Chapter 1, version A Dung_Nguyen_Thesis.pdf · T.D. Nguyen, A. Oloyede, Y.T. Gu, Stress Relaxation...

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Keywords

Atomic Force Microscopy

Cell biomechanics

Consolidation-dependent behaviour

Finite Element Analysis

Biomechanical indentation

Mechanical models

Mechanical properties

Osmotic pressure

Porohyperelastic model

Single living cells

Strain-rate dependent behaviour

Stress–relaxation behaviour

Thin-layer models

Viscoelastic properties

ii

Abstract

Living cells are the basic structural units existing in all known living organisms.

They perform several functions and metabolic activities within organs and tissue. It

is well known that cells are sensitive to variation in their mechanical and

physiological environments. Therefore, studying the mechanical properties and

behaviour of individual living cells can enhance knowledge of and insight into the

role of mechanical forces in supporting tissue regeneration or degeneration, leading

to new therapies and treatment.

Fluid-filled biological tissues respond differently to varying rates of loading. It

is hypothesised that living cells within their extracellular matrix would possess

similar behaviour. There is a lack of research, however, on the strain-rate dependent

mechanical properties of single living cells. Moreover, a number of studies in the

literature propose various mechanical models in cell biomechanics, such as the liquid

drop models, the solid models, the mixture theory and the consolidation theory.

Among these models, the consolidation theory, which in turn is extended to the so-

called porohyperelastic (PHE) model, has been used effectively and widely in tissue

engineering involving the articular cartilage and large arteries. The PHE model can

account for phenomena such as the swelling behaviour, the drag effect and the fluid-

solid interaction, and is believed to be a suitable model for living cell biomechanics.

Nevertheless, there have been few research attempts to use the PHE model for the

study of single living cells. As a result, the PHE model was investigated in the

present study in order to evaluate its capacity to elucidate the strain-rate dependent

mechanical properties of single living cells, namely, osteocytes, osteoblasts, and

chondrocytes using atomic force microscopy biomechanical testing and finite

element analysis modelling.

Firstly, the dependency of the mechanical deformation behaviour and

viscoelastic properties of single living cells on the strain-rate were investigated using

atomic force microscopy indentation and stress–relaxation experimental data,

respectively. The thin-layer models were used to obtain the suitable material

parameters. Secondly, the PHE model was utilised to simulate the strain-rate

dependent mechanical deformation and stress–relaxation behaviour of living cells

and extract the PHE material parameters. It is concluded that the living cells respond

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differently when subjected to different strain-rates, and that the PHE model can

capture the strain-rate dependent elastic behaviour as well as stress–relaxation

behaviour of living cells.

In addition, the effects of extracellular osmotic pressure on morphology and

mechanical properties in a typical cell type (i.e. chondrocytes) were investigated. It

was found that both hypoosmotic and hyperosmotic pressure affect the shape,

volume and mechanical properties of single living chondrocytes. Thus, it is

concluded that cells are sensitive to their osmotic environment, which may directly

change the cellular actin network structure and mechanical properties.

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List of Publications

Journal Articles (5)

1. T.D. Nguyen, Y.T. Gu, A. Oloyede, & W. Senadeera, Analysis of Strain-

rate-dependent Mechanical Behavior of Single Chondrocyte: A Finite

Element Study, International Journal of Computational Methods, 11,

1344005 (2014) 1-20.

2. T.D. Nguyen, Y.T. Gu, Exploration of Mechanisms Underlying the Strain-

Rate-Dependent Mechanical Property of Single Chondrocytes, Applied

Physics Letters, 104, 183701 (2014) 1-5.

3. T.D. Nguyen, A. Oloyede, Y.T. Gu, Stress Relaxation Analysis of Single

Chondrocytes Using Porohyperelastic Model Based on the AFM

Experiments, Theoretical and Applied Mechanics Letters, Accepted.

4. T.D. Nguyen, Y.T. Gu, Determination of Strain-rate-dependent Mechanical

Behavior of Living and Fixed Osteocytes and Chondrocytes Using AFM and

Inverse FEA, Journal of Biomechanical Engineering, 136, 101004 (2014) 1-

8.

5. T.D. Nguyen, A. Oloyede, Y.T. Gu, A Poroviscohyperelastic Model for

Numerical Analysis of Mechanical Behaviour of Single Chondrocyte,

Computer Methods in Biomechanics and Biomedical Engineering, In Press.

Conference Articles (3)

1. T.D. Nguyen, A. Oloyede, S. Singh, & Y.T. Gu (2014) Porohyperelastic

analysis of single osteocyte using AFM and inverse FEA. In Goh, James

(Ed.) IFMBE Proceedings: The 15th International Conference on Biomedical

Engineering, Springer International Publishing, Singapore, pp. 56-59.

2. T.D. Nguyen, Y.T. Gu (2013) Porohyperelastic analysis of single

chondrocyte using AFM and inverse FEA. In 5th Asia Pacific Congress on

Computational Mechanics & 4th International Symposium on Computational

Mechanics, 11-14 December 2013, Singapore.

3. T.D. Nguyen, Y.T. Gu, A. Oloyede, & W. Senadeera (2012)

Porohyperelastic analysis to explore mechanical properties of chondrocytes

using numerical modeling and experiments: a finite element study. In 4th

http://eprints.qut.edu.au/65802/






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International Conference on Computational Methods (ICCM 2012), 25-28

November 2012, Crowne Plaza, Gold Coast, QLD.

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Table of Contents

Keywords ................................................................................................................................................. i

Abstract ................................................................................................................................................... ii

List of Publications ................................................................................................................................ iv

Table of Contents ................................................................................................................................... vi

List of Figures ........................................................................................................................................ ix

List of Tables ...................................................................................................................................... xvii

List of Abbreviations ........................................................................................................................... xix

Statement of Original Authorship ........................................................................................................ xxi

Statement on Ethics Approval ............................................................................................................. xxii

Acknowledgements ............................................................................................................................. xxv

CHAPTER 1: INTRODUCTION ....................................................................................................... 1

1.1 Background .................................................................................................................................. 1

1.2 Research Problem ........................................................................................................................ 4

1.3 Research Aims and Objectives .................................................................................................... 6

1.4 Significance and Contribution ..................................................................................................... 7

1.5 Thesis Outline .............................................................................................................................. 7

1.6 Flowchart of this research ............................................................................................................ 9

CHAPTER 2: LITERATURE REVIEW ......................................................................................... 11

2.1 Introduction................................................................................................................................ 11 2.1.1 Cartilage and chondrocyte structure and properties ........................................................ 11 2.1.2 Swelling state in cartilage and chondrocyte ................................................................... 16 2.1.3 Structure and properties of bone cells............................................................................. 17

2.2 Mechanical models of living cells ............................................................................................. 21 2.2.1 Cortical shell-liquid core models (or liquid drop models) .............................................. 21 2.2.2 Solid models ................................................................................................................... 22 2.2.3 Mixture theory – based models ....................................................................................... 24 2.2.4 Consolidation models ..................................................................................................... 28

2.3 Experimental methods for living cells ....................................................................................... 32

2.4 Numerical techniques ................................................................................................................ 38

2.5 Summary and Implications ........................................................................................................ 40

CHAPTER 3: RESEARCH DESIGN ............................................................................................... 43

3.1 Introduction................................................................................................................................ 43

3.2 Atomic force microscopy experimental set-up and data post-processing .................................. 43

3.3 Materials and models ................................................................................................................. 46 3.3.1 Cell culturing and AFM sample preparation .................................................................. 46 3.3.2 Sample preparation for varying osmotic pressure environments .................................... 47 3.3.3 Confocal actin filament and vinculin staining and imaging ........................................... 47 3.3.4 Cell diameter measurement ............................................................................................ 48 3.3.5 Cell height measurement ................................................................................................ 49

3.4 Numerical models ...................................................................................................................... 52 3.4.1 Introduction of Finite Element Method .......................................................................... 52 3.4.2 FEA model used in this study ......................................................................................... 53 3.4.3 Inverse FEA method ....................................................................................................... 55

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CHAPTER 4: EXPLORATION OF STRAIN-RATE DEPENDENT MECHANICAL

DEFORMATION BEHAVIOUR OF SINGLE LIVING CELLS .................................................. 57

4.1 Introduction ................................................................................................................................ 57

4.2 AFM biomechanical indentation experiments ........................................................................... 58

4.3 Thin-layer elastic model ............................................................................................................ 59

4.4 PHE analysis of strain-rate dependent mechanical deformation behaviour of single cells ........ 60 4.4.1 PHE theory ..................................................................................................................... 61 4.4.2 Inverse FEA technique to estimate PHE material parameters ........................................ 65

4.5 Results and Discussions ............................................................................................................. 66 4.5.1 Cell diameter ................................................................................................................... 66 4.5.2 Cell height....................................................................................................................... 66 4.5.3 Comparison of elastic moduli among osteocytes, osteoblasts and chondrocytes ........... 67 4.5.4 Exploration of mechanisms underlying the dependency of mechanical

deformation behaviour of single living and fixed osteocytes, osteoblasts, and

chondrocytes on strain-rates ........................................................................................... 70 4.5.5 PHE analysis of strain-rate dependent mechanical behaviour of single living and

fixed osteocytes, osteoblasts and chondrocytes .............................................................. 73

4.6 Conclusion ................................................................................................................................. 86

CHAPTER 5: INVESTIGATION OF STRESS–RELAXATION BEHAVIOUR OF SINGLE

CELLS SUBJECTED TO DIFFERENT STRAIN-RATES ............................................................ 89

5.1 Introduction ................................................................................................................................ 89

5.2 AFM relaxation experiments ..................................................................................................... 90

5.3 Thin-layer viscoelastic model .................................................................................................... 91

5.4 PRI method ................................................................................................................................ 93

5.5 Results and Discussions ............................................................................................................. 94 5.5.1 Comparison of equilibrium moduli among living osteocytes, osteoblasts and

chondrocytes ................................................................................................................... 94 5.5.2 Viscoelastic properties of single living osteocytes, osteoblasts and chondrocytes

subjected to different strain-rates .................................................................................... 96 5.5.3 PHE analysis of strain-rate dependent relaxation behaviour of single cells ................. 101 5.5.3.1 Inverse FEA technique to estimate PHE material parameters ...................................... 101 5.5.3.2 PHE analysis results ..................................................................................................... 102

5.6 Conclusion ............................................................................................................................... 112

CHAPTER 6: EFFECT OF OSMOTIC PRESSURE ON THE MORPHOLOGY AND

MECHANICAL PROPERTIES OF SINGLE CHONDROCYTES ............................................. 115

6.1 Introduction .............................................................................................................................. 115

6.2 Materials and model ................................................................................................................. 116 6.2.1 Osmotic activity ............................................................................................................ 116 6.2.2 Methodology ................................................................................................................. 116

6.3 Results and Discussions ........................................................................................................... 117 6.3.1 Effect of extracellular osmotic pressure on chondrocyte morphology ......................... 117 6.3.2 Osmotic activity of single living chondrocytes............................................................. 120 6.3.3 Actin structural changes of chondrocytes when exposed to different osmotic

pressure conditions ....................................................................................................... 122 6.3.4 Effect of extracellular osmotic pressure on elastic property of single

chondrocytes ................................................................................................................. 125 6.3.4.1 AFM experimental results ............................................................................................ 125 6.3.4.2 PHE analysis of strain-rate dependent mechanical behaviour of single living

chondrocytes exposed to varying extracellular osmotic pressure conditions ................ 128 6.3.5 Dependency of relaxation behaviour of single chondrocytes on varying

extracellular osmotic pressure conditions ..................................................................... 131

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6.3.5.1 Comparison of the equilibrium moduli of chondrocytes when exposed to

solutions of varying osmolality .................................................................................... 132 6.3.5.2 Viscoelastic properties of single chondrocytes exposed to different osmotic

solutions ........................................................................................................................ 136

6.4 Conclusions.............................................................................................................................. 146

CHAPTER 7: CONCLUSION ........................................................................................................ 149

7.1 Conclusion ............................................................................................................................... 149 7.1.1 General conclusions ...................................................................................................... 149 7.1.2 Detailed conclusions ..................................................................................................... 150

7.2 Research limitations ................................................................................................................. 153

7.3 Future Research directions ....................................................................................................... 154 7.3.1 PHE analysis ................................................................................................................. 154 7.3.2 Further AFM biomechanical experiments on single cells ............................................ 154 7.3.3 Mechanical adhesiveness of single osteoblasts and chondrocytes ................................ 154

BIBLIOGRAPHY ............................................................................................................................. 157

APPENDICES ................................................................................................................................... 174 Appendix A Statistical parameters of curve fitting of AFM experimental force–

indentation curves at four different strain-rates of a typical single living and

fixed osteocyte, osteoblast and chondrocyte cell using thin-layer elastic model .......... 174

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List of Figures

Figure 1.1: Research flowchart

Figure 2.1: The disposition of chondrocytes in three zones of articular cartilage (i.e.

the surface, middle and deep zones). Reprinted from Biomaterials, 13(2),

Mow, V. C., Ratcliffe, A., Poole, A. R., Cartilage and diarthrodial joints as

paradigms for hierarchical materials and structures, Page 76, Copyright

1992, with permission from Elsevier

Figure 2.2: SEM images of chondrocyte morphology when the cells were exposed to

different osmotic stress. (A) In hypoosmotic medium; (B) In isoosmotic

medium, chondrocytes possessed a number of membrane ruffles and

microvilli; (C) In hyperosmotic medium. Scale bar = 10 µm. Reprinted

from Biophysical Journal, 82, Guilak, F., Erickson, G. R., Ting-Beall, H.

P., The effects of osmotic stress on the viscoelastic and physical properties

of articular chondrocytes, Page 723, Copyright 2002, with permission from

Elsevier

Figure 2.3: An illustration of two osteocytes (1) located in the lamellar bone of

calcified bone matrix (3). Two adjacent lamellae (2) with different

orientations of collagen fibre (7) are illustrated. The osteocyte cell bodies

are located in lacunae and are surrounded by a thin layer of un-calcified

matrix (4). The osteocytes’ processes (5) are housed in canaliculi (6).

Reprinted from Biophysical Journal, 27(3), Weinbaum, S., Cowin, S. C.,

Zeng, Y., A model for the excitation of osteocytes by mechanical loading-

induced bone fluid shear stresses, Page 342, Copyright 1994, with

permission from Elsevier

Figure 2.4: This figure presents the transitional cell types (during the second phase of

intramembranous ossification) between pre-osteoblasts and osteocytes

when osteoblast transform to osteocyte and their relationships to each

other. The pre-osteoblast layer is composed of proliferating cells. The

enlargement illustrates gap junction between the cell process of an

osteocyte and an embedding osteoblast. Arrow shows osteoid deposition

front; arrowhead presents mineralization front. 1. Pre-osteoblast, 2. Pre-

osteoblastic osteoblast, 3. osteoblast, 4. osteoblastic osteocyte (Type I pre-

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osteocyte), 5. osteoid-osteocyte (Type II pre-osteocyte), 6. Type III pre-

osteocyte, 7.young osteocyte, 8. old osteocyte. Reprinted from

Developmental Dynamics, 235(1), Franz-Odendaal, T. A., Hall, B. K.,

Witten, P. E., Buried alive: how osteoblasts become osteocytes, Page 178,

Copyright 2006, with permission from John Wiley and Sons

Figure 2.5: Models of linear viscoelasticity: (a) Maxwell, (b) Voigt and (c) SLS; and

(d) PHE model (where k, k1 and k2 are elastic constants, μ is a viscous

constant, and We is a strain energy density function of a hyperelastic

element)

Figure 2.6: Schematic representation of the three types of experimental technique

used to probe living cells; a) Atomic force microscopy (AFM) and (b)

magnetic twisting cytometry (MTC) are type A; (c) micropipette aspiration

and (d) optical trapping (d) are type B; (e) shear-flow and (f) substrate

stretching are type C. Reprinted by permission from Macmillan Publishers

Ltd: Nature Materials (Bao and Suresh 2003), Copyright 2003

Figure 2.7: A schematic view of the AFM method

Figure 2.8: Three different strategies to measure adhesion force using AFM. (a)

AFM cantilever is approached onto an adhered cell on substrate to

measure adhesion force between the cell and tip, (b) Cell attached to the

cantilever is brought into contact with another adhered cell (or a surface of

interest) to measure adhesion force between two cells (or between cell and

a surface of interest), (c) AFM cantilever is used to apply a shear force on

the cell until it’s detached to measure adhesion force between the cell and

substrate

Figure 3.1: (a) Nanosurf Flex AFM system; (b) AFM head

Figure 3.2: SEM image of colloidal probe cantilever SHOCONG-SiO2-A-5 used in

this study (the inset shows the real diameter of the bead)

Figure 3.3: Nikon A1R confocal microscope

Figure 3.4: Leica M125 light microscope

Figure 3.5: (a) JPK NanoWizard II AFM system; (b) CellHesion module; (c) AFM

head

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Figure 3.6: (a) SEM image of colloidal probe cantilever CP-PNPL-BSG-A-5 used for

the JPK–AFM system in this study (the inset shows the real diameter of

the bead – scale bar: 10 μm); (b) a living chondrocyte indented by a

colloidal probe cantilever (scale bar: 35 μm)

Figure 3.7: Cell height measurement procedure using AFM indentation (where h1,

and h2 are non-contact regions of force curves when indenting the cell and

substrate, and h is the cell’s height calculated as h = h2 – h1)

Figure 3.8: ABAQUS 6.9-1 software interface –1) Menu and toolbars; 2) Model tree;

and 3) Viewport

Figure 3.9: Boundary conditions of FEA model

Figure 4.1: Normalised deformation dependent hydraulic permeability of

chondrocytes used in the ABAQUS model in this study

Figure 4.2: Diameter and height distributions (normal) of osteocytes, osteoblasts and

chondrocytes

Figure 4.3: Trypan blue exclusion test of chondrocytes after AFM experiments – the

blue cytoplasm cell is dead (shown by a red circle)

Figure 4.4: Typical AFM experimental force–indentation curves at four different

strain-rates of a typical single living and fixed osteocyte, osteoblast and

chondrocyte cell (the Young moduli and the R2 values corresponding to

the strain-rates of 7.4, 0.74, 0.123 and 0.0123 s-1

are shown in the tables)

Figure 4.5: Young’s moduli of living and fixed osteocytes, osteoblasts and

chondrocytes subjected to four different strain-rates

Figure 4.6: FEA models of single (a) osteocyte, (b) osteoblast, and (c) chondrocyte

Figure 4.7: Experimental and PHE force–indentation curves of living and fixed

osteocytes, osteoblasts and chondrocytes at four different strain-rates (the

data are shown as mean values)

Figure 4.8: (a) von Mises stress, and (b) fluid pore pressure distributions of living

osteocytes after indentation at 7.4, 0.74, 0.123 and 0.0123 s-1

strain-rates

(the measurement unit in these figures is 106 Pa)

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osteoblasts after indentation at 7.4, 0.74, 0.123 and 0.0123 s-1

strain-rates



chondrocytes after indentation at 7.4, 0.74, 0.123 and 0.0123 s-1

strain-

rates (the measurement unit in these figures is 106 Pa)

Figure 5.1: AFM relaxation test diagram – A colloidal probe indented the cell using a

step displacement, which was then kept constant in order to study the

relaxation behaviour of the single cells

Figure 5.2: Equilibrium moduli Eequil (Pa) and 𝐸𝑌/𝐸𝑅 ratios of osteoblasts and

chondrocytes at four different strain-rates (the data are shown as mean ±

standard deviation)

Figure 5.3: Viscoelastic properties of osteocytes, osteoblasts and chondrocytes at

four different strain-rates (the data are shown as mean ± standard

deviation; Significant difference between cell types [p < 0.05] is indicated

by a corresponding coloured pentagon above the mechanical property)

Figure 5.4: Relaxation experimental data and thin-layer viscoelastic model fitted with

the curves of osteocytes, osteoblasts and chondrocytes subjected to four

different strain-rates (the data are shown as mean ± standard deviation)

Figure 5.5: AFM experimental data and PHE model force–indentation curves of a

typical living chondrocyte at (a) 7.4 s-1

, (b) 0.74 s-1

, (c) 0.123 s-1

, and (d)

0.0123 s-1

strain-rates

Figure 5.6: AFM stress–relaxation experimental data and thin-layer viscoelastic

model, PRI model and PHE model results for a typical living chondrocyte

at (a) 7.4 s-1

, (b) 0.74 s-1

, (c) 0.123 s-1

, and (d) 0.0123 s-1

strain-rates (the

fitting parameters for each model are shown in the corresponding coloured

texts)

Figure 5.7: von Mises stress (top) and fluid pore pressure (bottom) distributions – (a)

after indentation, and (b) after relaxation phase at 7.4 s-1

strain-rate (the

measurement unit in these figures is 106 Pa)

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strain-rate (the




strain-rate (the


Figure 5.10: von Mises stress (top) and fluid pore pressure (bottom) distributions –

(a) after indentation, and (b) after relaxation phase at 0.0123 s-1

strain-rate


Figure 5.11: Fluid pore pressure curves of a typical chondrocyte at (a) 7.4 s-1

, (b)

0.74 s-1

, (c) 0.123 s-1

, and (d) 0.0123 s-1

strain-rates extracted at the point

beneath the AFM tip

Figure 6.1: Diameter distributions of living chondrocytes exposed to 30, 100, 300,

450, 900 and 3,000 mOsm solutions

Figure 6.2: Height distributions of living chondrocytes exposed to 30, 100, 300, 450,

900 and 3,000 mOsm solutions

Figure 6.3: Chondrocyte volumes when exposed to 30, 100, 300, 450, 900 and 3,000

mOsm solutions (the data are shown as mean ± standard deviation; *p <

0.05 indicated that the volume was significantly changed)Figure 6.3:

Chondrocyte volumes when exposed to 30, 100, 300, 450, 900 and 3,000

mOsm solutions (the data are shown as mean ± standard deviation; *p <

0.05 indicated that the volume was significantly changed)

Figure 6.4: Ponder’s plot for the chondrocytes exhibiting a linear relationship

between the normalised cell volume and normalised extracellular medium

osmolality (the Ponder’s value was determined to be 0.5407; the data are

shown as mean ± standard deviation)

Figure 6.5: Confocal images of actin filaments of chondrocytes subjected to varying

osmotic pressure conditions from 30 to 3,000 mOsm osmolality (the cell’s

nucleus and F-actin are visualised in blue [DAPI] and red [568 phalloidin],

respectively)

xiv

Figure 6.6: Confocal images of focal adhesion distribution of chondrocytes at

varying osmotic pressure conditions

Figure 6.7: Young’s moduli of chondrocytes at four different strain-rates (7.4, 0.74,

0.123 and 0.0123 s-1

) when exposed to varying osmotic environments (30,

100, 300, 450, 900 and 3,000 mOsm)

Figure 6.8: FEA models of single chondrocytes exposed to (a) 30, (b) 100, (c) 300,

and (d) 3,000 mOsm solutions

Figure 6.9: Experimental and PHE force–indentation curves of typical single living

chondrocytes at four different strain-rates when exposed to four varying

osmotic pressure conditions (i.e. 30, 100, 300 and 3,000 mOsm)

Figure 6.10: Equilibrium modulus Eequil (Pa) of single living chondrocytes at varying

extracellular osmolality, namely, 30 and 100 mOsm (hypoosmotic

condition), 300 mOsm (isoosmotic condition) and 3,000 mOsm

(hyperosmotic condition) when subjected to different strain-rates (7.4,

0.74, 0.123 and 0.0123 s-1

) (the data are shown as mean ± standard

deviation; *p < 0.05 indicated the significant difference of the equilibrium

modulus in the osmotic pressure conditions compared to the control

condition)

Figure 6.11: 𝐸𝑌/𝐸𝑅 ratios of single living chondrocytes at varying extracellular

osmolality, namely, 30 and 100 mOsm (hypoosmotic condition), 300

mOsm (isoosmotic condition) and 3,000 mOsm (hyperosmotic condition)

when subjected to different strain-rates (7.4, 0.74, 0.123 and 0.0123 s-1

)

(the data are shown as mean ± standard deviation; *p < 0.05 indicated the

significant difference of the ratios in the osmotic pressure conditions

compared to the control condition)

Figure 6.12: Viscoelastic parameters, namely, relaxation modulus ER (Pa), relaxation

times under constant load 𝜏𝜎 (s), deformation 𝜏 (s), instantaneous

modulus E0 (Pa), and viscosity μ (log Pa.s) of single living chondrocytes at

varying extracellular osmolality – including 30 and 100 mOsm

(hypoosmotic condition), 300 mOsm (isoosmotic condition) and 3,000

mOsm (hyperosmotic condition) when subjected to 7.4 s-1

strain-rate (the

data are shown as mean ± standard deviation; *p < 0.05 indicated the

xv

significant difference in the viscoelastic parameters at the osmotic pressure

conditions compared to other conditions)







strain-rate (the

data are shown as mean ± standard deviation; *p < 0.05 indicated the









strain-rate










strain-rate




Figure 6.16: Relaxation experimental data and thin-layer viscoelastic model fitted

curves of living chondrocytes subjected to varying rates of loading (7.4,

0.74, 0.123 and 0.0123 s-1

) when exposed to four different osmotic

xvi

pressure conditions (i.e. 30, 100, 300 and 3,000 mOsm (the data are shown

as mean ± standard deviation)

xvii

List of Tables

Table 4-1: Diameters and heights of the osteocytes, osteoblasts and chondrocytes

Table 4-2: Young’s moduli (Pa) of living and fixed (using 4% paraformaldehyde)

osteocytes, osteoblasts and chondrocytes at four different strain-rates

Table 4-3 PHE material parameters of living and fixed osteocytes, osteoblast and

chondrocytes

Table 4-4: Volume strain of osteocytes, osteoblasts and chondrocytes subjected to

varying rates of loading (the measurement unit in these figures is 106 Pa)

Table 5-1: Equilibrium moduli 𝐸𝑒𝑞𝑢𝑖𝑙 (Pa) and 𝐸𝑌/𝐸𝑅 ratios of living osteocytes,

osteoblasts and chondrocytes at four different strain-rates

Table 5-2: Viscoelastic properties of living osteocytes, osteoblasts and chondrocytes

at four different strain-rates

Table 5-3: R2 and RMSE values of osteocytes, osteoblasts and chondrocytes at

different strain-rates when fitted with the thin-layer viscoelastic model

Table 5-4: PHE model material parameters and the poroelastic diffusion constant D

(µm2/s) of single living chondrocytes at four varying strain-rates

Table 5-5: Volume strain of chondrocytes after indentations and relaxation phases

when subjected to varying rates of loading

Table 6-1: Diameter (µm), height (µm), volume (µm3) and apparent membrane area

(µm2) of chondrocytes exposed to 30, 100, 300, 450, 900 and 3,000 mOsm

solutions

Table 6-2: Young’s modulus (Pa) of chondrocytes exposed to 30, 100, 300, 450, 900

and 3,000 mOsm solutions at four different strain-rates (7.4, 0.74, 0.123

and 0.0123 s-1

)

Table 6-3: PHE material parameters of living chondrocytes when exposed to four

varying extracellular osmotic pressure conditions

Table 6-4: Equilibrium moduli 𝐸𝑒𝑞𝑢𝑖𝑙 (Pa) and 𝐸𝑌/𝐸𝑅 ratios of living chondrocytes at

four different osmotic pressure conditions subjected to varying rates of

xviii

loading (7.4, 0.74, 0.123 and 0.0123 s-1

) (the data are shown as mean ±

standard deviation)

Table 6-5: Viscoelastic parameters, namely, relaxation modulus ER (Pa), relaxation

times under constant load 𝜏𝜎 (s), deformation 𝜏 (s), and viscosity μ (log

Pa.s) of living chondrocytes at four different osmotic pressure conditions

subjected to varying rates of loading (the data are shown as mean ±

standard deviation)

xix

List of Abbreviations

AAA abdominal aortic aneurysm

AFM Atomic Force Microscopy

ANOVA analysis of variance

CSK cytoskeleton

DEV deviatoric operator

DFL deflection

EPS extracellular polymer substance

FCD fixed charge density

FE Finite Element

FEA Finite Element Analysis

FEM Finite Element Method

hMSCs human mesenchymal stem cells

GAG glycosaminoglycan

MA micropipette aspiration

MSCs mesenchymal stem cells

MSE mean square error

MTC magnetic twisting cytometry

NaCl sodium chloride

NO nitric oxide

OA osteoarthritis

PBS Phosphate Buffered Saline

PCM pericellular matrix

PDL poly-D-lysine

PGA proteoglycan aggregates

PHE porohyperelastic

xx

PHEXPT porohyperelastic with the mass transport

PRI Poroelastic Relaxation Indentation

RMSE Root Mean Square Error

RP Reference Point

SEM Scanning Electron Microscope

SLS Standard Linear Solid

SnHS Standard Neo-Hookean Solid

XPT mass transport

xxi

Statement of Original Authorship

QUT Verified Signature

xxii

Statement on Ethics Approval

STATEMENT FROM THE CHAIR, QUT HUMAN RESEARCH ETHICS

COMMITTEE

Professor Michele Clark

xxiii

STATEMENT FROM THE AUTHOR

Trung Dung Nguyen

At the beginning of this project, the objective was mainly to study a suitable

mechanical model that can capture mechanical responses of single cells. As a result,

fixed cells were considered to investigate the model, and thus an ethical clearance

application was not submitted the QUT Human Research Ethics Committee (HREC)

for this project. However when our work was submitted to several academic journals,

the reviewers suggested to conduct experiments on living cells in order to accept our

results and proposed model. We then started to consider human living cells testing to

compare with fixed cells and cell line responses. The human cells tested were

obtained from Institute of Health and Biomedical Innovation (IHBI) which already

had an ethical clearance for collection of these cells from tissue of patients

undergoing knee surgery. The author acknowledges this careless mistake of not

instituting a variation to the existing approval or applying for an ethical clearance

before conducting the experiments as required by QUT. The author understands that

ethics is crucial and prerequisite in research and takes this as an experience for future

research. The author commits to follow all the QUT regulations and processes in

future studies.

STATEMENT FROM SUPERVISORS

Principal supervisor: Professor Yuantong Gu

Associate Supervisor: Professor Kunle Oloyede

In completing this thesis, extensive characterisation of fixed (dead) human

chondrocytes was initially conducted. To validate the results of these experiments,

live cells were tested to a limited extent. At the time the live cells were sourced, our

information was that our Centre’s ethics approval for experimenting on these cells

extended to their reported use in this thesis. It has since come to light that this was

mistaken and in breach of QUT’s ethics clearance/approval process for use of human

cells, and that a variation to the top-level ethics approval should have been submitted

for scrutiny and approval for their use in the experiments reported in this thesis. This

was not done and it is highly regretted by all our team members that were involved in

the work, although it was not our intention and there were some communication

issues between the different parties involved.

xxiv

Since completing the thesis, assessment and advice had been sought from the

QUT’s Office of Research Ethics and Integrity. This exercise has established that

while the conduct of our use of the live cells in our research breached the university

code of ethics. However, we do not have information on the patients or the specific

date and area from which the cells were harvested and the risk involved in the

application is low. In addition, the live human cells used in this thesis had an

institutional clearance code. We appreciate the judgement from the QUT Human

Research Ethics Committee (HREC), i.e. ‘that the study was conducted in an ethical

manner’.

The supervisory team, as well as the student, has learned a significant lesson

through this case. The supervisors commit that themselves and their team members

will comply with all the QUT regulations and processes in their future studies.

xxv

Acknowledgements

Firstly, I would like to express my utmost appreciation and acknowledgement to my

supervisor, Professor YuanTong Gu, who is exceedingly helpful and supportive

during my study. Without his valuable advice, support, and supervision, this research

would not be successfully done. Furthermore, I would like to thank Professor

Adekunle Oloyede, who is my associate supervisor, for his insightful guidance,

support, and encouragement throughout my study.

I would also acknowledge Queensland University of Technology (QUT) for

financial support, and high-performance computing facilities. In addition, I gratefully

acknowledge Central Analytical Research Facility (CARF) located at QUT and

Queensland node of the Australian National Fabrication Facility (ANFF) located at

The University of Queensland (UQ) for experimental support and assistance.

Moreover, Dr. Sanjleena Singh’s support and helpfulness are much appreciated

during my study.

I would also like to convey my love and thanks to my beloved families for their

continuously support, understanding and endless love without whom I would not

have accomplished my study. Last but not least, thanks to all members of my groups

for their helpfulness, support, encouragement, and joyfulness during my research.

Chapter 1:Introduction 1

Chapter 1: Introduction

1.1 BACKGROUND

Living cells are the basic units of structure and function in a living organism. They

are dynamic and perform several functions including metabolism, sensory detection,

growth, remodelling and apoptosis. A single biological cell is small (the size is

typically 1–100 µm) and composed of various components. For example, the

eukaryotic cell consists of a cell membrane, a cytoplasm – including the cytosol,

cytoskeleton (CSK) and various suspended organelles – and a nucleus. Among these,

the CSK is an important component when considering mechanical properties. The

CSK is composed of microtubules, actin filaments and other filaments. The CSK

stiffness is influenced by the mechanical and chemical environments such as cell-cell

and cell-extracellular matrix interactions. Hence, to understand the fundamental

processes of these biological materials, studies of mechanical properties and

responses as well as mechanochemical transduction in living cells and their

biomolecules are necessary (Bao and Suresh 2003).

Living cells in the human body are subjected to various mechanical stimuli

throughout life. Cells experience mechanical forces or deformation and transmit

mechanical signals into regulatory biological mechanisms (Bao and Suresh 2003).

These stresses and strains can result from both the external environmental and

internal physiological conditions. Experimental evidence has shown that cells are

sensitive to mechanical loading, and that the response of the cell plays an important

role in many aspects of cell physiology such as cell deformation, adhesion,

interaction, motility and signal transduction (Bao and Suresh 2003; Huang, Kamm

and Lee 2004; Lim, Zhou and Quek 2006). For example, the compression of the

extracellular matrix surrounding articular chondrocytes results in significant changes

in cell and nuclear volume and shape (Guilak 1995). Several studies have

demonstrated that many biological processes, such as growth, differentiation and

migration, are influenced by changes in cell shape and structural integrity (Lim,

Zhou and Quek 2006). It has been noted that the molecular structure of the

cytoskeleton and the cellular and sub-cellular elastic response have a connection with

2 Chapter 1:Introduction

human health and disease (Suresh et al. 2005). It is also known that the mechanical

environment of the cells is an important factor with a significant influence on the

health of the tissue (Guilak and Mow 2000). Many intact and pathological conditions

of cells are dependent on or regulated by their mechanical environment, and the

deformation behaviour of cells provide important information about their biological

and structural functions.

In the past few decades, several studies have identified the mechanical effects

within cells and molecules, and have established the connections between cell

structures, their mechanical responses and biological functions. The mechanical

response of the cell plays a significant role in many important aspects of cell

physiology such as cell deformation, adhesion, interaction, motility and signal

transduction. Therefore, a better understanding of the mechanical properties of living

cells and how they response to varying physiological conditions is an important first

step in investigating, understanding and potentially controlling the transmission,

distribution and conversion of mechanical signals into biological and chemical

responses within cells.

