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Nanobiomechanics of living cells: a review
Jinju Chen1,2
1
School of Mechanical and Systems Engineering, Newcastle University, Newcastle Upon Tyne NE1 7RU, UK
Arthritis Research UK (ARUK) Tissue Engineering Centre, Institute of Cellular Medicine, Newcastle University,
Newcastle Upon Tyne NE2 4HH, UK
2
rsfs.royalsocietypublishing.org
Review
Cite this article: Chen J. 2014 Nanobiomechanics of living cells: a review. Interface
Focus 4: 20130055.
http://dx.doi.org/10.1098/rsfs.2013.0055
One contribution of 8 to a Theme Issue
‘Nanobiomechanics of living materials’.
Subject Areas:
nanotechnology, biomechanics, biophysics
Keywords:
cell mechanics, nanobiomechanics,
nanoindentation, modelling
Author for correspondence:
Jinju Chen
e-mail: jinju.chen@ncl.ac.uk, jinju.chen82@
gmail.com
Nanobiomechanics of living cells is very important to understand cell–
materials interactions. This would potentially help to optimize the surface
design of the implanted materials and scaffold materials for tissue engineering. The nanoindentation techniques enable quantifying nanobiomechanics
of living cells, with flexibility of using indenters of different geometries. However, the data interpretation for nanoindentation of living cells is often
difficult. Despite abundant experimental data reported on nanobiomechanics
of living cells, there is a lack of comprehensive discussion on testing with
different tip geometries, and the associated mechanical models that enable
extracting the mechanical properties of living cells. Therefore, this paper
discusses the strategy of selecting the right type of indenter tips and the
corresponding mechanical models at given test conditions.
1. Introduction
The mechanical properties of living cells can affect their physical interactions with
their surrounding extracellular matrix [1,2], potentially influencing the process of
mechanical signal transduction in living tissues [3–7]. Alterations in cell properties
are of fundamental importance for a wide range of processes, and changes in cell
mechanics are associated with conditions such as osteoarthritis [8], asthma [9],
cancer [10], inflammation [11] and malaria [12]. The mechanical properties of
living cells have been quantified using various testing methods, such as micropipette aspiration [8,13], magnetic twisting cytometry [14], optical tweezers [15–17]
and nanoindentation [18–20]. Although there are some good review papers on
cell mechanics [21,22], they mainly focus on using micropipette aspiration techniques. From the perspective of cell mechanics, one should be aware of what is
measured with respect to particular techniques. For example, it is often observed
that the cell appears softer [23] during micropipette aspirations compared with cytocompression [24] or indentation with a large spherical tip [25]. During micropipette
aspirations, it was observed that the cytoskeleton can be disrupted [26,27]. In such a
case, there is no (or very limited) tensile stress in the actin fibres, which significantly
contributes to cell stiffness. Therefore, cell mechanics can be approximated, as cytosol reinforced with bundles of actin fibres (with diameter of 9–10 nm). The weight
concentration of actin fibres is 1–10% for non-muscle cells and 10–20% for muscle
cells, and the elastic modulus of these actin fibres is 1.3–2.5 GPa [28].
This paper will shed light on the nanoindentation techniques, because
investigation of mechanical properties of living cells at the nanometre (or submicrometre) scale is essential for understanding how cells interact with the
surrounding materials. Cells would sense and respond to the nanoscale (or
microscale) features on the materials surface. For example, when in contact
with implanted devices or scaffold materials, cells interact with nanoscale (or
submicroscale) surface features in topography [29,30] and surface chemistry
[31,32]. Therefore, the nanobiomechanics of the living cells is very important
for surface design of the implanted materials and the scaffold materials for
tissue engineering. In addition, it also helps us to improve the understanding
of cell interaction with nanoparticles, which is important for nanotoxicology
[33] and nanomedicine [34]. Compared with other measurement techniques,
nanoindentation has the advantage of in situ imaging of the indented cells
& 2014 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution
License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original
author and source are credited.
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(a)
2
drive plate 1
pick-up electrode
drive plate 2
(b)
photodiode
detector
three plate capacitor
tip
laser
cantilever
specimen
(c)
2.1. Nanoindenter apparatus and atomic force
microscope
Nanoindentation is also known as depth sensing indentation,
in which the indentation load–depth–time (P – d – t) profile
is recorded. It enables probing the mechanical properties at
the nanoscale or microscale. For such a small-scale indentation,
there are different approaches to take with respect to the testing
instrumentation. In general, we could divide them into nanoindenter apparatus and atomic force microscope (AFM).
The key difference between the commercial nanoindentation
apparatus and the AFM is on the different transducer operation
mechanisms: the former uses electrical capacitance gages
(figure 1a) or magnetic coils to directly drive the indenter
into the sample. When a voltage is applied, an electrostatic
force is generated between the pick-up electrode and drive
plates (as depicted in figure 1a), resulting in the movement of
the pick-up electrode between the drive plates. While the
AFM actuates the tip indirectly via the bending of a cantilever,
the AFM operates by measuring attractive or repulsive forces
between a tip and the sample, which causes vertical deflection
of the cantilever. To detect the displacement of the cantilever,
a laser is reflected at the back of the cantilever and collected
in a photodiode (figure 1b).
In addition to quasi-static loading, a dynamic drive signal
can be superposed with the force curve (figure 1c) in both the
nanoindenter apparatus [36] and the AFM [37]. This enables
measurement of the storage modulus, loss modulus and
phase angle, which can be converted to the instantaneous
modulus, equilibrium modulus and viscosity [38].
The nanoindenter apparatus allows better control of the
indentation force and displacement. The AFM offers the
unique advantages of applying very small indentation
forces (below 100 pN), but accurate calibration is not easy
[39]. Owing to ultra-high resolutions in force and displacement, AFM nanoindentation is particularly useful for
probing living cells and subcellular components such as the
cell membrane and cytoskeleton.
load
2. Experimental aspects
displacement
Figure 1. (a) Schematic of electrical capacitance gages that drives nanoindenter, (b) the bending of a cantilever that actuates the AFM tip and (c) dynamic
drive signal superposed with the force curve which enables dynamic mechanical
measurement during nanoindentation. (Online version in colour.)
