesis, Journal of Bone and Joint Surgery-American Volume, 74A(6)

(1992) 920-929.

[132] M.A. Vander Molen, H.J. Donahue, C.T. Rubin, K.J. McLeod, Osteoblastic

networks with deficient coupling: Differential effects of magnetic and electric

field exposure, Bone, 27(2) (2000) 227-231.

[133] F. Tabrah, M. Hoffmeier, F. Gilbert, S. Batkin, C.A.L. Bassett, Bone-density

changes in osteoporosis-prone women exposed to pulsed electromagnetic-

fields (pemfs), Journal of Bone and Mineral Research, 5(5) (1990) 437-442.

[134] M. De Mattei, A. Caruso, G.C. Traina, F. Pezzetti, T. Baroni, V. Sollazzo,

Correlation between pulsed electromagnetic fields exposure time and cell

proliferation increase in human osteosarcoma cell lines and human normal

osteoblast cells in vitro, Bioelectromagnetics, 20(3) (1999) 177-182.

[135] D.C. Fredericks, J.V. Nepola, J.T. Baker, J. Abbott, B. Simon, Effects of

pulsed electromagnetic fields on bone healing in a rabbit tibial osteotomy

model, Journal of Orthopaedic Trauma, 14(2) (2000) 93-100.

[136] P. Diniz, K. Shomura, K. Soejima, G. Ito, Effects of pulsed electromagnetic

field (PEMF) stimulation on bone tissue like formation are dependent on the

maturation stages of the osteoblasts, Bioelectromagnetics, 23(5) (2002) 398-

405.

[137] K.J. McLeod, L. Collazo, Suppression of a differentiation response in MC-

3T3-E1 osteoblast-like cells by sustained, low-level, 30 Hz magnetic-field

exposure, Radiation Research, 153(5) (2000) 706-714.

[138] Q.-H. Qin, Mechanics of Cellular Bone Remodeling: Coupled Thermal,

Electrical, and Mechanical Field Effects, CRC Press, 2013.

[139] M. Kroll, Parathyroid hormone temporal effects on bone formation and

resorption, Bulletin of Mathematical Biology, 62(1) (2000) 163-188.

[140] C. Rattanakul, Y. Lenbury, N. Krishnamara, D.J. Wolwnd, Modeling of bone

formation and resorption mediated by parathyroid hormone: response to

estrogen/PTH therapy, Biosystems, 70(1) (2003) 55-72.

[141] S.V. Komarova, R.J. Smith, S.J. Dixon, S.M. Sims, L.M. Wahl, Mathematical

model predicts a critical role for osteoclast autocrine regulation in the control

of bone remodeling, Bone, 33(2) (2003) 206-215.

197

[142] S.V. Komarova, Mathematical model of paracrine interactions between

osteoclasts and osteoblasts predicts anabolic action of parathyroid hormone

on bone, Endocrinology, 146(8) (2005) 3589-3595.

[143] L.K. Potter, L.D. Greller, C.R. Cho, M.E. Nuttall, G.B. Stroup, L.J. Suva, F.L.

Tobin, Response to continuous and pulsatile PTH dosing: A mathematical

model for parathyroid hormone receptor kinetics, Bone, 37(2) (2005) 159-

169.

[144] V. Lemaire, F.L. Tobin, L.D. Greller, C.R. Cho, L.J. Suva, Modeling the

interactions between osteoblast and osteoclast activities in bone remodeling,

Journal of Theoretical Biology, 229(3) (2004) 293-309.

[145] Y.N. Wang, Q.H. Qin, S. Kalyanasundaram, A theoretical model for

simulating effect of Parathyroid Hormone on bone metabolism at cellular

level, Journal of Molecular and Cellular Biomechanics., 6(2) (2009) 101-112.

[146] P. Pivonka, J. Zimak, D.W. Smith, B.S. Gardiner, C.R. Dunstan, N.A. Sims, T.

John Martin, G.R. Mundy, Model structure and control of bone remodeling:

A theoretical study, Bone, 43(2) (2008) 249-263.

[147] P. Pivonka, J. Zimak, D.W. Smith, B.S. Gardiner, C.R. Dunstan, N.A. Sims, T.

John Martin, G.R. Mundy, Theoretical investigation of the role of the RANK-

RANKL-OPG system in bone remodeling, Journal of Theoretical Biology,

262(2) (2010) 306-316.