With recent advances in nanotechnology, a number of new experimental

techniques for characterising and studying the mechanical behaviour of living cells

have been developed such as cell poker, particle tracking, magnetic twisting

cytometry (MTC), oscillatory magnetic twisting cytometry, atomic force microscope

(AFM), micropipette aspiration, cytoindenter, optical tweezers, atomic/molecular

force probes, micro-plate manipulators and optical stretchers (Lim, Zhou and Quek

2006). Among these techniques, AFM is a state-of-the-art experimental facility for

the high resolution imaging and mechanical testing of tissues, cells and artificial

surfaces both qualitatively and quantitatively (Touhami, Nysten and Dufrene 2003;

Rico et al. 2005; Zhang and Zhang 2007; Lin, Dimitriadis and Horkay 2007a;

Kuznetsova et al. 2007; Faria et al. 2008; Yusuf et al. 2012). The principle is to

indent the material/sample with a tip of microscopic dimensions which is attached to

a very flexible cantilever. The force is measured from the deflection of the cantilever

in order to obtain the force–indentation (F-δ) curve (Darling, Zauscher and Guilak

2006; Faria et al. 2008; Ladjal et al. 2009). This powerful tool is increasingly applied

in the study of cell responses to external stimuli such as mechanical and chemical


loading, and is therefore ideal for bridging the research gap in the understanding of

microscale responses of biological organisms.

The mechanical properties and responses of single cells have been studied

widely since it is believed that these properties play an important role in biophysical

and biological responses (Guilak 2000; Costa 2004). Understanding the mechanical

properties of single cells can provide insights into not only cell physiology and

pathology but also into how a cell physically interacts with its extracellular matrix

and how its properties influence the mechanotransduction process. Cellular

behaviour in response to external stimuli such as shear stress, fluid flow, osmotic

pressure and mechanical loading are among the topics that have been investigated

(Guilak, Erickson and Ting-Beall 2002; Ofek et al. 2010; Wu and Herzog 2006).

In this study, the author did not study the biological responses, but the

mechanical responses of living cells to different external stimuli. This is the

important and preliminary investigation to relate the biophysical and biological

responses with mechanical properties of single cells. As a result, most of the data are

mechanical properties of the cells. This research will be extended in the future

studies to understand the connection between mechanical properties and biological

functions of the cells.

In order to quantitatively characterise the mechanical properties and responses

of single living cells when undergoing external stimuli, a mechanical model with

appropriate parameters that can capture the observed phenomena from experiments

needs to be developed and applied. Generally, mechanical models of living cells are

derived from using either the continuum approach or the micro/nanostructural

approach (Lim, Zhou and Quek 2006). In the present research, the continuum

approach, which assumes that the cell comprises materials with certain continuum

material properties, is applied. This approach provides a straightforward method for

simulating and computing the mechanical properties of the living cells. Moreover, it

can provide details on the distribution of macroscopic stresses and strains induced in

the cell.

Several continuum mechanical models have been developed for the single cell

as well as other biological materials. One of them is the poroelastic field theory

which is fundamental for soil mechanics. The early studies in soil mechanics were

established by Biot and Terzaghi (Terzaghi 1943; Biot 1941). Terzaghi developed a


consolidation theory for one-dimensional confined compression and Biot extended

this theory for three-dimensional consolidation of clays and soils, thereby providing

the background for the poroelastic field theory. This theory was firstly used for wet

soil by Biot (Biot 1941), and later applied to soft tissues by Oloyede and colleagues

(Oloyede and Broom 1991; Oloyede, Flachsmann and Broom 1992). This poroelastic

model considers soft tissues as a porous material consisting of a pore fluid that

saturates the tissue and flows relative to the deformable porous elastic solid to

describe the transient response of the soft tissues. This continuum model was then

extended to account for the analysis of hyperelastic solids, namely non-linear

materials, in the poroelastic formulation to give a porohyperelastic (PHE) material

law (Simon and Gaballa 1989).

The PHE model has been widely used in tissue engineering applications, such

as cartilage (Nguyen 2005; Oloyede and Broom 1993b, 1994b, 1996; Oloyede,

Flachsmann and Broom 1992), large arteries (Simon, Kaufmann, McAfee and

Baldwin 1998; Geest et al. 2011) and the brain (Li, von Holst and Kleiven 2013).

Inasmuch as there are only a few analytical solutions available for particular

situations, numerical simulations are required. Thus, inverse finite element analysis

(FEA) has been utilised successfully in soft tissue structure studies.

1.2 RESEARCH PROBLEM

It is well-known that cells respond to their various physiological mechanical

environments wherein the cells are both the detectors and effectors (Charras and

Horton 2002). Physiological loads are usually applied at varying rates to achieve

optimal biomechanical and biochemical outcomes in the body. Various studies have

been conducted to investigate the effects of strain-rate on the mechanical responses

of biological tissues (Moo et al. 2012; Oloyede, Flachsmann and Broom 1992; Quinn

et al. 2001; Kaufmann 1996). These studies conclude that the strain-rate and

magnitude of loading greatly influences cell death (Kurz et al. 2001; Ewers et al.

2001). However, little research has been conducted to investigate the strain-rate

dependent mechanical deformation behaviour of single living cells. In addition, the

influence of strain-rate on the stress–relaxation behaviour of living cells has not yet

been studied. The understanding of strain-rate dependent responses of single cells


would provide insight into living cell health, in particular, and tissue dysfunction in

general.

It is reported in the literature that the response of tissues can be transformed

from fluid-dominant to purely elastic behaviour by changing the rate of loading

(Oloyede, Flachsmann and Broom 1992; Oloyede and Broom 1993a; Kaufmann

1996). It has been explained that the behaviour of fluid pore pressure under different

rates of loading accounts for the strain-rate dependent response of the tissues. It

would be expected that single cells exhibit similar strain-rate dependent mechanical

behaviour in fluid-filled biological tissues, since Moeendarbary et al. (Moeendarbary

et al. 2013) stated that “the rate of cellular deformation is limited by the rate at which

intracellular water can redistribute within the cytoplasm”. As a result, it is believed

that the cytoplasm of living cells behaves as a poroelastic material (Moeendarbary et

al. 2013; Zhou, Martinez and Fredberg 2013). Hence, the PHE model, which is an

extension of the poroelastic model, is considered to be a good candidate for

investigating the responses of a cell to external loading and other load-induced

stimuli. Although the PHE model has been used effectively and widely in tissue

engineering, there are very few works using the PHE model in single living cell

mechanics.

In order to bridge the gaps mentioned above, three main research problems are

addressed in this study:

1. Firstly, the strain-rate dependent mechanical deformation and relaxation

responses of single living cells are investigated by conducting AFM

indentation and stress–relaxation experiments, respectively. The fluid-

dominant load sharing deformation behaviour of single cells is elucidated.

2. Secondly, the role of intracellular fluid is studied further by investigating

the effect of osmotic pressure on the changes in living cell morphology

and mechanical properties.

3. Finally, the PHE constitutive model is combined with the inverse FEA

technique in order to study the strain-rate dependent responses of the cells

and to investigate the important role of intracellular fluid in living cell

responses.


1.3 RESEARCH AIMS AND OBJECTIVES

In this study, three different cell types from different tissue origins, namely,

osteocytes, osteoblasts and chondrocytes (which are CSK-rich eukaryotic cells), are

investigated. Osteocytes and osteoblasts are bone cells whereas chondrocytes are the

mature cells in cartilage tissues. These cells perform a number of functions within

cartilage and bone. In order to further our understanding of mechanical behaviour at

the microscale level, the AFM technique is applied to study the osteocyte, osteoblast

and chondrocyte with the objective of elucidating the role of strain-rate on the

mechanical deformation and relaxation responses of cells from hard and soft tissues

at the sub-microscale level. Thus, the objectives of this research are to:

1. Investigate the strain-rate dependent mechanical deformation behaviour of

single living cells (i.e. osteocytes, osteoblasts and chondrocytes) using

AFM biomechanical indentation testing. The thin-layer elastic model is

used to identify the elastic modulus of the cells at each of four strain-rates

tested.

2. Characterise the strain-rate dependent relaxation behaviour of single living

cells using AFM stress–relaxation testing. The thin-layer viscoelastic

model is utilised in order to determine the viscoelastic properties of single

living cells at varying rates of loading.

3. Study the effect of extracellular osmotic pressure on the morphology,

mechanical deformation and relaxation behaviour of single living

chondrocytes using AFM indentation and stress–relaxation testing. The

AFM experiments are conducted at four different strain-rates for each of

the osmotic solutions tested. The thin-layer elastic and viscoelastic models

are used to determine the elastic modulus and viscoelastic properties of the

chondrocytes at each of the four strain-rates when the cells are subjected to

different osmotic solutions.

4. Apply the PHE model coupled with the inverse FEA technique to simulate

both the strain-rate dependent mechanical deformation and relaxation

responses of single living cells in order to elucidate the role of intracellular

fluid.


1.4 SIGNIFICANCE AND CONTRIBUTION

To our knowledge, there is little research on the strain-rate dependent mechanical

responses of single cells and their mechanisms. Thus, this research provides insight

into how cells respond to varying mechanical rates of loading. Moreover, by

studying the responses of single living chondrocytes when exposed to solutions of

varying osmolality, the effect of the osmotic environment on cellular morphology

and mechanical properties is elucidated.

To date, there has been little research that uses the PHE model to study living

cell mechanics; thus, this study is one of the first to apply this model in the study of

single living cell mechanics. It should be noted, however, that the PHE model has

been applied in a variety of biomechanical studies yielding reasonable and acceptable

results (Oloyede and Broom 1991; Oloyede, Flachsmann and Broom 1992; Simon

and Gaballa 1989; Kaufmann 1996; Simon, Kaufmann, McAfee, Baldwin, et al.

1998). With this approach, the cells can be modelled and clear insights can be

obtained into their fluid-dominant deformation and swelling behaviour. Moreover,

the macroscale model created in this research provides information on the stress and

strain distributions induced on and in the cell. This information can be used as the

input in more accurate microscale or nanoscale simulations of the cell (i.e. nucleus

and cytoskeleton).

Therefore, overall, the research reported in this thesis provides a better

understanding of the mechanisms underlying the cellular responses to external

mechanical loadings and of the process of mechanical signal transduction in living

cells. It would help us to enhance knowledge of and insight into the role of

mechanical forces in supporting tissue regeneration or degeneration.

1.5 THESIS OUTLINE

This thesis comprises seven chapters. Chapter 1 presents the research background

and research problem. In addition, the research aims and objectives are discussed in

detail followed by an overview of the significance and contributions of this study.

Additionally, the thesis outline and flowchart are given in this chapter.

In Chapter 2, some previous works from literature which are related to this

research are reviewed. Information about the target cells such as the tissue origin and


structure is briefly outlined and the findings on those cells’ properties and behaviour

are discussed in detail. Several mechanical models that are commonly used for cell

biomechanics are discussed followed by a description of various experimental

techniques that can be used to characterise cells’ mechanical properties.

In Chapter 3, the materials and methodology used in this research are discussed

in detail. The sample preparations and AFM biomechanical experimental set-up are

introduced. Moreover, the single living cells’ dimension measurements are

discussed. Some information about the FEA model used in this study is also

presented.

In Chapter 4, the strain-rate dependent mechanical deformation behaviour of

single living cells is investigated using AFM indentation mechanical testing. In this

chapter, the thin-layer elastic model, which is utilised to characterise the elastic

modulus of living cells, is discussed in detail. The AFM experimental results and

calculated mechanical properties of the cells are given. The PHE theory and inverse

FEA technique are also presented in this chapter. The PHE coupled with inverse

FEA is then used to simulate the strain-rate dependent mechanical deformation

behaviour of single living cells.

In Chapter 5, investigation of the dependency of the relaxation behaviour of

single living cells on strain-rates is presented. The relaxation behaviour of each cell

type is evaluated by conducting AFM stress–relaxation testing. The thin-layer

viscoelastic model, which is used to determine the viscoelastic parameters of living

cells, is introduced in this chapter. In addition, the PHE coupled with inverse FEA is

used to simulate the strain-rate dependent relaxation response of single living cells.

In Chapter 6, the changes in the morphology and mechanical properties of

single living chondrocytes due to varying extracellular osmotic pressures are

investigated and discussed in detail. The thin-layer elastic and viscoelastic models

are applied to characterise the mechanical properties of chondrocytes at different

osmotic solutions. The results are discussed in this chapter. The PHE model, which is

similar to the one presented in Chapter 4, is used to study the effect of solution

osmolality on the hydraulic permeability of single living chondrocytes.

In Chapter 7, the major conclusions are presented. Additionally, possible

directions for future work are given. In particular, the mechanical adhesiveness


technique is introduced; it is believed to be a powerful tool that can be used in future

studies to investigate the adhesion strength between single cells and different

substrates as well as different proteins.

1.6 FLOWCHART OF THIS RESEARCH

Figure 1.1 presents a detailed flowchart of this research.

Figure 1.1: Research flowchart

Chapter 2: Literature Review 11

Chapter 2: Literature Review

2.1 INTRODUCTION

2.1.1 Cartilage and chondrocyte structure and properties

Cartilage is the flexible connective tissue found in many parts of human and animal

body such as nose, ear, elbow, knee, etc. Articular cartilage is the hyaline and

avascular tissue that covers the surfaces of the diarthrodial joints. Its function is to

provide a nearly frictionless bearing surface for the bones to transmit and distribute

mechanical loads and reduce the subchondral compressive stress within the join

(Mow, Ratcliffe and Poole 1992; Oloyede and Broom 1991; Oloyede, Flachsmann

and Broom 1992; Oloyede and Broom 1994b, 1996; Guilak, Erickson and Ting-Beall

2002). This mechanical property is due to the unique microstructure and composition

of articular cartilage. This tissue consists of fluid, collagen, proteoglycans and

chondrocytes, the single cell in this tissue. The state of constant turnover is

maintained in articular cartilage by the balance of the anabolic and catabolic

activities of the chondrocytes (Guilak, Erickson and Ting-Beall 2002). Cartilages

have different biomechanical properties in different species such as bovine, canine,

human, monkey and rabbit (Athanasiou et al. 1991).

This tissue is typically divided into four zones which are superficial, middle,

deep, and the calcified cartilage layer (Glenister 1976). Collagen and proteoglycan

content, collagen fiber orientation and cell morphology and density vary across these

zones. This renders articular cartilage inhomogeneous and anisotropic. For example,

the adult human patella-femoral groove has higher aggregate modulus i.e. 𝐻𝑎 =

1.237 ± 0.486 𝑀𝑃𝑎 in the direction parallel to the articular surface than in the

direction perpendicular to the surface i.e. 𝐻𝑎 = 0.845 ± 0.383 𝑀𝑃𝑎 (Jurvelin,

Buschmann and Hunziker 2003). It has also been proven that cartilage properties

such as modulus of elasticity and peak compressive stress are reduced when

proteoglycans were digested (Murakami et al. 2004). Researchers were also

interested in the behaviour of cartilage subjected to different loading rate varied from

impact velocity to very low strain rate (Oloyede, Flachsmann and Broom 1992). The

12 Chapter 2: Literature Review

authors conducted the compression tests on cartilage with and without the

subchondral bone at loading rate ranging from 10-5

s-1

to 103

s-1

and concluded that

the matrix stiffness increase dramatically in the “low” and “medium” strain-rate

regimes and reached a limiting value at “high” loading rate up to impact (Oloyede,

Flachsmann and Broom 1992).

Chondrocytes are cytoskeleton (CSK)-rich eukaryotic cells which are the

mature cells in cartilage tissues and perform a number of functions within the tissue.

In this study, the articular chondrocytes are investigated. It is well accepted that,

under physiological conditions, mechanical forces can regulate the metabolic activity

of chondrocytes in articular cartilage. During joint loading, the deformation of

cartilage is associated with significant changes in chondrocytes shape and volume

(Guilak, Ratcliffe and Mow 1995). These changes are believed to be involved in the

process of mechanotransduction by chondrocytes in articular cartilage, but the

specific mechanism of this phenomenon is not fully elucidated. In order to isolate the

mechanism by which chondrocytes transmit the mechanical signal into biochemical

response, it is important to identify the deformation of the cartilage as well as the

mechanical environment around the cells. However, the mechanical properties of

chondrocytes must be firstly known. Thus, a number of studies have been carried out

to determine the deformation behaviour and mechanical properties of chondrocytes

both in vivo and in vitro (Guilak et al. 1999; Guilak 2000; Guilak and Mow 1992).

The mechanical properties of these cells are significantly altered in the development

and progression of osteoarthritis (Guilak et al. 1999; Trickey, Lee and Guilak 2000;

Jones, Ting-Beall, et al. 1999).

There are a number of researches have been done to investigate the

chondrocytes’ properties. They were determined in situ, by embedding the cells in

agarose gel and compressing to different strains (Freeman et al. 1994), leading to the

conclusion that “the chondrocyte may be altering its intracellular composition by

cellular processes in response to mechanical loading”. A custom-designed computer-

controlled motorized loading apparatus was also created to study the deformation

behaviour of the chondrocyte in articular cartilage and its microenvironment under

transient loading (Chahine, Hung and Ateshian 2007). The authors observed that

significant strain amplification occurred in the microenvironment of the cell. In order

to have a better understanding of in situ deformation behaviour of chondrocytes,


several multi-scale numerical models have been developed to investigate the

biomechanical interactions between the chondrocyte and the extracellular matrix and

the influence of the cell and matrix properties on the local stress-strain state around

the cell (Wu, Herzog and Epstein 1999; Guilak and Mow 2000; Wu and Herzog

2000; Moo et al. 2012).

The mechanical properties of single chondrocytes have also been investigated

using different experimental techniques such as compression test (Leipzig and

Athanasiou 2005; Shieh, Koay and Athanasiou 2006; Shieh and Athanasiou 2006),

micropipette aspiration (Baaijens et al. 2005; Zhou, Lim and Quek 2005; Trickey et

al. 2006), and Atomic Force Microscopy (Darling, Zauscher and Guilak 2006;

Wozniak et al. 2010). The Young’s modulus and Poisson’s ratio were determined to

be around 0.61-2.7 kPa and 0.26-0.5, respectively for the chondrocytes collected

from different species such as heifers, steers, pigs and humans.

The chondrocytes are typically flat in the surface zone compared to spherical

shape in the middle/deep zone (Wu, Herzog and Epstein 1999) (see Figure 2.1). This

leads to position-dependent properties of the cell. For example, the superficial cells

have been shown to be stiffer than the middle/deep cells (Shieh and Athanasiou

2006; Guilak, Ratcliffe and Mow 1995). Their results are derived from compression

tests with step increase in pressure of the probe. The results have shown that at 15%

surface-to-surface tissue compression, around 14.8-15.7% local tissue strain was

observed in the middle and deep zones whereas 19.1% local strain was recorded in

the surface zone. Moreover, the cells at different zones also perform differently

during cyclic loading (Wu and Herzog 2006). The authors state that the depth-

dependent behaviour of the cells is influenced by the amplitude of cyclic loading.

Finally, the depth-dependent gradient in fixed charge density due to an increasing

fraction of proteoglycans from the surface to the deep zone of cartilage may also

affect the chondrocyte water transport response and properties. This fixed charge

density gradient caused the changing of extracellular osmolality in different zones

which may vary with loading conditions, growth and development, or disease

(Oswald et al. 2008). Oswald and colleagues also measured the water content in

chondrocyte which also varied from surface to deep zone.

The term “chondron” has been used to describe the chondrocyte together with

the enclosed pericellular matrix (PCM). This PCM is a distinct narrow tissue region


that surrounds the chondrocyte (Poole 1997, 1992; Poole, Flint and Beaumont 1987).

Several scientists have studied the mechanical properties of PCM as well as

chondron (Nguyen et al. 2009, 2010; Jones, Ping Ting-Beall, et al. 1999). It is noted

that the mechanical behaviour of the cells is altered because of the presence of PCM

(Guilak and Mow 1992). In addition, the modulus of isolated chondrons was

determined to be much larger than that of the chondrocytes, but still lower than that

of the extracellular matrix (ECM) of the cartilage (Nguyen et al. 2010; Guilak et al.

1999).

Figure 2.1: The disposition of chondrocytes in three zones of articular cartilage (i.e.

the surface, middle and deep zones). Reprinted from Biomaterials, 13(2), Mow, V.

C., Ratcliffe, A., Poole, A. R., Cartilage and diarthrodial joints as paradigms for

hierarchical materials and structures, Page 76, Copyright 1992, with permission from

Elsevier

The mechanical properties of the cells are also dependent on their condition

e.g. whether osteoarthritic or non-osteoarthritic condition. In particular, the Young’s

modulus was found to be 0.36 kPa for non-osteoarthritic and 0.5 kPa for

osteoarthritic chondrocytes (Trickey, Lee and Guilak 2000). Osteoarthritis may also

affect the osmolality of cartilage which in turn affects chondrocyte behaviour and

properties (Oswald et al. 2008). Also, the mechanical properties of PCM between

non-OA and OA chondrons were studied, with results demonstrating that the

Young’s modulus of the non-OA chondron is larger than that of the OA one. In

contrast, the non-OA chondron has lower permeability than the OA one, while no

significant difference in the Poisson’s ratio was found between them (Alexopoulos et

al. 2005; Jones et al. 1997).


Structurally, the chondrocyte mainly consists of nucleus, cytoskeleton and

cytoplasm. The process of mechanical signal transduction might be influenced by the

deformation of the nucleus of the chondrocyte (Guilak 1995). It has also been

suggested that the nuclei behave like a viscoelastic material and are stiffer and more

viscous than the whole cell (Guilak, Tedrow and Burgkart 2000; Ofek, Natoli and

Athanasiou 2009). In fact, chondrocyte nuclei have around 3 times larger the

instantaneous and equilibrium elastic moduli than those of the cytoplasm in normal

cells. Also, the nuclei have twice the viscosity than that of intact chondrocytes

(Guilak, Tedrow and Burgkart 2000). Moreover, the nucleus together with its

envelope is considered for modelling (Vaziri, Lee and Mofrad 2006; Vaziri and

Mofrad 2007). The nucleus envelop consists of three layers namely inner and outer

nuclear membranes and one thicker layer called nuclear lamina (Vaziri and Mofrad

2007). The membrane of the cell has also been considered together with the

cytoplasm (Zhang and Zhang 2007) to study the effect of membrane pre-stress on the

relation between the indentation force and depth. Additionally, it has been

demonstrated that the cell membrane affects water transport through and around

chondrocytes (Ateshian, Costa and Hung 2007).

It is known that the mechanical environment of the chondrocytes plays an

important role in influencing the health of the diarthrodial joint (Guilak and Mow

2000; Alexopoulos et al. 2005). Several studies have been performed to determine

the mechanical properties and response of chondrocytes to mechanical stimuli such

as compression (Caille et al. 2002; Ofek, Natoli and Athanasiou 2009; Guilak and

Mow 2000; Leipzig and Athanasiou 2005), direct shear (Ofek et al. 2010), aspirating

into micropipette (Baaijens et al. 2005), and under cyclic loading (Wu and Herzog

2006), etc. These studies employed both simulation and experimental methods to

explore the mechanical properties of the cells, which are shear modulus (Ofek et al.

2010), Poisson’s ratio (Trickey et al. 2006), etc. It is observed that chondrocytes

behave as an intrinsically viscoelastic solid-like material (Baaijens et al. 2005) and

its mechanical properties are position-dependent in articular cartilage i.e. on the

surface, in the middle and in the deep zones of cartilage (Wu and Herzog 2006) and

also non-uniform along the height itself (Ofek et al. 2010).

Moreover, various studies have been conducted to investigate the effects of

impact loading on cartilage damage and chondrocyte death (Ewers et al. 2001; Kurz


et al. 2001; Quinn et al. 2001). These researchers concluded that strain-rate and

magnitude of loading greatly influence chondrocyte death and that cell death

occurred mostly in the superficial zone of cartilage. A better understanding of the

strain-rate dependent behaviour of chondrocytes is arguably a significant

contribution that would provide insight into chondrocyte health in particular and

cartilage dysfunction in general.

2.1.2 Swelling state in cartilage and chondrocyte

When cartilage is under mechanical compression, the interstitial fluid flows out from

the tissue. This causes an increased concentration of macromolecules and fix-charged

density that alters the osmotic environment of the chondrocytes (Guilak 2000). Most

cells of the body respond to osmotic pressure by activating some processes to return

to its original state with their volume restored (Guilak, Erickson and Ting-Beall

2002). In order to characterise the influence of osmotic environment in mechanical

properties as well as the morphology of chondrocytes, the authors suspended the

cells in various media such as isoosmotic solution; hypoosmotic solution

(chondrocytes are exposed in deionised water) and hyperosmotic solution

(chondrocytes are suspended in NaCl solution). They observed that in hypoosmotic

medium, the cells swelled significantly with a smooth plasma membrane. This

swelling state resulted in cell diameter and volume increase. On the other hand,

chondrocytes exhibited dramatic shrinkage and decrease in cell volume with

concomitant increase in membrane ruffling when exposed to hyperosmotic medium

(Guilak, Erickson and Ting-Beall 2002) (see Figure 2.2). Moreover, the results have

shown that hypoosmotic pressure greatly decreased the instantaneous and

equilibrium elastic moduli and the apparent viscosity of the cell as compared to the

cell in isoosmotic condition. However, it is interesting to note that the hyperosmotic

pressure did not significantly affect chondrocyte properties.

As presented in (Tombs and Peacocke 1974), the osmotic pressure can be

determined for a three-component system with the assumption of ideal Donnan

equilibrium swelling conditions as:

𝜋 = 𝑅𝑇 (𝑐2

𝑀2+ 𝐵′𝑐2

2) (2.1)

where π is osmotic pressure; R is the universal gas constant; T is the temperature; B’

is the second virial coefficient representing the contribution to the total osmotic


pressure of the difference between the total molalities of small diffusible ions inside

and outside the membrane; c2 is the weight concentration of the macro-ion; and M2 is

its molar mass.

Figure 2.2: SEM images of chondrocyte morphology when the cells were exposed to

different osmotic stress. (A) In hypoosmotic medium; (B) In isoosmotic medium,

chondrocytes possessed a number of membrane ruffles and microvilli; (C) In

hyperosmotic medium. Scale bar = 10 µm. Reprinted from Biophysical Journal, 82,

Guilak, F., Erickson, G. R., Ting-Beall, H. P., The effects of osmotic stress on the

viscoelastic and physical properties of articular chondrocytes, Page 723, Copyright

2002, with permission from Elsevier

Nguyen (Nguyen 2005) utilised this relationship to develop a mathematical

model to account for physicochemical swelling and deformation-dependence of

cartilage deformation. This swelling behaviour of chondrocytes can be used to

determine their hydraulic permeability (McGann et al. 1988; Xu, Cui and Urban

2003; Wu, Lyu and Hsieh 2005; Kleinhans 1998).

2.1.3 Structure and properties of bone cells

It is well-known that osteocytes are the most plentiful cell type in bone, filling up

around 90-95% of all bone cells (around 20,000 to 80,000 cells per mm3 bone tissue)

(Kardas, Nackenhorst and Balzani 2013; Franz‐Odendaal, Hall and Witten 2006). In

vivo, osteocytes which are located inside ellipsoidal lacunae have round morphology

and numerous processes which are surrounded by a proteoglycan-rich bone fluid

space (see Figure 2.3 for more details) to act as the mechanosensors of the bone

(McCreadie and Hollister 1997), and thereby determine how cells respond to forces.

Osteocytes carry out mechanosensing function, producing nitric oxide (NO) in

response to stress to alter the activity of other cells for building and resorbing bone

(Burger et al. 1995; Burger and Klein-Nulend 1999; Aviral et al. 2006).


Figure 2.3: An illustration of two osteocytes (1) located in the lamellar bone of

calcified bone matrix (3). Two adjacent lamellae (2) with different orientations of

collagen fibre (7) are illustrated. The osteocyte cell bodies are located in lacunae and

are surrounded by a thin layer of un-calcified matrix (4). The osteocytes’ processes

(5) are housed in canaliculi (6). Reprinted from Biophysical Journal, 27(3),

Weinbaum, S., Cowin, S. C., Zeng, Y., A model for the excitation of osteocytes by

mechanical loading-induced bone fluid shear stresses, Page 342, Copyright 1994,

with permission from Elsevier

It was originally assumed that the load-bearing matrix directly results in strain

on cellular deformation. However, it was reported that the deformations of bone

matrix as a result of physiological loading are relatively small due to mineralization

of the extracellular matrix making the bone tissue significantly stiff (Cowin, Moss-

Salentijn and Moss 1991). Therefore, bone cells will not experience more than 0.2 to

0.4% unidirectional strain as a result of physiological loads (Rubin and Lanyon

1982). As a result, a different mechanism based on fluid flow has been proposed for

the sensitivity of osteocytes to mechanical loading (Weinbaum, Cowin and Zeng

1994; Klein-Nulend et al. 1995; Burger et al. 1995). It is known that mechanical

loading causes interstitial fluid flow through the canalicular network (Kufahl and

Saha 1990), and hence, it is hypothesized that this fluid flow through the canaliculi

provides the mechanism by which communicating osteocytes experience the very

small in vivo strains in the bone matrix (Weinbaum, Cowin and Zeng 1994; Klein-

Nulend et al. 1995; Burger et al. 1995). This mechanism shows that osteocytes are


very sensitive to very small fluid-induced shear stresses (Weinbaum, Cowin and

Zeng 1994) which stimulate the osteocytes to produce factors that regulate bone

metabolism (Klein-Nulend et al. 1995).

It has also been known, for around one and a half centuries, that osteocytes are

differentiated from osteoblasts which are bone forming cells (Franz‐Odendaal, Hall

and Witten 2006). The entire transformation process is clearly shown in Figure 2.4.

Briefly, the osteoblasts are differentiated from mesenchymal stem cells (MSCs),

secrete non-mineralized bone matrix (osteoid), then become osteoid osteocytes in

osteoid, and finally transform to mature osteocytes in mineralized bone matrix.

During this transformation process, the osteoblasts change their morphology from

cubical shape to the stellate shape of osteocytes (Cowin, Moss-Salentijn and Moss

1991; Franz‐Odendaal, Hall and Witten 2006; Palumbo, Palazzini and Marotti 1990;

Palumbo et al. 1990).

Figure 2.4: This figure presents the transitional cell types (during the second phase of

intramembranous ossification) between pre-osteoblasts and osteocytes when

osteoblast transform to osteocyte and their relationships to each other. The pre-

osteoblast layer is composed of proliferating cells. The enlargement illustrates gap

junction between the cell process of an osteocyte and an embedding osteoblast.

Arrow shows osteoid deposition front; arrowhead presents mineralization front. 1.

Pre-osteoblast, 2. Pre-osteoblastic osteoblast, 3. osteoblast, 4. osteoblastic osteocyte

(Type I pre-osteocyte), 5. osteoid-osteocyte (Type II pre-osteocyte), 6. Type III pre-

osteocyte, 7.young osteocyte, 8. old osteocyte. Reprinted from Developmental

Dynamics, 235(1), Franz-Odendaal, T. A., Hall, B. K., Witten, P. E., Buried alive:

how osteoblasts become osteocytes, Page 178, Copyright 2006, with permission from

John Wiley and Sons


It is well known that cells respond to the varying mechanical environments

imposed by normal physiological functions and diseases where the cells are both

detectors and effectors (Charras and Horton 2002). Cellular behaviour in response to

external stimuli such as shear stress, fluid flow, osmotic pressure and mechanical

loading have been investigated recently (Guilak, Erickson and Ting-Beall 2002; Ofek

et al. 2010; Wu and Herzog 2006). The results reveal that the mechanical properties

of cells are influenced by mechanical forces generated by the cytoskeleton structure,

interactions between neighbouring cells, and adhesion to substrates (Sugawara et al.

2008; Li et al. 1987; Ingber et al. 1994). Especially, the alterations of the mechanical

properties due to cytoskeletal changes affect cell growth, cell cycle progression and

gene expression (Sugawara et al. 2008; Ingber 1993; Ingber et al. 1995; Mooney et

al. 1992). As the mechanical properties of cells are related to physiologically

important processes, the investigation of these properties of living cells would yield

insight into the mechanisms involved in the functions and activities of living cells.

Understanding the importance of measurement of mechanical properties of

bone cells, several investigators have been attempted to identify the elastic modulus

of these living cells including osteocytes, and osteoblasts (Sugawara et al. 2008;

Rommel et al. 2008; Darling et al. 2008). Rommel et al. estimated the stiffness of

osteocytes of different morphologies using AFM. They reported that the flat adhered

osteocytes were stiffer than the round partially adhered cells (Rommel et al. 2008).

However, the flat cells exhibited an increase in fluorescence intensity, which is

proportional to the increase Nitric Oxide (NO), by only 17% compared to seven-fold

for the round cells. Thus, they concluded that even though the round cells are softer,

they seem more mechanosentitive than flat cells (Rommel et al. 2008). Another

research group investigated the mechanical properties of bone cells during the

process of changing from osteoblasts to osteocytes using AFM (Sugawara et al.

2008). Sugawara et al. concluded that the stiffness of bone cells reduced

continuously when the osteoblasts firstly transit to osteoid osteocytes and finally to

mature osteocytes. Also, osteoblasts had significantly higher focal adhesion area

compared to osteocytes (Sugawara et al. 2008). Some models have also been

developed and proposed by several investigators to study the mechanical behaviour

of osteocytes to different loads such as fluid drag, compressive force, etc.

(Weinbaum, Cowin and Zeng 1994; Lidan et al. 2001; Kardas, Nackenhorst and


Balzani 2013). Some common mechanical models for cell mechanics study will be

introduced in the next section.

2.2 MECHANICAL MODELS OF LIVING CELLS

As outlined in (Lim, Zhou and Quek 2006), there are two approaches to developing

mechanical models for living cells, namely, the continuum approach and

micro/nanostructural approach. The former is the focus in this study.

2.2.1 Cortical shell-liquid core models (or liquid drop models)

The cortical shell-liquid core models were first used to study the neutrophils in

micropipette aspiration. In the literature, a number of liquid drop models have been

developed including the Newtonian, the compound Newtonian, the shear thinning

and the Maxwell models. Evans and Kukan (Evans and Kukan 1984) studied the

large deformation response and recovery of granulocytes in micropipette aspiration

and observed that granulocytes were deformed continuously into micropipettes with

small diameters for suction pressures over a certain threshold and recovered to their

initial spherical shape upon release. They also proposed the concept that the

granulocyte membrane behaves like a “contractile surface carpet” under tension,

where the interior behaves like a highly viscous liquid. The Newtonian liquid drop

model was developed by Evans and Yeung (Evans and Yeung 1989b) for simulating

the passive flow of liquid-like spherical cells into a micropipette. This model

assumed the cell’s interior to be a homogeneous Newtonian viscous liquid and the

cell’s cortical shell to be an anisotropic viscous layer with persistent lateral tension.

In another of their work, they also determined the apparent viscosity and cortical

tension of blood granulocytes under micropipette aspiration assumptions (Evans and

Yeung 1989a).

If the Newtonian model is satisfactory for large deformations, Maxwell liquid

drop model can account for the small or initial deformation phase. Small

deformation and recovery properties of leukocytes have therefore been studied using

this model (Dong et al. 1988), where their model consists of prestressed cortical shell

containing a Maxwell fluid. However, many types of living cells such as eukaryotic

cells, chondrocytes and endothelial cells consist of several components, namely,

membrane, cytoplasm, and a nucleus with different properties. Thus, Newtonian and


Maxwell models are not totally valid for modelling these types of cells. The

compound liquid drop model was developed to address these types of cells

(Hochmuth et al. 1993; Dong, Skalak and Sung 1991).