The shape of cells can be spherical or spreading in morphology, depending on the physiological conditions and
microenvironment of the living cells and cell types. The
choice of appropriate AFM tips depends on cell morphology,
cell type and what is of interest (cellular mechanics or subcellular mechanics). This would provide useful guidelines
for designing experimental protocols.
2.2.1. Flat punch
Indentation of cells with a flat-ended cylindrical punch
(figure 2) is also known as cytoindentation [19]. In this case,
the size of the flat punch is much smaller than the cell. This
type of indenter is preferred for a very soft and fragile cellular
or subcellular structure. The advantage is that data interpretation is relatively straightforward because it avoids the
complication of determining the contact area. The contact
area is less likely to be affected by thermal drift or creep. The
drawback is the spatial resolution is relatively limited compared with the pointed indenter (e.g. pyramid and conical
tip); therefore, it is not suitable to characterize fine features.
In addition, there are also other practical concerns for using
flat punches, such as alignment, detection of contact point
and force concentration around edges. The tips are usually
made of silicon or glass.
2.2. Choice of appropriate atomic force microscope tips
There are various tip geometries that can be fitted with the
AFM cantilever for nanoindentation tests. The advantages
and disadvantages of these tips are discussed as follows.
2.2.2. Spherical tip
This type of indenter (figure 3) is also ideal for very soft and
fragile cellular or subcellular structures. This type of tip
Interface Focus 4: 20130055
AFM tip
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with a high resolution, a very good control of the probe position and loading (or unloading) speed, and the flexibility of
using different probe geometries (e.g. flat punch spherical,
pyramidal and conical). It also has the unique feature of mapping the measured mechanical properties over the investigated
surface of the sample [35]. However, data interpretation for
nanoindentation of living cells is often difficult.
Despite abundant experimental data reporting nanobiomechanics of living cells, there is a lack of comprehensive
discussion on testing with different tip geometries and
mechanical models.
Therefore, the goals of this study were (i) to present the
strategy of selecting the right type of indenter tips; (ii) to illustrate cell mechanics at different test conditions; (iii) to discuss
the mechanical models that enable extracting the mechanical
properties of living cells during nanoindentation.
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At a given penetration, the pyramid indenter (figure 4) yields a
much smaller contact area compared with the spherical and flat
punch tips. It is particularly useful to probe fine features such as
the cytoskeleton. Owing to the crystalline structure of silicon, it
can be easily etched at certain plane directions, enabling massive production of the probes. The drawback is that the sharp
edges may damage the fragile cell membrane or nuclear
membrane; therefore, it is not recommended for indentation
of living cells.
2.2.4. Conical tip
Similar to the pyramid tip, the conical tip (figure 5) yields a
much smaller contact area compared with the spherical and
flat punch tip. Compared with the pyramid tip, it is less
likely to cause damage in lateral directions because it does
not have sharp edges. It also circumvents complicated data
interpretation owing to coupling of anisotropic soft materials
and orientation of the pyramid tip. In principle, the semiincluded angle of the probe will not affect the measured
elastic or plastic properties, if the appropriate models are
used. But it affects the relationship between the yield strength
and hardness. At a given penetration, the deformation-affected
volume is related to the semi-included angle of the probe
[42,43]. Therefore, to eliminate the effect of the substrate or
the surrounding matrix, one may need to choose a tip with
smaller semi-included angle although the increased stress
intensity underneath the very sharp tip might cause puncture
of the cell membrane. The tip radius also contributes to
the effective deformation zone as discussed in [42,44]. But
this influence is not that significant if the tip radius is much
smaller than the penetration. When using this probe to do
indentation at shallow penetration, it would only sense localized properties mainly resulting from the cell membrane
with the underlying cortex or individual cytoskeleton. Sometimes, it may simply measure the bending stiffness of the
cell membrane. In such a case, it is unlikely to obtain the mechanical properties of the whole cell. For similar geometries such
as a cone (a cylindrical punch can be treated as cone with a
semi-apical angle of 908) and a pyramid, the effective strain
is a constant and related to the semi-apical angle (u) [41].
2.2.5. Extended atomic force microscope testing rigs
In recent years, another type of indentation, cytocompression
[45,46], has been widely used to assess the mechanical properties of single cells. In principle, this is an extended AFM
indentation on top of cytoindentation. The primary difference
in the deformation mechanisms for cytoindentation and cytocompression is in the relative size between the flat punch and
the cell. The former has a flat punch diameter well below that
3
terms
description
P
force
d
t
T
a
R
u
D
V
K
h
G
G1
G0
E
E1
E0
HA
g
displacement
time
normalized time constant
contact radius
tip radius
semi-included angle of indenter
diffusivity
Poisson’s ratio
permeability
viscosity
shear modulus
equilibrium shear modulus
instantaneous shear modulus
Young’s modulus
equilibrium elastic modulus
instantaneous elastic modulus
aggregate modulus
surface energy
of cell. The latter (figure 6) has a flat punch diameter exceeding
that of the cell. In such a case, data interpretation is the same
as in normal compression tests. The indenter (i.e. the flat
plate) is often made of glass, which enables in situ observation
of cell deformation [20,47].
2.2.6. Summary
Tables 1 and 2 summarize the effective contact radius, effective contact stress and representative strain for various tip
geometries. Where, dc is the contact depth.
Table 3 summarizes brief guidelines of recommended tip
geometries of cells with varied morphologies.
3. Mechanical modelling
Estimation of cell mechanics requires the use of analytical
models (or empirical models) of which there are two principal types, namely structure-based models and continuum
models. The former include tensegrity [48,49] and percolation
models [50], which consider cell mechanics to be dominant
by the collective discrete loading bearing element. The
latter include linear elastic [51 –53], hyperelastic [54 –56], poroelastic (also known as biphasic model) [57] and viscoelastic
models [8,58]. The continuum model may be interpreted as
load-bearing elements that are infinitesimally small relative
to the size of the cell.
3.1. Structure-based models
3.1.1. Tensegrity model
The tensegrity model is based on the use of isolated components
under compression inside a net of continuous tension, in such a
way that the compressed members (usually bars or struts) do
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2.2.3. Pyramid tip
Table 1. Terms and explanations.