[148] H. Wang, Q.H. Qin, FE approach with Green’s function as internal trial

function for simulating bioheat transfer in the human eye, Archives of

Mechanics, 62(6) (2010) 493-510.

[149] H. Wang, Q.H. Qin, Hybrid FEM with fundamental solutions as trial functions

for heat conduction simulation, Acta Mechanica Solida Sinica, 22(5) (2009)

487-498.

[150] Q.H. Qin, H. Wang, Matlab and C programming for Trefftz finite element

methods, CRC Press, 2008.

[151] Q.H. Qin, Trefftz finite element method and its applications, Applied

Mechanics Reviews, 58(5) (2005) 316-337.

[152] Q.H. Qin, Variational formulations for TFEM of piezoelectricity, International

Journal of Solids and Structures, 40(23) (2003) 6335-6346.

[153] Q. Qin, Hybrid-Trefftz finite element method for Reissner plates on an elastic

foundation, Computer Methods in Applied Mechanics and Engineering,

122(3-4) (1995) 379-392.

[154] Q. Qin, Hybrid Trefftz finite-element approach for plate bending on an elastic

foundation, Applied Mathematical Modelling, 18(6) (1994) 334-339.

[155] H. Weinans, R. Huiskes, H.J. Grootenboer, The Behavior of Adaptive Bone-

Remodeling Simulation-Models, Jounral of Biomechanics, 25(12) (1992)

1425-1441.

[156] D. Kardas, U. Nackenhorst, Studies on Bone Remodeling Theory Based on

Microcracks Using Finite Element Computations, PAMM, 9(1) (2009) 147-

148.

[157] C.L. Lin, Y.H. Lin, S.H. Chang, Multi-factorial analysis of variables

influencing the bone loss of an implant placed in the maxilla: Prediction

using FEA and SED bone remodeling algorithm, Jounral of Biomechanics,

43(4) (2010) 644-651.

[158] R. Hambli, H. Katerchi, C.-L. Benhamou, Multiscale methodology for bone

remodelling simulation using coupled finite element and neural network

198

computation, Biomechanics and Modeling in Mechanobiology, 10(1) (2011)

133-145.

[159] Q.H. Qin, S.W. Yu, An arbitrarily-oriented plane crack terminating at the

interface between dissimilar piezoelectric materials, International Journal of

Solids and Structures, 34(5) (1997) 581-590.

[160] Q.H. Qin, Y.W. Mai, S.W. Yu, Some problems in plane thermopiezoelectric

materials with holes, International Journal of Solids and Structures, 36(3)

(1999) 427-439.

[161] Q.H. Qin, Y.W. Mai, S.W. Yu, Effective moduli for thermopiezoelectric

materials with microcracks, International Journal of Fracture, 91(4) (1998)

359-371.

[162] Q.H. Qin, Y.W. Mai, A closed crack tip model for interface cracks

inthermopiezoelectric materials, International Journal of Solids and

Structures, 36(16) (1999) 2463-2479.

[163] Q.H. Qin, Y.W. Mai, Thermoelectroelastic Green's function and its application

for bimaterial of piezoelectric materials, Archive of Applied Mechanics,

68(6) (1998) 433-444.

[164] Q.H. Qin, Thermoelectroelastic Green's function for a piezoelectric plate

containing an elliptic hole, Mechanics of Materials, 30(1) (1998) 21-29.

[165] Q. Qin, Y. Mai, Crack growth prediction of an inclined crack in a half-plane

thermopiezoelectric solid, Theoretical and Applied Fracture Mechanics, 26(3)

(1997) 185-191.

[166] S. Scheiner, P. Pivonka, C. Hellmich, Coupling systems biology with

multiscale mechanics, for computer simulations of bone remodeling,

Computer Methods in Applied Mechanics and Engineering, 254(0) (2013)

181-196.

[167] J.B. Lian, G.S. Stein, Concepts of osteoblast growth and differentiation: basis

for modulation of bone cell development and tissue formation, Critical

Reviews in Oral Biology & Medicine, 3(3) (1992) 269-305.

[168] T.A. Owen, M. Aronow, V. Shalhoub, L.M. Barone, L. Wilming, M.S.

Tassinari, M.B. Kennedy, S. Pockwinse, J.B. Lian, G.S. Stein, Progressive

development of the rat osteoblast phenotype in vitro: Reciprocal relationships

in expression of genes associated with osteoblast proliferation and

differentiation during formation of the bone extracellular matrix, Journal of

Cellular Physiology, 143(3) (1990) 420-430.