2.2.2 Solid models

It has been reported that endothelial cells and chondrocytes behave as solid-like

materials (Caille et al. 2002; Guilak and Mow 2000). Hence, these cells can be

modelled using solid models including the incompressible elastic solid or

viscoelastic solid models. The salient feature of these models is that the whole cell is

usually assumed to be homogeneous without considering the cortical layer (Lim,

Zhou and Quek 2006). Among these, viscoelastic models are more commonly used

for modelling single living cells. There are several models of viscoelasticity such as

Maxwell, Voigt and the standard linear solid (SLS) which consist of springs and

dashpots (Fung 1965) (see Figure 2.5).

Figure 2.5: Models of linear viscoelasticity: (a) Maxwell, (b) Voigt and (c) SLS; and

(d) PHE model (where k, k1 and k2 are elastic constants, μ is a viscous constant, and

We is a strain energy density function of a hyperelastic element)

Linear viscoelasticity can be expressed in both forms e.g. the integral and

differential forms (Phan-Thien 2002), each of which has its own parameters. The

latter has been used frequently in cell mechanics literature.

In the differential form of linear viscoelasticity, the stress is expressed in terms

of strain history with three material constants (Zhou, Lim and Quek 2005):


𝐒 +𝜇

𝑘2�̇� = 𝑘1𝛆 + 𝜇 (1 +

𝑘1

𝑘2) �̇� (2.2)

𝛆 = ∇𝐮 + ∇𝐮𝑇�̇� = ∇𝐯 + ∇𝐯𝑇𝐒 = 𝛔 + 𝑝𝑰 (2.3)

where S is the deviatoric stress tensor, t is the current time, ε is the engineering strain

tensor, which is the same as the deviatoric component under the condition of

incompressibility, �̇� is the engineering strain rate tensor (the superpose dot denotes

differentiation with respect to time), k1 and k2 are two elastic constants, μ is a viscous

constant (see Figure 2.5c), u is the displacement field, v is the velocity field, σ is the

total stress tensor, p is the hydrostatic pressure and I is the unit tensor.

For the SLS model, the time-dependent shear relaxation and bulk moduli G(t)

are expressed as a one-term Prony series expansion as expressed below (ABAQUS

1996):

𝐺(𝑡) = 𝐺0[1 − 𝑔1(1 − 𝑒−𝑡/𝜆1)] (2.4)

𝐾(𝑡) = 𝐾0[1 − 𝑘1(1 − 𝑒−𝑡/𝜆1)] (2.5)

The relationship between the material constants and parameters are given in

Equation (2.3) as:

𝐺0 = 𝑘1 + 𝑘2𝑔1 =𝑘2

𝑘1+𝑘2𝜆1 =

𝜇

𝑘2 (2.6)

This model has been used widely to study the mechanical properties and

behaviour of not only chondrocytes but also its nucleus (Trickey, Lee and Guilak

2000; Darling, Zauscher and Guilak 2006; Vaziri and Mofrad 2007; Cheng,

Unnikrishnan and Reddy 2010; Sato et al. 1990) and a large variability in the results

was observed. For example, the Young’s modulus for non-osteoarthritic and

osteoarthritic chondrocytes was found to be 0.36 kPa and 0.50 kPa, respectively in

(Trickey, Lee and Guilak 2000) compared to the values of 0.65 kPa and 0.67 kPa in

(Jones, Ting-Beall, et al. 1999). The differences in the values obtained in the various

experiments are due to the time that isolated cells were in culture prior to testing.

Zhou et al. (Zhou, Lim and Quek 2005) proposed a nonlinear viscoelastic

model, namely, standard neo-Hookean solid (SnHS) model for large deformation

analysis of living cells. This model replaces the linear elastic elements by Neo-

Hookean hyperelastic elements, with the constitutive law as a simple hyperelastic


relationship, where the strain energy density function of this incompressible material

is:

𝑈 =𝐺0

2(𝐼1 − 3) (2.7)

where G0 is the shear modulus, I1 is the deviatoric strain invariant, defined as:

𝐼1 = 𝜆12 + 𝜆2

2 + 𝜆32 (2.8)

with 𝜆1, 𝜆2 and 𝜆3 are principal stretches. The deviatoric part S of the Cauchy stress

tensor is:

𝐒 = 𝐺0 (𝐁 −1

3𝐼1 ∙ 𝐈)

𝐁 = 𝐅 ∙ 𝐅𝑇 𝐅 =∂𝐱

∂𝐗

(2.9)

where S is the deviatoric part of the Cauchy stress tensor, G0 is the shear modulus, F

is the deformation gradient of the current configuration x relative to the initial

configuration X, and B is the left Cauchy-Green strain tensor.

This SnHS viscoelastic model is an extension of the SLS viscoelastic

fomulation. The deviatoric part of the Cauchy stress tensor is:

𝐒(𝑡) = 𝐒0(𝑡) + SYM [∫�̇�(𝑠)

𝐺0𝐅𝑡

−1𝑡

0(𝑡 − 𝑠) ∙ 𝐒0(𝑡 − 𝑠) ∙ 𝐅𝑡(𝑡 − 𝑠)𝑑𝑠] (2.10)

𝐅𝑡(𝑡 − 𝑠) =𝜕𝐱(𝑡 − 𝑠)

𝜕𝐱(𝑡)

where Ft(t − s) is the deformation gradient of the configuration x(t − s) at time t − s,

relative to the configuration x(t) at time t, and S0(t) represents the instantaneous

stress caused by the deformation, which can be computed using Equation (2.9),

SYM[·] denotes the symmetric part of a matrix.

2.2.3 Mixture theory – based models

The cortical shell-liquid core models and solid models described above treat the cell

as a single phase material. However, it is known that cytoplasm consists of both the

solid contents and interstitial fluid (Leterrier 2001). The biphasic model is an

approach that treats the cell as constituting two separate phases. Several researchers

have utilised this model to study musculoskeletal cell mechanics, especially single

chondrocytes and their interaction with the extracellular cartilage matrix (Mow et al.

1980; Guilak and Mow 2000; Alexopoulos et al. 2005; Wu, Herzog and Epstein


1999; Ofek, Natoli and Athanasiou 2009). The solid phase was treated as linearly

elastic and non-dissipative and the fluid phase as an incompressible viscous/non-

viscous and non-dissipative fluid. The stresses in the two phases can be described as

(Mow et al. 1980; Lim, Zhou and Quek 2006):

𝜎𝑠 = −𝜙𝑠𝑝𝐼 + 𝜆𝑠𝑡𝑟(휀)𝐼 + 2𝜇𝑠휀

𝜎𝑓 = −𝜙𝑓𝑝𝐼 (2.11)

where s and f denote the stresses in the solid phase and in the interstitial fluid,

respectively; s and s are the first and the second Lamé constants for the solid

phase; I is the identity tensor; is the Caucy’s infinitesimal strain tensor; p is the

fluid pressure; s and

f represent the solid and fluid volumetric fractions,

respectively (where 1s s ).

This biphasic theory was used to study the biomechanical interactions between

the chondrocyte and its extracellular matrix (Guilak and Mow 2000). The authors

assumed that the cell is a continuum homogeneous mixture of a solid phase

comprising cytoskeleton and proteins, and a fluid phase comprising cytosol – water

with dissolved proteins and ions. They also assumed that these phases are

incompressible and that the cell membrane does not influence the mechanical

behaviour of the cell at small strains. In order to study the influence of the

chondrocyte and tissue properties on the local stress-strain environment, Guilak and

Mow (Guilak and Mow 2000) developed a biphasic multi-scale finite element model

for the mechanical environment of a single chondrocyte within the cartilage

extracellular matrix. They concluded that the elastic properties of the chondrocyte

were important factors in determining the biomechanical interactions between the

cell and its matrix. Also, the presence of a pericelluar matrix may play a significant

role in defining the mechanical environment of the cell.

The mechanical properties of this narrow pericellular matrix were again

considered. Its properties were measured using micropipette aspiration coupled with

a linear biphasic finite element model. The properties of intact PCM were compared

with that of the osteoarthritic PCM to determine the biomechanical changes of the

latter (Alexopoulos et al. 2005). The results revealed significant differences in

Young’s modulus and permeability between non-OA and OA PCM but not the

Poisson’s ratios.


The biphasic model has also been used to study the stress–relaxation behaviour

of articular cartilage in compression (Wang, Hung and Mow 2001). In their model,

they consider the inhomogeneous property of cartilage by accounting for the depth-

dependent aggregate modulus. They concluded that the mechanical environment

inside the cartilage was regulated significantly when the inhomogeneity is considered

and that the inhomogeneous aggregate modulus should be incorporated into the

biphasic theory (Wang, Hung and Mow 2001). Besides that, Wilson and his

colleagues have improved the biphasic model by incorporating collagen-fibril

structure (Wilson et al. 2004). They developed the viscoelastic collagen fibrils in 13

different orientations at arbitrary points in the matrix to investigate the stresses and

strains in the collagen network. These stresses and strains were believed to reflect the

damage of collagen which is likely to be one of the earliest signs of osteoarthritic

cartilage degeneration (Wilson et al. 2004).

The biphasic model has been compared with other models in the literature.

Leipzig and Athanasiou (Leipzig and Athanasiou 2005) compared the biphasic

model with elastic and viscoelastic models to obtain the material properties of single

chondrocyte through unconfined creep compression. They concluded that the

biphasic model is not the best one to model the compression of chondrocyte whereas

the viscoelastic model may be able to capture the creep response of chondrocytes to

unconfined compression. This conclusion is identical to Baaijens et al. (Baaijens et

al. 2005). They used the viscoelastic model; purely biphasic model (the stress tensor

for the solid phase is described by the compressible Neo-Hookean model); and

biphasic viscoelastic model (the stress tensor for the solid phase is described by the

two-mode viscoelastic model). They found that the purely biphasic model cannot

capture the observed creep behaviour of chondrocyte, while a viscoelastic or biphasic

viscoelastic model can predict more precisely the chondrocyte response. However,

both intrinsic viscoelastic mechanisms (e.g. solid-solid interactions) and biphasic

mechanisms (e.g. fluid-solid interactions) may influence the overall response of

chondrocytes to mechanical loads (Trickey et al. 2006).

This biphasic model has been used widely to study the mechanical properties

and deformation behaviour of cartilage tissue (Ateshian et al. 1997). In a previous

study (Mow et al. 1990), the investigators assumed the permeability is constant

(independent of deformation) and obtained different results that were compared with


experiments. They also observed a large deviation in the permeability coefficient and

this leads to development of an exponential function for the deformation-dependent

permeability coefficient (Lai, Mow and Roth 1981; Ateshian et al. 1997). They

concluded that the frictional drag from the relative motion between solid and fluid

phases is the most important factor accounting for the viscoelastic properties of the

cartilage in compression (Lai, Mow and Roth 1981).

As stated by Lai et al. (Lai, Hou and Mow 1991), when the unloaded cartilage

specimen is put in NaCl solutions at constant temperature, the tissue’s dimensions

decrease exponentially with increasing NaCl concentration. This descent reaches an

asymptote at a high concentration, e.g., 2.5 M NaCl. Hence, cartilage is in a swollen

state at its physiological state of 0.15 M NaCl, with the swelling pressure resisted by

the elastic stress in the collagen-proteoglycan solid matrix. To have a clearer insight

of such phenomenon, Lai et al. (Lai, Hou and Mow 1991) has proposed a triphasic

mathematical model which consists of the two fluid-solid phases and an ion phase,

and is a further development of biphasic mixture theory proposed by Mow et al.

(Mow et al. 1980):

𝜎 = −𝑃𝐼 − 𝑇𝑐𝐼 + 𝜆𝑠𝑡𝑟(𝐸)𝐼 + 2𝜇𝑠𝐸 (2.12)

where σ is the stress in the tissue’s matrix (solid, fluid phases and ions); −𝑇𝑐𝐼 is the

chemical-expansion stress in the solid phase; and other parameters are defined in

Equation (2.11)

In this theory, a more complicated formula for Donnan pressure Tc was added

into the constitutive equation of the biphasic model to account for osmotic effect.

They assumed that the fixed charge density (FCD) along the proteoglycan aggregates

(PGA)’s glycosaminoglycan (GAG) chains is unchanged, and the counter-ions are

the cations of a single salt of the bathing solution e.g. NaCl. In 1998, a modified

triphasic model was proposed by Gu et al. (Gu, Lai and Mow 1998) which is

composed of n+2 components (1 charged solid phase, 1 noncharged solvent phase,

and n ion species) to model the mechano-electrochemical behaviour of charged-

hydrated soft tissues with multi-electrolytes. They concluded that there are three

types of forces involved in the transport of ions and solvent through such materials:

1) a mechanochemical force; (2) an electrochemical force; and (3) an electrostatic

force. Nguyen (Nguyen 2005) showed that the governing equations of biphasic and


triphasic models are actually similar to that of the classic consolidation model for soil

mechanics (Terzaghi 1943), and more particularly the generalised form of biphasic

model is identical to Biot’s poroelastic theory (Biot 1941; Biot 1972).

However, there are several limitations of biphasic model that need to be

addressed. Firstly, Harrigan (Harrigan 1987) stated that the “molecular-level mixing

[of cartilage] makes the definition of phases within the tissue meaningless, and the

only reasonable phase to define is a single phase in the cartilage as a whole”.

Secondly, it was proven that, for the linear biphasic model, the ratio of the

instantaneous stress to the equilibrium stress as determined by the biphasic model

cannot be larger than 3/(2(1 + )) ∈ ⟨1,1.5⟩, where is the Poisson’s ratio of the

solid phase (Armstrong, Lai and Mow 1984; Miller 1998). This limitation was

demonstrated again by the unconfined compression test of chondroepiphysis in the

“fast” loading-rate regime (Brown and Singerman 1986). Finally, Brown and

Singerman reported that the biphasic model “is seemingly incapable of capturing a

very substantial portion of the transient component of the response [of cartilage] in

the case of “slow” loadings…” (Brown and Singerman 1986). Therefore, it was

believed that the consolidation approach which is mentioned below is suitable to

study the functional relationships between the individual components of soft

biological materials (Oloyede and Broom 1993a).

2.2.4 Consolidation models

The classical consolidation theory is commonly used for the behaviour of a porous

solid saturated with pore fluid such as soils and clays (Terzaghi 1943). The general

theory of three-dimensional consolidation was developed by Biot (Biot 1941) with

several assumptions such as the isotropy of soils, linearity of stress-strain relations

and small strains deformation. Moreover, the water contained in the pores is

incompressible and may contain air bubbles. This water flows through the porous

skeleton and can be described using Darcy’s law. Biot later developed a theory of

finite deformation of porous solids to account for non-linear problems (Biot 1972).

Biot’s theory has been applied to many engineering problems including soil

mechanics (Sherwood 1993) and biomechanics (Meroi, Natali and Schrefler 1999;

Simon 1992; Nguyen 2005; Weinbaum, Cowin and Zeng 1994; Moeendarbary et al.

2013).


This model is also called the poroelastic model and has been used extensively

in tissue engineering applications especially articular cartilage (Oloyede and Broom

1991; Oloyede, Flachsmann and Broom 1992; Oloyede and Broom 1993b, 1994b)

and large arteries (Simon, Kaufmann, McAfee, Baldwin, et al. 1998; Simon,

Kaufmann, McAfee and Baldwin 1998). Oloyede et al. (Oloyede, Flachsmann and

Broom 1992) discovered that the cartilage stiffness increased as the strain-rate

increased in the low strain-rate regime, but that this stiffness reached an asymptotic

value with increasing strain rate. They also concluded that the response of this tissue

is transformed from the fluid-dominant to purely elastic behaviour by changing the

rate of loading (Oloyede, Flachsmann and Broom 1992; Oloyede and Broom 1993a).

Whereas articular cartilage behaved as a hyperelastic material at high strain-rates, it

responded with consolidation-dependent behaviour at low strain-rates. The authors

concluded that fluid is dominant in the strain-rate dependent behaviour of cartilage.

After that, Oloyede and Broom (Oloyede and Broom 1993b) developed a physical

model to describe the behaviour of cartilage based on consolidation theory. They

have compared the behaviour of their model of the sponge with the cartilage

deformation and concluded that the model could demonstrate the effect of

permeability on a consolidating non-linear elastic matrix of the cartilage. Another

interesting feature was (Oloyede and Broom 1994b) that the faster the rate of

decrease in radial permeability of the cartilage relative to the axial one, the longer it

takes for radial consolidation to be completed.

This poroelastic model has also been used to investigate the relaxation

behaviour of the samples (Hu et al. 2010; Hu et al. 2011; Chan et al. 2012). Recently,

a method called Poroelastic Relaxation Indentation (PRI), which is discussed in

detail in Chapter 5, was firstly proposed to study the poroelastic relaxation behaviour

of hydrogels (Hu et al. 2010; Hu et al. 2011; Chan et al. 2012; Yu, Sanday and Rath

1990). Lately, this model has been applied in cell biomechanics studies

(Moeendarbary et al. 2013).

In order to characterise and predict the behaviour of finite strain and non-linear

structures, porohyperelastic (PHE) theory was developed as an extension of the

poroelastic theory (Simon and Gaballa 1989). The details of this theory are described

clearly by (Simon 1992; Simon, Kaufmann, McAfee and Baldwin 1998; Kaufmann


1996; Laible et al. 1994). The field equations of porohyperelasic theory are

summarised below.

Kinematics:

A material point is initially at Xi, and time t0 and finally at xi at time t. The

solid’s total displacements are ui = ui (Xj, t)= xi – Xi, the velocities are �̇�𝑖 = 𝑑𝑢𝑖/𝑑𝑡

and the accelerations are �̈�𝑖 = 𝑑�̇�𝑖/𝑑𝑡. The motion of fluid is described using an

average fluid displacement Ui. The corresponding pore fluid relative displacements,

velocities, and accelerations are 𝑤𝑖 = 𝑛(𝑈𝑖 − 𝑢𝑖); �̇�𝑖 = 𝑛(�̇�𝑖 − �̇�𝑖); and �̈�𝑖 =

𝑛(�̈�𝑖 − �̈�𝑖), respectively. The Lagrangian descriptions for these equations are

�̃�𝑖 = 𝐽𝜕𝑋𝑖/𝜕𝑥𝑗𝑤𝑗; �̇̃�𝑖 = 𝐽𝜕𝑋𝑖/𝜕𝑥𝑗�̇�𝑗; and �̈̃�𝑖 = 𝐽𝜕𝑋𝑖/𝜕𝑥𝑗�̈�𝑗, where J is the

volumetric strain.

The current volume dV corresponds to the reference volume dVo. Assuming

that the material is saturated by the fluid, the porosity is 𝑛 = 𝑑𝑉𝑓/𝑑𝑉 and assuming

an incompressible solid, 𝑛 = 1 − 𝐽−1(1 − 𝑛0). Here dVf is the volume of fluid in dV

and the original porosity of the material is 𝑛0 = 𝑑𝑉0𝑓

/𝑑𝑉0. The void ratio e = n/(1 -

n). The volumetric strain J = dV/dV0 = det(Fij). Then the overall density 𝜌 =

𝑑𝑚/𝑑𝑉 = (1 − 𝑛)𝜌𝑠 + 𝑛𝜌𝑓 where 𝜌𝑠 = 𝑑𝑚𝑠/𝑑𝑉𝑠and 𝜌𝑓 = 𝑑𝑚𝑓/𝑑𝑉𝑓.

The deformation gradient Fij = dxi/dXj, and Finger's strain 𝐻𝑖𝑗 = 𝐹𝑖𝑘−1𝐹𝑗𝑘

−1.

Green's strain is Eij = (FkiFki – δij)/2 and �̇�𝑖𝑗 =1

2(

𝜕𝑥𝑘

𝜕𝑋𝑖

𝜕�̇�𝑘

𝜕𝑋𝑗+

𝜕𝑥𝑘

𝜕𝑋𝑗

𝜕�̇�𝑘

𝜕𝑋𝑖). The Lagrangian

relative fluid volumetric strain are: 𝜍̃ =𝜕�̃�𝑘

𝜕𝑋𝑘, 𝜍̃̇ =

𝜕�̇̃�𝑘

𝜕𝑋𝑘. Deviatoric invariants are

𝐼1̅ = 𝐽−2/3𝐼1and 𝐼2̅ = 𝐽−4/3𝐼2 with strain invariants 𝐼1 = 3 + 2𝐸𝑘𝑘 and 𝐼2 = 3 +

4𝐸𝑘𝑘 + 2(𝐸𝑖𝑖𝐸𝑗𝑗 − 𝐸𝑖𝑗𝐸𝑖𝑗).

Momentum Conservation Equations:

𝜕𝑇𝑖𝑗

𝜕𝑋𝑗+ 𝐽𝜌(𝑏𝑖 − �̈�𝑖) − 𝜌𝑓 𝜕𝑥𝑖

𝜕𝑋𝑗�̈̃�𝑗 = 0 (2.13)

where 𝑏𝑖 are body forces, 𝑇𝑖𝑗 = 𝐽𝜎𝑚𝑗𝜕𝑋𝑖

𝜕𝑥𝑚 is the first Piola-Kirchhoff total stress.

A generalised Dacy’s law:

𝜕𝜋𝑓

𝜕𝑋𝑖+ 𝜌𝑓 𝜕𝑥𝑗

𝜕𝑋𝑖(𝑏𝑗 − �̈�𝑗) −

1

𝑛

𝜕𝑥𝑘

𝜕𝑋𝑖

𝜕𝑥𝑘

𝜕𝑋𝑗�̈̃�𝑗 = �̃�𝑖𝑗

−1�̇̃�𝑗 (2.14)


where the fluid stress given by 𝜋𝑓 = −(fluid pressure), �̃�𝑖𝑗 is the symmetric

permeability tensor referred to reference configuration as �̃�𝑖𝑗 = 𝐽𝜕𝑋𝑖

𝜕𝑥𝑚𝑘𝑚𝑛

𝜕𝑋𝑗

𝜕𝑥𝑛. The

anisotropic permeability is kij.

Conservation of (incompressible) solid and (incompressible) fluid mass is a

constraint of the form:

𝜕�̇̃�𝑖

𝜕𝑋𝑘+ 𝐽𝐻𝑘𝑙�̇�𝑘𝑙 = 0 (2.15)

Constitutive law:

There are two material properties required, namely, the drained effective strain

energy density function 𝑊𝑒 = 𝑊𝑒(𝐸𝑖𝑗), and the hydraulic permeability �̃�𝑖𝑗.

𝑊𝑒defines the “effective” Cauchy stress, 𝜎𝑖𝑗𝑒 , in:

𝜎𝑖𝑗 = 𝜎𝑖𝑗𝑒 + 𝜋𝑓𝛿𝑖𝑗 , 𝜎𝑖𝑗

𝑒 = 𝐽−1𝐹𝑖𝑚𝑆𝑚𝑛𝑒 𝐹𝑗𝑛 (2.16)

where the pore fluid stress is 𝜋𝑓 = −(fluid pressure); 𝑆𝑖𝑗𝑒 = 𝐽𝐹𝑖𝑚

−1𝜎𝑚𝑛𝑒 𝐹𝑗𝑛

−1 is the

effective second Piola-Kirchhoff stress derived from 𝑊𝑒 as:

𝑆𝑖𝑗 = 𝑆𝑖𝑗𝑒 + 𝐽𝜋𝑓𝐻𝑖𝑗 , 𝑆𝑖𝑗

𝑒 =𝜕𝑊𝑒

𝜕𝐸𝑖𝑗 (2.17)

where 𝑊𝑒 and 𝜋𝑓 are indeterminate, subject to the mass constraint shown in

Equation (2.15).

This theory was used to identify the material properties of large arteries with

the assumption of isotropic materials (Simon, Kaufmann, McAfee, Baldwin, et al.

1998). Following this, the model was applied in understanding the local

biomechanical environment in abdominal aortic aneurysm (AAA)

(Ayyalasomayajula, Vande Geest and Simon 2010). Moreover, PHE model has been

demonstrated to be suitable for gaining insight into complex stress sharing between

the fluid and solid phases of articular cartilage (Oloyede and Broom 1994a; Oloyede

and Broom 1993a).

The PHE theory was extended to include transport and swelling effects in soft

tissue (Simon et al. 1996). This theory works for linear, small-strain and isotropic

materials which include specific convection and chemical effects. In their works, the

soft tissue structures were considered to consist of deformable, porous elastic

skeleton (solid phase) that are saturated with a mobile pore fluid (fluid phase) that


flows through the pores of the solid phase. The “third phase” is a mobile species

which can move in or with the interstitial fluid (Simon et al. 1996; Simon, Kaufman,

et al. 1998; Rigby, Park and Simon 2004). This model was called a porohyperelastic-

transport-swelling (PHETS) which is capable of simulating coupled deformation,

stress, mobile water flux, albumin flux and swelling behaviour of soft tissues (Simon,

Kaufman, et al. 1998).

Geest et al. (Geest et al. 2011) coupled the porohyperelastic (PHE) and mass

transport (XPT) models, leading to the PHEXPT model to account for the mass

transport of a single neutral species in a soft tissue. They then used the commercial

FEA software ABAQUS (ABAQUS Inc., USA) to solve the Eulerian PHE Finite

Element Method (FEM) and the Lagrangian XPT FEM separately and utilized

Fortran program to couple the two FEM results (Geest et al. 2011).

2.3 EXPERIMENTAL METHODS FOR LIVING CELLS

To date, there are several experimental methods developed to study the mechanical

behaviour of living cells such as micropipette aspiration, AFM indentation,

cytoindentation and MTC (Trickey, Lee and Guilak 2000; Darling, Zauscher and

Guilak 2006; Jones, Ting-Beall, et al. 1999). These can be classified into three

categories as shown in Figure 2.6 (Bao and Suresh 2003).

The first classification is local probing of a portion of the cell. Atomic force

microscopy (AFM) and MTC belong to this category (Bao and Suresh 2003). In

AFM, a sharp probe attached at the free end of a flexible cantilever is used to

generate a local deformation on the cell (Figure 2.6a). The applied force can be

calculated from the deflection of the cantilever. There is a novel biomechanical

testing technique similar to AFM, namely, cytoindentation that has also been

developed for measuring the intrinsic mechanical properties of single cells (Shin and

Athanasiou 1999, 1997). In MTC, magnetic beads are attached to a cell and a

magnetic field generates a twisting moment on the beads to apply deformation on a

portion of the cell (Figure 2.6b) (Bao and Suresh 2003).

The second category is mechanical loading of an entire cell. Micropipette

aspiration and optical tweezers or laser trap belong to this type (Bao and Suresh

2003). In micropipette aspiration, a suction pressure is applied through the

micropipette which is placed on the surface of the cell to apply deformation on it


(Figure 2.6c). By measuring the projection length of cells inside the pipette, the

response of the cell is studied. This technique has been used widely to study the

mechanical properties of single cells and their nuclei (Sato et al. 1990; Baaijens et al.

2005; Vaziri and Mofrad 2007). In order to determine the material parameters, some

theoretical models have been proposed, e.g. the half-space theory. For the

micropipette aspiration of the SLS viscoelastic model, the viscoelastic parameters

were determined from the experimental data with use of the half-space theory with

an applied uniform pressure from the micropipette (Trickey, Lee and Guilak 2000).

This theory was proposed by (Sato et al. 1990) to account for the viscoelastic

response of the cell as:

𝐿(𝑡)

𝑅𝑝=

𝛷𝑝∆𝑃

2𝜋𝑘1[1 + (

𝑘1

𝑘1+𝑘2− 1) 𝑒−𝑡/𝜆1] 𝐻(𝑡) (2.18)

where H(t) is the Heaviside function, L is the projection length, RP is the pipette

radius, 𝛷𝑝 is a function of the ratio of the pipette wall thickness to the pipette radius,

𝛷𝑝 = 2.0 − 2.1 when the ratio is equal to 0.2–1.0.

Figure 2.6: Schematic representation of the three types of experimental technique

used to probe living cells; a) Atomic force microscopy (AFM) and (b) magnetic

twisting cytometry (MTC) are type A; (c) micropipette aspiration and (d) optical

trapping (d) are type B; (e) shear-flow and (f) substrate stretching are type C.

Reprinted by permission from Macmillan Publishers Ltd: Nature Materials (Bao and

Suresh 2003), Copyright 2003


In optical tweezers technique, a dielectric bead of high refractive index and a

laser beam are used to create an attraction force between them. The bead is pulled

towards the focal point of the trap (Figure 2.6d) (Bao and Suresh 2003).

The third type is simultaneous mechanical stressing of a cohort of cells. Shear-

flow method and substrate stretching belong to this type (Bao and Suresh 2003).

Shear-flow experiments are conducted by using either a cone-and-plate viscometer or

a parallel-plate flow chamber (Figure 2.6e). In substrate stretching, a thin-sheet

polymer substrate on which cells are cultured is deformed while maintaining the

cell’s viability in vitro to examine the effects of mechanical loading on cell

morphology, phenotype and injury (Figure 2.6f). The substrate is coated with ECM

molecules for cell adhesion (Bao and Suresh 2003).

Among these techniques, AFM has emerged as a state-of-art experimental

facility for high resolution imaging of tissues, cells and any surfaces at the nanometer

or sub-nanometer scale as well as for probing mechanical properties of the samples

both qualitatively and quantitatively (Touhami, Nysten and Dufrene 2003; Rico et al.

2005; Zhang and Zhang 2007; Lin, Dimitriadis and Horkay 2007a; Kuznetsova et al.

2007; Faria et al. 2008). It can be used to study various soft materials (Radmacher,

Fritz and Hansma 1995; Dimitriadis et al. 2002). AFM has been used in a variety of

cells such as cancer cells (Sokolov 2007; Li et al. 2008; Faria et al. 2008; Cross et al.

2008), stem cells (Ladjal et al. 2009), bacterial cells (Deupree and Schoenfisch 2008;

Zhang et al. 2011), osteocytes (Rommel et al. 2008), chondrocytes (Darling,

Zauscher and Guilak 2006; Darling et al. 2007), etc.

It was invented in 1986 (Binnig, Quate and Gerber 1986), and can be operated

in different environments e.g. in air and liquid media. Its principle is based on

interaction force detection between a sharp probe, known as AFM tip, and the

sample’s surface. This tip is attached onto a very flexible cantilever which has

triangular or rectangular shape. The normal and lateral deflections of the cantilever

are detected by an optical system of detection. Laser light is reflected from the top of

the cantilever and detected by a photodiode (Figure 2.7). The AFM tip is landed on

or close to the sample’s surface. While scanning over the surface, AFM system

collects the deflection of the cantilever to map the three-dimensional morphology of

the surface of interest. Due to the flexibility of cantilever, it can detect the surface

with nanometer or sub-nanometer precision.


Figure 2.7: A schematic view of the AFM method

AFM experiments can be performed in different modes depending on the

nature of the interaction between the tip and sample surface. These modes include

contact mode AFM techniques (e.g. force modulation, lateral force microscopy and

force-curve analysis) or by phase imaging in the tapping mode AFM (intermittent,

semi contact) (Kuznetsova et al. 2007).

AFM is a powerful and high precision technique to probe the mechanical

properties of samples (Radmacher 1997; Rico et al. 2005; Sirghi 2010). Its principle

is to indent the cell with a tip of microscopic dimension and the force is measured

from the deflection of the cantilever to obtain the force-indentation (F-δ) curve

(Darling, Zauscher and Guilak 2006; Faria et al. 2008; Ladjal et al. 2009). The

Young’s modulus of the sample is extracted from this curve by using Hertzian

models from the continuum mechanics of contacts which were widely used in AFM

(Hertz 1881; Sneddon 1965; Johnson 1987). These models describe the indentation

of a rigid indenter (AFM tip) into an infinitely extending deformable elastic half

space (sample surface) with the assumption of negligible tip-surface adhesion

(Touhami, Nysten and Dufrene 2003). The force-indentation depth relationships are

given for two tip geometries e.g. a conical and a paraboloid indenter (Touhami,

Nysten and Dufrene 2003):

𝐹𝑐𝑜𝑛𝑒 =2

𝜋tan 𝛼

𝐸

1−2 𝛿2 (2.19)

𝐹𝑝𝑎𝑟𝑎𝑏𝑜𝑙𝑜𝑖𝑑 =4

3

𝐸

1−2𝑅1/2𝛿3/2 (2.20)


where α is the half-opening angle of a conical tip, R is the radius of curvature of a

spherical or paraboloid indenter, E is the Young’s modulus or tensile elastic modulus

of the materials and is its Poisson’s ratio.

This Hertzian theory has two major assumptions which are linear elasticity and

infinite sample thickness. Unfortunately, these two assumptions may lead to

significant error (Dimitriadis et al. 2002). Therefore, Dimitriadis et al. have proposed

a modified Hertzian model called thin-layer model to account for the finite thickness

of soft materials. In their works, they considered the indentation with spherical tips

on finite thickness samples which are both bonded and not bonded to the substrate

(Dimitriadis et al. 2002). Inasmuch as our single living cells are relatively small/thin

compared to the indenter size, this modified model, namely, thin-layer model is used

in this study and is presented in detail in Chapter 4.

Recently, AFM has been dominant in the study of the mechanical properties of

soft materials such as cells (Touhami, Nysten and Dufrene 2003; Li et al. 2008; Faria

et al. 2008; Darling et al. 2008; Darling et al. 2007). However, data post-processing

to identify the tip-sample interaction points from the force-indentation curves

remains one of the most challenging tasks, especially, for biological materials. A

number of analysis techniques have been utilized by researchers to determine the

pertinent, linear elastic portion of dataset and identify the Young’s modulus by fitting

the data with a contact mechanics model (Lin, Dimitriadis and Horkay 2007b). The

simplest fitting method is to visually (manually) inspect the force-indentation curves,

and find the contact point and eliminate the non-contact regions. Nevertheless, this

method may cause subjective, poor and biased results since the contact point is

determined without considering its effect on the quality of fit. As reported in

previous study (Dimitriadis et al. 2002), choosing the incorrect contact point may

cause significant error in the estimated Young’s modulus of the samples. Thus, an

automatic AFM force curve analysis algorithm was proposed by Lin et al. (Lin,

Dimitriadis and Horkay 2007b) to find the contact point and estimate elastic modulus

automatically and precisely. This algorithm is used in this study following

implementation in MATLAB R2013a (The MathWorks, Inc.).

AFM has also been used to measure the adhesion force of cells on different

biomaterial surfaces (Marcotte and Tabrizian 2008; Simon and Durrieu 2006; Franz

and Puech 2008). There are three different strategies to measure adhesion force using


AFM investigated in the literature. In the first approach, the cantilever indented the

cell with a certain indentation depth before retracting from the cell (see Figure 2.8a).

The adhesion force between the AFM tip and adhered cell is measured from the

force-indentation curve. This method was used to measure the adhesion forces

between the bacterial cell and tip at different surface positions on the cell as well as

at cell-cell interface of a developing biofilm (Fang, Chan and Xu 2000). The authors

concluded that the adhesion force at cell-cell interface was higher than that at the

bacterium surface, which is most likely because of the accumulation of extracellular

polymer substance (EPS).

In the second approach, the AFM tip is firstly coated with a single cell and then

brought into contact with either a surface of interest (Razatos 2001; Bowen et al.

1998) or another cell that adhered onto a substrate (Benoit et al. 2000; Krieg et al.

2008) until a set-point contact force is achieved (see Figure 2.8b). After a certain

contact time, the single cell coated AFM cantilever is retracted and a force-distance

curve is recorded. The cell-surface or cell-cell adhesion force is measured from this

curve.