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would be particularly useful if the elastic properties of the
materials were to change with strain. As the effective strain
is related to the ratio of the contact radius and the tip radius,
it enables determining the stress–strain curves of the indented
materials. The typical spherical tip is made of glass which is
easy to manufacture. Similar to the flat punch probe, the
spherical glass probe is less likely to cause damage to cells
[40]. Again, this may not be good for probing fine features.
The typical radius of the probe for indentation of cells is
2.5–10 mm. The representative strain for the spherical tip is
the ratio of the contact radius over the effective tip radius [41].
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Table 2. The summary of the effective contact radius, effective contact stress and representative strain for various tip geometries.
effective contact radius (ac)
sphere
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Rdc d2c
cylindrical punch
cone
pyramid
representative strain
P
pa2c
ac
R
0.1433cos2u þ 0.205cosu þ 0.0191a
triangle-base
Interface Focus 4: 20130055
R
dc tanu
rffiffiffiffiffiffiffiffi
pffiffi
3 3
dc tanu
p
rffiffiffiffi
4
dc tanu
p
square-base
a
effective contact stress
This empirical expression is based on best fitting of the raw data in [41], see appendix A.
Table 3. The summary of a brief guideline of recommended tip geometries for cells with varied morphologies.
flat punch
spherical tip
rounded cell
3
3
spreading cell
soft cell
3
3
3
3
stiff cell
subcellular structure
pyramid tip
conical tip
3
3
3
2R
2R
q
Figure 5. Schematic of nanoindentation by a conical tip.
Figure 2. Schematic of nanoindentation by a flat-ended cylindrical punch.
2R
Figure 3. Schematic of nanoindentation by a spherical tip.
Figure 6. Schematic of cytocompression of a spherical cell.
Figure 4. Schematic of nanoindentation by a pyramid tip.
not touch each other and the pre-stressed, tensioned cables [59],
as shown in (figure 7). This model was coined by Buckminster
Fuller in the 1960s. Such a concept was then introduced by
Ingber [48] to cell mechanics. It assumes that a cell stabilizes
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tip geometry
4
its structure by incorporating compression-resistant elements
to resist the global pull of the contractile cytoskeleton [48].
This simple stick and string tensegrity model predicts that a
cell appears round when unattached (owing to the internal tension) or attached with a very soft substrate, and spreads out
when attached to a stiff substrate. All these agree well with
experimental observations [48,61].
3.1.2. Percolation models
The percolation theory describes the behaviour of connected
clusters in a random manner. It was introduced in
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difficult to quantify the mechanical properties of living cells
or subcellular structures.
x
y
Figure 7. A schematic tensegrity model of cell structure for a spherical cell [60].
3.2.1. Elastic model
If the tests were performed slowly such that the cell reaches
equilibrium, then it is reasonable to use elastic models. At
relatively small deformation, a simple linear elastic model
may be used to find the Young modulus of the cell (E). Evidence has been shown that this simple elastic model can still
reveal useful insights of cell mechanics such as the stiffness
ratio of the nucleus over the cytoplasm [51].
If the cell undergoes large deformation, then it may reach
the nonlinear elastic region. The neo-Hookean (NH) model,
also known as the Gaussian model, is one of the most
widely used nonlinear elastic (or hyperelastic) models
owing to its simplicity. For example, it has been successfully
used to model the deformation of single cells in cytocompression tests [66]. For incompressible materials, the NH model
has the following energy function [67]
W ¼ C1 ðl21 þ l22 þ l23 3Þ;
Figure 8. A schematic percolation model of cell structure. The smaller interior
cube representing the nucleus is supported by the pre-stressed cytoskeletal
network [50].
mathematics and then it was applied to materials science.
Recently, such a theory was introduced to describe the cell
structure and its mechanics [50]. The percolation cluster
shown in figure 8 contains substructures of tensegrity on a
small scale. These tensegrity substructures are likely to contribute to inherent tension in the cytoskeleton. Owing to the
random nature of their interconnection, percolation networks
are so flexible that they can easily adapt to the dynamic
conditions that affect cells [50].
3.1.3. Summary of structure-based models
The percolation and tensegrity models of the cytoskeleton are not mutually exclusive, but complementary. As
commented in [50], it is possible that during evolution
certain locally ordered tensegrity-type structures may have
emerged from more randomly interconnected percolation
structures. These structure-based models are very successful
at explaining a range of physical observations on cell mechanics such as cell spreading [62], cell migration [63], cell
detachment [64] and mechanosensation [65], etc. But it is
ð3:1Þ
where W is the strain energy density, lj ( j ¼ 1,2,3) are the three
principal stretch ratios. The Young modulus E is given by
E ¼ 6C1 :
ð3:2Þ
Other more sophisticated hyperealstic models have also been
developed, such as the polynominal and Ogden models. The
strain energy function for the former is given by the flowing
equation [68]:
!j
N
X
1
1
1
i
2
2
2
W¼
ð
l
þ
l
þ
l
3Þ
þ
þ
3
: ð3:3Þ
Cij 1
2
3
l21 l22 l23
iþj¼1
The Young modulus E is given by
E ¼ 6ðC10 þ C01 Þ:
ð3:4Þ
The Ogden model can describe a wide range of strainhardening characteristics, and it takes the following form as
described in previous studies [67– 69]:
W¼
n
X
2m
i
i¼1
a2i
ðla1 i þ la2 i þ la3 i 3Þ;
ð3:5Þ
where ai is strain-hardening (or stiffening) exponent. The
constants mi are related to the initial Young modulus E, by
E¼3
n
X
i¼1
mi :
ð3:6Þ
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z
Despite neglecting microstructural features, continuum models
enable quantifying the mechanical properties of cells under
various conditions that could provide essential information of
cellular subpopulations [25], disease [18], malignant transformation and cell–materials interactions [2]. However, it is worth
pointing out that there is no universal mechanical model available to quantify the mechanical properties of living cells at
various physiological and microenvironmental conditions,
because living cells can dynamically adapt to their environment. The feasible methodology is to choose appropriate
models according to testing conditions (or microenvironment)
for a given cell type. Therefore, this paper will focus on the discussion of the feasibilities of various continuum models at
given test conditions.
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3.2. Continuum models
5
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This model has been adopted to describe cell mechanics when
the chondrocyte is embedded in various hydrogel scaffolds [2].