[169] M. Hartig, U. Joos, H.P. Wiesmann, Capacitively coupled electric fields

accelerate proliferation of osteoblast-like primary cells and increase bone

extracellular matrix formation in vitro, Eur Biophys J, 29(7) (2000) 499-506.

[170] N. Bohr, Atomic physics and human knowledge, Courier Dover Publications,

2010.

[171] H.M. Shapiro, Input-output models of biological systems: formulation and

applicability, Computers and Biomedical Research, 2(5) (1969) 430-445.

[172] V.Z. Marmarelis, Identification of nonlinear biological systems using Laguerre

expansions of kernels, Annals of Biomedical Engineering, 21(6) (1993) 573-

589.

[173] M.J. Korenberg, I.W. Hunter, The identification of nonlinear biological

systems: Volterra kernel approaches, Annals of Biomedical Engineering,

24(2) (1996) 250-268.

[174] E.F. Keller, L.A. Segel, Model for chemotaxis, Journal of Theoretical Biology,

30(2) (1971) 225-234.

199

[175] T. Hillen, K. Painter, Global existence for a parabolic chemotaxis model with

prevention of overcrowding, Advances in Applied Mathematics, 26(4) (2001)

280-301.

[176] R.W. Li, N.T. Kirkland, J. Truong, J. Wang, P.N. Smith, N. Birbilis, D.R.

Nisbet, The influence of biodegradable magnesium alloys on the osteogenic

differentiation of human mesenchymal stem cells, Journal of Biomedical

Materials Research Part A 102(12) (2014) 4346-4357.

[177] A. Arifin, A.B. Sulong, N. Muhamad, J. Syarif, M.I. Ramli, Material

processing of hydroxyapatite and titanium alloy (HA/Ti) composite as

implant materials using powder metallurgy: a review, Materials & Design, 55

(2014) 165-175.

[178] Z. Ma, Z. Mao, C. Gao, Surface modification and property analysis of

biomedical polymers used for tissue engineering, Colloids and Surfaces B:

Biointerfaces, 60(2) (2007) 137-157.

[179] D.D. Deligianni, N.D. Katsala, P.G. Koutsoukos, Y.F. Missirlis, Effect of

surface roughness of hydroxyapatite on human bone marrow cell adhesion,

proliferation, differentiation and detachment strength, Biomaterials, 22(1)

(2000) 87-96.

[180] T.J. Webster, C. Ergun, R.H. Doremus, R.W. Siegel, R. Bizios, Enhanced

functions of osteoblasts on nanophase ceramics, Biomaterials, 21(17) (2000)

1803-1810.

[181] T.J. Webster, R.W. Siegel, R. Bizios, Osteoblast adhesion on nanophase

ceramics, Biomaterials, 20(13) (1999) 1221-1227.

[182] R. Huang, S. Lu, Y. Han, Role of grain size in the regulation of osteoblast

response to Ti–25Nb–3Mo–3Zr–2Sn alloy, Colloids and Surfaces B:

Biointerfaces, 111 (2013) 232-241.

[183] M. Vandrovcová, L. Bacakova, Adhesion, growth and differentiation of

osteoblasts on surface-modified materials developed for bone implants,

Physiological Research, 60(3) (2011) 403-417.

[184] D.L. Elbert, J.A. Hubbell, Surface Treatments of Polymers for

Biocompatibility, Annual Review of Materials Science, 26(1) (1996) 365-

394.

[185] K. Anselme, Osteoblast adhesion on biomaterials, Biomaterials, 21(7) (2000)

667-681.

[186] K.C. Dee, T.T. Andersen, R. Bizios, Osteoblast population migration

characteristics on substrates modified with immobilized adhesive peptides,

Biomaterials, 20(3) (1999) 221-227.

[187] K. Anselme, L. Ploux, A. Ponche, Cell/Material Interfaces: Influence of

Surface Chemistry and Surface Topography on Cell Adhesion, Journal of

Adhesion Science and Technology, 24(5) (2010) 831-852.

[188] A. Stevens, H.G. Othmer, Aggregation, blowup, and collapse: the ABC's of

taxis in reinforced random walks, SIAM Journal on Applied Mathematics,

57(4) (1997) 1044-1081.

[189] Wheeless' Textbook of Orthopaedics, Duke University Medical Center's

Division of Orthopedic Surgery, Data Trace Internet Publishing, 1996.