Figure 2.8: Three different strategies to measure adhesion force using AFM. (a)

AFM cantilever is approached onto an adhered cell on substrate to measure adhesion

force between the cell and tip, (b) Cell attached to the cantilever is brought into

contact with another adhered cell (or a surface of interest) to measure adhesion force

between two cells (or between cell and a surface of interest), (c) AFM cantilever is

used to apply a shear force on the cell until it’s detached to measure adhesion force

between the cell and substrate

In the last strategy, the AFM tip is used in the contact mode to apply the shear

lateral force on the cell until it detaches (see Figure 2.8c). This method has been used

successfully to investigate the adhesion force of various bacterial cells such as


Staphylococcus aureus (Boyd et al. 2002), Enterococcus faecalis (Sénéchal, Carrigan

and Tabrizian 2004), P. aeruginosaand S. Aureus (Whitehead et al. 2006), etc. The

advantage of this technique is that it utilizes the contact scanning mode that is

available in any AFM system compared to other techniques that may require some

special facilities (Huang et al. 2003).

2.4 NUMERICAL TECHNIQUES

While a number of experimental techniques have been developed and applied in

biomechanics studies, they still do not provide a comprehensive analysis of a single

cell’s response to short- and long-term mechanical forces/loads. The reason is that it

is very difficult to experiment on a single living cell in the real biological

environment. In addition, the experiments do not show the dynamic progression of

responses in a mechanical event. Numerical simulations therefore offer significant

benefits towards gaining insights into biological processes and predicting outcomes,

particularly when used with advanced experimental studies to calibrate model

parameters and identify appropriate model assumptions.

Recently, numerical techniques have been developed and applied to many

engineering problem in various fields including biotechnology. They have proven to

have numerous advantages and to be a useful tool for scientists to accomplish their

objectives in an easier and more effective way. Numerical simulations are potential

methods to explore mechanical properties of single living cells. Among these, Finite

Element Method (FEM) is one of the most commonly used numerical methods. This

method has been applied widely to study the mechanical properties and behaviour of

tissues and single cells as well as their components (Zhou, Lim and Quek 2005;

Vaziri and Mofrad 2007; Vaziri, Gopinath and Deshpande 2007; Simon, Wu and

Evans 1983; Guilak and Mow 1992; Zhao, Wyss and Simmons 2009).

The mechanical properties of isolated cells were extensively investigated in the

literature. Several experiment methods have been utilised to predict the material

parameters of biological cells such as magnetic twisting cytometry (MTC), atomic

force microscopy (AFM) and micropipette aspiration (MA) and FEM is a useful

method together with experiments to understand the effect of various constituents in

the cell (Cheng, Unnikrishnan and Reddy 2010). Simulation results are also

compared with those of theory models to study whether both techniques estimate the


identical material parameters (Zhao, Wyss and Simmons 2009) and it is shown that

both the analytical standard half-space model and inverse FE method fit well with the

experimental data.

Simon and colleagues have used FEM to study various soft tissues such as eye,

arteries, intervertebral disc, etc., (Simon, Wu and Evans 1983; Simon and Gaballa

1989; Simon 1992; Laible et al. 1994; Simon et al. 1996; Simon, Kaufmann,

McAfee, Baldwin, et al. 1998; Simon, Kaufmann, McAfee and Baldwin 1998;

Simon, Kaufman, et al. 1998; Rigby, Park and Simon 2004; Simon and Durrieu

2006; Ayyalasomayajula, Vande Geest and Simon 2010; Geest et al. 2011; Laible et

al. 1993). Besides, this method has been applied effectively in cell biomechanics. A

number of investigators have studied both isolated and in situ cells biomechanical

properties and their microenvironments. For instance, multiscale finite element (FE)

models with application of biphasic models have been proposed to study the

interaction between chondrocytes and their extracellular matrix subjected to external

mechanical stimuli (Guilak and Mow 2000; Chahine, Hung and Ateshian 2007; Moo

et al. 2012).

Ateshian et al. (Wu and Herzog 2000; Ateshian, Costa and Hung 2007) have

simulated the compression of chondrocyte cells to study their location- and time-

dependent mechanical behaviour as well as intracellular transport mechanisms.

Another experimental technique, i.e. micropipette aspiration, has also been used for

single chondrocytes. While those techniques presented above tested the whole cells,

AFM is used to locally indent a single cell and probe its mechanical properties using

a flexible cantilever. Researchers have used several mechanical models such as

hyperelastic, viscoelastic and biphasic to determine chondrocytes’ mechanical

properties from micropipette aspiration experiments (Zhou, Lim and Quek 2005;

Baaijens et al. 2005; Trickey et al. 2006). FE models have also been developed to

simulate this technique as presented in the literature (Ladjal et al. 2009; Charras and

Horton 2002). A review of the simulation of several experimental techniques has

been presented in a previous study (Cheng, Unnikrishnan and Reddy 2010).

Also, the effects of cell constituents on their properties have been studied such

as the membrane (Zhang and Zhang 2007) and nucleus (Vaziri, Gopinath and

Deshpande 2007). Isolated cell nuclei have been studied to determine their

mechanical properties (Vaziri, Lee and Mofrad 2006; Vaziri and Mofrad 2007).


Moreover, its contribution on properties and behaviour of the cell has also been

investigated (Caille et al. 2002; Ofek, Natoli and Athanasiou 2009; McGarry 2009).

The properties of PCM and chondrons were also investigated by several researchers

(Alexopoulos et al. 2005; Nguyen et al. 2010).

The interaction between cells and their substrates has also been studied using

FEM. The reorientation of endothelial cells subjected to both uniaxial and biaxial

cyclic stretch of the substrate was investigated and compared with experimental

results (McGarry, Murphy and McHugh 2005; Wang et al. 2001). Another important

aspect of cell behaviour, namely adhesion force, has also been examined with FEM.

Both simulation and cytodetachment experiments were considered in the literature

and the results showed that FEM simulation can capture experiments very well and

be used to study the effect of focal adhesion on cells’ adhesion (Huang et al. 2003;

McGarry and McHugh 2008).

2.5 SUMMARY AND IMPLICATIONS

From the literature review performed, several points of interact to the current work

are summarized below:

An abundance of living cells such as blood cells, endothelial cells,

osteocytes, chondrocytes, stem cells, etc. have been studied to explore

their mechanical properties and responses to mechanical loading in vivo as

well as in vitro conditions. The external stimuli might include

compression, indentation, shear, etc. However, little research has been

conducted to investigate the strain-rate dependent mechanical response of

single living cells and their mechanisms.

A number of continuum mechanical models with suitable parameters have

been developed to best fit experimentally observed phenomena in order to

study cells’ mechanical properties and behaviour. In particular, the cortical

shell-liquid core models have been used effectively and extensively for

single white blood cells, which were assumed to be liquid-like materials.

Additionally, solid models have been utilized to study the mechanical

properties of eukaryotic cells, which were assumed to be solid-like

materials. However, the single living cells consist of both solid and liquid

components, which require more complicated mechanical models. The


PHE model has been used widely in biomechanics and is considered to be

one of the most suitable models for living cells since it can account for the

interaction between fluid and solid phases and swelling behaviour under

osmotic pressure. However, to the best of our knowledge, there has been

very little work using the PHE model for osteocytes, osteoblasts, and

chondrocytes.

Stress–relaxation behaviour of single living cells has been widely studied

in the literature. There is little work, however, to study its dependence on

strain-rate. The most common mechanical model used to capture

relaxation behaviour of single cells is viscoelastic model which could not

consider the effect of intracellular fluid. It is hypothesized that the PHE

model would be more suitable to capture this behaviour. However, there is

a lack of investigation applying the PHE model to capture the relaxation


The effect of extracellular osmotic pressure on single cell morphology and

mechanical properties has been studied in the literature. However, the

effect of osmotic pressure on single cell strain-rate dependent mechanical

deformation and relaxation behaviour has not been investigated. Moreover,

the variations of PHE material parameters, especially, the hydraulic

permeability of living cells with media osmolality has not been considered.

There are plenty of advanced experimental techniques used to

mechanically probe cells with forces and displacement such as

micropipette aspiration, cytoindenter, AFM, etc. Among these methods,

AFM has shown various advantages in single cell biomechanics such as

high resolution imaging, probing mechanical properties, adhesive strength

measurements, etc. It can also be used to study the stress–relaxation

behaviour of the cells.

These techniques could provide a better understanding of the mechanisms

underlying the mechanical behaviour of living cells that are difficult to

explain using pure experimentation. FEA is probably the most commonly

used method in recent years.

Therefore, in this study the AFM biomechanical indentation and stress–

relaxation experiments will be conducted to investigate the strain-rate dependent


mechanical deformation and relaxation behaviour, respectively, of single cells i.e.

osteocytes, osteoblasts, and chondrocytes. The PHE model will be combined with

FEM as a simulation tool to explore this behaviour and to study the contribution of

intracellular fluid to single living cells’ responses.

Next, the effect of extracellular osmotic pressure on the morphology, elastic

modulus, and relaxation behaviour of single living chondrocytes will also be studied

using AFM experiments. Moreover, the changes of the hydraulic permeability of

single living chondrocytes with solution osmolality will be investigated using the

PHE model in this study. It is expected that changes to these properties and

behaviour are due to the intracellular fluid of chondrocytes.

Chapter 3:Research Design 43

Chapter 3: Research Design

3.1 INTRODUCTION

In order to further understand the fundamental mechanisms underlying the responses

of single cells (i.e. osteocytes, osteoblasts and chondrocytes) at certain strain-rates,

strain-rate dependent mechanical deformation and relaxation behaviour are

investigated in this study. Most importantly, the suitable constitutive law for the

mechanical behaviour of single cells is unknown. As a result, the aim of this study is

to explore the strain-rate dependent behaviour of single cell types using AFM and

inverse FEA.

In this chapter, the details of AFM indentation testing procedures are presented

in Section 3.2. The cell culturing protocol and sample preparations for AFM

biomechanical testing are presented in Section 3.3.1. The osmotic solutions and

confocal imaging sample preparations are presented in Sections 3.3.2 and 3.3.3,

respectively. Next, the cells’ diameter and height measurement techniques are shown

in Sections 3.3.4 and 3.3.5. Finally, the FEA model development and inverse FEA

method are presented in Section 3.4.

3.2 ATOMIC FORCE MICROSCOPY EXPERIMENTAL SET-UP AND

DATA POST-PROCESSING

The AFM used in this study was a Nanosurf FlexAFM (Nanosurf AG, Switzerland),

which is mounted on a Leica DM IRB (Leica Microsystems) (see Figure 3.1(a)). The

Nanosurf C3000 software provided by the manufacturer was used to conduct AFM

experiments. One of the advantages of the Nanosurf FlexAFM system is that the

cantilever holder has an alignment system; thus, it is not necessary to adjust the laser

light manually compared to other AFM systems. Figure 3.1(b) shows the AFM head

and attached cantilever holder. However, the requirement was that the cantilever

used should have alignment grooves to be used in this system. The AFM operational

procedure is presented step-by-step in this section.

44 Chapter 3:Research Design

To start the experiments, the “Laser align” button in the “Acquisition” tab in

the software was used to check the laser light signal. The good signal is when the

open rhomb symbol is in the green area of the bar and when the green dot is within a

grey rectangle.

(a)

(b)

Figure 3.1: (a) Nanosurf Flex AFM system; (b) AFM head

Leica light

microscope

AFM head

Vibration

isolator

Camera

Cantilever

holder


After checking the laser light, the next important step is to determine the spring

constant of the cantilevers used. A colloidal probe SHOCONG-SiO2-A-5 (AppNano)

cantilever was used in the experiment. Figure 3.2 shows the scanning electron

microscope (SEM) image of the colloidal probe cantilever used. The inset is the

enlarged area of the cantilever end where the colloidal probe is located. The colloidal

probe had a diameter of 5 µm and its spring constant was around 0.224–0.3114 N/m

as obtained using the thermal noise fluctuations prior to indentation testing. It was

measured by selecting the “Thermal tunning” button in the “Acquisition” tab. The

software automatically determines and displays the spring constant on the right-hand

side of the window.

Finally, the sensitivity of the cantilever needs to be identified. Inasmuch as

deflection of the cantilever was detected by a photodiode, the signal obtained was in

electrical units (i.e. volts) in this AFM system. However, in order to measure the

force applied by the cantilever, the cantilever deflection signal should be in length

units (e.g. nanometres). The coefficient that converts volt units into nanometre units

is called the sensitivity of the cantilever, and the finding of this coefficient is called

the sensitivity calibration of the cantilever. The principle is to indent the hard surface

which can be treated as rigid material (e.g. Petri disk surface in this study) and to

record the height–deflection curve.

Figure 3.2: SEM image of colloidal probe cantilever SHOCONG-SiO2-A-5 used in

this study (the inset shows the real diameter of the bead)

Thus, the “Approach” button in the “Acquisition” tab in the software was

firstly used to land the tip on the surface. The “Set-point” was roughly selected to be


5 nN. Note that the set-point value is the deflection of the cantilever that is

maintained by the feedback, so that the force between the tip and sample is kept

constant. The tip was brought into contact with the sample until the deflection of the

cantilever reached the set-point value. After the tip and sample were in contact, the

cantilever sensitivity was calibrated using the “Spectroscopy” tab at the bottom of

the screen. The indentation was conducted on the Petri disk and the height–deflection

curves were recorded. Note that only the approach curve was used and that the

cantilever sensitivity calibration was conducted in the same liquid environment with

that of the sample in order to obtain the most accurate results.

At the end of the AFM experiments, the samples were incubated in a 1:200

dilution of trypan blue (GIBCO, Invitrogen Corporation, Melbourne, Australia). This

dye exclusion test was conducted to help determine whether or not the tested cells

were living during the experiments. The samples were observed under a light

microscope in order to check the viability of the single cells (the living cells have a

clear cytoplasm whereas the dead cells have a blue cytoplasm).

In order to determine the Young’s moduli of the chondrocytes, a program was

developed using Matlab R2013a (MathWorks, Inc.) based on the automatic AFM

force curve analysis algorithm proposed by Lin et al. (Lin, Dimitriadis and Horkay

2007b). The developed program used throughout this study.

A program was also developed using Matlab R2013a (MathWorks, Inc.) to

estimate the viscoelastic and poroelastic relaxation properties of single cells by

fitting the AFM relaxation experimental data with either the thin-layer viscoelastic

model function or poroelastic relaxation indentation (PRI) function. (This procedure

is discussed in detail in Chapter 5 and 6.)

3.3 MATERIALS AND MODELS

3.3.1 Cell culturing and AFM sample preparation

Three types of cells were considered in this study, namely, osteocytes, osteoblasts

and chondrocytes. The cell culturing and sample preparation was similar for all three

cell types. The primary osteoblasts and chondrocytes, which were given from

Institute of Health and Biomedical Innovation (IHBI), QUT, Brisbane, Australia, and

the MLO-Y4 osteocytes were cultured using Dulbecco’s Modified Eagle’s Medium


(low glucose) (GIBCO, Invitrogen Corporation, Melbourne, Australia) supplemented

with 10% fetal bovine serum (HyClone, Logen, UT) and 1% penicillin and

streptomycin (P/S) (GIBCO, Invitrogen Corporation, Melbourne, Australia). After

culturing for a week until the cells were confluent, they were detached using 0.5%

trypsin (Sigma-Aldrich). They were seeded onto a cultured Petri dish coated with

poly-D-lysine (PDL) (Sigma-Aldrich) for 1–2h. Cells were placed on the PDL

surface to form a strong attachment while keeping their morphology round. One of

the samples for each cell type was fixed using 4% paraformaldehyde (Sigma-

Aldrich) for 20 minutes before changing it to phosphate buffered saline (PBS)

(Sigma-Aldrich). All the samples were stored in a refrigerator at -4 0C before the

experiments. Biomechanical testing was conducted at room temperature. All of the

cells tested are Passage 1–2 cells.

3.3.2 Sample preparation for varying osmotic pressure environments

In order to study the effect of extracellular osmotic pressure on the elastic and

viscoelastic mechanical properties of single cells, several hyperosmotic and

hypoosmotic testing solutions were created. Chondrocyte is the target cell type

investigated in this study. The other cell types will be considered in future works.

Firstly, the isoosmotic solution was made by adding 0.9 g of sodium chloride (NaCl)

in 100 ml of deionised water. This solution has an osmolality of approximately 300

mOsm. Then, NaCl and deionised water were added to this isoosmotic solution in

order to achieve three hyperosmotic (i.e. varying osmolality of 450, 900 and 3,000

mOsm) and two hypoosmotic (i.e. varying osmolality of 100 and 30 mOsm) testing

solutions, respectively. The cells were firstly suspended in a culturing medium and

seeded on a PDL-coated cultured Petri dish for one hour to allow the cells to attach.

After that, the culturing medium was changed to hyperosmotic, hypoosmotic, and

control solutions for 30 mins to expose the cells to the osmotic challenge before

testing or fixation. All the testing was conducted at room temperature. All of the cells

tested are Passage 1–2 cells.

3.3.3 Confocal actin filament and vinculin staining and imaging

The chondrocytes were trypsinised with 0.5% trysin. Then, they were seeded onto

22× 22 mm glass coverslip slides and allowed to attach for one hour. After that, the

attached cells were gently washed with PBS three times before being fixed with 4%


paraformaldehyde for 20 minutes. The samples were then washed again with PBS

and thereafter permeabilised with 0.1% Triton X100 (Sigma-Aldrich) in PBS for 1

minute. After another wash with PBS, the samples were then incubated in a 1:100

dilution of DAPI1 and Alexa Fluor 568 phalloidin (GIBCO, Invitrogen Corporation,

Melbourne, Australia) for 15 minutes in order to observe the chondrocytes’ nuclei

and actin filament network, respectively. The samples were also exposed to

monoclonal anti-vinculin (Sigma-Aldrich) for 15 minutes in order to observe the

focal adhesion area of the cells. The samples were then washed one more time before

being imaged on a confocal laser microscope (Nikon A1R confocal, Nikon, Japan)

(see Figure 3.3) using a 40x Nikon oil immersion objective lens.

Figure 3.3: Nikon A1R confocal microscope

3.3.4 Cell diameter measurement

In order to develop an FEA model, several important parameters need to be

determined, one of which is the cells’ diameter. In this study, the diameters of single

living osteocytes, osteoblasts and chondrocytes were measured using the Leica M125

light microscope (Leica Microsystems) (see Figure 3.4). Firstly, the cells were fixed

using the protocol presented in Section 3.3.1. Then, the blue dye was added to the

1 DAPI (4',6-diamidino-2-phenylindole)


sample for 5 minutes for better visualisation. Finally, the sample was mounted on the

sample stage in order to observe it using a Leica 10x lens. Note that only the round

cells were picked for measurement, and the diameter was the average of the

horizontal and vertical diameters. The diameters of the chondrocytes when exposed

to varying extracellular osmotic pressures were also calculated using this technique.

Figure 3.4: Leica M125 light microscope

3.3.5 Cell height measurement

The second important parameter that needs to be determined is the cells’ heights or

thicknesses. One of the techniques to measure this parameter is to use the AFM

system, as proposed by Ladjal et al. (Ladjal et al. 2009) and illustrated in Figure 3.7.

In this study, the heights of the living osteocytes, osteoblasts and chondrocytes were

measured using this technique. The heights of the single living chondrocytes when

subjected to varying osmotic pressures were also evaluated using this method.

Note that the maximum displacement of the piezoelectric scanner in Z-

direction of the Nanosurf FlexAFM system is limited to only 10 μm, another AFM

system in another institute was used to measure the heights of the cells. The AFM

system used was a JPK NanoWizard II AFM (JPK Instruments, Germany) that was

mounted on a Zeiss light microscope. Figure 3.5(a) shows the AFM head and the


light microscope. This AFM system is located at the Australian Institute for

Bioengineering and Nanotechnology, the University of Queensland.

(a) (b)

(c)

Figure 3.5: (a) JPK NanoWizard II AFM system; (b) CellHesion module; (c) AFM

head

Zeiss light

microscope

AFM head

Cantilever holder

Camera


The maximum range of the Z-piezoelectric scanner is 15 µm, which was not

suitable for our work. Thus, an external module was used, namely, the CellHesion

(see Figure 3.5(b)), which helped to increase the indentation range in the AFM

system up to 100 µm. The cantilevers in the JPK system are different to the Nanosurf

system and do not have the alignment grooves. Therefore, the laser light needed to be

aligned manually. Figure 3.5(c) presents the AFM head and attached cantilever

holder. Firstly, the laser light and cantilever images were observed using a digital

camera for better visualisation. Next, the laser light was manually aligned to be

exactly on top of the end of the cantilever at which the spherical tip is located. The

best position of the laser light is when the laser signal value is maximum. Other

operational procedures such as the spring constant measurement and the cantilever

approach are similar to our original system.

A triangular colloidal probe CP-PNPL-BSG-A-5 (NanoAndMore GmbH)

cantilever was used in the experiment. The diameter of the colloidal probe was 5 µm

and its spring constant was determined to be 0.0217 N/m using the thermal noise

fluctuations before the indentation testing. Figure 3.6(a) shows the SEM image of the

colloidal probe cantilever used. Figure 3.6(b) presents a typical living chondrocyte

indented by a colloidal probe cantilever.

Figure 3.6: (a) SEM image of colloidal probe cantilever CP-PNPL-BSG-A-5 used for

the JPK–AFM system in this study (the inset shows the real diameter of the bead –

scale bar: 10 μm); (b) a living chondrocyte indented by a colloidal probe cantilever

(scale bar: 35 μm)

Firstly, the Zeiss light microscope was utilised to locate the AFM tip and the

cells in order to bring the tip to above the central area of the cells before the

indentations were performed on the cells. Several positions were measured around


the central area and the maximum value of the deflection of the AFM cantilever was

recorded to ensure that the tip measured the (relative) highest point of the cells. The

indentation was then performed on the adjacent area of the substrate to obtain the

height–deflection curves. Finally, both curves were processed using the JPK SPM

data processing software (Version 4.4.23) (JPK Instruments, Germany) (JPK-

Instruments 2011). Next, the contact points were determined to identify h1 for the

cell and h2 for the substrate (see Figure 3.7). Finally, the cell’s height was calculated

as h = h2 – h1.

Figure 3.7: Cell height measurement procedure using AFM indentation (where h1,

and h2 are non-contact regions of force curves when indenting the cell and substrate,

and h is the cell’s height calculated as h = h2 – h1)

3.4 NUMERICAL MODELS

3.4.1 Introduction of Finite Element Method

Numerical modelling is a well-established technique for predicting the behaviour of

complicated systems. Its principle is to transform a complicated problem into a set of

discrete forms with mathematical steps. The behaviour of these individual

“elements” is known, from which the original problem domain is rebuilt in order to

investigate its behaviour. The problem will then be solved on a computer and finally

the physical process will be visualised according to the requirements of the analysts

(G. R. Liu 2005; Zienkiewicz and Taylor 2000). With the development of computer

technology, engineering problems can be solved promptly even if the number of

discrete elements is very large. The advantage of numerical modelling is that, once


the model is set up and established, a complicated problem may be solved effectively

using the numerical models. The models provide a method for predicting a system’s

response to a range of stimuli and phenomena without subjecting the system to actual

conditions. This is important when test conditions are difficult, unethical, expensive

or dangerous to create. Numerical modelling can therefore be a key mechanism for

exploring the effect of, for example, new therapies for diseases.

Numerical simulations are potential methods to explore the mechanical

properties of single living cells and FEA is the most commonly used approach. This

method was firstly used in solid mechanics to solve the problems of stress analysis,

and has since been applied in many other engineering problems including fluid flow

analysis, thermal analysis and transportation. The FEA determines the approximate

results on the distribution of the field variables in the problem domain (Liu and Quek

2003). A number of studies in the literature have used this method as a simulation

tool to study the mechanical behaviour of different types of cells such as red blood

cells and chondrocytes. This technique has also been used to validate the

experimental results on the mechanical properties of single living cells. A number of

FEA commercial software products are available, such as ABAQUS, NASTRAN

and ANSYS, to support the analysis. Due to the advantages mentioned above, FEA is

used in this study.

3.4.2 FEA model used in this study

The FEA commercial software package used in this study was the ABAQUS 6.9-1

(ABAQUS Inc., USA). The interface of this software, including the menu, toolbars,

model tree and the main viewport, is shown in Figure 3.8. This software enables the

user to create a model directly or to import it from other modelling software. This

software also supports a number of material constitutive models (e.g. hyperelastic,

viscoelastic, PHE models).

Based on the cell diameter and height for each cell type as measured using the

methods presented above in Sections 3.3.4 and 3.3.5, FEA models were developed

and analysed using the ABAQUS software package in this study. More details of

these models are presented in Chapter 4. One of our models is shown in the viewport

(see Figure 3.8) in the “Assembly” module where all the individual parts are

assembled. Both the cells and the AFM colloidal probe are spherical; therefore, the


axisymmetric part models were used in this study in order to save computational

cost. After creating a model, other important steps are to create the initial and

boundary conditions. The boundary conditions are presented in Figure 3.9 below.

Figure 3.8: ABAQUS 6.9-1 software interface –1) Menu and toolbars; 2) Model tree;

and 3) Viewport

Our samples comprise both solid and fluid constituents; therefore, three initial

conditions, namely, the void ratio, saturation and fluid pore pressure, need to be

considered. The void ratio and saturation initial conditions were assumed to be “4”,

and “1”, respectively, in this study. The initial void ratio used in this study is similar

with that of previous work (Moo et al. 2012). This ratio means that the fluid volume

fraction of the cell is around 80%. The initial condition of saturation used in this

study means that the cell is fully saturated with fluid. In addition, the fluid pore

pressure was initially assumed to be “0” because the osmotic pressure within the

cells is not considered in this study.

The boundary conditions are also very important for finite element analysis.

The FEA model in this study possessed the following four boundary conditions:

All six degrees of freedoms are fixed at the reference point (RP) of the

substrate part of the FEA model (i.e. the “ENCASTRE” symmetric

boundary condition is used in the ABAQUS software).

Inasmuch as the axisymmetric part is used, the “XSYMM” symmetric

boundary condition in the ABAQUS software is assigned at the middle

plane of the cells. This boundary condition fixes four degrees of freedom.

1

3 2


Inasmuch as the initial fluid pore pressure within the cells is “0”, the fluid

pore pressure boundary condition of “0” is also assigned on the membrane

of the cell. This simulates the fluid flow when there is a pressure gradient

developed within the cell during deformation.

The AFM tip is prescribed with a displacement of around 1.8–3.0 µm at

the RP to simulate the indentation of the AFM experiments.

Figure 3.9: Boundary conditions of FEA model

It is reported in the literature that the maximum membrane area of single cells

is around 200–240% of the initial area, and it is presumed that the cellular membrane

consists of many folds and ruffles which unfold during deformation (Evans and

Kukan 1984; Evans and Yeung 1989a; Tran-Son-Tay et al. 1991; Guilak, Erickson

and Ting-Beall 2002) (see Figure 2.2(b) and Chapter 6, Section 6.3.1 for details).

The folds and ruffles help the cells to withstand large deformations without exerting

significant stress on the membrane. Therefore, researchers have concluded that the

cell membrane does not contribute to the mechanical properties of the cells at small

strains. As a result, the cell membrane is not considered in the FEA models used in

this study.

3.4.3 Inverse FEA method

To the best of our knowledge, there are few/no analytical solutions in the literature

for determining PHE material parameters. Therefore, the inverse FEA method was


employed in this study to estimate these parameters from the AFM experimental

results. The general procedure of the inverse FEA technique is as follows:

After creating the FEA models, the initial PHE material parameters are

assumed.

The AFM indentation simulation is analysed, and the reaction force is

extracted at the RP of the AFM tip. The simulation result is compared to

that of the AFM experiment in order to estimate the error.

If the error is large, the PHE parameters are then accordingly modified and

the simulation is conducted again.

This process is iteratively repeated until the error is reasonably small in

order to identify the cells’ PHE parameters.

Details of the material parameters’ estimation procedure and the results are

presented in Chapters 4 and 5.

Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 57

Chapter 4: Exploration of Strain-Rate

Dependent Mechanical

Deformation Behaviour of Single

Living Cells

4.1 INTRODUCTION

The characterisation of the strain-rate dependent mechanical deformation behaviour

of single cells using AFM biomechanical indentation experiments is presented in this

chapter. AFM indentation experiments at four different strain-rates were conducted

and their force–indentation curves were extracted and analysed. The Young’s moduli

of the cells were then estimated by using the thin-layer elastic model to study the

dependency of the elastic moduli of single cells on the strain-rates.

In research reported in the literature, the PHE model has been used effectively

and extensively to capture this behaviour for various fluid-filled biological tissues.

However, there is lack of research using this model for single cell biomechanics.

Therefore, this chapter investigates the application of the PHE model coupled with

the inverse FEA technique to study strain-rate dependent mechanical deformation


Therefore, in this chapter, some information about the AFM experiments is

firstly provided in Section 4.2. Then, the details of the thin-layer elastic model are

presented in Section 4.3. Details of the PHE field theory and inverse FEA technique

are presented in Section 4.4.1 and 4.4.2, respectively. After that, the diameter and

height measurements of the three cell types are presented in Sections 4.5.1 and 4.5.2,

respectively. The AFM experimental data and the determined elastic moduli of the

three cell types are then shown in Sections 4.5.3 and 4.5.4. Next, the application of

the PHE model on the study of living osteocytes, osteoblasts and chondrocytes is

shown in Section 4.5.5 and the PHE material parameters among these cell types are

compared. Several conclusions are then presented in Section 4.6.

58 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells

4.2 AFM BIOMECHANICAL INDENTATION EXPERIMENTS

The colloidal probe was used in this study because Dimitriadis et al. (Dimitriadis et

al. 2002) proved that the smallest radius of the bead should be 𝑅𝑚𝑖𝑛 =ℎ

12.8 in order

to prevent the tips from prompting local strains that exceed the material linearity

regime. In our study, ℎ𝑚𝑎𝑥 ≈ 17 μm, which means 𝑅𝑚𝑖𝑛 = 1.33 μm. Dimitriadis et

al. also concluded that the results were more accurate when using the microspheres

of either 2 or 5 µm radius rather than the sharp pyramidal tips as the probe tips for

thin samples (thickness ≤ 5 µm). Harris and Charras (Harris and Charras 2011) also

reported that spherical-tipped cantilevers measured cellular elasticity correctly,

whereas pyramidal tips overestimated it. Moreover, at the same indentation depth,

when using the 2.5 µm radius spherical indenter, the applied stress on the samples

may be reduced by around 15 to 100 times compared to the sharp conical indenter

with the half opening angle varied from 35° to 15° (Hu et al. 2010). In the case of the

indentation where the contact radius is equal to the bead radius, the ratio between the

surface contact area of the bead and the cell surface area was determined to be

around 9%, which is suitable to indent the samples without prompting large local

strains on the cells. Therefore, in this study, the colloidal probe with a radius of

around 2.5 µm was used. This size probe has also been widely used for single cell

biomechanical testing (Darling, Zauscher and Guilak 2006; Ladjal et al. 2009; Li et

al. 2008).

In our experiments, the position of the cantilever was firstly adjusted so that the

colloidal probe lined up with the central (nuclear) region of the cell by using the

Leica light microscope. This central (nuclear) region was the only point tested as it

was believed that indenting the central region can avoid errors due to the indenting

bias regions. Each single cell was then indented at each of the four different strain-

rates, namely, 7.4, 0.74, 0.123 and 0.0123 s-1

. The indentation testing was conducted

by controlling the absolute displacements of the piezoelectric scanner in Z-direction.

Thus, the force set-point threshold was not used in our study. Firstly, the cell was

indented to a maximum strain of around 10–15% of the cell’s diameter

(corresponding to a displacement of approximately 0.65 µm for osteocytes; 1.4 µm

for osteoblasts; and 2.4 µm for chondrocytes). The force–indentation curves were

59

Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 59

then obtained and pre-processed using the SPIP 6.2.8 software (Image Metrology

A/S, Denmark).

One-way analysis of variance (ANOVA) was employed in this study to

identify the significant differences in mechanical properties among the cell types at

each of the four strain-rates using the statistical software Minitab 16.1.1 (Minitab

Inc., 2010) with statistical significance reported at the 95% confidence level (p <

0.05).

4.3 THIN-LAYER ELASTIC MODEL

As discussed in Chapter 2, Dimitriadis et al. (Dimitriadis et al. 2002) developed a

modified Hertzian model – called the thin-layer model – to account for the thickness

of the sample in AFM indentation testing. In their model, the relationship between

the applied force F and the indentation δ for spherical tips is:

𝐹 =4𝐸𝑌

3(1−2)𝑅

1

2𝛿3

2 [1 −2𝛼0

𝜋𝜒 +

4𝛼02

𝜋2 𝜒2

−8

𝜋3 (𝛼03 +

4𝜋2

15𝛽0) 𝜒3 +

16𝛼0

𝜋4 (𝛼03 +

3𝜋2

5𝛽0) 𝜒4] (4.1)

where 𝜒 = √𝑅𝛿/ℎ, h is the thickness of the sample, the constants 𝛼0 and 𝛽0 are

functions of the material Poisson’s ratio given below, and EY and R are the

Young’s modulus and the radius of the rigid indenter, respectively. It is worth noting

that the term outside the bracket is the elastic Hertzian expression for the indentation

with spherical tips of a semi-infinite sample and the ones inside the bracket are

correction terms to consider the finite thickness of the sample. This relationship will

simplify to the Hertzian solution when the thickness h becomes very large. One of

the advantages of this model is that it is valid for all Poisson’s ratio values.

Moreover, it is shown that the stiffness of the sample is maximum for incompressible

materials. This solution can account for the bonded property between the material

and substrate where the only difference is the constants 𝛼0 and 𝛽0 which depend

differently on . For the un-bonded interaction between the material and substrate,

these constants are given as follows (Dimitriadis et al. 2002):

𝛼0 = −0.3473−2

1−, 𝛽0 = −0.056

5−2

1− (4.2)


When the material is bonded to the substrate, these constants are shown as

follows:

𝛼0 = −1.2876−1.4678+1.34422

1− (4.3)

𝛽0 =0.6387−1.0277+1.51642

1− (4.4)

It is observed that there are two variables (i.e. EY and ) in Equation (4.1).

Researchers have concluded that the measured properties change by less than 20%

when varying the Poisson’s ratio from 0.3 to 0.5 (Darling et al. 2008) and that it is

reasonable to assume incompressibility for most biological samples. As a result, the

relationship between the applied force F and indentation δ when the sample is not

bonded to the substrate becomes:

𝐹 =16𝐸𝑌

9𝑅1/2𝛿3/2[1 + 0.884𝜒 + 0.781𝜒2

+0.386𝜒3 + 0.0048𝜒4] (4.5)

and when the sample is bonded to the substrate:

𝐹 =16𝐸𝑌

9𝑅1/2𝛿3/2[1 + 1.133𝜒 + 1.283𝜒2

+0.769𝜒3 + 0.0975𝜒4] (4.6)

𝑅 = (1

𝑅𝑡𝑖𝑝+

1

𝑅𝑐𝑒𝑙𝑙)

−1

(4.7)

where 𝜒 = √𝑅𝛿/ℎ; h, F, EY, R, and 𝛿 are the cell’s height, applied force, Young’s

modulus, relative radius (Rtip = 2.5 μm in this study), and indentation, respectively.