6
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3.2.2. Poroelastic model
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The poroelastic model attributes the time-dependence to
the flow of a fluid through an elastic (or viscoelastic)
porous solid. Such a model was first proposed by Biot [70]
and was based on the assumptions of linearity between
the stress (sij, p) and the strain (e ij, r) and reversibility of
the deformation process. With the respective addition of the
scalar quantities p and r to the stress and strain group, the
linear constitutive relations can be obtained by extending
the known elastic expressions. The most general form for isotropic material constitutive behaviour response is described
as follows [70]:
sij
dij p
1
1
1ij ¼
dij skk þ
ð3:7Þ
6G 9K
3H
2G
Figure 9. Schematic of the generalized Maxwell model.
and
r¼
skk
p
þ ;
3H M
ð3:8Þ
where K and G are bulk and shear modulus of the drained
elastic solid. The parameters H and M characterize the
coupling between solid and fluid stress and strain.
This model was originally developed for soil mechanics
and it was then applied to describe the mechanics of hydrogels [71] and tissues such as bone [72] and cartilage [73]. Very
recently, it has also been applied to cell mechanics [24]. When
it comes down to cell mechanics, the material properties of
interest include the shear modulus G (or aggregate modulus
HA), and Poisson’s ratio of the solid matrix and Darcy permeability k. The permeability could be analogous to the
diffusion coefficient, but driven by the mechanical gradient
instead of the chemical gradient.
During stress relaxation, it gives [74]
PðtÞ Pð1Þ
¼ gðtÞ;
Pð0Þ Pð1Þ
ð3:9Þ
where P(1) and P(0) signify the force at infinite time (t ¼ 1)
and t ¼ 0, respectively.
Where the normalized time t is given by
t¼
Dt
;
a2
ð3:10Þ
where a is the contact radius and t is time.
The diffusivity D is given by
D¼
2ð1 vÞ Gk
;
1 2v h
ð3:11Þ
where v, G, k and h are Poisson’s ratio, shear modulus
and permeability of the solid matrix and viscosity of the
solvent, respectively.
The equilibrium Young modulus E is given by
E ¼ 2Gð1 þ vÞ:
ð3:12Þ
The relation between the aggregate modulus and the
Young modulus is given by
HA ¼
Eð1 vÞ
:
ð1 þ vÞð1 2vÞ
ð3:13Þ
This poroelastic model is quite similar to the biphasic
theory of the mixture of an incompressible solid and
incompressible fluid which was independently developed
by Mow et al. [75] and Bowen [76]. In addition, by replacing
the elastic media with viscoelastic media to account for the
intrinsic viscoelastic properties of the actin filaments, a
more complex model ( poro-viscoelastic) [77] can be used to
describe soft tissue or cell mechanics. When the cell is
exposed to an ionic solution, the ionic charge also contributes
to the mechanical responses of cells. In such a case, a third
phase (i.e. the ionic phase) can be included in the constitutive
equation, which is called the triphasic model [78].
3.2.3. Spring-dashpot viscoelastic models
The basic premise of viscoelasticity of this type is that it
replaces some elastic springs with a time-dependent dashpot
as shown in figure 9. These spring-dashpot models usually
refer to viscoelastic models. In a general manner, the empirical Prony series [79] has been used to describe the material’s
time-dependent constitutive response in the experimental
time domain.
For stress relaxation, the relaxation shear modulus G(t) in
an empirical Prony series is given by
X
t
GðtÞ ¼ G1 þ
;
ð3:14Þ
Gi exp ti
where G1 is the equilibrium shear modulus and the instanP
taneous shear modulus G0 ¼ G1 þ Gi or creep, the creep
compliance J(t) in an empirical Prony series is given by
X
t
;
ð3:15Þ
J(t) ¼ C0 Ci exp ti
where the parameters Ci are associated with compliance
values (inverse shear modulus).
The stress relaxation modulus and the creep compliance
are not explicitly inverse in the time domain but are in the
Laplace domain (i.e. GðsÞJðsÞ ¼ s2 ).
The equilibrium shear modulus (G1) and the instantaneous shear modulus (G0) can be determined by the
following equations:
G1 ¼
1
C0
ð3:16Þ
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Conical tip: the force–displacement relation is given by [84]
and
1
P :
C0 Ci
ð3:17Þ
P¼
4Gtanu 2
d;
pð1 vÞ
ð4:3Þ
where u is the semi-included angle of the conical indenter.
4.2. Poroelastic– nanoindentation models
3.2.4. Power-law rheology
Flat punch: the force– time (P 2 t) relation is given by [74]
Pð0Þ ¼ 8Gda;
ð4:4Þ
PðtÞ Pð1Þ
Pð0Þ Pð1Þ
pffiffiffi
¼ 1:304 expð tÞ 0:304expð0:254tÞ:
ð4:5Þ
and
gðtÞ ¼
Spherical tip: the force –time relation is given by [74]
Pð0Þ ¼
16
Gda;
3
ð4:6Þ
and
pffiffiffi
gðtÞ ¼ 0:491 expð0:908 tÞ þ 0:509expð1:679tÞ:
3.2.5. Summary
The advantage of the elastic models is that they eliminate
the number of elastic parameters, which are easy for empirical curve fitting and numerical simulations. It was argued
that poroelastic models provide physical constants related
to the material microstructure, compared with the empirical spring-dashpot viscoelastic models. However, it must be
pointed out that poroelastic models homogenize the whole
structure and may not be able to predict structure reorganization of the cell during external stimuli (e.g. cell migration
and cell spreading) compared with those structure-based
models (e.g. tensegrity model). Therefore, one should also
treat structure-based models and the continuum models as
complementary to each other.
4. Nanoindentation models
As mentioned earlier, only the nanoindentation models (continuum based) for the tips (i.e. flat punch, spherical and
conical tip) suitable for testing living cells will be discussed
in detail. Once the AFM deflection –displacement curves are
converted to force –displacement curves, indentation theories
would be applicable. The models below are based on
assumptions of small deformation and negligible tip –cell
adhesions. More detailed discussion is provided in §5,
when these assumptions are invalid.
4.1. Elastic nanoindentation models
Conical tip: the force –time relation is given by [74]
Pð0Þ ¼ 4Gda;
pffiffiffi
gðtÞ ¼ 0:493 expð0:822 tÞ þ 0:507expð1:348tÞ;
8 G pffiffiffiffi 3=2
P¼
Rd ;
31 v
where R are the radius of the spherical indenter.