[190] J. Park, S. Bauer, K.A. Schlegel, F.W. Neukam, K. von der Mark, P. Schmuki,

TiO2 Nanotube Surfaces: 15 nm—An Optimal Length Scale of Surface

Topography for Cell Adhesion and Differentiation, Small, 5(6) (2009) 666-

671.

200

[191] S. Minagar, J. Wang, C.C. Berndt, E.P. Ivanova, C. Wen, Cell response of

anodized nanotubes on titanium and titanium alloys, Journal of Biomedical

Materials Research Part A, 101A(9) (2013) 2726-2739.

[192] M. Sato, A. Aslani, M.A. Sambito, N.M. Kalkhoran, E.B. Slamovich, T.J.

Webster, Nanocrystalline hydroxyapatite/titania coatings on titanium

improves osteoblast adhesion, Journal of Biomedical Materials Research Part

A, 84A(1) (2008) 265-272.

[193] E.K.F. Yim, K.W. Leong, Significance of synthetic nanostructures in dictating

cellular response, Nanomedicine: Nanotechnology, Biology and Medicine,

1(1) (2005) 10-21.

[194] B. Stevens, Y. Yang, A. Mohandas, B. Stucker, K.T. Nguyen, A review of

materials, fabrication methods, and strategies used to enhance bone

regeneration in engineered bone tissues, Journal of Biomedical Materials

Research Part B: Applied Biomaterials: An Official Journal of The Society

for Biomaterials, The Japanese Society for Biomaterials, and The Australian

Society for Biomaterials and the Korean Society for Biomaterials, 85(2)

(2008) 573-582.

[195] K.J. Painter, T. Hillen, Volume-filling and quorum-sensing in models for

chemosensitive movement, The Quarterly Journal of Mechanics and Applied

Mathematics, 10(4) (2002) 501-543.

[196] S. Chen, C.Y. Lee, R.W. Li, P.N. Smith, Q.H. Qin, Modelling osteoblast

adhesion on surface-engineered biomaterials: optimisation of nanophase grain

size, Computer Methods in Biomechanics and Biomedical Engineering, 20(8)

(2017) 905-914.

[197] D.J. Muehsam, A.A. Pilla, A Lorentz model for weak magnetic field

bioeffects: Part I—Thermal noise is an essential component of AC/DC effects

on bound ion trajectory, Bioelectromagnetics, 30(6) (2009) 462-475.

[198] C.L.M. Bauréus Koch, M. Sommarin, B.R.R. Persson, L.G. Salford, J.L.

Eberhardt, Interaction between weak low frequency magnetic fields and cell

membranes, Bioelectromagnetics, 24(6) (2003) 395-402.

[199] J. Miyakoshi, Effects of static magnetic fields at the cellular level, Progress in

Biophysics and Molecular Biology, 87(2) (2005) 213-223.

[200] S. Yamaguchi-Sekino, M. Sekino, S. Ueno, Biological effects of

electromagnetic fields and recently updated safety guidelines for strong static

magnetic fields, Magnetic Resonance in Medical Sciences, 10(1) (2011) 1-10.

[201] V. Lednev, Possible mechanism for the influence of weak magnetic fields on

biological systems, Bioelectromagnetics, 12(2) (1991) 71-75.

[202] J. Zhang, C. Ding, L. Ren, Y. Zhou, P. Shang, The effects of static magnetic

fields on bone, Progress in Biophysics and Molecular Biology, 114(3) (2014)

146-152.

[203] T. Lemaire, E. Capiez-Lernout, J. Kaiser, S. Naili, V. Sansalone, What is the

importance of multiphysical phenomena in bone remodelling signals

expression? A multiscale perspective, Journal of the Mechanical Behavior of

Biomedical Materials, 4(6) (2011) 909-920.

[204] R.C. Riddle, H.J. Donahue, From streaming-potentials to shear stress: 25 years

of bone cell mechanotransduction, Journal of Orthopaedic Research, 27(2)

(2009) 143-149.

[205] S. Ghodbane, A. Lahbib, M. Sakly, H. Abdelmelek, Bioeffects of static

magnetic fields: oxidative stress, genotoxic effects, and cancer studies,

BioMed Research International, 2013 (2013).

201

[206] P.J. Hore, Are biochemical reactions affected by weak magnetic fields?,

Proceedings of the National Academy of Sciences, 109(5) (2012) 1357-1358.