Inasmuch as our samples are single living cells where the heights/thicknesses

are quite thin, the modified Hertzian model (i.e. the so-called thin-layer model)

proposed by Dimitriadis et al. (Dimitriadis et al. 2002) was used. Additionally,

Inasmuch as the single cells were attached on the substrate, the equation for the

bonded sample and substrate as shown in Equation (4.6) was used.

4.4 PHE ANALYSIS OF STRAIN-RATE DEPENDENT MECHANICAL

DEFORMATION BEHAVIOUR OF SINGLE CELLS

In this section, the PHE model is used to fit the AFM indentation experimental data

at four varying strain-rates in order to study the strain-rate dependent mechanical

61


deformation behaviour of single living cells. The PHE material parameters are also

compared among the three cell types.

4.4.1 PHE theory

The PHE model has already been described in detail in Chapter 2. This section

presents the PHE theory again with simpler governing equations for the case where

there are no body forces. The PHE theory was developed as an extension of the

poroelastic theory (Simon and Gaballa 1989) to characterise and predict the large

deformation and non-linear responses of structures. With respect to cell studies, this

theory assumes that the living cell is a continuum consisting of an incompressible

hyperelastic porous solid skeleton, saturated by an incompressible mobile fluid.

While the solid and fluid constituents are incompressible, the whole cell is

compressible because of the loss of fluid during deformation. The theory has been

applied in many engineering fields including soil mechanics (Sherwood 1993) and

biomechanics (Simon 1992; Meroi, Natali and Schrefler 1999; Nguyen 2005; Olsen

and Oloyede 2002), with the theoretical details extensively presented by several

authors (Simon 1992; Simon et al. 1996; Simon, Kaufmann, McAfee, Baldwin, et al.

1998; Simon, Kaufmann, McAfee and Baldwin 1998; Kaufmann 1996). The field

equations for the isotropic form of this theory are summarised in this section:

Conservation of linear momentum:

𝜕𝑇𝑖𝑗

𝜕𝑋𝑗= 0 (4.8)

where 𝑇𝑖𝑗 is the first Piola–Kirchhoff stress.

Conservation of (incompressible) solid and (incompressible) fluid mass:

𝜕�̇̃�𝑖

𝜕𝑋𝑘+ 𝐽𝐻𝑘𝑙�̇�𝑘𝑙 = 0 (4.9)

where 𝐻𝑘𝑙, J, �̇̃�𝑖, and �̇�𝑖𝑗 are the Finger’s strain, volume strain of the material,

Lagrangian fluid velocity and rate of Green’s strain, respectively.

Two material properties are required in the PHE constitutive law, namely, the

drained effective strain energy density function, 𝑊𝑒, and the hydraulic permeability,

�̃�𝑖𝑗. 𝑊𝑒 defines the “effective” Cauchy stress, 𝜎𝑖𝑗𝑒 , as:

𝜎𝑖𝑗 = 𝜎𝑖𝑗𝑒 + 𝜋𝑓𝛿𝑖𝑗 , 𝜎𝑖𝑗

𝑒 = 𝐽−1𝐹𝑖𝑚𝑆𝑚𝑛𝑒 𝐹𝑗𝑛 (4.10)


𝑆𝑖𝑗 = 𝑆𝑖𝑗𝑒 + 𝐽𝜋𝑓𝐻𝑖𝑗 , 𝐻𝑖𝑗 = 𝐹𝑖𝑚

−1𝐹𝑗𝑚−1, 𝑆𝑖𝑗

𝑒 =𝜕𝑊𝑒

𝜕𝐸𝑖𝑗 (4.11)

where 𝜋𝑓 is the pore fluid stress = – (pore fluid pressure); and 𝑆𝑖𝑗𝑒 , and 𝐻𝑖𝑗 are the

second Piola-Kirchhoff stress and Finger's strain, respectively. It is interesting to

note that 𝑊𝑒 in the PHE model is equivalent to the classical strain energy density

function for a compressible material due to the relative fluid motion based on the

classical hyperelastic theory.

Conservation of fluid mass (Darcy’s law):

�̃�𝑖𝑗𝜕𝜋𝑓

𝜕𝑋𝑖= �̇̃�𝑗 (4.12)

where �̃�𝑖𝑗 is the hydraulic permeability.

Note that all the tilde signs above represent the Lagrangian form of the field

equations.

For simplicity, the neo-Hookean strain energy density function was used in this

study (Brown et al. 2009; ABAQUS 1996) as follows:

𝑊𝑒 = 𝐶1(𝐼1̅ − 3) +1

𝐷1(𝐽 − 1)2 (4.13)

where 𝐼1̅ = 𝐽−2/3𝐼1 is the first deviatoric strain invariant, and C1 and D1 are the

material constants.

The “effective” Cauchy stress then becomes:

𝜎𝑖𝑗𝑒 = 2𝐽−1𝐷𝐸𝑉 (

𝜕𝑊𝑒

𝜕𝐼1̅�̅�𝑖𝑗) +

𝜕𝑊𝑒

𝜕𝐽𝛿𝑖𝑗 (4.14)

where DEV = the deviatoric operator, and �̅�𝑖𝑗 = �̅�𝑖𝑘�̅�𝑗𝑘, and �̅�𝑖𝑗 = 𝐽−1/3𝐹𝑖𝑗. Note that

the bar signs represent the deviatoric parts.

Inasmuch as the soft tissues and cells are undergoing large deformation, the

influence of strain on permeability may not be negligible. Thus, the strain-dependent

permeability k is expressed as (Holmes and Mow 1990):

𝑘 = 𝑘0 [Ф0

𝑠 Ф𝑓

(1−Ф0𝑠 )Ф𝑠]

𝑒𝑥𝑝[𝑀(𝐼𝐼𝐼 − 1)/2] (4.15)

where Ф𝑠 and Ф𝑓 are the instantaneous volumetric fraction of the solid and fluid

components, respectively; 𝑘0 and Ф0𝑠 are the permeability and the volumetric fraction

63


of the solid component in the original state, respectively; III is the third principal

invariant of the Cauchy–Green deformation tensor for the solid component; and M

and are the material constants.

In order to use the PHE model in the finite element simulations, it is more

convenient to have the permeability k that is dependent on void ratio 𝑒 = Ф𝑓/Ф𝑠.

Thus, the isotropic permeability is expressed as a function of e, k(e) by using the

relations Ф𝑠 = Ф0𝑠/𝐼𝐼𝐼 and Ф𝑓 + Ф𝑠 = 1 (Wu and Herzog 2000):

𝑘 = 𝑘0 (𝑒

𝑒0)

𝑒𝑥𝑝 {𝑀

2[(

1+𝑒

1+𝑒0)

2

− 1]} (4.16)

where k0 is the initial permeability, e0 is the initial void ratio, and and M are the

non-dimensional material parameters.

The hydraulic permeability of the osteocytes and osteoblasts was assumed to

be constant and homogeneous. The initial void ratio, which is the ratio of the volume

of fluid to the volume of solid component, was assumed to be 𝑒0 = 4. Note that the

void ratio, e, relates to porosity n, that is, the volume of the matrix occupied by fluid

by: 𝑒 = 𝑛/(1 − 𝑛). The hydraulic permeability of the chondrocyte was assumed to

be deformation-dependent as shown in Equation (4.16) above.

The initial void ratio, e0, of the chondrocytes was also assumed to be e0 = 4.

The material parameters and M were determined to be 0.0848 and 4.638,

respectively, in Holmes (Holmes 1986), and were used by several researchers (Wu

and Herzog 2000; Holmes and Mow 1990; Moo et al. 2012). Figure 4.1 presents the

normalised strain-dependent permeability used in the ABAQUS model in this study.

The volume strain of the cell is given by:

𝐽 =𝑑𝑉

𝑑𝑉0=

1+𝑒

1+𝑒0 (4.17)

where V and V0 are the deformed and undeformed volume of the material,

respectively.

The Hertzian model is based on the theory of linear elasticity – it can only

capture the linear stress-strain relationship and can only be applied to cases in which

the contact radius is small compared to the radius of the indenter. It has been proven

to be able to capture the behaviour of materials at small deformations but not for

biological soft tissues. Thus, non-linear elastic contact models based on hyperelastic


strain energy functions have been developed (Lin, Dimitriadis and Horkay 2007a;

Lin et al. 2009). Researchers have used the stress-strain relations based on the

incompressible Mooney–Rivlin theory to derive the force–indentation relation, using

microspheres as indenters, as given next.

Figure 4.1: Normalised deformation dependent hydraulic permeability of

chondrocytes used in the ABAQUS model in this study

The Mooney–Rivlin equation (Treloar 1975):

𝜎 = 2𝐶1(𝜆 − 𝜆−2) + 2𝐶2(1 − 𝜆−3) (4.18)

where 𝜎 is the stress, 𝜆 is the extension ratio, and 𝐶1 and 𝐶2 are constants.

The indentation stress 𝜎∗ is equal to the mean contact pressure:

𝜎∗ =𝐹

𝜋𝑎2 (4.19)

The contact radius 𝑎(Lin, Dimitriadis and Horkay 2007a):

𝑎 = √𝑅𝛿 (4.20)

where R is the radius of the sphere, and 𝛿 is the indentation depth.

The indentation strain is defined as (Lee et al. 1998; Lin et al. 2009):

휀∗ = 0.2𝑎

𝑅 (4.21)

65


The strain factor of 0.2 was identified experimentally by Tabor (Tabor 1951).

Replacing 𝜎 with −𝜎∗ and 𝜆 with (1 − 휀∗) in Equation (4.18) yields the following

equation (Lin et al. 2009):

𝐹 = 2𝐶1𝜋 (𝑎5−15𝑅𝑎4+75𝑅2𝑎3

5𝑅𝑎2−50𝑅2𝑎+125𝑅3) + 2𝐶2𝜋 (𝑎5−15𝑅𝑎4+75𝑅2𝑎3

−𝑎3+15𝑅𝑎2−75𝑅2𝑎+125𝑅3) (4.22)

The Young’s modulus of material is determined by:

𝐶1 + 𝐶2 =10𝐸0

9𝜋(1−2) (4.23)

where 𝐸0 is the initial Young’s modulus and = 0.5 for the incompressible materials.

Note that inasmuch as the neo-Hookean strain energy density function is used in this

study, the 𝐶2 value in Equations (4.22) and (4.23) is zero:

𝐹 = 2𝐶1𝜋 (𝑎5−15𝑅𝑎4+75𝑅2𝑎3

5𝑅𝑎2−50𝑅2𝑎+125𝑅3) (4.24)

𝐶1 =10𝐸0

9𝜋(1−2) (4.25)

4.4.2 Inverse FEA technique to estimate PHE material parameters

In total, there are three material parameters of PHE models that need to be estimated,

namely, C1, D1 and k0. In order to determine these necessary material parameters in

this study, the following inverse FEA procedure was employed:

At a high level of strain-rate, it is well accepted that the cell is in an

undrained state and behaves as an incompressible hyperelastic material.

Thus, the AFM force–indentation data were fitted with Equation (4.24) in

order to determine the C1 parameter in Equation (4.13) corresponding to

this strain-rate.

At a low level of strain-rate, it is known that the cell is in a drained state

and behaves as a compressible elastic material. Thus, the inverse FEA was

conducted with the C1 determined in the previous step in order to identify

the D1 parameter in Equation (4.13) corresponding to this strain-rate.

Finally, the inverse FEA was conducted in order to determine the initial

permeability k0 in Equation (4.16) so that the FEA model results agree

well with the experimental data at all four strain-rates.


4.5 RESULTS AND DISCUSSIONS

4.5.1 Cell diameter

Cell diameter was measured using the method presented above in Section 3.3.4,

leading to 5.86 ± 1.8 µm (n = 42), 9.62 ± 1.94 µm (n = 62) and 16.99 ± 2.041 µm (n

= 54) for the osteocytes, osteoblasts and chondrocytes, respectively. Figure 4.2

shows the normal distribution of the diameters of the three cell types tested.

4.5.2 Cell height

The cells’ heights, as calculated using the technique presented above in Section

3.3.5, were measured to be 4.14 ± 1.48 µm (n = 36), 5.70 ± 1.29 µm (n = 36) and

15.59 ± 3.47 μm (n = 60) for the osteocytes, osteoblasts and chondrocytes,

respectively. Figure 4.2 presents the normal distribution of the cells’ heights. These

diameters and heights were subsequently used to create the FEA models of the single

cells, as discussed in detail in section 4.5.5. The diameters and heights of osteocytes,

osteoblasts and chondrocytes are summarised in Table 4-1.

Figure 4.2: Diameter and height distributions (normal) of osteocytes, osteoblasts and

chondrocytes

67


Table 4-1: Diameters and heights of the osteocytes, osteoblasts and chondrocytes

Diameter (µm) Height (µm)

Osteocyte 5.86 ± 1.80 4.14 ± 1.48

Osteoblast 9.62 ± 1.94 5.70 ± 1.29

Chondrocyte 16.99 ± 2.041 15.59 ± 3.47

4.5.3 Comparison of elastic moduli among osteocytes, osteoblasts and

chondrocytes

As mentioned in Chapter 3, after the AFM experiments, the trypan blue exclusion

test of living cells was conducted in order to check the viability of the cells. An

image of living chondrocytes incubated in the trypan blue dye, which was captured

using a light microscope, is presented in Figure 4.3. The blue cytoplasm cell in this

image (shown by a red circle) indicates that the cell is dead. The rest of the cells with

clear cytoplasm are living chondrocytes. This exclusion test indicated that most of

the chondrocytes were still alive and only a few cells were identified as dead after the

AFM experiments. This result revealed that the experiments were conducted mostly

on living cells and that the results are reliable. Similar results were obtained for the

other cell types.

Figure 4.3: Trypan blue exclusion test of chondrocytes after AFM experiments – the

blue cytoplasm cell is dead (shown by a red circle)


As presented in Chapter 3, the AFM force–indentation data was post-processed

and fitted with Equation (4.6) using a Matlab program to estimate the Young’s

moduli of the single cells at each of the four strain-rates tested, namely, 7.4, 0.74,

0.123 and 0.0123 s-1

. This is the characterising parameter that provided a comparison

of the elastic stiffness of the single living osteocyte, osteoblast and chondrocyte cells

subjected to varying rates of loading. Figure 4.4 presents the force–indentation

curves at each of the four strain-rates of a typical living and fixed osteocyte,

osteoblast and chondrocyte. The estimated Young’s moduli and their corresponding

R2 values at these strain-rates are also shown in this figure (see Appendix for details

of the statistical parameters including p-values, Root Mean Square Errors (RMSEs),

and R2 values). The thin-layer elastic model clearly fitted very well with the AFM

indentation data for all three cell types with regard to high values of R2

(see Figure

4.4). Similar results were also obtained for the remaining cell populations. Thus, it

was concluded that the thin-layer elastic model, which can account for sample

thickness, could be used to extract the elastic properties of the cells.

The calculated Young’s moduli of both the living and fixed cells of the three

cell types (i.e. osteocytes, osteoblasts and chondrocytes) at each of the four strain-

rates are presented in Figure 4.5 and Table 4-2. It was observed that the standard

deviations of Young’ moduli of all three cell types are large. Such large divergence

has been formerly ascribed to biological variability of different patients and intrinsic

inhomogeneity of the cells (Jones, Ting-Beall, et al. 1999). Moreover, differences in

culture time and harvested zones may have further contributed to inconstancy in the

measurements. In total, 39 living and 30 fixed osteocytes, 40 living and 33 fixed

osteoblasts and 43 living and 34 fixed chondrocytes were tested. The ANOVA

statistical analysis was conducted to evaluate the significant differences in the elastic

stiffness among the cell types tested.

It was interesting to note that the living osteoblasts in this study were stiffer

than the living chondrocytes at all strain-rates (p < 0.05). These results are similar to

results reported in the literature (Darling et al. 2008). The osteoblasts exhibited a

reduction in the Young’s moduli from 3,523 ± 2,374 Pa to 1,397 ± 2,044 Pa, whereas

the Young’s moduli of the chondrocytes decreased from 1,642 ± 890 Pa to 629 ± 493

Pa when the strain-rate was reduced from 7.4 s-1

to 0.0123 s-1

. In addition, the living

osteocytes had similar stiffness, which decreased from 3,569 ± 2,251 Pa to 1,342 ±

69


1,761 Pa with decreasing strain-rates, compared to the living osteoblasts at all strain-

rates. The cells differ in regard to the tissues of origin (the chondrocytes are cartilage

cells and the osteocytes and osteoblasts are bone cells), and it has been hypothesised

in the literature that the mechanical properties of cells may be used as biomarkers of

their extracellular matrix or phenotype (Darling et al. 2008). The fixed cells in this

study also exhibited similar elastic stiffness differences.

Figure 4.4: Typical AFM experimental force–indentation curves at four different

strain-rates of a typical single living and fixed osteocyte, osteoblast and chondrocyte

cell (the Young moduli and the R2 values corresponding to the strain-rates of 7.4,

0.74, 0.123 and 0.0123 s-1

are shown in the tables)


Table 4-2: Young’s moduli (Pa) of living and fixed (using 4% paraformaldehyde)

osteocytes, osteoblasts and chondrocytes at four different strain-rates

Strain-rates 7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

Osteocytes

Living

(n = 39)

3,569 ±

2,251

1,969 ±

2,548

1,426 ±

1,527

1,342 ±

1,761

Fixed

(n = 30)**

17,371 ±

14,337

14,079 ±

12,525

13,879 ±

11,526

12,271 ±

10,557

Osteoblasts

Living

(n = 40)

3,523 ±

2,374

2,181 ±

1,892

1,771 ±

1,976

1,397 ±

2,044

Fixed

(n = 33)**

20,064 ±

8,476

16,240 ±

7,510

15,056 ±

7,781

13,443 ±

7,401

Chondrocytes

Living

(n = 43)*

1,642 ±

890

1,216 ±

822

944 ±

704

629 ±

493

Fixed

(n = 34)**

6,891 ±

3,327

5,303 ±

2,587

4,664 ±

2,319

4,251 ±

2,118

* p < 0.05 demonstrated that the Young’s moduli of the living chondrocytes were significantly smaller than those of the living

osteoblasts at all strain-rates.

** p < 0.001 demonstrated that the Young’s moduli of the fixed cells were significantly larger than those of the living cells at

all strain-rates for all cell types.

4.5.4 Exploration of mechanisms underlying the dependency of mechanical

deformation behaviour of single living and fixed osteocytes, osteoblasts,

and chondrocytes on strain-rates

It was observed in the experiments that both the living and fixed cells had similar

mechanical behaviour, in which the cells became more flexible with the decrease of

the strain-rate for all cell types. Not surprisingly, this strain-rate dependent

deformation behaviour of single chondrocytes is similar to that of articular cartilage

tissue (Oloyede and Broom 1991; Oloyede, Flachsmann and Broom 1992). It is

noted that the Young’s modulus of the living chondrocytes at a low strain-rate of

0.0123 s-1

determined in our study is consistent with the findings reported in the

literature (Darling, Zauscher and Guilak 2006; Darling et al. 2008). In addition, it is

noted that the Young’s modulus of the osteoblasts at a high strain-rate of 0.74 s-1

71


estimated in this study is similar to the results in Darling et al.’s work (Darling et al.

2008).

Figure 4.5: Young’s moduli of living and fixed osteocytes, osteoblasts and

chondrocytes subjected to four different strain-rates

In this study, the author mostly focused on the mechanical properties and

responses of individual cells and the mechanisms underlying these responses. It is

hypothesized that both solid and liquid phases of living cells play important role in

their mechanical responses. Thus, the solid phase of living cells is assumed to be the

CSK, including actin filaments, intermediate filaments and microtubules, and the rest

of the cell is assumed to be the fluid phase. It is believed that the mechanisms

underlying the strain-rate dependent mechanical behaviour of the single cells can be

attributed to both the viscoelasticity of the cellular CSK and the intracellular fluid

(Trickey et al. 2006; Oloyede, Flachsmann and Broom 1992). Thus, in order to study

only the effect of the intracellular fluid on the mechanical behaviour of single cells,

without loss of generality, fixed cells should be used. This is because it has been

found that the solid CSK of fixed cells is stable (Svitkina 2010), and thus its role is

not as important as the role of the intracellular fluid in the mechanical response to

external loadings. Thus, by investigating the strain-rate dependent mechanical

properties of the fixed cells and comparing them to the properties of the living cells,

the effect of the viscoelasticity of the CSK can be decoupled; this helps to shed light

on the mechanisms underlying the dependency on the strain-rate behaviour.

Figure 4.5 shows the (log Pa) average and standard deviation values of the

Young’s moduli of the living and fixed osteocytes, osteoblasts and chondrocytes at


four different strain-rates. It was observed that all the cell types had similar

behaviour whereby the fixed cells were stiffer at higher strain-rates. It can be

explained that, at a high strain-rate, the intracellular fluid does not move relative to

the solid skeleton due to the low permeability of the cell, as it is unable to escape

quickly from the matrix and gets trapped within the cell. This renders the cell

virtually incompressible because both the fluid and solid constituents are

incompressible. Therefore, the cell displays an almost classical elastic mechanical

deformation response. On the other hand, the fixed cells were softer with decreasing

strain-rates corresponding to a reduced Young’s moduli (see Figure 4.5). This is

because the intracellular fluid plays a dominant role and is able to exude from the

cell matrix during indentation at these relatively low strain-rates. Since the fluid has

left the cell, the cell undergoes a net volume change and is therefore compressible.

This is called the consolidation-dependent deformation behaviour. It is interesting to

note that the Young’s moduli of the fixed cells decreased dramatically with

decreasing strain-rates of 7.4 to 0.123 s-1

and reached an asymptotic/limiting value at

0.0123 s-1

(see Figure 4.5). At such low strain-rates, the intracellular fluid can freely

move through the solid CSK with very low resistance. Thus, it is believed that the

strain-rate dependent mechanical property of fixed cells is mainly governed by their

intracellular fluid which plays an important role in cell biomechanics (Moeendarbary

et al. 2013).

From Figure 4.5, it can be clearly observed that the living cells had similar

behaviour, whereby their stiffness reduced with decreasing strain-rate (Figure 4.5

and Table 4-2). However, the living cells were significantly softer than the fixed cells

at all strain-rates (p < 0.001). This can be explained by the fact that “the cross-linking

of proteins by paraformaldehyde” during fixation process stabilizes the cellular

structures (Yamane et al. 2000; Vegh et al. 2011) which makes the fixed cells much

stiffer than the living ones (Ladjal et al. 2009). This explanation is supported by

Jungmann et al.’s finding that the actin networks reflect the elasticity of the cells

(Jungmann et al. 2012). As observed in Figure 4.5, the elastic moduli of the living

cells reduced almost linearly with decreasing strain-rates without reaching a plateau

value at 0.0123 s-1

compared to the fixed cells. It is likely that this was because the

cellular CSK reorganised or unbound its cross-linkers and deform to respond to the

external loadings during deformations (Lieleg et al. 2011), indicating the important

73


role of CSK at low strain-rates. This explanation is reasonable because Chahine et al.

reported that no significant remodelling of actin and intermediate filaments was

observed during repetitive loading at strain-rates within the intermediate range as

used in our study (Chahine et al. 2013). From the above discussion, it can be

concluded that the intracellular fluid is an important factor in controlling cellular

mechanical behaviour at high strain-rates, whereas the cellular CSK shows its

predominant effect on the living cells’ mechanical behaviour at relatively low strain-

rates.

4.5.5 PHE analysis of strain-rate dependent mechanical behaviour of single

living and fixed osteocytes, osteoblasts and chondrocytes

As discussed in previous section, both the CSK and intracellular fluid play important

roles in the strain-rate dependent mechanical behaviour of cells. In addition, it is

widely known that the cell membrane is a porous and semi-permeable membrane

allowing certain substances to infiltrate the cell while keeping out other substances in

order to protect the interior of the cell (Yeagle 1989). Thus, it is believed that the

cytoplasm of the living cells behaves as a poroelastic material (Moeendarbary et al.

2013; Zhou, Martinez and Fredberg 2013), and that the cytoplasm in fixed cells also

behaves this way, as observed in this study. This continuum model has been

extended to include the hyperelastic response of the non-linear solid skeleton leading

to the PHE material model. The PHE model considers the cell to consist of an

incompressible hyperelastic porous solid skeleton, saturated by an incompressible

mobile fluid. This model, which can account for non-linear behaviour, fluid-solid

interaction and rate-dependent drag effects, is potentially a good candidate for

investigating the responses of a cell to external loading and other load-inducing

stimuli (Nguyen et al. 2014). During attempts to apply the PHE model to single cells

in this study, it was observed that both the solid and fluid material parameters

affected the performance of the model in simulating the strain-rate dependent

behaviour. Thus, it was believed that the PHE model could also be used to

investigate the effect of both the CSK and the intracellular fluid on the strain-rate

dependent mechanical deformation behaviour of single cells. At the same time, this

model has other advantages including the advantages that all well-developed

hyperelastic constitutive relationships can be utilised (such as the neo-Hookean,

Mooney–Rivlin and Fung-Mooney relationships) and that the PHE constitutive law


has been integrated in commercial finite element software (e.g. ABAQUS). Although

the PHE model has been widely and effectively utilised in tissue engineering at the

macroscale such as articular cartilage modelling (Oloyede and Broom 1991, 1996)

and in other poroelastic tissues (Kaufmann 1996; Simon, Kaufmann, McAfee and

Baldwin 1998; Rigby, Park and Simon 2004; Ayyalasomayajula, Vande Geest and

Simon 2010), its application in the modelling of the single living cell is, to date,

significantly limited.

In order to investigate the performance of the PHE model when applied to

single osteocytes, osteoblasts and chondrocytes, the FEA models for living and fixed

cells based on the ABAQUS 6.9-1 software (ABAQUS Inc., USA) were developed

using the PHE model. The average diameters and heights of each cell type were used

to create the FEA models that are shown in Figure 4.6. It is noted that the difference

between the chondrocytes’ height and diameter was less than 9% due to the short cell

culture time in this study (~1 h). Experimental results reported in the literature

(Huang et al. 2003; Ladjal et al. 2009; Chahine et al. 2013) have shown similar

results. In other words, the cell is nominally to a sphere. Thus, in these cases, it is

assumed that the cells are spherical whereby the cell height is equal to the diameter

(Darling et al. 2007; Nguyen et al. 2010) and this dimension was used in the FEA

model of the single chondrocyte in this study. On the other hand, the differences

between the height and diameter of the osteocytes and osteoblasts were around 30%;

thus, the FEA models of the single osteocyte and osteoblast are not a sphere in order

to account for the cell height. In addition, the average height of each cell type was

used to extract the Young’s modulus using the thin-layer elastic model discussed

above. These cell height measurements were also used to create the FEA model,

especially the models for the osteocytes and osteoblasts.

The AFM nano-indentation experiment was simulated with this model. Both

the cell and AFM tip are spherical; therefore, axisymmetric geometry and element-

approximation were assumed, thereby saving computational cost (ABAQUS 1996).

An 8-node quadratic pore fluid/stress (i.e. CAX8RP) was used in PHE model in this

study to simulate the consolidation-dependent mechanical behaviour of single cells.

The model consists of a cell with a diameter of 5.9 µm, 9.6 µm and 17 µm

corresponding to the osteocytes, osteoblasts and chondrocytes, respectively. The

cells were indented with a colloidal probe with a diameter of 5 µm at different

75


loading rates in order to observe the effect of strain-rate on the cells’ biomechanical

response.

(a) (b)

(c)

Figure 4.6: FEA models of single (a) osteocyte, (b) osteoblast, and (c) chondrocyte

The inverse FEA approach was employed to estimate the PHE material

parameters for each cell type (see Section 4.4.2 for the procedure). Table 4-3 shows

these parameters of both the living and fixed cells for each cell type. The C1 values


of the living osteocytes, osteoblasts and chondrocytes were determined to be 2,062 ±

1,255 Pa, 1,924 ± 1,253 Pa, and 706.6 ± 384.7 Pa, respectively. This finding

indicated that the instantaneous stiffness of the osteocytes and osteoblasts in this

study was similar and the C1 values of these cells were larger than in the

chondrocytes (p < 0.05). These results are consistent with the results for the highest

strain-rate (i.e. 7.4 s-1

) presented in Section 4.5.3.

Table 4-3 PHE material parameters of living and fixed osteocytes, osteoblast and

chondrocytes

PHE

parameters C1 (Pa)

D1 (10-3

1/Pa)

Initial

permeability k0

(109

µm4/N.s)

Initial void

ratio e0

Osteocytes

Living

(n = 39)

2,062 ±

1,255

52.00 ±

22.00 259.42 ± 363.09 4

Fixed

(n = 30)

9,147 ±

7,292

0.95 ±

0.70 50.90 ± 166.70 4

Osteoblasts

Living

(n = 40)

1,924 ±

1,253

32.70 ±

51.20 398.70 ± 873.00 4

Fixed

(n = 33)

10,518 ±

4,372

1.04 ±

0.70 74.60 ± 205.90 4

Chondrocytes

Living

(n = 43)

707 ±

385*

17.50 ±

17.80 20.90 ± 22.00* 4

Fixed

(n = 34)

3,221 ±

1,569

2.10 ±

5.80 6.73 ± 9.48 4

* p < 0.05 indicated that the living chondrocytes had smaller PHE material parameters compared to the living osteocytes, and

osteoblasts

In the studies reported in the literature, the cell permeability is assumed to be the

same as the permeability of the extracellular matrix (Ateshian, Costa and Hung 2007)

or to be 5–6 orders of magnitude smaller than the matrix (Wu and Herzog 2000; Moo

et al. 2012). There is a lack of research to experimentally estimate the permeability

77


of single cells, which is one of the most interesting and important parameters in cell

biomechanics. Thus, one of the advantages of the approach in this study is that the

permeability of the cells can be estimated based on AFM indentation testing at

various strain-rates. This study is one of the first to calculate cell permeability for a

wide range of strain-rates.

In this study, the hydraulic permeability of the living osteocytes and osteoblasts

(259.42 ± 363.09×109

µm4/N.s and 398.7 ± 873×10

9 µm

4/N.s, respectively) was

found to be similar and larger than that of the living chondrocytes (20.9 ± 22×109

µm4/N.s) (p < 0.05). It was again confirmed that the mechanical properties of the

cells may be used as biomarkers of their extracellular matrix. It is noted that the

permeability of the osteoblasts in this study was around three orders of magnitude

smaller than that reported in previous works (i.e. 1.18 ± 0.65 ×1014

µm4/N.s) (Shin

and Athanasiou 1999, 1997). This can be explained by several factors: the cells

tested in the previous works were osteoblast-like cells (MG63 osteosarcoma cell

line), while primary osteoblast cells were investigated in this study; the number of

cells tested in the previous works was only 10 cells compared to 40 cells in this

study; and the investigators in the previous works calculated the permeability from

the cytoindentation experimental data at only one strain-rate, whereas it was

estimated from a wide range of strain-rates (from 7.4 s-1

to 0.0123 s-1

) in the present

study.

It is also interesting to note that the permeability of the living cells was larger

than that of the fixed ones (i.e. 259.42 ± 363.09×109

µm4/N.s vs 50.9 ± 166.7×10

9

µm4/N.s for the osteocytes; 398.7 ± 873×10

9 µm

4/N.s vs 74.6 ± 205.9×10

9 µm

4/N.s

for the osteoblasts; and 20.9 ± 22×109

µm4/N.s vs 6.73 ± 9.48×10

9 µm

4/N.s for the

chondrocytes) (p < 0.05). This was hypothetically because of the effect of the cellular

CSK, which might alter its structure during deformation for living cells. It is

hypothesised that the CSK structure alteration might help the intracellular fluid to be

easily distributed and transported during deformation in living cells compared to

fixed cells. This helps the living cells to respond to various external stimulations.

The difference in cell permeability can also be explained by the D1 values

presented in Table 4-3. This parameter was estimated using the ABAQUS software,

and is the inverse of the bulk modulus. It was noticed that the D1 values of the living

cells were larger (or, on the other hand, the bulk moduli were smaller) than those of


the fixed cells (i.e. 0.0252 ± 0.022 Pa-1

vs 0.000953 ± 0.0007 Pa-1

for the osteocytes;

0.0327 ± 0.0512 Pa-1

vs 0.001036 ± 0.0007 Pa-1

for the osteoblasts; and 0.0175 ±

0.0178 Pa-1

vs 0.0021 ± 0.0058 Pa-1

for the chondrocytes) (p < 0.05). It means that

the living cells are more compressible than the fixed cells. This finding further

supports the hypothesis that the CSK structure is altered, causing more intracellular

fluid transportation and loss during deformation.

Figure 4.7 presents the AFM experimental data at four strain-rates and the

corresponding PHE simulation results (the data are shown in mean values).

Interestingly, the results in Figure 4.7 showed that the PHE simulation results agreed

well with the AFM experimental data at all four strain-rates tested. This indicates

that the PHE model can capture the strain-rate dependent mechanical deformation

behaviour of both living and fixed cells.

It is worth noting that the simulation results at the highest strain-rate (i.e. 7.4 s-

1) and lowest strain-rate (i.e. 0.0123 s

-1) were similar to those of the neo-Hookean

incompressible and compressible hyperelastic models, respectively. These results

support the hypothesis that single living cells behave as incompressible and

compressible elastic materials at high and low strain-rates, respectively. At the

intermediate strain-rates (i.e. 0.74 and 0.123 s-1

) in this study, the cells exhibited the

consolidation-dependent behaviour whereby the effect of the intracellular fluid is

dominant. This is similar to the behaviour of other fluid-filled biological tissues that

have been the subject of investigation in prior research (Oloyede and Broom 1993a;

Oloyede and Broom 1994a; Simon, Kaufmann, McAfee, Baldwin, et al. 1998;

Simon, Kaufmann, McAfee and Baldwin 1998). Therefore, it can be concluded that

the PHE constitutive model is a promising constitutive model to simulate the strain-

rate dependent properties and other behaviour e.g. relaxation behaviour of single

cells.

In order to have a better understanding of how cells respond to varying rates of

loading (namely, 7.4, 0.74, 0.123, and 0.0123 s-1

), the von Mises stress and fluid pore

pressure distribution of the living osteocytes, osteoblasts and chondrocytes were

extracted as shown in Figure 4.8–4.10. Furthermore, the volume strains of the cells

were also calculated, as shown in Table 4-4, using Equation (4.17). It was observed

from the results presented in Figure 4.8–4.10 that the maximum von Mises stress

reduced only slightly, whereas the fluid pore pressure decreased dramatically when

79


the strain-rate decreased from 7.4 s-1

to 0.0123 s-1

for the living cells. The fixed cells

also expressed similar behaviour (these data are not shown). It was observed that the

fluid pore pressure was relatively small at the lowest strain-rate, causing the cell to

behave as a compressible elastic material. These results suggest that the intracellular

fluid plays an important role in consolidation-dependent behaviour of single cells.