ð4:2Þ
ð4:9Þ
where a is the contact radius, and t is the normalized time
constant as defined earlier.
A simpler poroelastic model to describe cell mechanics has
been presented in [85,86], which considers fluid propagates
through a cell owing to a local pressure increase in a twodimensional manner. This model requires other techniques to
determine the pore size and viscosity, rather than relying on
the analysis of the force–time–displacement curve alone. For
pore size, it can be estimated by hindered tracer particle diffusion experiments, and for viscosity it can be estimated by a
nanoparticle diffusion experiment [85,86].
4.3. Viscoelastic-nanoindentation models
When viscoelastic models were adopted, experimentalists used
either the stress relaxation (for tests performed under displacement control) or creep (for tests performed under force control)
period to determine viscoelastic parameters. It is advantageous
to use stress relaxation or creep for data analysis because they
circumvent the possible complexity in nonlinear changes
during ramping.
Flat punch: the following force–time and the displacement–
time relations can be obtained by incorporating the viscoelastic
model into equation (3.16)
Stress relaxation:
ðt
4R
ddðtÞ
PðtÞ ¼
Gðt tÞdðtÞ
d t;
3ð1 vÞ
dt
ð4:1Þ
where R and n are the radius of the flat punch and Poisson’s
ratio, respectively.
Spherical tip: the force–displacement relation is given by
the Hertz elastic model [83]
ð4:8Þ
and
Flat punch: the force–displacement (P 2 d) relation is given
by [82]
4GR
P¼
d;
ð1 vÞ
ð4:7Þ
ð4:10Þ
0
and
Creep:
dðtÞ ¼
ðt
3ð1 vÞ
dPðtÞ
Jðt tÞ
d t:
4R
dt
ð4:11Þ
0
More rigorous analytical solutions were given by [87], which
consider that the elastic components may have different
Poisson’s ratios.
Interface Focus 4: 20130055
Power-law rheology (GðtÞ tn , where n is a positive constant) has also been adopted to describe the mechanics of
certain cells. For example, it is found that n ¼ 0.2 for lung epithelial cells during stress relaxation [80]. The physical basis of
this power law may be related to molecular adjustment of the
cytoskeleton matrix, which is similar to soft glassy materials
close to the glass transition [80]. Such a model may work
better for those cells that do not have a strong cytoskeleton
structure. For example, such a power-law rheology model
has also been applied to neutrophils and macrophages [81].
rsfs.royalsocietypublishing.org
G0 ¼
7
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Spherical tip: the force –time and the displacement –time
relations are given by [88,89]
ð4:12Þ
0
and
continuum models
elastic modulus (kPa)
biphasic model
spring-dashpot model
1.38 + 0.46a
1.48 + 0.35
a
Creep:
d3=2 ðtÞ ¼
ðt
3ð1 vÞ
dPðtÞ
pffiffiffiffi
Jðt tÞ
d t:
dt
8 R
ð4:13Þ
Note that the original aggregate modulus value of HA (2.58 kPa) for
biphasic model is converted to the equilibrium modulus based on equation
(3.13), with Poisson’s ratio of 0.38 [25].
Conical tip: the force–time and the displacement –time
relations are given by [90]
Stress relaxation:
ðt
4tanu
ddðtÞ
PðtÞ ¼
Gðt tÞdðtÞ
dt;
pð1 vÞ
dt
ð4:14Þ
0
Table 5. The comparison of elastic moduli (determined by elastic contact
model) for osteoblastic cells (spread morphology) measured by pyramidal
and spherical AFM tip.
elastic parameters
pyramidal tip
spherical tip
E
14 kPa [102]
3.18 kPa [103]
and
Creep: d2 ðtÞ ¼
ðt
pð1 vÞ
dPðtÞ
Jðt tÞ
d t:
4tanu
dt
ð4:15Þ
0
For complicated load –time or displacement– time histories, Boltzmann hereditary integrals can be used to find full
P – d – t profiles [58,91].
5. Discussion
It must be pointed out that most researchers compared
measured cell mechanical properties across published results
without analysing the experimental techniques related to controlling factors on cell mechanics. These factors include tip
geometry, indentation penetration, cell morphology, cell type
and mechanical models. Therefore, it is essential to discuss
how these factors affect the measured cell mechanical properties.
5.1. Cell mechanics determined by different models
Depending on the models to be adopted, it may affect the
values of determined mechanical properties of cells. Table 4
summarizes the elastic properties of bovine articular chondrocytes determined by the biphasic and spring-dashpot
viscoelastic models. It can be seen that these two models
give similar results. However, the cell actually experiences
large deformation, which was not considered in both models.
Although many researchers adopted viscoelastic models to
describe the time-dependent behaviour of cell mechanics
[8,25,26,92], other researchers found that poroelastic models are
more appropriate in their experimental test protocols [86,93].
However, it has been shown that neither the poroelastic model
nor the viscoelastic model is capable of capturing the complete
mechanical responses of cells during creep or stress relaxation
[94,95]. A recent paper published in Nature Materials [95] has
demonstrated that the poroelastic model captures the timedependent mechanics of cells at a short-time scale (less than
0.5 s), but the viscoelastic model appears to work better at a
long-time scale [95]. This may suggest that, at a short-time
scale, it is fluid diffusion that governs the time-dependent behaviour of cells. At a long-time scale, it is likely that the collective
viscoelastic behaviour of the polymer-like materials inside cell
and cytoskeleton dynamics governs the time-dependent
behaviour of the cell. Therefore, a further improved model,
poro-viscoelastic model presented in [77,96–98] may be more
appropriate. If the cell is exposed to an osmotic environment,
then it passively swells or shrinks. The poroelastic (biphasic)
model is limited to describe the effective materials parameters
that vary with extracellular osmolality [99]. The triphasic
model [78] has the capability to better describe the mechanochemical coupling by introducing both mechanical and
chemical parameters in the governing equations, which could
better describe the mechanochemical equilibrium of the cells [99].