[207] H.B. Murray, B.A. Pethica, A follow-up study of the in-practice results of

pulsed electromagnetic field therapy in the management of nonunion

fractures, Orthopedic Research and Reviews, 55 (2016) 67-72.

[208] X. Bai, D. Miao, D. Panda, S. Grady, M.D. McKee, D. Goltzman, A.C.

Karaplis, Partial rescue of the Hyp phenotype by osteoblast-targeted PHEX

(phosphate-regulating gene with homologies to endopeptidases on the X

chromosome) expression, Molecular Endocrinology, 16(12) (2002) 2913-

2925.

[209] Q.H. Qin, Y.N. Wang, A mathematical model of cortical bone remodeling at

cellular level under mechanical stimulus, Acta Mechanica Sinica, 28(6)

(2012) 1678-1692.

202

Appendix

203

#An example of programming SMF data in MATLAB

clear all close all

% raw data raw_x=[20 26 32 56 97]'; raw_y=[2600 2500 2800 1900 2000]'; raw_e=[200 100 450 350 200]';

%N N=500; %R R=(0:200)';

%del_t del_t=60; %final_t final_t=60000;%1000 steps %t vector t=(0:del_t:final_t)'; %h h=1/N; %xi i=(0:N)'; x=i*h; %cellular sensitivity of ti a=0.17+0.11*exp(-0.027*R); %molecular sensitivity of ti d=1.37-1.29*exp(-0.02*R); %initial cell density A=1800;

%rescaling: e.g. final projected time = n*(del_t*coe)-

>myscale=del_t*coe final_projected_time=8; myscale=final_projected_time/length(t);

%preallocation p=zeros(size(t,1),size(x,1)); g=zeros(size(t,1),size(x,1)); f=zeros(size(t,1),size(x,1));

%t=0 initialisation %pi(0) var=1./d;

%% clear p g f p{1}=zeros(length(var),length(x)); g{1}=zeros(length(var),length(x)); f{1}=zeros(length(var),length(x)); for i=1:length(var) p{1}(i,:)=sqrt(var(i)/2./pi).*exp(-0.5.* var(i)*x.^2); end %g and f for i=1:length(x) switch 1 case i==1 j=3;

204

g{1}(:,i)=p{1}(:,j+1).*(1-a.*p{1}(:,j+2))+p{1}(:,j-

1).*(1-a.*p{1}(:,j-2)); f{1}(:,i)=(1-p{1}(:,j+1)).*(1-a.*p{1}(:,j-

1))+p{1}(:,j+1).*(1-a.*p{1}(:,j+2))+p{1}(:,j-1).*(1-a.*p{1}(:,j-

2))+(1-p{1}(:,j-1)).*(1-a.*p{1}(:,j+1)); case i==2 j=3; g{1}(:,i)=p{1}(:,j+1).*(1-a.*p{1}(:,j+2))+p{1}(:,j-

1).*(1-a.*p{1}(:,j-2)); f{1}(:,i)=(1-p{1}(:,j+1)).*(1-a.*p{1}(:,j-

1))+p{1}(:,j+1).*(1-a.*p{1}(:,j+2))+p{1}(:,j-1).*(1-a.*p{1}(:,j-

2))+(1-p{1}(:,j-1)).*(1-a.*p{1}(:,j+1)); case i==length(x)-1 j=length(x)-2; g{1}(:,i)=p{1}(:,j+1).*(1-a.*p{1}(:,j+2))+p{1}(:,j-

1).*(1-a.*p{1}(:,j-2)); f{1}(:,i)=(1-p{1}(:,j+1)).*(1-a.*p{1}(:,j-

1))+p{1}(:,j+1).*(1-a.*p{1}(:,j+2))+p{1}(:,j-1).*(1-a.*p{1}(:,j-

2))+(1-p{1}(:,j-1)).*(1-a.*p{1}(:,j+1)); case i==length(x) j=length(x)-2; g{1}(:,i)=p{1}(:,j+1).*(1-a.*p{1}(:,j+2))+p{1}(:,j-

1).*(1-a.*p{1}(:,j-2)); f{1}(:,i)=(1-p{1}(:,j+1)).*(1-a.*p{1}(:,j-

1))+p{1}(:,j+1).*(1-a.*p{1}(:,j+2))+p{1}(:,j-1).*(1-a.*p{1}(:,j-

2))+(1-p{1}(:,j-1)).*(1-a.*p{1}(:,j+1)); case i>2 && i<length(x)-1 g{1}(:,i)=p{1}(:,i+1).*(1-a.*p{1}(:,i+2))+p{1}(:,i-