Figure 4.7: Experimental and PHE force–indentation curves of living and fixed

osteocytes, osteoblasts and chondrocytes at four different strain-rates (the data are

shown as mean values)


7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

(a)


osteocytes after indentation at 7.4, 0.74, 0.123 and 0.0123 s-1

strain-rates (the


81


7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

(b)

(Continue) Figure 4.8: (a) von Mises stress, and (b) fluid pore pressure distributions

of living osteocytes after indentation at 7.4, 0.74, 0.123 and 0.0123 s-1

strain-rates



7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

(a)


osteoblasts after indentation at 7.4, 0.74, 0.123 and 0.0123 s-1

strain-rates (the


83


7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

(b)


of living osteoblasts after indentation at 7.4, 0.74, 0.123 and 0.0123 s-1

strain-rates



7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

(a)


chondrocytes after indentation at 7.4, 0.74, 0.123 and 0.0123 s-1

strain-rates (the


85


7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

(b)


of living chondrocytes after indentation at 7.4, 0.74, 0.123 and 0.0123 s-1

strain-rates



Table 4-4: Volume strain of osteocytes, osteoblasts and chondrocytes subjected to

varying rates of loading (the measurement unit in these figures is 106 Pa)

Osteocytes Osteoblasts Chondrocytes

Living

7.4 s-1

0.95 0.93 0.97

0.74 s-1

0.88 0.84 0.92

0.123 s-1

0.84 0.76 0.86

0.0123 s-1

0.82 0.71 0.81

Fixed

7.4 s-1

0.98 0.97 0.97

0.74 s-1

0.95 0.92 0.91

0.123 s-1

0.93 0.87 0.88

0.0123 s-1

0.91 0.83 0.86

It can be explained that at high strain-rates, because the rise time for

indentation is much shorter than the timescale for the movement of fluid through the

cell height, the intracellular fluid does not have enough time to respond to the

mechanical loading and thereby is blocked inside the cells. On the other hand, the

fluid can freely exude from within the cells at low strain-rates because the time

needed for the fluid to diffuse is shorter than the indentation timescale. This causes

the fluid pore pressure to be higher at a high rate of loading compared to a low rate.

It was observed that the volume strains of all cell types at the highest strain-rate

(i.e. 7.4 s-1

) were very close to “1”, indicating that the cells were nearly

incompressible. The volume strains then reduced when the strain-rate decreased,

demonstrating that more fluid loss occurred and thereby that the cells were more

compressible at low strain-rates than at high ones. In addition, the volume strains of

the fixed cells were larger than those of the living cells, demonstrating that the fixed

cells were more compressible than the living cells. This can be explained by the

effect of the CSK structure alteration during indentation.

4.6 CONCLUSION

The strain-rate dependent mechanical deformation behaviour of single osteocytes,

osteoblasts and chondrocytes were investigated in this study using AFM indentation

87


testing. Both living and fixed cells were studied to explore the mechanisms

underlying the strain-rate dependent behaviour. The PHE model combined with the

inverse FEA technique was used to investigate the strain-rate dependent mechanical

deformation of single cells. Several conclusions were drawn as follows:

The thin-layer elastic model was utilised to determine the Young’s moduli

of single living cells, namely, osteocytes, osteoblasts and chondrocytes

from AFM indentation experimental data at four different strain-rates. The

results show that this model, which can account for the thin thickness of

the samples, can be used to characterise the elastic properties of living

cells.

The results reveal that both living and fixed cells had similar mechanical

deformation behaviour, whereby their stiffness reduced with decreasing

strain-rate. By comparing the mechanical properties and behaviour of

living and fixed cells, it was concluded that the fixed cells’ strain-rate

dependent behaviour is mainly governed by their intracellular fluid, which

is called consolidation-dependent deformation behaviour. On the other

hand, in regard to the behaviour of living cells, the intracellular fluid plays

an important role at high strain-rates and the contribution of the cellular

CSK network is dominant at relatively low strain-rates.

It was found that the fixed cells were stiffer than the living cells at all

strain-rates and in all the cell types tested. These results are consistent with

those published in the literature. This is because the paraformaldehyde

molecules form cross-linking of proteins during fixation process.

The PHE model was used to investigate the strain-rate dependent

mechanical deformation behaviour of single living and fixed osteocytes,

osteoblasts and chondrocytes. It was found that the osteocytes and

osteoblasts in this study had larger instantaneous stiffness and hydraulic

permeability than the chondrocytes for both living and fixed cells. In

addition, the hydraulic permeability of the living cells was larger than that

of the fixed cells. This finding indicates that the cellular CSK network of

living cells alters its structure so that the intracellular fluid can be more

easily distributed and transported compared to fixed cells.


Finally, the procedures and results reported in this chapter open a new avenue

for the analysis of the mechanical deformation behaviour of osteocytes, osteoblasts

and chondrocytes as well as other similar types of cells (such as stem cells and

bacterial cells).

Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 89

Chapter 5: Investigation of Stress–

Relaxation Behaviour of Single

Cells Subjected to Different

Strain-Rates

5.1 INTRODUCTION

As well as the elastic stiffness, the relaxation behaviour of single living cells is also

of interest to various researchers when studying cell mechanics. It is believed that

these properties are important in biophysical and biological responses (Guilak 2000;

Costa 2004). In Chapter 4, the strain-rate dependent mechanical deformation

behaviour of single living osteocyte, osteoblast and chondrocyte cells were

investigated. In this chapter, the dependency of the relaxation behaviour of these

single cells on strain-rates is investigated. AFM stress–relaxation experiments at

varying rates of loading were conducted. In Section 5.2, the AFM relaxation testing

scheme is presented. The AFM relaxation data of the osteocytes, osteoblasts and

chondrocytes were fitted with the thin-layer viscoelastic model, which is presented in

Section 5.3, at four different strain-rates in order to extract viscoelastic properties.

The PRI model, which is discussed in detail in Section 5.4, is used to fit with the

AFM stress–relaxation data.

It is widely argued in the literature that the relaxation behaviour of single cells

is governed by both the viscoelasticity of the cellular CSK and by the intracellular

fluid (Trickey et al. 2006; Chan et al. 2012). Therefore, these two effects are

investigated using the PHE model coupled with the inverse FEA approach, as

presented in Section 5.5.3. The inverse FEA techniques used for these two

investigations are shown in Sections 5.5.3.1. All the results are discussed in Section

5.5, and conclusions are drawn in Section 5.6.

90 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates

5.2 AFM RELAXATION EXPERIMENTS

In this study, AFM was used to conduct stress–relaxation testing. The osteocytes,

osteoblasts and chondrocytes were firstly indented with different strain-rates. After

the indentation, the cantilever’s displacement was kept constant for 60 seconds

instead of allowing retraction of the cantilever. The cantilever’s deflection was

recorded while the cantilever’s chip was kept constant in order to study the relaxation

behaviour of the cells (see Figure 5.1).

As can be seen in Figure 5.1, two datasets were obtained in this study, namely,

the indentation data and the stress–relaxation data. The former set was extracted

during the indentation phase and was used to estimate the elastic properties of the

single cells (as comprehensively presented in Chapter 4). The latter set was obtained

from the AFM stress–relaxation experiment, which is considered in this chapter. In

this case, the force–time curves rather than the force–indentation curves of each cell

type at four different strain-rates were extracted. These curves were then curve-fitted

with the thin-layer viscoelastic model in order to investigate the long-term

mechanical behaviour of single living cells subjected to different strain-rates. Note

that besides the Young’s moduli as already calculated in Chapter 4, the equilibrium

moduli of the single cells are also extracted in this chapter using the force–

indentation data at the end of the 60 seconds relaxation.

Figure 5.1: AFM relaxation test diagram – A colloidal probe indented the cell using a

step displacement, which was then kept constant in order to study the relaxation

behaviour of the single cells

91

Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 91

5.3 THIN-LAYER VISCOELASTIC MODEL

Darling et al. developed and derived a viscoelastic solution for small indentations of

an isotropic, incompressible sample with a hard, spherical tip in order to determine

the viscoelastic properties of single living cells using stress–relaxation experimental

data (Darling, Zauscher and Guilak 2006). They utilised the Hertzian equation and

basic elastic and viscoelastic solutions to develop their solution as shown in the

following equations:

𝐹 =4𝐸𝑌𝑅1/2

3(1−2)𝛿3/2 (5.1)

where F is the applied force, EY is the Young’s modulus, 𝑅 = (1

𝑅𝑡𝑖𝑝+

1

𝑅𝑐𝑒𝑙𝑙)

−1

is the

relative radius, is the Poisson’s ratio, and 𝛿 is the indentation.

𝜎 = 2𝐺(𝑡)휀 (5.2)

(1 + 𝜏d

d𝑡) 𝜎 = 𝐸𝑅 (1 + 𝜏𝜎

d

d𝑡) 휀 (5.3)

The final viscoelastic solution, which is expressed in the time domain, is given

as (Darling, Zauscher and Guilak 2006):

𝐹(𝑡) =4𝐸𝑅

3(1−)𝑅1/2𝛿0

3/2(1 +

𝜏𝜎−𝜏𝜀

𝜏𝜀𝑒−𝑡/𝜏𝜀) (5.4)

where ER is the relaxation modulus, and 𝜏𝜎 and 𝜏 are the relaxation times under

constant load and deformation, respectively.

Darling et al. later applied the modified Hertzian model to develop and derive a

viscoelastic solution – the so-called thin-layer viscoelastic model – in order to

account for the finite thickness of the sample. The principle is similar to the one

mentioned above, with the only difference being that the thin-layer (modified

Hertzian) solution is used instead of the traditional Hertzian solution (Darling et al.

2007). This model is used to develop a mathematical expression of stress–relaxation

response for the well-known standard linear solid (SLS) viscoelastic model. By using

both the thin-layer model solution (Dimitriadis et al. 2002) and the stress–relaxation

model (Darling, Zauscher and Guilak 2006), Darling et al. proposed the thin-layer

viscoelastic model to determine the three parameters that describe a cell’s stress–

relaxation response as an SLS (a Kelvin spring-dashpot in parallel with another

Kelvin spring element). The final model is applicable for small indentations of an


isotropic, incompressible sample bonded to the substrate with finite thickness with a

hard, spherical tip:


3(1−)𝑅1/2𝛿0

3/2(1 +

𝜏𝜎−𝜏𝜀

𝜏𝜀𝑒−𝑡/𝜏𝜀) 𝐶

𝐶 = [1 −2𝛼0

𝜋𝜒 +

4𝛼02

𝜋2 𝜒2 −8

𝜋3 (𝛼03 +

4𝜋2

15𝛽0) 𝜒3

+16𝛼0

𝜋4 (𝛼03 +

3𝜋2

5𝛽0) 𝜒4] (5.5)

In order to investigate the viscoelastic property of single cells subjected to

different strain-rates, this model is used in the present study to determine the

viscoelastic properties of osteocytes, osteoblasts and chondrocytes for each of the

four strain-rates. Similar to our previous investigation, the cells are assumed to be

incompressible. Thus, this model solution becomes:


3𝑅1/2𝛿0

3/2(1 +

𝜏𝜎−𝜏𝜀

𝜏𝜀𝑒−𝑡/𝜏𝜀) [1 + 1.133𝜒 + 1.283𝜒2 +

0.769𝜒3 + 0.0975𝜒4] (5.6)

By fitting the Equation (5.6) with relaxation force–time curves, the three parameters

of the SLS viscoelastic model are determined as:

𝑘1 = 𝐸𝑅 (5.7)

𝑘2 = 𝐸𝑅 (𝜏𝜎−𝜏𝜀

𝜏𝜀) (5.8)

𝜇 = 𝐸𝑅(𝜏𝜎 − 𝜏 ) (5.9)

where 𝑘1, and 𝑘2 are Kelvin spring elements and 𝜇 is a damper element. The

instantaneous modulus and Prony constant can be calculated as follows:

𝐸0 = 𝐸𝑅 (1 +𝜏𝜎−𝜏𝜀

𝜏𝜀) (5.10)

𝑔1 =𝑘2

𝑘1+𝑘2=

𝜏𝜎−𝜏𝜀

𝜏𝜎 (5.11)

Note that the Prony constant is used to determine the stress–relaxation properties

of single cells as expressed in the shear relaxation modulus G(t):

𝐺(𝑡) = 𝐺0[1 − 𝑔1(1 − 𝑒−𝑡/𝜏𝜀)] (5.12)

where 𝐺0 is the instantaneous shear modulus.

93


5.4 PRI METHOD

A number of recent works have been conducted to investigate the relaxation

behaviour of the poroelastic material, hydrogel (Chan et al. 2012; Hu et al. 2011; Hu

et al. 2010). The gels are made of a cross-linked polymer network and a species of

mobile solvent molecules. These materials, which are used widely in various

technological applications such as fuel cells, drug delivery and bioengineering, are

poroelastic and involve the coupled deformation of the network and the exuding of

the solvent. The performance of these gels is known to closely correspond to their

ability to regulate the transport of small molecules such as solvents, in what is called

the poroelastic relaxation process. This is one of the two primary relaxation

processes: the second one is the viscoelastic relaxation process which involves

conformational changes of the polymer network. By applying a theory of

poroelasticity, the PRI method (Hu et al. 2010; Hu et al. 2011; Chan et al. 2012; Yu,

Sanday and Rath 1990) was developed to investigate the poroelastic stress–relaxation

process of hydrogels. The calculations in the PRI method are described in this

section (Chan et al. 2012).

The contact radius a is related to indentation δ by:

𝑎 = √𝑅𝛿 ∙ 𝑓𝑎(√𝑅𝛿/ℎ) (5.13)

where R and h are the colloidal probe radius and sample thickness, respectively, and

𝑓𝑎(√𝑅𝛿/ℎ) accounts for deviation from the Hertz mechanics of the contact radius,

and is shown as:

𝑓𝑎(√𝑅𝛿/ℎ) =1.41(√𝑅𝛿/ℎ)

2+0.57(√𝑅𝛿/ℎ)+0.5

(√𝑅𝛿/ℎ)2

+0.49(√𝑅𝛿/ℎ)+0.5 (5.14)

The instantaneous load response (Pi) of the sample right after the indentation is

shown as:

𝑃𝑖 =16

3𝑅1/2𝛿3/2𝐺 ∙ 𝑓𝑃(√𝑅𝛿/ℎ) (5.15)

where G is the shear modulus, and 𝑓𝑃(√𝑅𝛿/ℎ) accounts for the deviation from the

Hertz load and is expressed as:

𝑓𝑃(√𝑅𝛿/ℎ) =2.36(√𝑅𝛿/ℎ)

2+0.82(√𝑅𝛿/ℎ)+0.46

(√𝑅𝛿/ℎ)+0.46 (5.16)


The equilibrium or long-time load response (Pf) is defined as follows:

𝑃𝑓 =8𝑅1/2𝛿3/2𝐺

3(1−)∙ 𝑓𝑃(√𝑅𝛿/ℎ) (5.17)

where is the Poisson’s ratio of the material.

The final result exhibits a dimensionless load relaxation function g which

relates to a dimensionless relaxation time 𝐷𝑡/𝑅𝛿:

𝑃(𝑡)−𝑃𝑓

𝑃𝑖−𝑃𝑓= 𝑔(𝐷𝑡/𝑅𝛿, √𝑅𝛿/ℎ) (5.18)

𝑔(𝐷𝑡/𝑅𝛿, √𝑅𝛿/ℎ) = 𝑒𝑥𝑝(−𝛼(𝐷𝑡/𝑅𝛿)𝛽) (5.19)

where D is the diffusion coefficient. The parameters α and β are functions of the

√𝑅𝛿/ℎ ratios and are represented by polynomials given as:

𝛼 = 1.15 + 0.44(√𝑅𝛿/ℎ) + 0.89(√𝑅𝛿/ℎ)2

−0.42(√𝑅𝛿/ℎ)3

+ 0.06(√𝑅𝛿/ℎ)4 (5.20)

𝛽 = 0.56 + 0.25(√𝑅𝛿/ℎ) + 0.28(√𝑅𝛿/ℎ)2

−0.31(√𝑅𝛿/ℎ)3

+ 0.1(√𝑅𝛿/ℎ)4

− 0.01(√𝑅𝛿/ℎ)5 (5.21)


5.5.1 Comparison of equilibrium moduli among living osteocytes, osteoblasts

and chondrocytes

In this study, the single cell is assumed to be a solid, homogenous and viscoelastic

material. After stress–relaxation testing, the equilibrium moduli Eequil of the three cell

types were determined by using Equation (4.6) and the force measurement after 60

seconds. These moduli, together with the Young’s and relaxation moduli ratios

𝐸𝑌/𝐸𝑅 for each cell type at four strain-rates are presented in Figure 5.2 and Table

5-1. The large variability in cell’s properties was also observed. Firstly, the

equilibrium moduli of each of the three cell types were compared among four strain-

rates tested. It was observed that, although the Eequil were slightly different when the

strain-rate decreased, the variation was not statistically significant for all three cell

types.

95


Table 5-1: Equilibrium moduli 𝐸𝑒𝑞𝑢𝑖𝑙 (Pa) and 𝐸𝑌/𝐸𝑅 ratios of living osteocytes,

osteoblasts and chondrocytes at four different strain-rates

Osteocytes

(n = 39)

Osteoblasts

(n = 40)

Chondrocytes

(n = 43)

𝐸𝑒𝑞𝑢𝑖𝑙 (Pa)

7.4 s-1

460.56 ± 1,220.70 431.09 ± 718.74 470.80 ± 435.51

0.74 s-1

431.00 ± 586.29 467.88 ± 669.23 482.59 ± 388.10

0.123 s-1

369.48 ± 465.08 528.05 ± 683.77 514.67 ± 467.87

0.0123 s-1

389.94 ± 725.94 502.91 ± 569.96 457.99 ± 490.15

𝑙𝑜𝑔(𝐸𝑌

/𝐸𝑅)

7.4 s-1

1.13 ± 0.44 0.86 ± 0.46 0.70 ± 0.26*

0.74 s-1

0.72 ± 0.18 0.68 ± 0.43 0.53 ± 0.25*

0.123 s-1

0.65 ± 0.18♯ 0.48 ± 0.14 0.42 ± 0.15*

0.0123 s-1

0.59 ± 0.35 0.41 ± 0.30 0.36 ± 0.27*

* p < 0.05 indicates that the chondrocytes had smaller EY

/ER

ratios than the osteocytes at all strain-rates.

♯ p < 0.05 indicates that the osteocytes had larger EY

/ER

ratios than the osteoblasts at 0.123 1/s strain-rates.

Secondly, the equilibrium moduli were compared among the three cell types at

each of the four strain-rates. The difference in Eequil for the three cell types was not

statistically significant across all the applied strain-rates. Thus, it was observed that

the living osteocytes, osteoblasts and chondrocytes in this study exhibited similar

long-term elastic stiffness.

Furthermore, the Young’s and relaxation moduli ratios 𝐸𝑌/𝐸𝑅 of the three cell

types were calculated and compared to each other (see Table 5-1 and Figure 5.2). It

can be seen that the osteocytes in this study exhibited higher EY/ER ratios compared

to the chondrocytes at all the tested strain-rates (p < 0.05, see Figure 5.2 and Table

5-1). In addition, the osteocytes showed moduli ratios that were similar to those of

the osteoblasts except at the 0.123 s-1

strain-rate. These results revealed that different

cell types possess different relaxation behaviour. The data in Figure 5.2 are presented

as logarithmic values for clearer illustration. Furthermore, it was observed that the

EY/ER ratios reduced with decreasing strain-rates for all three cell types in this study,

indicating that the rate at which living cells relax is likely dependent on the rate of

loading.


Figure 5.2: Equilibrium moduli Eequil (Pa) and 𝐸𝑌/𝐸𝑅 ratios of osteoblasts and

chondrocytes at four different strain-rates (the data are shown as mean ± standard

deviation)

5.5.2 Viscoelastic properties of single living osteocytes, osteoblasts and

chondrocytes subjected to different strain-rates

In order to study the effect of strain-rate on the relaxation behaviour of osteocytes,

osteoblasts and chondrocytes, the viscoelastic properties of these cells at varying

strain-rates were studied. After indentation, the cells were allowed to relax for 60

seconds and the force–time curves were recorded and analysed. The stress–relaxation

data (i.e. the force–time curves) were fitted with the thin-layer viscoelastic model

function (i.e. Equation (5.6) in Section 5.2) to estimate the viscoelastic material

parameters, namely, ER, 𝜏𝜎 and 𝜏 for each cell type from which the other parameters

were calculated using Equations (5.7)–(5.11). These viscoelastic parameters are

presented in Figure 5.3 and Table 5-2 for all three cell types at the four different

strain-rates. These parameters provide a characterising comparison of the viscoelastic

properties of living osteocyte, osteoblast and chondrocyte cells subjected to different

strain-rates.

In Figure 5.3, the significant differences in each viscoelastic property between

the cell types are indicated by a corresponding coloured pentagon above that

property. It was observed that the osteoblasts in this study had larger relaxation and

instantaneous moduli (i.e. ER and E0) than the moduli of the chondrocytes at all

strain-rates. Additionally, the viscosity µ showed significant difference between

these two cell types in this study at the strain-rates of 7.4, 0.74 and 0.123 s-1

(p <

0.05, see Figure 5.3 and Table 5-2). However, the osteoblasts did not exhibit

significant difference in the Prony constant g1 compared to the chondrocytes. These

97


results are similar to those reported by Darling et al. (Darling et al. 2008) who

applied a strain-rate close to the strain-rate of 0.74 s-1

. In addition, the osteocytes in

this study exhibited similar viscoelastic properties to the osteoblasts. Interestingly,

the Prony constant g1 of the osteocytes at 7.4 s-1

and 0.0123 s-1

strain-rates were

significantly larger than those of the chondrocytes (see Figure 5.3).

Figure 5.3: Viscoelastic properties of osteocytes, osteoblasts and chondrocytes at

four different strain-rates (the data are shown as mean ± standard deviation;

Significant difference between cell types [p < 0.05] is indicated by a corresponding

coloured pentagon above the mechanical property)

It is seen in the results in Table 5-2 that the relaxation moduli ER of the

osteocytes, osteoblasts and chondrocytes were unchanged with decreasing strain-

rates. This is consistent with the equilibrium moduli discussed in the previous

section. On the other hand, the instantaneous moduli E0 of these cells slightly (i.e.

not significant) reduced with decreasing strain-rates.


Table 5-2: Viscoelastic properties of living osteocytes, osteoblasts and chondrocytes

at four different strain-rates

See Figure 5.3 for the significant differences in viscoelastic properties among the three cell types.

From our data it can be justifiably hypothesised that the viscoelastic properties

of single cells are dependent on the applied loading rates or strain-rates. It is possible

Osteocytes

(n = 39)

Osteoblasts

(n = 40)

Chondrocytes

(n = 43)

ER (Pa)

7.4 s-1

559.86 ± 582.88 714.18 ± 1,004.60 370.09 ± 339.38

0.74 s-1

637.27 ± 714.90 743.73 ± 998.51 358.37 ± 254.43

0.123 s-1

458.46 ± 476.88 710.53 ± 906.25 343.28 ± 239.89

0.0123 s-1

308.08 ± 303.65 538.78 ± 616.32 280.11 ± 199.91

E0 (Pa)

7.4 s-1

1,588.90 ± 1,273.80 1,352.52 ± 1,243.15 748.12 ± 473.19

0.74 s-1

1,481.55 ± 1,626.35 1,253.34 ± 1,251.21 645.1 ± 376.63

0.123 s-1

1,220.73 ± 1,590.88 1,154.44 ± 1,202.12 572.47 ± 355.35

0.0123 s-1

1,128.56 ± 1,245.26 883.93 ± 973.46 462.21 ± 324.69

𝜏𝜎 (s)

7.4 s-1

20.62 ± 27.33 8.35 ± 10.28 3.37 ± 1.75

0.74 s-1

38.82 ± 128.87 6.45 ± 5.20 3.92 ± 4.29

0.123 s-1

36.48 ± 66.31 5.55 ± 3.46 4.27 ± 2.42

0.0123 s-1

115.78 ± 211.62 28.66 ± 100.56 29.67 ± 115.01

𝜏 (s)

7.4 s-1

4.78 ± 3.87 2.66 ± 2.94 1.45 ± 0.80

0.74 s-1

5.47 ± 3.64 2.90 ± 2.60 1.73 ± 0.69

0.123 s-1

12.37 ± 22.66 3.03 ± 2.36 2.36 ± 0.98

0.0123 s-1

10.71 ± 5.58 5.24 ± 6.58 4.45 ± 7.06

𝑔1

7.4 s-1

0.64 ± 0.20 0.59 ± 0.24 0.52 ± 0.18

0.74 s-1

0.52 ± 0.17 0.51 ± 0.18 0.45 ± 0.16

0.123 s-1

0.48 ± 0.22 0.46 ± 0.14 0.39 ± 0.15

0.0123 s-1

0.53 ± 0.27 0.40 ± 0.19 0.34 ± 0.20

99


to argue that the discrepancy in the mechanical properties of single cells reported in

the literature (Darling et al. 2008; Chahine et al. 2013; Darling, Zauscher and Guilak

2006; Wozniak et al. 2010) is a consequence of this characteristic. Furthermore, a

close scrutiny of the Prony constant g1 revealed that the amount by which the cell

stiffness was reduced, as shown in Equation (5.11) was dependent on the strain-rates.

This dependence accords well with the response observed in whole tissue (Oloyede,

Flachsmann and Broom 1992; Oloyede and Broom 1993a), suggesting that the fluid-

dominated load sharing describing the deformation of soft biological tissues is

continuous from the microscale cellular level right through to the macroscale level.

Figure 5.4 presents the AFM stress–relaxation data shown as the mean ±

standard deviation at different applied strain-rates for the osteocytes, osteoblasts and

chondrocytes and their corresponding fitted curves using the thin-layer viscoelastic

model as mentioned above. The R2 values of these curves are also presented in

Table 5-3. The data from this analysis revealed that the thin-layer viscoelastic

model can be utilised for characterising the viscoelastic properties of single living

cells with reasonable accuracy. However, during our attempt to determine the

viscoelastic properties using the thin-layer viscoelastic model, it was noticed that this

model could not provide a good fit to the experimental data in some cases, especially

at highest strain-rate (i.e. 7.4 s-1

). This may be the reason for the higher Prony

constant g1 of the osteocytes at 0.0123 s-1

than at the other strain-rates.

It was observed from the AFM stress–relaxation testing results that there were

two phases in the force–time curves. In the first phase, a sudden drop of applied force

takes place immediately after the indention, which lasts for a few seconds (see the

strain-rate of 7.4 s-1

in Figure 5.4). In the second phase, following the first phase, the

applied force gradually reduces and reaches an asymptotic value. These two phases

are called the transient and equilibrium phases, respectively, in this study. It is

hypothesised that the occurrence of these phases is due to the effects of both the

cellular CSK network and the intracellular fluid. The thin-layer model, however,

assumed that the material behaves as a homogenous solid material, whereas the cells

comprise both fluid and solid components. As discussed in Chapter 4, both the CSK

and intracellular fluid govern the mechanical behaviour of single cells. Thus, this

might be one of the limitations of this model.


Figure 5.4: Relaxation experimental data and thin-layer viscoelastic model fitted with

the curves of osteocytes, osteoblasts and chondrocytes subjected to four different

strain-rates (the data are shown as mean ± standard deviation)

Table 5-3: R2 and RMSE values of osteocytes, osteoblasts and chondrocytes at

different strain-rates when fitted with the thin-layer viscoelastic model

As a result, this model cannot give the best fit to the experimental data on

single living chondrocytes in two possible ways, particularly at the highest strain-rate

in this study. The model is capable of either only capturing the transient phase of the

Strain-rate (s-1

) Parameters Osteocytes

(n = 39)

Osteoblasts

(n = 40)

Chondrocytes

(n = 43)

7.4 RMSE 0.70 0.69 0.25

R2 0.88 0.88 0.91

0.74 RMSE 0.52 0.53 0.19

R2 0.91 0.87 0.89

0.123 RMSE 0.26 0.30 0.13

R2 0.96 0.93 0.92

0.0123 RMSE 0.16 0.19 0.11

R2 0.97 0.95 0.91

101


AFM stress–relaxation data (and not the equilibrium phase of the AFM stress–

relaxation data as shown in the 7.4 s-1

strain-rate curves in Figure 5.4) or only

capturing the equilibrium phase of the data, but not the sudden drop of applied force

from its maximum value, which might be due to the effect of the intracellular fluid,

in the first few seconds of the stress–relaxation behaviour as shown in the work of

Darling et al. (Darling et al. 2007; Darling, Zauscher and Guilak 2006). Thus, it can

be concluded that the thin-layer viscoelastic model can only provide best fits to the

experimental data at relatively low strain-rates where the instantaneous F0 and

equilibrium Fequil force ratio is less than 4.

Another drawback is that three parameters need to be determined in this model,

of which the solutions are significantly influenced by the initial conditions. This may

lead to significant error if the initial conditions are not carefully chosen. Thus,

additional experience and skill may be required to select the right initial conditions

when attempting to conduct the curve fitting. This disadvantage may limit the use of

this model.

5.5.3 PHE analysis of strain-rate dependent relaxation behaviour of single cells

In this section, the application of the PHE model in simulating the relaxation

behaviour of single cells is investigated. This model combined with the inverse FEA

technique was used to investigate the dependence of the relaxation behaviour of

single living cells on the rate of loading. In the current study, only single living

chondrocytes were investigated. The other cell types will be considered in future

studies.

The results were then compared to the results of the thin-layer viscoelastic

model and PRI method (discussed in detail in Section 5.4) in order to investigate the

application of the PHE model for relaxation behaviour simulation.

5.5.3.1 Inverse FEA technique to estimate PHE material parameters

Similar to the previous investigation in this study, the inverse FEA together with the

PHE model was performed as reported in this section. An FEA model similar to the

model in Figure 4.6(c) was developed, in which the single chondrocyte was indented,

followed by 60 seconds relaxation. In this model, the chondrocyte is incompressible

at the indentation phase, and then compressible in the relaxation phase. The neo-

Hookean hyperelastic constitutive law shown in Equation (4.13) was also used in this


investigation. The simulation was conducted and the reaction forces were then

extracted and compared to the experimental data in order to determine the cell’s

mechanical properties, namely, C1, D1 and k0 in Equations (4.13) and (4.16) using the

inverse FEA procedure as follows:

From the force–indentation curves during the indentation phase in which the

applied force reached its maximum value, the AFM force–indentation data

was fitted with Equation (4.24) in order to determine the C1 parameter in

Equation (4.13) because the cell was assumed to be incompressible. Note

that the C2 value was zero because the neo-Hookean constitutive law was

used.

From the force–time curves during the equilibrium state at the relaxation

phase where the applied force obtained its asymptotic value, the inverse

FEA was conducted and the results compared to the AFM experimental

results in order to determine the D1 parameter in Equation (4.13).

From the force–time curves during the transient state at the relaxation phase

in which the applied force reduced dramatically, the inverse FEA was

conducted in order to determine the initial permeability k0 in Equation

(4.16).

The volume strain of the cell was also calculated using Equation (4.17)

based on the void ratios of the cell at the end of each phase in order to study

the effect of the pore fluid pressure within the cell.

5.5.3.2 PHE analysis results

As discussed above, the hydrogels exhibit two different processes when undergoing

stress–relaxation, namely, the viscoelastic and poroelastic relaxation processes. By

applying the PRI, researchers have successfully captured the poroelastic relaxation

process of hydrogels.

It was observed that the single living cells had a similar structure to the

hydrogels mentioned above. Thus, it is hypothesised that the relaxation behaviour of

single cells is also governed by both the viscoelasticity of the cellular CSK network

and by the intracellular fluid (i.e. poroelastic relaxation process). Both the thin-layer

viscoelastic model and the PRI method discussed in this chapter can capture the

relaxation behaviour of living cells. However, the PRI method can only capture the

103


poroelastic relaxation process, and the thin-layer viscoelastic model assumes the cells

behave as solid-like materials and has several limitations as discussed above. Thus,

the development of a mechanical model that can overcome these limitations is

necessary in order to study the stress–relaxation behaviour of single living cells.

In Section 4.5.5, it was already demonstrated that the PHE model could

accurately capture the strain-rate dependent mechanical deformation behaviour of

single living cells, which is governed by both solid and fluid constituents. It is

hypothesised that this model can also capture the relaxation behaviour of living cells.

The PHE model was applied to simulate the stress–relaxation behaviour of

single living chondrocytes at four different strain-rates. Table 5-4 shows the material

parameters of the PHE model determined using the procedure mentioned above.

Figure 5.5 and Figure 5.6 present the performance of the PHE model in capturing the

mechanical deformation behaviour of the chondrocytes during indentation and the

stress–relaxation behaviour of the chondrocytes, respectively, at four different strain-

rates. Figure 5.5 presents the AFM force–indentation experimental data and PHE

simulation results of a typical chondrocyte. In order to investigate the application of

the PHE model compared to the thin-layer viscoelastic model and the PRI method,

together with the AFM force–time relaxation data, the PHE simulation results and

fitted curves using the thin-layer viscoelastic model and PRI method are also

presented in Figure 5.6. The estimated parameters for all three models and their R2

values are also presented in these figures.

Figure 5.5: AFM experimental data and PHE model force–indentation curves of a

typical living chondrocyte at (a) 7.4 s-1

, (b) 0.74 s-1

, (c) 0.123 s-1

, and (d) 0.0123 s-1

strain-rates


Table 5-4: PHE model material parameters and the poroelastic diffusion constant D

(µm2/s) of single living chondrocytes at four varying strain-rates

7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

C1 (Pa) 720.82 ±

392.62

539.05 ±

374.57

424.82 ±

318.73

272.37 ±

191.26

D1 (10-3

1/Pa) 33.80 ±

55.10

13.00 ±

14.00

9.20 ±

8.80

7.70 ±

8.30

Initial permeability k0

(109

µm4/N.s)

8.72 ±

13.20

3.74 ±

5.59

1.00 ±

1.24

0.30 ±

0.21

Initial void ratio e0 4 4 4 4

D (µm2/s) 2.31 ± 1.42 1.34 ± 1.17 0.83 ± 0.82 0.68 ± 2.29

It was observed that, at the highest strain-rate (i.e. 7.4 s-1

), both the thin-layer

viscoelastic model and the PHE model captured the sudden drop of the applied force

in the transient phase of relaxation behaviour (see Figure 5.6 (a)). However, the thin-

layer viscoelastic model did not capture the gradual reduction of applied force after

the transient phase, whereas the PHE model captured this accurately. Similarly, the

PRI model captured the equilibrium phase of relaxation behaviour very well but not

the transient phase. Among these models, the PHE model is the only one that could

effectively capture the stress–relaxation behaviour of the single living chondrocytes

at both the transient and equilibrium phases. This was demonstrated by a much

higher R2 value of the PHE model compared to that of the other two models.

Similarly, at the strain-rate of 0.74 s-1

, the thin-layer model captured the sudden

drop in the transient phase of relaxation, but not the gradual reduction of applied

force at the equilibrium phase. The PHE and PRI models, however, captured both of

these phases very well as demonstrated by much higher R2 values of the PHE and

PRI models compared to that of the thin-layer viscoelastic model (see Figure 5.6 (b)).

On the other hand, at the strain-rate of 0.123 s-1

, all three models captured the

stress–relaxation behaviour of the chondrocytes very well, corresponding to their

high R2 values (see Figure 5.6 (c)). These results suggest that the thin-layer

viscoelastic and PRI models can only capture the stress–relaxation behaviour at low

105


strain-rates, whereas the PHE model can capture the chondrocyte behaviour at a wide

range of strain-rates.

Figure 5.6: AFM stress–relaxation experimental data and thin-layer viscoelastic

model, PRI model and PHE model results for a typical living chondrocyte at (a) 7.4

s-1

, (b) 0.74 s-1

, (c) 0.123 s-1

, and (d) 0.0123 s-1

strain-rates (the fitting parameters for

each model are shown in the corresponding coloured texts)

At the lowest strain-rate (i.e. 0.0123 s-1

) all three models captured the stress–

relaxation behaviour of the chondrocytes very well as demonstrated their high R2

values (see Figure 5.6 (d)), which were similar to those at the 0.123 s-1

strain-rate.