It must be pointed out that, in the poroelastic (biphasic)
model and triphasic model, small deformation and incompressible solid matrix are assumed. In a real test or physiological
loading, cells may experience large deformation. In addition,
the solid matrix in cells may not be incompressible. It would
be very complex to include all these in the constitutive equations of poroelastic models and triphasic models. However,
for viscoelastic models, it is straightforward to account for
the large deformation and compressibility of the solids via the
hyperviscoelastic model.
The above-mentioned models deal with static loading.
Indeed, oscillatory measurement associated with the AFM
can be used to determine the viscoelastic properties of the
living cell. More details can be found in [37,100].
5.2. Cell mechanics determined by different indenters
As discussed in §2, at a given force, the effective deformation
zone varies significantly with tip geometry. Consequently,
this can affect the measured mechanical properties of cells.
In general, when using a pyramid tip with low penetration,
the results are more likely to be affected by spatial heterogeneity, as observed in [101]. Table 5 summarizes the comparison
of elastic moduli (determined by elastic contact model) for
osteoblastic cells (spread morphology) measured by a pyramidal and spherical tip. It has been shown that elastic
moduli measured by a pyramidal tip are about four times
higher compared with those measured by a spherical tip.
When using a sharp conical or pyramid tip for spreading
cells, it senses the region near an actin filament, as observed
by Hoh et al. [104]. In such a case, the probed elastic response
Interface Focus 4: 20130055
0
8
rsfs.royalsocietypublishing.org
Stress relaxation:
pffiffiffiffi ðt
8 R
dd3=2 ðtÞ
PðtÞ ¼
Gðt tÞ
d t;
3ð1 vÞ
dt
Table 4. Elastic properties of bovine articular chondrocytes determined by
the biphasic and spring-dashpot models [24].
Downloaded from http://rsfs.royalsocietypublishing.org/ on October 2, 2016
(a)
(b)
9
(c)
rsfs.royalsocietypublishing.org
10 mm
Table 6. The comparison of elastic moduli (determined by elastic contact
model) for lung epithelial cells measured by a pyramidal and a flat punch tip.
elastic
parameters
pyramid [100]
flat punch (average
radius 0.53 mm)
[105]
E
0.53 + 0.11 kPa
0.5 kPa
may be significantly affected by this stiff filament (with
elastic modulus on the order of GPa).
Table 6 summarizes the comparison of elastic moduli
(determined by the elastic contact model) for lung epithelial
cells measured by a pyramidal and a flat punch tip. In this
test, large penetration of 3 mm was applied via a pyramid tip,
and the indentations were not far from the nucleus. In that
case, it probes the overall cell mechanics. That is why it yields
a similar result to that determined by a flat punch tip. However,
it should be pointed out that the thin-layer effect was not
considered in the original paper. Table 7 summarizes the
comparison of viscoelastic properties for a single bovine
chondrocyte using cytoindentation and cytocompression. The
indenter diameter for cytoindentation is 5 mm [106], which is
almost half of the cell diameter. The penetration for cytoindentation is similar to the cell radius; therefore, it is expected that
it senses the properties of the whole cell instead of local cellular
properties. In such a case, it would be expected that the elastic
properties determined by cytoindentation are similar to those
by cytocompression. As shown in table 7, the equilibrium
moduli and viscosity determined by these two indenters are
quite similar. However, the instantaneous modulus determined
by cytoindentation is over three times higher than cytocompression. This may suggest that indenter size might affect the
instantaneous mechanical responses of cells.
5.3. Cell morphology on cell mechanics
As cells are living materials, they respond to mechanical,
physical and chemical stimuli. Cells will change their
morphology after being removed from their native environment. Depending on the physical properties or surface
chemistry of the substrate that they are seeded on, cells may
spread along the substrate or retain a spherical shape.
Figure 10 displays a typical example that shows the substrate
stiffness regulates the shape of fibroblasts. Table 8 summarizes
the mechanical properties of the different cell types and the
change in their morphology. In that case, a borosilicate
glass sphere (5 mm diameter) probe was used for the AFM
nanoindentation [107].
Table 7. The comparison of viscoelastic properties for single bovine
chondrocyte (spherical morphology) using cytoindentation and cytocompression.
viscoelastic
parameters
cytoindentation
[106]
cytocompression
[46]
E1 (kPa)
1.09 + 0.40
1.48 + 0.35
E0 (kPa)
h (kPa s)
8.00 + 4.41
1.50 + 0.92
2.47 + 0.85
1.92 + 1.80
For all the above-presented models, they assume a sample
with infinite thickness and width compared with indentation
depth. When probing the spreading cells, if the indentation size
is comparable to the local thickness, then it invalidates these
assumptions. Therefore, the estimation of cell mechanical
properties will be affected by the stiffness of the underlying
substrate. In such a case, the thin-layer model proposed by
Darling et al. [108] could be applicable. This model introduces
a geometry factor to describe the force–displacement relation.
For example, for a spherical tip, the original elastic model can
be modified as follows:
P¼
4 E pffiffiffiffi 3=2
Rd f1 ðxÞ;
3 1 v2
ð5:1Þ
where f1(x) is geometrical correction factor which is given by
4a2
4a2
2a0
4p2
f1 ðxÞ ¼ 1 b0 x3
x þ 20 x2 20 a30 þ
p
p
p
15
ð5:2Þ
2
16a0
3p
3
4
þ 4 a0 þ
b x ;
p
5 0
and
pffiffiffiffiffiffi
Rd
;
x¼
h
ð5:3Þ
where h is the cell thickness,
1:2867 1:4678v þ 1:3442v2
1v
ð5:4Þ
0:6387 1:0277v þ 1:5164v2
:
1v
ð5:5Þ
a0 ¼ and
b0 ¼
This geometry factor f1(x) can be easily incorporated into the
above-mentioned viscoelastic models.
For the thin-layer-based poroelastic model, empirical correction parameters can be obtained based on finite-element
simulation [109]. For example, the following empirical
Interface Focus 4: 20130055
Figure 10. The morphology of fibroblast cultured on polyacrylamide gels with varied stiffness: (a) 10.4 kPa, (b) 3.31 kPa and (c) 0.56 kPa.
Downloaded from http://rsfs.royalsocietypublishing.org/ on October 2, 2016
Table 8. Comparison of viscoelastic properties for AFM indentation (with a spherical tip) of osteoblasts, chondrocyte, adipocytes, adipose-derived adult stem
cells (ADAS) and mesenchymal stem cells (MSCs) in spread and spherical morphologies [107].
