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i
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
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International Conference on Computational Methods (ICCM 2012), 25-28
November 2012, Crowne Plaza, Gold Coast, QLD.
vi
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
vii
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
viii
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
ix
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-
x
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
xi
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)
xii
Figure 4.9: (a) von Mises stress, and (b) fluid pore pressure distributions of living
osteoblasts 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 4.10: (a) von Mises stress, and (b) fluid pore pressure distributions of living
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)
xiii
Figure 5.8: von Mises stress (top) and fluid pore pressure (bottom) distributions – (a)
after indentation, and (b) after relaxation phase at 0.74 s-1
strain-rate (the
measurement unit in these figures is 106 Pa)
Figure 5.9: von Mises stress (top) and fluid pore pressure (bottom) distributions – (a)
after indentation, and (b) after relaxation phase at 0.123 s-1
strain-rate (the
measurement unit in these figures is 106 Pa)
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 measurement unit in these figures is 106 Pa)
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)
Figure 6.13: 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 0.74 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)
Figure 6.14: 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 0.123 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)
Figure 6.15: 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 0.0123 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)
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
Chapter 1:Introduction 3
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
4 Chapter 1:Introduction
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
Chapter 1:Introduction 5
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.
6 Chapter 1:Introduction
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.
Chapter 1:Introduction 7
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
8 Chapter 1:Introduction
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
Chapter 1:Introduction 9
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,
Chapter 2: Literature Review 13
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
14 Chapter 2: Literature Review
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).
Chapter 2: Literature Review 15
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
16 Chapter 2: Literature Review
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
Chapter 2: Literature Review 17
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).
18 Chapter 2: Literature Review
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
Chapter 2: Literature Review 19
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
20 Chapter 2: Literature Review
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
Chapter 2: Literature Review 21
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
22 Chapter 2: Literature Review
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):
Chapter 2: Literature Review 23
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
24 Chapter 2: Literature Review
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
Chapter 2: Literature Review 25
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.
26 Chapter 2: Literature Review
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
Chapter 2: Literature Review 27
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
28 Chapter 2: Literature Review
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).
Chapter 2: Literature Review 29
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
30 Chapter 2: Literature Review
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)
Chapter 2: Literature Review 31
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
32 Chapter 2: Literature Review
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
Chapter 2: Literature Review 33
(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
34 Chapter 2: Literature Review
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.
Chapter 2: Literature Review 35
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)
36 Chapter 2: Literature Review
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
Chapter 2: Literature Review 37
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
38 Chapter 2: Literature Review
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
Chapter 2: Literature Review 39
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).
40 Chapter 2: Literature Review
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
Chapter 2: Literature Review 41
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
behaviour of single living cells.
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
42 Chapter 2: Literature Review
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
Chapter 3:Research Design 45
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
46 Chapter 3:Research Design
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
Chapter 3:Research Design 47
(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%
48 Chapter 3:Research Design
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)
Chapter 3:Research Design 49
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
50 Chapter 3:Research Design
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
Chapter 3:Research Design 51
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
52 Chapter 3:Research Design
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
Chapter 3:Research Design 53
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
54 Chapter 3:Research Design
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
Chapter 3:Research Design 55
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
56 Chapter 3:Research Design
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
behaviour of single living cells.
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)
60 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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)
62 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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
64 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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.
66 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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)
68 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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)
70 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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
72 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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
74 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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
76 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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
78 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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)
80 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
7.4 s-1
0.74 s-1
0.123 s-1
0.0123 s-1
(a)
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)
81
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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
(the measurement unit in these figures is 106 Pa)
82 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
7.4 s-1
0.74 s-1
0.123 s-1
0.0123 s-1
(a)
Figure 4.9: (a) von Mises stress, and (b) fluid pore pressure distributions of living
osteoblasts 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)
83
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 83
7.4 s-1
0.74 s-1
0.123 s-1
0.0123 s-1
(b)
(Continue) Figure 4.9: (a) von Mises stress, and (b) fluid pore pressure distributions
of living osteoblasts 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)
84 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
7.4 s-1
0.74 s-1
0.123 s-1
0.0123 s-1
(a)
Figure 4.10: (a) von Mises stress, and (b) fluid pore pressure distributions of living
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)
85
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 85
7.4 s-1
0.74 s-1
0.123 s-1
0.0123 s-1
(b)
(Continue) Figure 4.10: (a) von Mises stress, and (b) fluid pore pressure distributions
of living 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)
86 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells
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
Chapter 4: Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living Cells 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.
88 Chapter 4:Exploration of Strain-Rate Dependent Mechanical Deformation Behaviour of Single Living 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
92 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
isotropic, incompressible sample bonded to the substrate with finite thickness with a
hard, spherical tip:
𝐹(𝑡) =4𝐸𝑅
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:
𝐹(𝑡) =8𝐸𝑅
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
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 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)
94 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
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 RESULTS AND DISCUSSIONS
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
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 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.
96 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
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
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 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.
98 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different 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
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 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.
100 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
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
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 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
102 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
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
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 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
104 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different 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
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 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
106 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
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
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 107
(a) (b)
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)
108 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
Figure 5.8: von Mises stress (top) and fluid pore pressure (bottom) distributions – (a)
after indentation, and (b) after relaxation phase at 0.74 s-1
strain-rate (the
measurement unit in these figures is 106 Pa)
109
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 109
(a) (b)
Figure 5.9: von Mises stress (top) and fluid pore pressure (bottom) distributions – (a)
after indentation, and (b) after relaxation phase at 0.123 s-1
strain-rate (the
measurement unit in these figures is 106 Pa)
110 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
(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
measurement unit in these figures is 106 Pa)
111
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 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
112 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
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
Chapter 5: Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates 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
114 Chapter 5:Investigation of Stress–Relaxation Behaviour of Single Cells Subjected to Different Strain-Rates
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 RESULTS AND DISCUSSIONS
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.
118 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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
120 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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)
122 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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
124 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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.
126 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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
128 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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
130 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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.
132 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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
134 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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)
136 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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.
138 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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)
30 mOsm 100 mOsm 300 mOsm 3,000 mOsm
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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)
30 mOsm 100 mOsm 300 mOsm 3,000 mOsm
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.
140 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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 significant difference in the viscoelastic parameters at the osmotic
pressure conditions compared to other conditions)
141
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 141
Figure 6.13: 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
0.74 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)
142 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
Figure 6.14: 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
0.123 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)
143
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 143
Figure 6.15: 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
0.0123 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)
144 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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)
146 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 6: Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes 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
148 Chapter 6:Effect of Osmotic Pressure on the Morphology and Mechanical Properties of Single Chondrocytes
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
Chapter 7:Conclusion 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.
152 Chapter 7:Conclusion
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
Chapter 7:Conclusion 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.
154 Chapter 7:Conclusion
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
Chapter 7:Conclusion 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
Bibliography 175
Table A-2: Statistical parameters of a typical living and fixed osteoblast cell at four
different strain-rates
Strain-rate (s-1
) Parameters Living Fixed
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
) Parameters Living Fixed
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