However, it was noticed during the simulation that the PHE model did not capture

the relaxation behaviour at this lowest strain-rate with very high accuracy compared

to the other strain-rates. This might be due to the influence of the intracellular fluid,

which is inferior at this low strain-rate. As discussed in Chapter 4 (Section 4.5.4), at

such low strain-rates, the intracellular fluid can freely move through the cellular CSK

with very low resistance at the indentation phase, leading to a relatively small fluid


pore pressure gradient. As a result, at the relaxation phase, the relaxation behaviour

is not governed by the intracellular fluid, but is mainly governed by the remodelling

of the cellular CSK. Therefore, it can be concluded that the stress–relaxation

behaviour of single living chondrocytes at this low strain-rate is mainly contributed

by the cellular CSK network.

Finally, it can be seen from the results presented in Table 5-4 that both the

poroelastic diffusion constant and the hydraulic permeability of the single living

chondrocytes reduced with decreasing strain-rates (p < 0.05). This might be because

the intracellular fluid volume fraction and the fluid pore pressure gradient of

chondrocytes after the indentation phase are higher with higher strain-rates. This is

similar to the results reported by Moeendarbary et al. (Moeendarbary et al. 2013)

who found that the diffusion constant reduced and the cells relaxed at lower rates

with decreasing fluid fractions. Thus, it can be concluded that the relaxation

behaviour of chondrocytes is dependent on strain-rates. In order to understand how

chondrocytes exhibit stress–relaxation behaviour, the von Mises stress and pore

pressure distributions of a typical chondrocyte at four different strain-rates were

extracted, and are shown in Figure 5.7–5.10.

The results are shown for the instant after the indentation and relaxation phases

for each strain-rate tested. It is interesting to note that the von Mises stress reduced

slightly during the relaxation phase, whereas the fluid pressure decreased

significantly. This can be explained by the fact that, after the indentation phase, the

intracellular fluid is blocked inside the cell due to the low permeability of the cell;

this causes the pore pressure to increase. After that, during the relaxation phase,

because of the fluid pore pressure gradient inside the cell, the intracellular fluid starts

to flow out, causing the cell to become softer. Additionally, at low strain-rates, some

of the intracellular fluid exudes out from the cell during the indentation phase,

causing a lower fluid pore pressure than at high strain-rates. At the end of the

relaxation phase, the cell is in an equilibrium condition wherein the fluid pore

pressure reaches a relatively small value for all four strain-rates.

107


(a) (b)



strain-rate (the measurement

unit in these figures is 106 Pa)




strain-rate (the


109


(a) (b)



strain-rate (the



(a) (b)

Figure 5.10: von Mises stress (top) and fluid pore pressure (bottom) distributions –

(a) after indentation, and (b) after relaxation phase at 0.0123 s-1

strain-rate (the


111


In order to obtain a clearer illustration, the fluid pore pressure was extracted at

the point beneath the tip (as shown in Figure 5.11) for all four strain-rates tested. It

was observed that the fluid pore pressure increased to a maximum value (i.e. around

320, 170, 61 and 19 Pa for 7.4, 0.74, 0.123 and 0.0123 s-1

strain-rates, respectively),

immediately after the indentation phase, and then significantly decreased to a

limiting low value (i.e. almost zero) at the end of the relaxation phase. At that point,

the chondrocyte reached its equilibrium condition wherein the fluid pore pressure

was equal to the extracellular pressure. It was observed that the significant reduction

of fluid pore pressure resulted in a significant decrease of applied force in the

relaxation phase.

Figure 5.11: Fluid pore pressure curves of a typical chondrocyte at (a) 7.4 s-1

, (b)

0.74 s-1

, (c) 0.123 s-1

, and (d) 0.0123 s-1

strain-rates extracted at the point beneath the

AFM tip

Furthermore, the volume strains of a typical chondrocyte were also measured

using Equation (4.17) by determining the void ratios of the cell. The strains were

measured at the instant after indentation and after the relaxation phase when the


chondrocytes were subjected to four different strain-rates. The results are presented

in Table 5-5. It was observed that the volume strain of the chondrocyte was larger

after indentation compared to the volume strain after the relaxation phase at all

strain-rates. This finding suggested that the whole chondrocyte was compressible due

to the fluid flux during the relaxation phase. Thus, it can be concluded that, even

though both the solid and liquid components are incompressible, the whole cell is

compressible because of the fluid loss.

Table 5-5: Volume strain of chondrocytes after indentations and relaxation phases

when subjected to varying rates of loading

7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

After indentation 0.99 0.99 0.98 0.97

After relaxation 0.87 0.91 0.93 0.94

As a result, it can be concluded that the PHE model is suitable for capturing the

stress–relaxation behaviour of living chondrocytes. It is hypothesised that the PHE

model is a potential model to capture the relaxation behaviour of other cell types

which will be considered in future studies. Additionally, as presented in Section

4.5.5, this model can also capture the strain-rate dependent mechanical deformation

behaviour of living cells. Therefore, it can be concluded that the PHE model, which

is developed from the poroelastic theory, would be a potential mechanical

constitutive model for single cell biomechanics (Moeendarbary et al. 2013; Nguyen

et al. 2014; Nguyen and Gu 2014).

5.6 CONCLUSION

In this study, the strain-rate dependent relaxation behaviour of single living cells,

namely, osteocytes, osteoblasts and chondrocytes was investigated using AFM

stress–relaxation testing. The thin-layer viscoelastic model was applied to determine

the viscoelastic properties of all three cell types at four different strain-rates. The

PHE model combined with the inverse FEA technique was used to investigate the

strain-rate dependent relaxation behaviour of single living cells. The following

conclusions can be reported:

113


When using the thin-layer viscoelastic model, the osteocytes and

osteoblasts in this study showed similar viscoelastic properties to each

other, and exhibited larger properties than the chondrocytes. Darling et al.

also reported similar results to ours at the 0.74 s-1

strain-rate (Darling et al.

2008).

From the results obtained in the experiment reported in this chapter,

together with the results presented in Chapter 4, it can be concluded that

both the elastic and viscoelastic properties of single living osteocytes,

osteoblasts and chondrocytes are dependent on strain-rates, which is

similar to the dependence of other fluid-filled biological tissues. This

finding suggests that the deformation of soft biological tissues, which is

described as fluid-dominated load sharing, is continuous from the

microscale cellular level to the macroscale level.

During the attempt to curve fit the thin-layer viscoelastic model to the

AFM stress–relaxation experimental data in this study, it was observed

that the model could not capture the relaxation behaviour of the single cell

at high strain-rates, that is, where the F0/Fequil ratios were relatively large.

Therefore, it can be concluded that the thin-layer viscoelastic model can

only capture the strain-rate dependent relaxation behaviour of single living

cells with good results when the F0/Fequil ratio is less than 4.

The PHE model and PRI method were used to estimate the hydraulic

permeability and poroelastic diffusion constant, respectively, of the

chondrocytes at varying strain-rates. It was found that both of the

parameters reduced with decreasing strain-rates, indicating that the

relaxation behaviour of single living chondrocytes is dependent on the

strain-rate. This might be because the volume fraction of intracellular fluid

is higher at higher strain-rates.

By comparing the performance of the PHE model with the performance of

the thin-layer viscoelastic model and PRI method, the PHE model was

demonstrated to effectively capture both the transient and equilibrium

phases in the relaxation behaviour of the living chondrocytes (whereas the

other two methods only captured one of the relaxation phases). Thus, the


results suggest that the PHE model can precisely capture the stress–

relaxation behaviour of single chondrocytes.

By using the PHE model, it was observed that the intracellular fluid

exudes from the cell during the relaxation phase because of the gradient of

the fluid pore pressure. This causes volume loss or compressibility of the

chondrocytes.

From the results presented in this and previous chapters, it can be

concluded that both the mechanical deformation and relaxation behaviour

of single living cells are dependent on the rate of loading and that the

intracellular fluid plays an important role in cellular mechanical properties

and responses to external mechanical stimuli. Moreover, it can be

concluded that the PHE model can capture not only the strain-rate

dependent mechanical deformation behaviour of single living

chondrocytes, but also the stress–relaxation behaviour.

Although the chondrocyte was the only cell type used for the PHE model

analysis in this study, it is believed that this model can also be applied

effectively to any other cell types which have similar structures and

behaviour.

Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 115

Chapter 6: Effect of Osmotic Pressure on

the Morphology and Mechanical

Properties of Single

Chondrocytes

6.1 INTRODUCTION

As discussed in Chapters 4 and 5, the intracellular fluid plays an important role in

cellular mechanical behaviour. Therefore, in this chapter, the effect of intracellular

fluid is investigated further by studying the mechanical behaviour of single living

cells exposed to different osmotic pressures. Chondrocyte is the only cell type that is

considered in this study. The other cell types will be investigated in future works.

It is well-known that single living cells are sensitive to their physicochemical

environment which influences their metabolic activity and their structure and

properties. Most cells of the body respond to osmotic pressure by activating some

processes. The mechanisms may include the organisation of the CSK network and

provocation of several transporters in the membrane to stimulate the mobilisation of

osmotically active solutes (Sarkadi and Parker 1991). In particular, chondrocytes

change their shape and volume due to the increased negatively fixed-charge density

when the cartilage loses water during deformation. Moreover, it has been reported

that the disruption of the collagen network in the early stage of osteoarthritis causes

the increase of water content of the cartilage which in turn leads to a reduction of the

pericellular osmolality of the chondrocytes (Maroudas et al. 1985). Thus, the

characterisation of the mechanical properties of chondrocytes subjected to varying

osmotic pressures, as investigated in this study, would provide a better understanding

of chondrocyte mechanotransduction.

In this study, single living chondrocytes were exposed to different osmotic

solutions, in which the mechanical properties were estimated. The thin-layer elastic

model (as presented in Chapter 4, Section 4.3) and the thin-layer viscoelastic model

116 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes

(as presented in Chapter 5, Section 5.3) were used to determine the elastic and

viscoelastic properties, respectively, of the chondrocytes. Sections 6.3 and 6.4

present the results and conclusions. The PHE model is also used to study the effect of

extracellular osmotic pressure on the hydraulic permeability of single chondrocytes,

as presented in Section 6.3.4.2.

6.2 MATERIALS AND MODEL

6.2.1 Osmotic activity

In this study, the cell volumes, V, which are normalised in isoosmotic conditions V0,

at different osmolality of extracellular mediums are fitted with a linear model (the so-

called Boyle-Van’t Hoff model) of the ideal osmotic swelling behaviour (Lucke and

McCutcheon 1932; Ting-Beall, Needham and Hochmuth 1993; Guilak, Erickson and

Ting-Beall 2002). With the assumption that the osmotic activity is constant inside the

cell, this model relates the normalised cell volume, V/V0, to the osmolality of the

suspending medium P (normalised to the osmolality of the isoosmotic condition) as

follows:

𝑉/𝑉0 = (𝑅/𝑃) + (1 − 𝑅) (6.1)

where R is the Ponder’s value representing the chemical activity of the intracellular

fluid compared to the isoosmotic condition.

6.2.2 Methodology

The following steps were taken in the investigation:

Six samples of varying osmolality, namely, 30, 100, 300, 450, 900 and

3,000 mOsm, of single living chondrocytes were prepared based on the

procedure presented in Chapter 3 (Section 3.3.2). The 3,000 mOsm

solution was studied because most of the intracellular fluid is removed. As

a result, we can study only the effects of the solid phase of the cells. With

this method, the important role of each phase in cellular mechanical

responses can be investigated.

The diameters and heights of the living chondrocytes when exposed to

varying osmotic pressures were measured using the technique presented in

Chapter 3 (Sections 3.3.4 and 3.3.5, respectively), to investigate the effect

117

Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 117

of extracellular osmotic pressure on the morphology of chondrocytes. The

Boyle-Van’t Hoff model was also used to study the osmotic activity of the

cells.

Next, the F-actin filament structure changes of the living chondrocytes

with varying osmotic pressures were studied using a confocal laser

microscope. Details on the sample preparation for the confocal imaging

and the microscope used were presented in Chapter 3 (Section 3.3.3).

The AFM indentation experiments were then conducted on each of the six

samples at four varying strain-rates, namely, 7.4, 0.74, 0.123 and 0.0123 s-

1. The thin-layer elastic model presented in Chapter 4 (Section 4.3) was

used to study the osmotic pressure-dependent elastic stiffness of the living

chondrocytes at each of the four strain-rates tested.

The AFM stress–relaxation experiments were conducted on each of the

samples at four varying strain-rates, namely, 7.4, 0.74, 0.123 and 0.0123 s-

1. The thin-layer viscoelastic model presented in Chapter 5 (Section 5.3)

was used to study the dependence of the relaxation behaviour of the

chondrocytes on the extracellular osmotic pressure at each of the four

strain-rates.


6.3.1 Effect of extracellular osmotic pressure on chondrocyte morphology

In this study, a total of six solutions comprising two hypoosmotic (i.e. 30 and 100

mOsm), one isoosmotic (i.e. 300 mOsm) and three hyperosmotic (i.e. 450, 900 and

3,000 mOsm) solutions were investigated. The sample preparation was presented in

Chapter 3 (Section 3.3.2). The chondrocyte diameter and height at six different

osmotic solutions were determined using the techniques presented in Sections 3.3.4,

and 3.3.5, respectively. The results are shown in Figure 6.1, Figure 6.2 and Table

6-1. It is observed that the ratios between the height and diameter of the

chondrocytes were equal to “1” in such a short cell culture time that the chondrocytes

were assumed to be spherical. Thus, the volumes and apparent membrane areas of

the chondrocytes were also calculated, as shown in Table 6-1 and Figure 6.3, for

each osmolality.


As presented in Table 6-1, the chondrocytes underwent swelling corresponding

to significant increase in diameter (i.e. from 16.99 ± 2.041 µm to 30.312 ± 4.493 µm

(p < 0.001), apparent membrane area (i.e. from 919.77 ± 217.65 µm2

to 2,948.28 ±

927.47 µm2

(p < 0.001) and volume (i.e. from 2,677.1 ± 937.39 µm3

to 15,568.59 ±

7,801.56 µm3 (p < 0.001) when exposed to the hypoosmotic solutions.

Table 6-1: Diameter (µm), height (µm), volume (µm3) and apparent membrane area

(µm2) of chondrocytes exposed to 30, 100, 300, 450, 900 and 3,000 mOsm solutions

Osmolality

(mOsm) Diameter (µm) Height (µm) Volume (µm

3)

Membrane

area (µm2)

30 30.31 ± 4.50

(n = 38)*

22.88 ± 3.11

(n = 30)*

15,568.59 ±

7,801.56*

2,948.28 ±

927.47*

100 22.04 ± 1.90

(n = 38)*

18.8 ± 3.25

(n = 41)*

5,729.21 ±

1,552.07*

1,537.00 ±

271.17*

300 16.99 ± 2.04

(n = 54)

15.59 ± 3.47

(n = 60)

2,677.10 ±

937.39

919.77 ±

217.65

450 15.01 ± 1.69

(n = 51)*

14.26 ± 3.23

(n = 42)

1,840.70 ±

670.26*

717.06 ±

167.69*

900 12.75 ± 2.54

(n = 41)*

12.36 ± 1.90

(n = 31)*

1,223.72 ±

883.25*

530.84 ±

232.27*

3,000 12.13 ± 1.56

(n = 53)

11.95 ± 2.44

(n = 39)

982.19 ±

405.29

469.97 ±

125.02

*p < 0.05 indicated that the diameter, height and volume were significantly changed when the chondrocytes were exposed to

different osmotic solutions

Similarly, significant decreases in diameter and volume indicated that the cells

were shrinking when exposed to hyperosmotic solutions, excluding the one with the

highest osmolality (i.e. 3,000 mOsm). In fact, the chondrocytes’ diameter, membrane

area and volume significantly reduced to 12.75 ± 2.54 µm (p < 0.001), 530.84 ±

232.27 µm2

(p < 0.001) and 1,223.72 ± 883.25 µm3

(p < 0.001), respectively, and

then slightly decreased to 12.13 ± 1.56 µm (p = 0.147), 469.97 ± 125.02 µm2

(p =

119


0.107) and 982.19 ± 405.29 µm3

(p = 0.081). The possible reason is that most of the

intracellular fluid had been lost when the cells were subjected to the 900 mOsm

solution. These results suggest that the osmotic environment greatly influences the

morphology of the chondrocytes. The height of the chondrocytes exhibited similar

changes with the varying osmolality except for the case of 450 mOsm hyperosmotic

pressure where the cells did not significantly change the height compared to the

isoosmotic condition (p = 0.053).

Figure 6.1: Diameter distributions of living chondrocytes exposed to 30, 100, 300,

450, 900 and 3,000 mOsm solutions

The cellular apparent membrane area increased on average by a factor of 3.21

when the chondrocytes were subjected to the hypoosmotic condition of 10 mOsm

relative to the isoosmotic condition in this study. This result suggests that

chondrocytes have a significantly large membrane area in the control condition

which is consistent with previously published work (Guilak, Erickson and Ting-Beall

2002). The reason suggested by the previous authors was because the cellular

membrane consists of many folds and ruffles that can be seen by observing the SEM


image of chondrocytes in the isoosmotic state (Guilak, Erickson and Ting-Beall

2002) (see Figure 2.2(b)). Thus, it is reasonable to suggest that the chondrocytes can

withstand large deformations without resulting in large stress on the cell membrane.

Moreover, this finding can further support the hypothesis that the mechanical

properties of living chondrocyte cells are not influenced by the membrane (Guilak,

Erickson and Ting-Beall 2002). Our FEA models shown in Figure 4.6 are

demonstrated to be logical since the effect of the cell membranes is neglected.

Figure 6.2: Height distributions of living chondrocytes exposed to 30, 100, 300, 450,

900 and 3,000 mOsm solutions

6.3.2 Osmotic activity of single living chondrocytes

It was observed from the results in Table 6-1 and Figure 6.3 that the single living

chondrocytes changed their volume when exposed to varying osmotic pressure

conditions. It was also found that the normalised cell volume was linearly related to

the inverse of the osmolality of the extracellular medium (referred to as the Boyle-

Van’t Hoff relationship) (see Figure 6.4), indicating that the chondrocytes behave as

121


osmometers (Guilak, Erickson and Ting-Beall 2002). Furthermore, the Ponder’s

value was determined to be around 0.5407, which is also defined as the volume

fraction (i.e. 54.07%) of osmotically active intracellular water relative to the cell

volume. This fraction is close to the figure of 61% determined in a previous study

using the same method (Guilak, Erickson and Ting-Beall 2002) and to the range of

58–62% determined in another study applying Kedem–Katchalsky equations

(Oswald et al. 2008).

Figure 6.3: Chondrocyte volumes when exposed to 30, 100, 300, 450, 900 and 3,000

mOsm solutions (the data are shown as mean ± standard deviation; *p < 0.05

indicated that the volume was significantly changed)

Figure 6.4: Ponder’s plot for the chondrocytes exhibiting a linear relationship

between the normalised cell volume and normalised extracellular medium osmolality

(the Ponder’s value was determined to be 0.5407; the data are shown as mean ±

standard deviation)


6.3.3 Actin structural changes of chondrocytes when exposed to different

osmotic pressure conditions

Confocal images of single living chondrocytes were also obtained in this study at six

different osmotic pressure conditions in order to study the effect of extracellular

osmotic pressure on the cells’ actin filament structure. The confocal images of the

actin filament and focal adhesion distribution are shown in Figure 6.5 and Figure 6.6,

respectively. The sample preparation and facility used were discussed in Chapter 3

(Section 3.3.3 and Figure 3.3, respectively).

It was observed that, in the control condition (i.e. isoosmotic stress), the actin

filament network was distributed mainly in a thin region at the cortex of the cells (see

the middle left image in Figure 6.5). When the chondrocytes were subjected to the

hypoosmotic pressure conditions, the actin filament network was dispersed

throughout the cells and did not show local distribution at the cortex (see the top two

images in Figure 6.5). This might have been due to the dissociation of the actin

cortex when the cell volume increased, which is consistent with previous published

results (Guilak, Erickson and Ting-Beall 2002). Although the mechanisms

underlying this behaviour have not been fully discovered, previous investigators have

suggested that it might be because of the transient increase in the intracellular

concentration of Ca2+

(Erickson and Guilak 2001; Guilak, Erickson and Ting-Beall

2002; Richelme, Benoliel and Bongrand 2000).

On the other hand, the actin filament network was not altered significantly

when the chondrocytes were exposed to the first two hyperosmotic pressure

conditions (i.e. 450 and 900 mOsm), compared to the control condition (see the

middle right and bottom left images in Figure 6.5). This finding is consistent with the

results reported by Guilak et al. (Guilak, Erickson and Ting-Beall 2002). However, it

is worth noting that when the hyperosmotic pressure was increased to 3,000 mOsm

in this study, the CSK network showed a similar structure to that of the hypoosmotic

conditions where the actin filament distributed evenly within the cells (see the

bottom right image in Figure 6.5). This is an interesting finding indicating that the

hyperosmotic pressure does affect the actin filament structure when the solution

osmolality is relatively large.

It is widely known that the CSK network governs the elastic property of single

living cells. Hence, it is hypothesised that these actin filament structural changes

123


with solution osmolality may also alter the mechanical properties of the

chondrocytes, as discussed in the next section.

Figure 6.5: Confocal images of actin filaments of chondrocytes subjected to varying

osmotic pressure conditions from 30 to 3,000 mOsm osmolality (the cell’s nucleus

and F-actin are visualised in blue [DAPI] and red [568 phalloidin], respectively)

10 µm 10 µm

10 µm 10 µm

5 µm 5 µm

30 mOsm 100 mOsm

300 mOsm 450 mOsm

900 mOsm 3,000 mOsm


Figure 6.6: Confocal images of focal adhesion distribution of chondrocytes at

varying osmotic pressure conditions from 30 to 3,000 mOsm osmolality

10 µm 10 µm

10 µm 10 µm

5 µm 5 µm

30 mOsm 100 mOsm

300 mOsm 450 mOsm

900 mOsm 3,000 mOsm

125


The focal adhesions of the chondrocytes at different osmotic pressure

conditions were also imaged, as shown above in Figure 6.6. It was observed that the

focal adhesions of the living chondrocytes were distributed evenly throughout the

cells in all the osmolality solutions tested. However, in order to quantitatively

characterise the effect of osmotic pressure on the adhesion behaviour of

chondrocytes, cell mechanical adhesiveness should be taken into account. This can

be experimentally quantified using AFM, and will be considered in our future

studies.

6.3.4 Effect of extracellular osmotic pressure on elastic property of single

chondrocytes

6.3.4.1 AFM experimental results

In this study, the biomechanical properties of single living chondrocytes exposed to

six solutions of varying osmolality were quantified. Each sample was tested in AFM

indentation experiments at four different strain-rates (i.e. 7.4, 0.74, 0.123 and 0.0123

s-1

), which were similar to the experiments conducted in the earlier part of the study

(see Chapter 4 for details).

In order to investigate the changes in mechanical properties, the thin-layer

elastic model (as shown in Chapter 4, Section 4.3) was used in this part of the study

to estimate the Young’s moduli of the living chondrocytes at each of the four

different strain-rates, namely, 7.4, 0.74, 0.123 and 0.0123 s-1

, when exposed to

hyperosmotic and hypoosmotic solutions (see Chapters 3 and 4 for more details of

the AFM set-up and the theoretical model). The measured results are shown in Table

6-2 and Figure 6.7.

Firstly, it is interesting to note that the single living chondrocytes also

exhibited strain-rate dependent mechanical deformation behaviour when subjected to

hyperosmotic and hypoosmotic solutions, whereby the stiffness of the cells reduced

when the rate of loading decreased (see Table 6-2). This finding suggests that the

strain-rate dependent behaviour of the cells is consistent with varying biochemical

conditions and plays an important role in cellular response. This study is to

investigate the mechanical properties of chondrocytes at varying rates of loading and

varying extracellular osmotic environments.


As presented in Figure 6.7, the living chondrocytes expressed similar Young’s

modulus changes and behaviour at each strain-rate when exposed to varying osmotic

environments. When the cells were subjected to hypoosmotic solutions (i.e. 30 and

100 mOsm), the stiffness of the chondrocytes reduced significantly compared to the

chondrocytes in the control condition (i.e. 300 mOsm) (p < 0.05, Table 6-2) at all

strain-rates tested. In addition, the stiffness of the single chondrocytes significantly

reduced when exposed to the hypoosmotic solution of 30 mOsm compared to the

stiffness when exposed to another hypoosmotic solution (i.e. 100 mOsm). As

presented in the previous section and Figure 6.5, it can be concluded that the

dissociation and redistribution of the actin filament network might explain the

reduction of the chondrocytes’ stiffness.

Figure 6.7: Young’s moduli of chondrocytes at four different strain-rates (7.4, 0.74,

0.123 and 0.0123 s-1

) when exposed to varying osmotic environments (30, 100, 300,

450, 900 and 3,000 mOsm)

On the other hand, the chondrocytes exhibited more complicated mechanical

properties when exposed to the hyperosmotic solutions. The cells did not show

127


significant difference in elastic modulus when the environment osmolality changed

from 300 to 900 mOsm (see Figure 6.7). These results are consistent with those

reported in previous research (Guilak, Erickson and Ting-Beall 2002). Guilak et al.

concluded that the hypoosmotic pressure significantly reduced the elastic modulus of

single living chondrocytes whereas the hyperosmotic pressure did not significantly

affect the Young’s moduli of the cells compared to the isoosmotic condition. It is

noted that the maximum osmolality tested in the previous study was around 466

mOsm. In this study, the hyperosmotic pressure was increased to even higher

osmolality (around 900 and 3,000 mOsm). It is interesting to note that the

hyperosmotic pressure did not have a significant effect on the chondrocytes at up to

900 mOsm. As discussed in the previous section, the chondrocytes exhibited a

similar actin filament structure at these hypoosmotic conditions as in the isoosmotic

condition, leading to insignificant changes in the elastic modulus of the cells.

Table 6-2: Young’s modulus (Pa) of chondrocytes exposed to 30, 100, 300, 450, 900

and 3,000 mOsm solutions at four different strain-rates (7.4, 0.74, 0.123 and 0.0123

s-1

)

7.4 s-1

0.74 s-1

0.123 s-1

0.0123 s-1

30 mOsm

(n = 30)

367.70 ±

318.14*

301.33 ±

309.63*

225.27 ±

214.91*

156.11 ±

154.42*

100 mOsm

(n = 42)

1,078.22 ±

637.49*

711.25 ±

566.56*

537.63 ±

379.00*

392.76 ±

236.41*

300 mOsm

(n = 43)

1,641.55 ±

889.56

1,215.52 ±

822.26

944.13 ±

704.17

628.89 ±

493.35

450 mOsm

(n = 37)

1,710.68 ±

1,429.43

1,163.40 ±

988.46

822.49 ±

738.93

643.46 ±

564.85

900 mOsm

(n = 30)

1,729.81 ±

1,121.49

1,288.22 ±

912.17

985.38 ±

851.06

672.10 ±

604.02

3,000 mOsm

(n = 30)

2,804.76 ±

2,648.00*

2,275.53 ±

2,395.30*

1,901.17 ±

2,191.66*

1,805.65 ±

2,041.68*

* p < 0.05 indicated that the Young’s modulus of the chondrocytes significantly changed when the cell was exposed to varying

osmotic pressure conditions


The living chondrocytes’ stiffness, however, was significantly increased when

the cells were subjected to the highest solution osmolality (3,000 mOsm) in this

study (p < 0.05, Table 6-2). As presented in the previous section, the alteration of the

cellular CSK network in this hyperosmotic pressure condition might be the

explanation for an increase in the Young’s moduli of the chondrocytes. These

findings suggest that the actin filament conformation may reflect the changes in the

chondrocytes’ mechanical properties. This is an interesting and novel finding which

will be investigated in our future work to better understand the mechanisms leading

to this behaviour. Based on the results reported in this section, it can be concluded

that the extracellular osmotic pressure significantly alters the elastic stiffness of

single living chondrocytes.

6.3.4.2 PHE analysis of strain-rate dependent mechanical behaviour of single

living chondrocytes exposed to varying extracellular osmotic pressure

conditions

In this section, the effect of extracellular osmotic pressure on the PHE material

parameters (especially the hydraulic permeability) of the single chondrocytes is

investigated. Moeendarbary et al. (Moeendarbary et al. 2013) reported that the

poroelastic diffusion constant of the cells decreased with decreases in the fluid

volume fraction. It is hypothesised that the hydraulic permeability of single living

chondrocytes also changes when exposed to varying osmotic pressure conditions.

Therefore, the PHE model coupled with the inverse FEA technique was also applied

in this study to investigate the dependence of the hydraulic permeability of

chondrocytes on extracellular osmotic pressure.

As discussed in the previous section, the chondrocytes’ properties significantly

changed when the cells were exposed to all the hypoosmotic solutions tested

compared to their properties in the isoosmotic condition. Only one hyperosmotic

solution (3,000 mOsm), however, affected the cells’ properties. As a result, for

simplification, only four solutions, comprising two hypoosmotic solutions (i.e. 30

and 100 mOsm), one isoosmotic solution (i.e. 300 mOsm) and one hyperosmotic

solution (i.e. 3,000 mOsm) are investigated in this section without lack of generality.

The technique presented in Section 4.4.2 was applied in this part of the study to

estimate the PHE material parameters of the living chondrocytes exposed to 30, 100,

129


300 and 3,000 mOsm conditions. The AFM indentation biomechanical testing data at

four different strain-rates were used in this investigation. The diameters of the

chondrocytes presented in Section 6.3.1 and in Table 6-1 were used to develop the

FEA models of the cells shown in Figure 6.8. The chondrocytes were assumed to be

spherical at four different osmotic solutions because the differences between

diameters and heights of the cells are negligibly small.

(a) (b)

(c) (d)

Figure 6.8: FEA models of single chondrocytes exposed to (a) 30, (b) 100, (c) 300,

and (d) 3,000 mOsm solutions


Table 6-3 presents the PHE material parameters of the chondrocytes when

exposed to four different osmotic solutions. It was observed that the C1 values

increased with increasing solution osmolality. This finding suggested that the

instantaneous modulus of the living chondrocytes was altered when the cell was

exposed to varying osmotic pressure conditions, which was similar to the results at

the highest strain-rate (i.e. 7.4 s-1

) reported in previous section. Moreover, it is

interesting to note that the hydraulic permeability of the chondrocytes was

significantly increased when the cells were exposed to the hypoosmotic solutions

(i.e. 30 and 100 mOsm) compared to the isoosmotic condition. These findings are

consistent with those reported in a previous investigation (Moeendarbary et al. 2013)

in which the authors reported that the diffusion constant increased when the

intracellular fluid volume fraction increased. In contrast, the chondrocytes did not

experience a significant change in hydraulic permeability when exposed to the

hyperosmotic solution (i.e. 3,000 mOsm) (p = 0.786). As presented in the previous

section (Section 6.3.4) and above in this section, it can be concluded that a volume

increase of chondrocytes increases the hydraulic permeability but reduces the

Young’s modulus and that a volume decrease of chondrocytes leads to an increase of

the Young’s modulus and unchanged hydraulic permeability.

Table 6-3: PHE material parameters of living chondrocytes when exposed to four

varying extracellular osmotic pressure conditions

Osmolality C1 (Pa) D1 (10

-3

1/Pa)

Initial permeability

k0 (109 µm

4/N.s)

Initial void

ratio e0

30 mOsm 181.56 ±

148.21

129.00 ±

201.00 102.54 ± 140.53* 4

100 mOsm 584.67 ±

253.87

29.40 ±

38.40 63.53 ± 87.96* 4

300 mOsm 706.60 ±

384.70

17.50 ±

17.80 20.90 ± 22.00 4

3,000 mOsm 1,483.80 ±

1,348.10

17.60 ±

33.20 18.76 ± 39.07 4

* p < 0.05 indicated that the hydraulic permeability of the living chondrocytes was significantly increased when exposed to the

hypoosmotic solutions compared to the isoosmotic condition.

131


Figure 6.9 presents the AFM experimental data at four strain-rates and the PHE

simulation results of typical chondrocytes when exposed to four different osmotic

solutions. It can be seen that the PHE model was able to effectively capture the

consolidation-dependent behaviour of the chondrocytes when exposed to varying

extracellular osmotic pressure conditions. Thus, it can be concluded again that the

PHE constitutive model is a promising constitutive model to simulate the strain-rate

dependent properties and other behaviour of single cells.

Figure 6.9: Experimental and PHE force–indentation curves of typical single living

chondrocytes at four different strain-rates when exposed to four varying osmotic

pressure conditions (i.e. 30, 100, 300 and 3,000 mOsm)

6.3.5 Dependency of relaxation behaviour of single chondrocytes on varying

extracellular osmotic pressure conditions

In this section, the relaxation behaviour of single living chondrocytes when exposed

to different osmotic pressure conditions is investigated. Similar to the procedures

reported in Chapter 5, the AFM stress–relaxation testing was conducted on

chondrocytes at different osmolality for 60 seconds after the indentation, and the

force–time curves were extracted.


Similar to the procedure reported in previous section, only four osmotic

solutions were tested for simplification. For each solution osmolality condition, the

AFM experiments were conducted at four different rates of loading, namely, 7.4,

0.74, 0.123 and 0.0123 s-1

, in order to investigate the effect of the solution osmolality

on the relaxation behaviour of the single living chondrocytes.

The AFM set-up and AFM experimental diagram were the same as presented

in Chapter 3 (Section 3.2) and Chapter 5 (Figure 5.1), respectively. The resulting

experimental data were then post-processed using SPIP 6.2.8 software (Image

Metrology A/S, Denmark). Next, the acquired force–time curves were fitted with the

thin-layer viscoelastic model (i.e. Equation (5.6) presented in Chapter 5, Section 5.3)

in order to determine the viscoelastic properties of the chondrocytes at each of the

four osmotic solutions when subjected to different strain-rates. Even though it was

found earlier in the study that this model cannot give good results at high strain-rates,

it still helped us to investigate the effect of osmotic pressure on the chondrocytes’

relaxation behaviour.

6.3.5.1 Comparison of the equilibrium moduli of chondrocytes when exposed

to solutions of varying osmolality

Similar to the procedure reported in Chapter 4, the equilibrium moduli of single

living chondrocytes at varying osmotic pressure conditions when subjected to

different rates of loading were determined using the force–indentation data at the end

of the 60 seconds relaxation to curve fit with Equation (4.6). The results are shown in

Figure 6.10 and Table 6-4. Firstly, the equilibrium moduli Eequil of the chondrocytes

were compared among the strain-rates tested in each of the solutions. The one-way

ANOVA statistical analysis was conducted, and the results showed that there were

no significant changes of the Young’s moduli when the rate of loading varied for

each of the four solutions tested. Therefore, similar to the previous finding, it can be

concluded that the chondrocytes respond differently to varying rates of loading but

always return to the same long-term or equilibrium condition even when the cells are

under different extracellular osmotic pressure conditions.