11.6 + 33.9
1.70 + 3.13
112.2 + 450
5.80 + 14
ADAS cell
MSC
1.7 + 1.1
2.3 + 2.1
3.81 + 15.4
4.23 + 14.4
20.2 + 47.8
20.5 + 128
osteoblast
0.60 + 0.78
2.1 + 3.7
10.3 + 21.4
chondrocyte
ADAS cell
0.45 + 0.42
0.37 + 0.31
0.91 + 1.3
1.6 + 2.6
4.5 + 8.9
8.8 + 16.6
MSC
0.52 + 0.60
Osteoblast
chondrocyte
5.3 + 16.1
46.3 + 140
Note that the values of E0 and viscosity h were recalculated based on the raw data in [107].
model is for spherical indenter [109]:
gðtÞ ¼
PðtÞ Pð1Þ
¼ expða1 tb1 Þ;
Pð0Þ Pð1Þ
a1 ¼ 1:15 þ 0:44x þ 0:89x2 0:42x3 þ 0:06x4 ;
ð5:6Þ
ð5:7Þ
and
b1 ¼ 0:56 þ 0:25x þ 0:28x2 0:31x3 þ 0:1x4 0:01x5 : ð5:8Þ
It needs to be pointed out that depending on where the indentation is located, one needs to use different thin-layer models
(i.e. non-adhered thin-layer model [110] or adhered thin-layer
model [111,112]). For example, it was found that when indentation was made at a thin region relatively near the edge of the
fibroblast, the adhered thin-layer model is more appropriate
[37]. Although when indenting a thin region further from the
edge of the fibroblast, the non-adhered thin-layer model may
be more suitable [37]. One needs to be aware that all these
thin-layer models are valid for cells with approximately uniform
thickness (i.e. well-spread). Otherwise, computational models
are required, which will be discussed in §5.7. To minimize the
thin-layer effect, a tip with a smaller half-angle is preferred [113].
When indenting a specimen with finite thickness and
finite width, there is complex coupling between the stiffening
effect, caused by the underlying substrate, and the softening
effect, caused by a free edge in the lateral direction. A typical
example is to use a spherical tip indenting into a spherical
cell at a large deformation. Thus, another model has been
presented to deal with this coupling effect [58].
The following empirical geometry factor f2(x) can then be
used [58]:
f2 ðxÞ ¼ 1 þ ax;
ð5:9Þ
a ¼ 0:66 þ 0:917 v 2:479 v2 :
ð5:10Þ
and
This geometrical factor can be incorporated into the viscoelastic models [58], and it has been successfully applied to
extract viscoelastic properties of spherical chondrocytes
adherent to a glass plate.
5.4. A special case: cell trapped inside a well
Microfabricated well arrays have been proposed to trap cells
before nanoindentation [113–115]. This would enable automatic indentation for high-throughput screening methods
that may prove useful for cells sorting. All the existing contact
Figure 11. The schematic of AFM indentation of an isolated chondrocyte by a
spherical probe.
Table 9. The comparison of elastic moduli (determined by elastic contact
model) for the same murine NIH3T3 fibrobalst cell line (spread
morphology) cultured on different substrates indented by a spherical tip.
elastic
parameters
collagen-I-coated
glass
uncoated
glass
E
1– 2 kPa [117]
0.6 kPa [37]
Table 10. The comparison of elastic moduli (determined by elastic contact
model) for adult bovine chondrocytes (spherical morphology) are placed on
a tissue culture plastic and on an uncoated glass, determined by
cytocompression.
elastic
parameters
tissue culture
plastic
uncoated
glass
E
2.55 kPa [24]
1.17 kPa [46]
mechanics-based models only deal with indented materials
placed on a flat surface (i.e. the bottom surface is constrained
vertically), free from lateral constraint. Little work has been
carried out for indented materials placed within a well
(figure 11); in that case, the assumptions in the classic Hertz
model and thin-layer models have been violated. This problem
has recently been dealt with by Chen [116]. Interestingly, it
introduces a similar correction parameter to f2(x).
5.5. Substrate modulation
The substrate physics may affect the cell mechanical properties, even if the cell maintains the same morphology.
Table 9 summarizes the comparison of elastic moduli (determined by the elastic contact model) for the same murine
NIH3T3 fibrobalst cell line (spread morphology) cultured
on different substrates indented by a spherical tip. Table 10
Interface Focus 4: 20130055
4.5 + 2.3
1.0 + 1.6
spread
E1 (kPa)
rsfs.royalsocietypublishing.org
h (kPa s21)a
cell type
spherical
a
E0 (kPa)a
cell morphology
10
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11
P
loading
unloading
d
5.6. Cell –tip adhesion
Some degree of tip –cell adhesion may be unavoidable in
nanoindentation. This would be affected by the composition
and surface chemistry of the tip. There are different models
that incorporate tip –sample adhesion, such as the Johnson –
Kendall– Roberts (JKR) [118], Derjaguin –Muller– Toporov
[119] and Maugis–Dugdale [120] models. Among which,
the JKR model is appropriate for the indentation of relatively
compliant materials with probes of relatively large radii and
strong adhesive forces. This would be preferred to compliant
materials such as cells, in particular for a spherical indenter in
contact with spherical cells [121].
During unloading, one may observe the negative force
resulting from the surface adhesion, as shown in figure 12.
Such a pull-out force is related to the surface energy ( g) at
the cell –tip interface and is given by [83]
3
Ppull ¼ pRg:
2
ð5:11Þ
In such a case, one needs to replace the force (P) in all the
equations above by P þ Ppull. More detailed discussions
about the adhesive contact during AFM indentation has
been summarized by Lin & Horkay [122].
5.7. Inverse finite-element analysis
All the previous mechanical models work properly when the
irregularity of cell shape is negligible. For example, when
indenting a spread cell (with non-uniform thickness) with a
conical tip at small depth, the effective-deformed region is
mainly confined within the dotted region, as shown in
figure 13. In that case, the cell can be assumed a regular
shape with a single radius, so that the previous models in
§4 are applicable. However, at greater penetration, the effective deformation zone spreads out, and the irregularity of cell
shape becomes important. In that case, one should treat the
previous thin-layer model with caution, because the non-uniform thickness and the change of curvature becomes
important. Therefore, three-dimensional finite-element analysis (FEA) with the true cell shape is necessary. Figure 14
displays a flowchart of inverse FEA to determine the mechanical properties of cells during nanoindentation. In that case,
the three-dimensional model a of cell and its nucleus model
can be reconstructed from the sliced images obtained by
confocal microscopy [123].