1).*(1-a.*p{1}(:,i-2)); f{1}(:,i)=(1-p{1}(:,i+1)).*(1-a.*p{1}(:,i-

1))+p{1}(:,i+1).*(1-a.*p{1}(:,i+2))+p{1}(:,i-1).*(1-a.*p{1}(:,i-

2))+(1-p{1}(:,i-1)).*(1-a.*p{1}(:,i+1)); end end %g and f along with time index k for k=1:size(t,1)-1 for i=1:length(x) switch 1 case i==1 j=3;

g{k}(:,i)=p{k}(:,j+1).*(1-a.*p{k}(:,j+2))+p{k}(:,j-

1).*(1-a.*p{k}(:,j-2)); f{k}(:,i)=(1-p{k}(:,j+1)).*(1-a.*p{k}(:,j-

1))+p{k}(:,j+1).*(1-a.*p{k}(:,j+2))+p{k}(:,j-1).*(1-a.*p{k}(:,j-

2))+(1-p{k}(:,j-1)).*(1-a.*p{k}(:,j+1)); case i==2 j=3; g{k}(:,i)=p{k}(:,j+1).*(1-a.*p{k}(:,j+2))+p{k}(:,j-

1).*(1-a.*p{k}(:,j-2)); f{k}(:,i)=(1-p{k}(:,j+1)).*(1-a.*p{k}(:,j-

1))+p{k}(:,j+1).*(1-a.*p{k}(:,j+2))+p{k}(:,j-1).*(1-a.*p{k}(:,j-

2))+(1-p{k}(:,j-1)).*(1-a.*p{k}(:,j+1)); case i==length(x)-1 j=length(x)-2; g{k}(:,i)=p{k}(:,j+1).*(1-a.*p{k}(:,j+2))+p{k}(:,j-

1).*(1-a.*p{k}(:,j-2)); f{k}(:,i)=(1-p{k}(:,j+1)).*(1-a.*p{k}(:,j-

1))+p{k}(:,j+1).*(1-a.*p{k}(:,j+2))+p{k}(:,j-1).*(1-a.*p{k}(:,j-

2))+(1-p{k}(:,j-1)).*(1-a.*p{k}(:,j+1)); case i==length(x)

205

j=length(x)-2; g{k}(:,i)=p{k}(:,j+1).*(1-a.*p{k}(:,j+2))+p{k}(:,j-

1).*(1-a.*p{k}(:,j-2)); f{k}(:,i)=(1-p{k}(:,j+1)).*(1-a.*p{k}(:,j-

1))+p{k}(:,j+1).*(1-a.*p{k}(:,j+2))+p{k}(:,j-1).*(1-a.*p{k}(:,j-

2))+(1-p{k}(:,j-1)).*(1-a.*p{k}(:,j+1)); case i>2 && i<length(x)-1 g{k}(:,i)=p{k}(:,i+1).*(1-a.*p{k}(:,i+2))+p{k}(:,i-

1).*(1-a.*p{k}(:,i-2)); f{k}(:,i)=(1-p{k}(:,i+1)).*(1-a.*p{k}(:,i-

1))+p{k}(:,i+1).*(1-a.*p{k}(:,i+2))+p{k}(:,i-1).*(1-a.*p{k}(:,i-

2))+(1-p{k}(:,i-1)).*(1-a.*p{k}(:,i+1)); end

p{k+1}(:,i)=(h.^2.*p{k}(:,i)+del_t.*g{k}(:,i))./(h.^2+del_t.*f{k}(:,

i)); end

end

clear N N=zeros(length(t),length(d)); for i=1:length(t)

%Trapezoidal numerical integration b1=repmat(transpose(A.*2.*pi.*x(1:end-

1)),length(d),1).*p{i}(:,1:end-1);

b2=repmat(transpose(A.*2.*pi.*x(2:end)),length(d),1).*p{i}(:,2:end); N(i,:)=sum(h.*(b2+b1)/2,2);

end

myh=surf(N); set(myh,'EdgeColor','none') figure plot(N(300,:),'k-') hold on errorbar(raw_x,raw_y,raw_e,'b.','MarkerSize',20);

Documents

Prediction of Bone Cell Probability Distribution in Weak ......To Bobin Xin, Cheuk Yu Lee, Haiyang Zhou and Shuang Zhang, thank you for your endless encouragement, concern, support,