133


Table 6-4: Equilibrium moduli 𝐸𝑒𝑞𝑢𝑖𝑙 (Pa) and 𝐸𝑌/𝐸𝑅 ratios of living chondrocytes at

four different osmotic pressure conditions subjected to varying rates of loading (7.4,

0.74, 0.123 and 0.0123 s-1

) (the data are shown as mean ± standard deviation)

30 mOsm 100 mOsm 300 mOsm 3,000 mOsm

𝐸𝑒𝑞𝑢𝑖𝑙

(Pa)

7.4 s-1

22.95 ±

53.86*

224.34 ±

409.51*

470.80 ±

435.51

540.66 ±

836.16

0.74 s-1

39.93 ±

67.07*

390.19 ±

483.32

482.59 ±

388.10

844.82 ±

1,061.21*

0.123 s-1

56.91 ±

72.35*

401.01 ±

386.57

514.68 ±

467.87

791.22 ±

855.50

0.0123 s-1

67.38 ±

74.10*

426.00 ±

464.50

457.99 ±

490.16

936.21 ±

1,142.58*

𝑙𝑜𝑔(𝐸𝑌

/𝐸𝑅)

7.4 s-1

1.16 ± 0.55** 1.16 ± 0.54** 0.70 ± 0.26 0.74 ± 0.29

0.74 s-1

0.90 ± 0.40** 0.71 ± 0.30** 0.53 ± 0.25 0.57 ± 0.29

0.123 s-1

0.65 ± 0.39** 0.51 ± 0.19** 0.42 ± 0.15 0.50± 0.28

0.0123 s-1

0.42 ± 0.31 0.41 ± 0.28 0.36 ± 0.27 0.33 ± 0.27

* p < 0.05 indicated that the equilibrium modulus of the living chondrocytes was significantly reduced when exposed to

hypoosmotic and hyperosmotic conditions compared to isoosmotic condition.

** p < 0.05 indicated that the EY

/ER

ratio of the living chondrocytes was significantly increased when exposed to hypoosmotic

conditions compared to isoosmotic condition.

Secondly, the equilibrium moduli of the chondrocytes when exposed to varying

osmotic pressure conditions were compared with each other at each of the four

strain-rates in order to study the effect of osmotic pressure on changes in the

mechanical properties of living chondrocytes. The one-way ANOVA statistical

analysis was also conducted to investigate whether the equilibrium modulus was

significantly changed. It was observed from the results presented in Figure 6.10 that

when the chondrocytes were exposed to hypoosmotic pressure conditions, the

equilibrium modulus was significantly reduced, especially at the lowest osmolality

condition. When subjected to 100 mOsm, the equilibrium modulus decreased

significantly only at the highest strain-rate (i.e. 7.4 s-1

) compared to the response at

the control condition. However, when the extracellular osmolality was further

decreased to 30 mOsm, the chondrocytes expressed a significant reduction of


equilibrium modulus (p < 0.05) compared not only to the 100 mOsm hypoosmotic

condition but also to the isoosmotic condition of 300 mOsm. This can be explained

by the fact that the chondrocytes swell or the fluid volume fraction is larger when the

cells are exposed to the hypoosmotic condition, thereby the cells are softer at the

equilibrium condition.

Figure 6.10: Equilibrium modulus Eequil (Pa) of single living chondrocytes at varying

extracellular osmolality, namely, 30 and 100 mOsm (hypoosmotic condition), 300

mOsm (isoosmotic condition) and 3,000 mOsm (hyperosmotic condition) when

subjected to different strain-rates (7.4, 0.74, 0.123 and 0.0123 s-1

) (the data are

shown as mean ± standard deviation; *p < 0.05 indicated the significant difference of

the equilibrium modulus in the osmotic pressure conditions compared to the control

condition)

On the other hand, when the cells were subjected to the hyperosmotic pressure

condition (3,000 mOsm in this study), the equilibrium moduli significantly increased

at 0.74 and 0.0123 s-1

strain-rates, corresponding to p < 0.05 using the ANOVA

analysis. Therefore, it can be concluded that the osmotic pressure, in either

hyperosmotic or hypoosmotic conditions, significantly alters the equilibrium

135


modulus of single living chondrocytes. These findings also suggest that the

intracellular fluid plays an important role in the chondrocytes’ properties which

might be altered with changes of extracellular environment. In order to gain a better

understanding, the Young’s and relaxation modulus ratios EY/ER of the cells at four

different osmotic pressure conditions subjected to four varying rates of loading were

determined, as shown in Figure 6.11 and Table 6-4. The ratios are shown in

logarithmic values for clearer illustration. These ratios of the chondrocytes at

hypoosmotic and hyperosmotic conditions corresponding to each of the four strain-

rates were compared with those at the isoosmotic condition using the ANOVA

analysis.

Figure 6.11: 𝐸𝑌/𝐸𝑅 ratios of single living chondrocytes at varying extracellular

osmolality, namely, 30 and 100 mOsm (hypoosmotic condition), 300 mOsm

(isoosmotic condition) and 3,000 mOsm (hyperosmotic condition) when subjected to

different strain-rates (7.4, 0.74, 0.123 and 0.0123 s-1

) (the data are shown as mean ±

standard deviation; *p < 0.05 indicated the significant difference of the ratios in the

osmotic pressure conditions compared to the control condition)


It was observed that these ratios did not significantly change when the

chondrocytes were exposed to the hyperosmotic solution (i.e. 3,000 mOsm). This

was due to both the Young’s and equilibrium moduli increasing, leading to the ratios

remaining unchanged. On the other hand, when the cells were exposed to the

hypoosmotic solutions (i.e. 30 and 100 mOsm), the ratios significantly increased (p <

0.05) (see Figure 6.11 and Table 6-4). This is an interesting finding because both the

Young’s moduli (see Section 6.3.3) and the equilibrium moduli of the living

chondrocytes reduced, whereas their ratios increased when exposed to hypoosmotic

pressure conditions. This finding suggests that the rate of softening of chondrocytes

is significantly affected when the cells are exposed to a hypoosmotic solution. On the

other hand, hyperosmotic pressure does not influence the softening behaviour of

living chondrocytes. Additionally, it was observed from the results in Table 6-4 that

the 𝐸𝑌/𝐸𝑅 ratios of the living chondrocytes at all four solution osmolality reduced

with decreased strain-rates. From the results presented in this part of the study, it can

be concluded that the extracellular osmotic pressure significantly influences not only

the morphology but also the mechanical properties of single living chondrocytes.

6.3.5.2 Viscoelastic properties of single chondrocytes exposed to different

osmotic solutions

Similar to the procedure reported in Chapter 5, in order to investigate the relaxation

behaviour of single living chondrocytes, the AFM stress–relaxation experimental

data were fitted with the thin-layer viscoelastic theoretical model (Section 5.3,

Equation (5.6)) to determine the viscoelastic parameters. The viscoelastic parameters

are the relaxation modulus, ER (Pa), and the relaxation times under constant load and

deformation, 𝜏𝜎 (s) and 𝜏 (s), respectively, from which other parameters were then

calculated using Equations (5.7)–(5.9). These are the parameters to characterise the

viscoelastic properties of living chondrocytes at varying osmotic pressure conditions.

In each of the four solutions, namely 30, 100, 300 and 3,000 mOsm, these parameters

of the chondrocytes were determined at four varying rates of loading, namely, 7.4,

0.74, 0.123 and 0.0123 s-1

. The results are shown in Table 6-5 and Figure 6.12–6.13

for the four different strain-rates. The asterisk in these figures indicates the

significant difference in a viscoelastic property in one osmotic solution compared to

the other solutions (p < 0.05).

137


At the highest strain-rate (i.e. 7.4 s-1

), the viscoelastic parameters of the single

living chondrocytes including the relaxation modulus ER (Pa), relaxation time under

constant load, 𝜏𝜎 (s) and deformation, 𝜏 (s), instantaneous modulus E0 (Pa) and

viscosity μ (log Pa.s) at four varying osmotic pressures are shown in Figure 6.12. It

was observed that all the viscoelastic properties, except the relaxation time, of the

living chondrocytes under constant load were unchanged when exposed to the 100

mOsm hypoosmotic solution. The relaxation modulus, instantaneous modulus and

relaxation time under constant load and viscosity, however, were significantly

reduced when exposed to the 30 mOsm hypoosmotic pressure compared to both the

isoosmotic condition and the 100 mOsm hypoosmotic solution. These findings are

similar to those previously reported by Guilak et al. (Guilak, Erickson and Ting-

Beall 2002) in which the viscoelastic properties of chondrocytes were revealed to be

significantly changed when exposed to 153 mOsm solution.

On the other hand, when the chondrocytes were exposed to hyperosmotic

pressure (i.e. 3,000 mOsm), both the relaxation times and viscosity were significantly

increased compared to those at the isoosmotic condition. This finding is in contrast

with the findings in previously reported work (Guilak, Erickson and Ting-Beall

2002) in which the viscoelastic properties of chondrocytes were determined to be

unchanged when exposed to hyperosmotic pressure. This might be because the

previous authors tested a hyperosmotic solution of only 466 mOsm compared to

3,000 mOsm in this study.

For a strain-rate of 0.74 s-1

, the viscoelastic parameters of the single living

chondrocytes at four varying osmotic pressure conditions are shown in Figure 6.13

above. It was observed that the viscoelastic properties of the living chondrocytes at

hypoosmotic solutions exhibited similar changes to those at the strain-rate of 7.4 s-1

.

However, when the chondrocytes were exposed to hyperosmotic pressure, besides

the relaxation times, both the relaxation and instantaneous moduli were also

significantly increased whereas these moduli were insignificantly increased at the 7.4

s-1

strain-rate. In addition, the viscosity of the cells was only slightly increased (not a

statistically significant difference) at this strain-rate compared to a significant

increase at the 7.4 s-1

strain-rate.


Table 6-5: Viscoelastic parameters, namely, relaxation modulus ER (Pa), relaxation

times under constant load 𝜏𝜎 (s), deformation 𝜏 (s), and viscosity μ (log Pa.s) of

living chondrocytes at four different osmotic pressure conditions subjected to

varying rates of loading (the data are shown as mean ± standard deviation)


𝐸𝑅 (Pa)

7.4 s-1

38.54 ±

44.58

238.65 ±

324.18

370.09 ±

339.38

513.88 ±

689.53

0.74 s-1

51.57 ±

49.75

286.34 ±

340.95

351.79 ±

253.62

798.81 ±

982.30

0.123 s-1

55.27 ±

55.98

309.93 ±

326.11

343.28 ±

239.89

935.01 ±

1,224.22

0.0123 s-1

61.08 ±

65.34

270.76 ±

261.07

291.22 ±

197.78

900.41 ±

1,162.75

𝜏𝜎 (s)

7.4 s-1

24.41 ±

36.33

19.66 ±

46.07

3.37 ±

1.75

6.37 ±

6.77

0.74 s-1

13.15 ±

20.31

4.07 ±

3.44

3.85 ±

4.31

7.52 ±

9.96

0.123 s-1

32.79 ±

68.18

4.66 ±

4.90

4.27 ±

2.42

32.94 ±

117.99

0.0123 s-1

29.37 ±

52.44

7.55 ±

12.68

5.18 ±

4.84

29.50 ±

111.12

𝜏 (s)

7.4 s-1

2.55 ± 3.57 1.79 ± 3.509 1.45 ± 0.80 2.63 ± 3.15

0.74 s-1

2.13 ± 1.81 1.45 ± 1.17 1.71 ± 4.31 4.00 ± 6.04

0.123 s-1

4.44 ± 4.14 2.14 ± 2.00 2.36 ± 0.98 4.53 ± 4.76

0.0123 s-1

7.18 ± 6.60 2.80 ± 1.51 3.28 ± 2.37 5.55 ± 6.85

139


(Continued) Table 6-5: Viscoelastic parameters, namely, relaxation modulus

ER (Pa), relaxation times under constant load 𝜏𝜎 (s), deformation 𝜏 (s), and viscosity

μ (log Pa.s) of living chondrocytes at four different osmotic pressure conditions

subjected to varying rates of loading (the data are shown as mean ± standard

deviation)


𝐸0 (Pa)

7.4 s-1

122.17 ±

88.25

782.16 ±

700.31

748.12 ±

473.19

1,055.97 ±

1,205.67

0.74 s-1

182.70 ±

128.42

623.00 ±

535.50

622.33 ±

349.11

1,296.37 ±

1,347.27

0.123 s-1

129.86 ±

96.65

536.69 ±

524.12

572.47 ±

355.35

1,278.32 ±

1,393.21

0.0123 s-1

106.48 ±

94.84

461.64 ±

364.59

434.42 ±

301.63

972.96 ±

1,077.63

μ (log

Pa.s)

7.4 s-1

2.10 ± 0.39 2.43 ± 0.46 2.55 ± 0.41 2.79 ± 0.58

0.74 s-1

2.21 ± 0.46 2.30 ± 0.69 2.53 ± 0.37 2.77 ± 0.81

0.123 s-1

2.24 ± 0.56 2.34 ± 0.62 2.57 ± 0.42 2.95 ± 0.74

0.0123 s-1

2.14 ± 0.71 2.50 ± 0.46 2.39 ± 0.53 2.69 ± 0.93

The viscoelastic parameters of the single living chondrocytes at four varying

osmotic pressure conditions when subjected to the 0.123 s-1

strain-rate were shown in

Figure 6.14 above. It was observed that the viscoelastic properties including the

relaxation and instantaneous moduli, the relaxation time under constant load and the

viscosity of the living chondrocytes at hypoosmotic solutions exhibited similar

changes to those at the strain-rates of 7.4 and 0.74 s-1

. However, it is interesting to

note that the relaxation time under constant deformation was also significantly

increased when the cells were exposed to the 30 mOsm hypoosmotic pressure

compared to both the isoosmotic condition and the 100 mOsm hypoosmotic solution.

Furthermore, it is worth noting that all the viscoelastic parameters of the living

chondrocytes were significantly larger when the cells were subjected to the

hyperosmotic solution compared to the isoosmotic condition.



times under constant load 𝜏𝜎 (s), deformation 𝜏 (s), instantaneous modulus E0 (Pa),

and viscosity μ (log Pa.s) of single living chondrocytes at varying extracellular

osmolality – including 30 and 100 mOsm (hypoosmotic condition), 300 mOsm


7.4 s-1

strain-rate (the data are shown as mean ± standard deviation; *p < 0.05

indicated the significant difference in the viscoelastic parameters at the osmotic

pressure conditions compared to other conditions)

141







0.74 s-1










0.123 s-1




143







0.0123 s-1





Finally, for the lowest strain-rate of 0.0123 s-1

, the viscoelastic parameters of

the living chondrocytes when exposed to four different osmotic solutions were

shown in Figure 6.15 above. It was observed that the relaxation and instantaneous

moduli and the viscosity were significantly reduced whereas the relaxation times

under constant load and the deformation of the living chondrocytes were

significantly increased, when the cells were exposed to hypoosmotic solutions

compared to the control condition. These changes in the viscoelastic properties’ were

similar to those observed at the strain-rate of 0.123 s-1

as reported above. When the

chondrocytes were exposed to hyperosmotic extracellular stress, only the relaxation

and instantaneous moduli were significantly increased at this strain-rate.

In summary, it was observed that the relaxation and instantaneous moduli of

the single living chondrocytes were significantly reduced at all the tested strain-rates

when the cells were exposed to the hypoosmotic solutions. In contrast, these moduli

were greatly increased when the cells were exposed to the hyperosmotic solution.

These findings suggest that the stiffness of the cells is influenced by extracellular

osmotic pressure at all the strain-rates tested in this study. On the other hand, the

relaxation time under constant deformation and viscosity showed significant changes

only at some strain-rates when the cells were exposed to either the hypoosmotic

condition or the hyperosmotic condition. The results reveal the important role of

intracellular fluid in influencing single cells’ properties and behaviour, and that the

relaxation behaviour of chondrocytes is altered when the cells are exposed to varying

extracellular osmotic pressure conditions.

Furthermore, as observed in the results reported in Table 6-5, the relaxation

moduli ER of the chondrocytes were unchanged with decreasing strain-rates, which

was similar to the behaviour of the equilibrium moduli Eequil (see Section 6.3.5.1).

Additionally, the instantaneous moduli E0 of the cells were reduced with decreasing

strain-rates at all four osmolality. However, when the chondrocytes were exposed to

30 and 3,000 mOsm at the highest strain-rate (i.e. 7.4 s-1

), these moduli were smaller

than those at the 0.74 s-1

strain-rate. This might be due to the limitations of the thin-

layer viscoelastic model as already discussed in Chapter 5 (Section 5.5.2).

Finally, it can be concluded that both the mechanical deformation and the

relaxation behaviour of single living chondrocytes are significantly influenced by

their physicochemical environment. Furthermore, it is hypothesised in the literature

145


that besides the mechanical deformation of articular cartilage tissue, the changes in

the osmotic environment of chondrocytes in situ due to the compression of the

extracellular matrix might have a significant influence on the cellular CSK network

structure and the mechanical properties of the cells (Guilak, Erickson and Ting-Beall

2002; Guilak and Mow 2000).

Figure 6.16 shows the AFM stress–relaxation experimental data of the living

chondrocytes at four different strain-rates for each of the four solutions and their

corresponding fitted thin-layer viscoelastic model curves. The data were averaged

from the results of all the cells tested and are shown as mean ± standard deviation. It

is noted that the model is able to capture the relaxation behaviour of living

chondrocytes especially at low strain-rates. It is also noted that there are two phases

in the relaxation behaviour of the chondrocytes, namely, a transient phase and an

equilibrium phase. These findings are similar to those presented in Chapter 5.

Figure 6.16: Relaxation experimental data and thin-layer viscoelastic model fitted

curves of living chondrocytes subjected to varying rates of loading (7.4, 0.74, 0.123

and 0.0123 s-1

) when exposed to four different osmotic pressure conditions (i.e. 30,

100, 300 and 3,000 mOsm (the data are shown as mean ± standard deviation)


6.4 CONCLUSIONS

In this chapter load was applied with the AFM, and the force–indentation and force–

time characteristics under the various strain-rates were logged in order to investigate

the mechanical deformation and relaxation behaviour of chondrocytes when exposed

to different extracellular osmotic pressures including hypoosmotic, isoosmotic and

hyperosmotic solutions. The thin-layer elastic and viscoelastic models (as presented

in Chapter 4, Section 4.3 and Chapter 5, Section 5.3) were applied to determine the

mechanical elastic and viscoelastic properties of single living chondrocytes at four

different strain-rates for each of the osmotic solutions tested. Several conclusions

were drawn as follows:

The results in this study revealed that the hypoosmotic pressure increased

the diameter, height and volume of the living chondrocytes and the

hyperosmotic pressure reduced the diameter, height and volume of the

living chondrocytes. Based on the confocal images of the chondrocytes, it

was also found that the solution osmolality altered the actin filament

network structure of the chondrocytes. These results suggest that the

extracellular osmotic pressure affects the morphology of living

chondrocytes. Moreover, the volume fraction of the osmotically active

intracellular water relative to the cell volume was determined to be

54.07%, which is similar to the results that published in the literature.

By using the AFM indentation testing at four different strain-rates (similar

to those presented in Chapter 4), the changes in the mechanical elastic

properties of the chondrocytes when subjected to six osmotic solutions

(comprising two hypoosmotic, one isoosmotic and three hyperosmotic

solutions) were investigated. The thin-layer elastic model was applied to

determine the Young’s modulus of the single chondrocytes for each case.

The results showed that both hypoosmotic extracellular osmotic pressure

conditions caused a significant reduction in the chondrocyte stiffness.

These results are in line with those previously reported (Guilak, Erickson

and Ting-Beall 2002). The elastic property of the chondrocytes, however,

exhibited a more complicated trend when the cells were exposed to

hyperosmotic solutions. The chondrocytes did not show significant change

in Young’s modulus when exposed up to 900 mOsm, which is consistent

147


with Guilak et al.’s results (Guilak, Erickson and Ting-Beall 2002).

However, when the osmolality was increased to 3,000 mOsm, the

chondrocytes’ elastic moduli were significantly increased. To the best of

our knowledge, this is an interesting result that has not been published to

date. It might be due to the significant change in the cellular actin filament

network at this solution compared to the other hyperosmotic solutions.

The PHE model was used to study the effect of extracellular osmotic

pressure on the PHE material parameters of chondrocytes, especially the

hydraulic permeability. As discussed above, only the solution of 3,000

mOsm affected the chondrocytes’ properties; thus, this was the only

hyperosmotic pressure condition considered in this investigation. It was

found that the hypoosmotic pressure reduced the elastic stiffness and

increased the hydraulic permeability, whereas the hyperosmotic pressure

increased the elastic stiffness and kept the hydraulic permeability of

chondrocytes unchanged. This might have been due to the changes in the

intracellular fluid volume fraction when the cells were exposed to different

solution osmolality.

It was found that the PHE model can accurately capture the consolidation-

dependent behaviour of both living and fixed cells. Therefore, it is

reported that the PHE model is a suitable mechanical constitutive model

for single cell mechanics.

Based on the AFM stress–relaxation testing, the relaxation behaviour of

the living chondrocytes when exposed to four osmotic solutions

(comprising two hypoosmotic solutions, one isoosmotic solution and one

hyperosmotic solution) was investigated. The thin-layer viscoelastic model

was utilised to extract the viscoelastic properties of the single living

chondrocytes for each case. It was found that the hypoosmotic pressure

significantly affected most of the viscoelastic properties of the

chondrocytes at all four strain-rates. On the other hand, the chondrocytes’

relaxation behaviour was significantly influenced at only some strain-rates

when exposed to the hyperosmotic solution. This is in contrast with

previous published results (Guilak, Erickson and Ting-Beall 2002), which


might be due to the much higher solution osmolality tested in this study

compared to the solution osmolality tested in the previous work.

These findings suggest that the extracellular osmotic pressure which is

either hypoosmotic or hyperosmotic significantly alters not only the

morphology but also the mechanical properties of single living

chondrocytes indicating the important role of the intracellular fluid in the

cells. In addition, it is hypothesised that the change in the osmotic

environment of chondrocytes in situ caused by the compression of the

cartilage extracellular matrix might influence the cellular actin filament

structure and the mechanical properties of the cells.

Chapter 7:Conclusion 149

Chapter 7: Conclusion

7.1 CONCLUSION

7.1.1 General conclusions

In this research, the mechanical properties of single living cells were investigated

using both experiments and numerical modelling. Some of the general conclusions

are summarised in this section.

The single cells investigated in this study exhibited strain-rate dependent

mechanical properties that are similar to those observed in other fluid-filled

biological tissue, such as articular cartilage (Nguyen 2005; Oloyede and Broom

1993b, 1994b, 1996; Oloyede, Flachsmann and Broom 1992) and large arteries

(Simon, Kaufmann, McAfee and Baldwin 1998; Geest et al. 2011).

A number of models were considered in this study, including the thin-layer

viscoelastic and PHE models, to study the relaxation behaviour of living cells. From

the experimental and simulation results reported in this study, it was found that the

thin-layer viscoelastic model gives good results only at low strain-rates whereas the

PHE model provides good results at high strain-rates and reasonable results at low

strain-rates. The PHE model can also precisely capture the mechanical deformation

behaviour of single cells. Therefore, this model is likely a promising model for single

cell mechanics studies. It should be studied in order to further improve its

performance, and as such, be able to consider other behaviour such as swelling

behaviour, mass transport, etc.

The methodology proposed in this study provides appropriate mechanical

models for investigating the mechanical properties of single cells subjected to

various mechanical stimuli. The investigation of the behaviour of single cells in

varying conditions (including the magnitude and rate of loading) helps to elucidate

the deformation mechanisms underlying cellular responses to external mechanical

loadings and the process of mechanotransduction in living cells.

150 Chapter 7:Conclusion

Moreover, the adhesiveness of single living cells is also of our interest. It is

hypothesised that different cell types have different adhesive strength and that the

external stimuli (e.g. varying extracellular environment) and/or the diseases (e.g.

cancers) may alter the adhesiveness of the cells. Thus, a comprehensive investigation

of mechanical adhesiveness of living cells will be conducted in our future studies.

7.1.2 Detailed conclusions

The research in this thesis focused on three main areas: the strain-rate dependent

mechanical deformation behaviour of single cells, the strain-rate dependent

relaxation behaviour of single cells, and the effect of extracellular osmotic pressure

on the morphology and mechanical properties of single living chondrocytes. The

conclusions drawn in each of these areas are summarised as follows:

1. Strain-rate dependent mechanical deformation behaviour of single cells

The mechanical properties of single cells: The results demonstrated that

all the tested cell types responded similarly with respect to the patterns of

their force–indentation curves, whereby the cells’ elastic stiffness reduced

with decreasing strain-rates. Moreover, it was found that the thin-layer

model which can account for sample thickness can capture AFM

indentation data very well.

Exploration of the mechanisms underlying the strain-rate dependent

mechanical deformation behaviour: Comparing the mechanical

behaviour of living and fixed cells, it was found that the intracellular fluid

effect was predominant at high strain-rates. On the other hand, at relatively

low strain-rates compared to the impact velocity, the fluid exited the

matrix gradually over time with the result that the cellular CSK was able to

reorganise, unbind its cross-linkers and deform, leading to a time-

dependent non-linear stiffness that was lower than that produced at loading

velocity close to impact. Thus, it is reported in this study that the

intracellular fluid governs the cellular mechanical behaviour at high strain-

rates, whereas the cellular CSK network plays a dominant role in the

mechanical behaviour of single cells at relatively low strain-rates.

Combined PHE model and inverse FEA technique to simulate strain-

rate dependent mechanical deformation behaviour of single living

151


cells: The results revealed that the instantaneous modulus and hydraulic

permeability of the living osteocytes and osteoblasts in this study were

larger than those of the living chondrocytes. Moreover, the hydraulic

permeability of the living cells was significantly larger than that of the

fixed cells, leading to the conclusion that the cellular CSK network of

living cells might alter its structure during deformation to help the

intracellular fluid be distributed and exuded easily within the cells.

2. Strain-rate dependent relaxation behaviour of single cells

AFM stress–relaxation experimental data investigation: The AFM

experimental results showed that two phases, referred to as the transient

and equilibrium phases in this study, occur during the relaxation of single

cells. In the transient phase, the applied force reduces dramatically from its

maximum value immediately after the indentation. This is followed by a

gradual reduction of the applied force in the equilibrium phase.

The relaxation behaviour of living cells: By fitting the AFM relaxation

experimental data with the thin-layer viscoelastic model, it was found that

the relaxation behaviour of living cells is also dependent on the strain-rate.

However, it was observed that this model gives a good fit only when the

F0/Fequil ratio is less than 4.

Investigation of the strain-rate dependent relaxation behaviour of

living chondrocytes using the PHE model combined with the inverse

FEA method: The PHE model and PRI method were used to determine

the hydraulic permeability and poroelastic diffusion constant of living

chondrocytes, respectively. It was found that both of these parameters of

living chondrocytes reduced with decreasing strain-rates. This can be

explained by the fact that the fluid volume fraction is higher with higher

strain-rates. The PHE simulation results were also compared with the

results from the thin-layer viscoelastic model and PRI method. It was

demonstrated that the PHE model was the only model that could capture

both the transient and equilibrium phases of relaxation behaviour. The

other two models could only capture one of the two phases.


In summary, the results obtained from the first two investigations revealed that

single living cells possess strain-rate dependent elastic and viscoelastic properties

which are similar to such properties of other fluid-filled biological tissue. These

findings suggest that the consolidation-dependent deformation behaviour is

continuous from the cellular level to the tissue level.

3. Effect of extracellular osmotic pressure on the morphology and mechanical

properties of single living chondrocytes

The effect of osmotic pressure on the morphology and actin filament

structure of chondrocytes: The diameter, height and volume of the

chondrocytes at solutions of varying osmolality were measured. The

results showed that the hypoosmotic pressure increased these dimensions

of the chondrocytes (i.e. the cells were swollen) and the hyperosmotic

pressure decreased these dimensions of the chondrocytes (i.e. the cells

were decreased in size). The confocal images of the chondrocytes at six

varying osmolality were obtained to study the effect of osmotic pressure

on the actin filament structure. It was observed that the actin filament

network of the cells was significantly altered when the chondrocytes were

exposed to different solution osmolality.

The effect of osmotic pressure on the elastic stiffness of chondrocytes:

It was found that the Young’s moduli of the chondrocytes were

significantly reduced when the cells were exposed to hypoosmotic

pressure at all the strain-rates tested. On the other hand, the elastic moduli

of the chondrocytes were largely increased when the cells were exposed to

hyperosmotic pressure. It is hypothesised that the actin filament network

plays an important role in this behaviour because it is well-known that the

cellular CSK network governs the elastic property of single living cells. In

addition, the mechanical properties of the chondrocytes still include the

strain-rate dependent property when the cells are exposed to different

osmotic solutions.

Combined PHE model and inverse FEA technique to investigate the

effect of extracellular osmotic pressure on the hydraulic permeability

of the living chondrocytes: As reported, it was found that hypoosmotic

153


pressure reduced the elastic stiffness and increased the permeability of the

living chondrocytes. On the other hand, the hyperosmotic pressure

increased only the elastic stiffness of the cells. This might have been

because of the effect of the intracellular fluid, whose volume fraction is

different in solutions of varying osmolality.

The effect of osmotic pressure on the viscoelastic properties of

chondrocytes: The results revealed that the viscoelastic properties of the

living chondrocytes changed significantly when the cells were exposed to

either extracellular hypoosmotic or hyperosmotic pressure. Briefly, it can

be concluded that the osmotic environment of chondrocytes may influence

the cells’ morphology, actin filament network structure and mechanical

properties.

In summary, from the results presented in the main parts of this investigation, it

can be concluded that the PHE model can capture the strain-rate dependent

behaviour and is a suitable mechanical constitutive model for cell biomechanics. In

addition, it can be concluded that the intracellular fluid is an important factor in

governing living cells’ mechanical properties and behaviour.

7.2 RESEARCH LIMITATIONS

The following limitations in the study are noted:

In this study, the single living cells were studied over a short culture time

(i.e. 1–2 hours only). The changes in the mechanical properties of the cells

over a longer culturing time should be considered.

The PHE model used in this study did not consider the osmotic pressure

within the cells and did not consider the mass transport. Thus, this model

needs to be further improved in order to give more comprehensive results

and to consider other aspects e.g. swelling behaviour, and mass transport.

The study did not include experiments to investigate the mechanisms

underlying the different mechanical properties of different cell types.

Only healthy living cells were considered in this study. Diseased cells

should be studied and compared with normal ones in order to investigate

the effect of disease on the mechanical properties of single living cells.


7.3 FUTURE RESEARCH DIRECTIONS

7.3.1 PHE analysis

As discussed in Chapter 5 (Section 5.5.3), the PHE model was applied in this study

to investigate the relaxation behaviour of single living chondrocytes only. Thus,

future researchers could utilise this mechanical constitutive model to investigate the

changes in PHE material parameters (especially the hydraulic permeability) of other

cell types such as osteocytes and osteoblasts at varying strain-rates.

Moreover, this model could also be used in future works to determine the

strain-rate dependent relaxation behaviour of chondrocytes when exposed to different

extracellular osmotic pressure conditions and where the intracellular fluid volume

fraction is varied. This would help to characterise the important roles of solid and

fluid components within the cells as well as the solid-fluid interaction.

7.3.2 Further AFM biomechanical experiments on single cells

AFM experiments could be performed on living cells with various culture times in

order to study the changes in mechanical properties with time and morphology.

These experiments could also be conducted on defective cells in order to estimate

their mechanical properties and compared them to the properties of healthy cells in

an attempt to investigate the possible effect of disease on cell properties. Moreover,

healthy cells with some chemical treatments (such as cytochalasin and latrunculin)

could also be tested in future works in order to study the contributions of different

cellular components to the mechanical properties of living cells.

7.3.3 Mechanical adhesiveness of single osteoblasts and chondrocytes

Cell adhesion is of interest in many fields including biomaterials, cancer cell studies,

etc. (Okegawa et al. 2004; Bačáková et al. 2004; Pignatello 2013). It is well-known

that cell adhesion strength is different with different substratum material and

topography (Singhvi, Stephanopoulos and Wang 1994; Bacakova et al. 2011; Zhu et

al. 2004). A better understanding of cell adhesion would open an insight into therapy

in such as tissue engineering, and cancer treatment. As discussed in Chapter 2,

besides conducting mechanical probing testing, AFM can also be used in future

works to evaluate the detachment force of the samples. In the literature, there are

155


three strategies that have been effectively performed to characterise the adhesive

strength of single cells (as discussed in Section 2.3 and Figure 2.8 in detail).

As presented in Chapter 6 (Section 6.3.3), the adhesion distribution of single

living chondrocytes was not significantly changed when the cells were exposed to

different environment osmotic pressure conditions. However, the mechanical

adhesiveness of the cells at varying extracellular osmolality needs to be

quantitatively evaluated in order to study the effect of osmotic pressure on the

adhesion behaviour of chondrocytes. Thus, the third strategy presented in Figure

2.8(c) would be the most suitable technique in future investigations. Although this

technique has been used widely for bacterial cells, there is lack of research applying

this method to eukaryotic cells in the literature.

As discussed above, this technique may open a new avenue for investigating

the adhesion processes of single living cells. Thus, future works could:

Study the adhesive strength of different single living cell types in order to

investigate the variation of adhesiveness among cell types.

Study the adhesive strength of single living cells after varying seeding

times (2 hours, 6 hours, 24 hours, etc.).

Characterise the mechanical adhesiveness of single cells when exposed to

different extracellular osmotic pressure conditions.

Investigate the adhesion process between living cells and different

substrate materials.

Study the interaction between living cells and other proteins by seeding the

cells on different protein-coated substrates.

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Appendices

Appendix A

Statistical parameters of curve fitting of AFM experimental force–indentation

curves at four different strain-rates of a typical single living and fixed osteocyte,

osteoblast and chondrocyte cell using thin-layer elastic model

Table A-1: Statistical parameters of a typical living and fixed osteocyte cell at four

different strain-rates

Strain-rate (s-1

) Parameters Living Fixed

7.4

p-value 4.56×10-15

9.57×10-14

RMSE 0.22 1.03

R2 0.9926 0.9878

0.74

p-value 1.26×10-15

5.12×10-14

RMSE 0.11 0.73

R2 0.9951 0.9833

0.123

p-value 3.13×10-16

2.34×10-14

RMSE 0.06 0.52

R2 0.9964 0.9681

0.0123

p-value 3.19×10-16

1.56×10-15

RMSE 0.06 0.12

R2 0.9951 0.9968

175

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Table A-2: Statistical parameters of a typical living and fixed osteoblast cell at four

different strain-rates

Strain-rate (s-1


7.4

p-value 9.59×10-15

1.01×10-15

RMSE 1.23 0.41

R2 0.9889 0.9995

0.74

p-value 3.74×10-15

2.84×10-16

RMSE 0.74 0.21

R2 0.9770 0.9998

0.123

p-value 4.39×10-16

3.09×10-16

RMSE 0.26 0.22

R2 0.9924 0.9997

0.0123

p-value 1.61×10-16

6.85×10-16

RMSE 0.16 0.33

R2 0.9968 0.9991

176 Bibliography

Table A-3: Statistical parameters of a typical living and fixed chondrocyte cell at

four different strain-rates

Strain-rate (s-1


7.4

p-value 1.71×10-16

7.97×10-15

RMSE 0.31 1.36

R2 0.9988 0.9791

0.74

p-value 7.33×10-17

1.88×10-15

RMSE 0.22 1.15

R2 0.9987 0.9830

0.123

p-value 2.55×10-17

3.23×10-16

RMSE 0.14 0.48

R2 0.9991 0.9960

0.0123

p-value 9.46×10-18

6.59×10-16

RMSE 0.08 0.64

R2 0.9994 0.9876

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