In addition, in order to obtain the mechanical properties
of a subcellular structure of living cells without mechanical
or chemically separating them from living cells, FEA is also
essential. For example, finite-element simulations have been
Figure 13. Schematic of indenting a spread cell (with non-uniform thickness)
with a conical tip at small depth; in that case, the cell can be assumed as
regular shape.
used to do inverse analysis to obtain the mechanical properties of cytoplasm [2,51,54], nucleus elastic properties [2,51,54]
or the pericellular matrix [114]. As living materials, the external force may trigger passive stress generation in the cell. FEA
can be used to investigate how the passive stress may affect
the elastic modulus by treating it as residual stress [124].
5.8. Summary of strategy of selecting tip geometry
and mechanical models
Prior to discussing the strategy to select tip geometry and
size, it is good to recall the cellular structure. A eukaryotic
cell (excluding the red blood cell) is composed of cell membrane, cytoplasm (including cytosol, cytoskeleton and other
suspended organelles) and nucleus (including nucleus membrane, nucleoplasm, a structure layer nuclear lamina, nucleus
and other suspended organelles). In this paper, we discuss
the design indentation protocols and the associated indentation models with regard to the cell morphology, because cell
morphology will affect the boundary conditions of the indentation models. In addition, cell morphology can be regulated
by the substrate as shown in figure 10. Cell type is also
important, and we will generally classify the cell as a cell
with a strong cytoskeleton (such as osteoblast, chondrocyte
and muscle cells), medium strong cytoskeleton (such as fibroblast cell, epithelial cell and endothelial cell) and weak
cytoskeleton (such as fat cell, neutrophils, blebbing cell and
mesenchymal stem cell).
If one is interested in the mechanics of the cell membrane
(with thickness of approx. 7 nm), then a very small indentation (blow 1 nm) should be made with a conical or pyramid
tip with a small semi-included angle. This is due to the fact
that the tip with a smaller semi-included angle could restrict
the effective deformation zone. A simple mechanical model
such as an elastic model is acceptable.
If one is interested in the mechanics of cytoplasm, a small
indentation, less than 10% of the vertical distance from the
Interface Focus 4: 20130055
Figure 12. Schematic of force – displacement curve for a typical adhesive
indentation. (Online version in colour.)
rsfs.royalsocietypublishing.org
summarizes the comparison of elastic moduli (determined by
elastic contact model) for adult bovine chondrocytes (spherical morphology) placed on tissue culture plastic and on
uncoated glass, probed by cytocompression. In both cases,
it is evident that the substrates can significantly regulate
cell mechanical properties even if the cell shape is retained.
Similar behaviour of the matrix modulating cell mechanics was also found in a three-dimensional scaffold. For
example, when a bovine chondrocyte cell line were seeded
in different scaffolds, they maintained a spherical morphology, but different mechanical properties with regard to the
three-dimensional scaffold [2].
Downloaded from http://rsfs.royalsocietypublishing.org/ on October 2, 2016
12
nanoindentation
finite-element model
estimated mechanical
properties
match
simulated
indentation response
deformed cell shape by
simulation
(a)
(b)
d
d
Figure 15. Schematic of indentation of (a) a spherical and (b) spread cell.
nucleus (i.e. less than 0.1d as shown in figure 15) should be
made with a conical tip. The suitable penetration depth
varies with the semi-included angle. Alternatively, one can
locate the indentation near the edge, away from the nucleus.
The poroelastic model is more appropriate. It is also recommended to use a cone with a big angle to prevent high
strain-induced nonlinear elasticity. For a typical commercial
AFM probe, the tip radius is 4–10 nm. One may need to
use the modified models for a blunt cone or truncated pyramid as discussed by Lin & Horkay [122], when penetration is
comparable to the tip radius.
If one is interested in the mechanics of the whole cell
(mixed responses of cytoplasm and nucleus), then a bigger
indentation (more than 0.1d) should be made with a flat
punch or a spherical tip. For a well-spread cell (i.e. thickness
is small), a conical tip with a big semi-apical angle is appropriate. In this case, poroelastic, viscoelastic and more complex
poro-viscoelastic models are appropriate for the cells with a
strong or medium strong cytoskeleton. If the cell undergoes
large deformation, then the hyperviscoelastic model is more
appropriate [66], particularly for cells with a strong cytoskeleton. The power-law rheology model may be more suitable
for cells with a weak cytoskeleton.
The guidelines about the relative penetration above are a
rough estimation based on a coating/substrate system [44];
some dedicated computational modelling is required to further
reveal this rule of thumb to better guide the experimentalists to
design test protocols.
If cells are exposed to osmotic pressure, then the poroelastic model (or poro-viscoelastic models) should be used. If the
ionic charge is important, then a triphasic model is more
appropriate [78,125]. If penetration is comparable to the cell
thickness (typically for a spread cell), then a thin-layer model
in combination with a viscoelastic/poroelastic model should
be used. When the contact radius is comparable to cell width
and thickness, the model proposed by Chen & Lu [58]
should be used to account for coupling of the free edge effect
and the thin-layer effect.
6. Conclusion and perspectives
The exploration of nanobiomechanics of living cells helps
us to understand a range of processes such as disease
progression and cell –materials interactions. This provides
essential information for cellular therapy and tissue engineering. A range of mechanical models has been discussed. The
structural models have the advantage in predicting how
cells rearrange their structure (e.g. cell spreading and cell
migration) according to external stimuli. The continuum
models enable quantifying the mechanical properties in
cells under various conditions that could provide essential
information of cellular subpopulations, disease, malignant
transformation and cell –materials interactions.
It is almost impossible to get an accurate universal physical model to quantify the mechanical properties of cells
during indentation, owing to the complex cell structure and
the complicated mechanotransduction mechanisms (e.g. via
cytoskeleton and nuclear lamina). Therefore, it is more realistic to use the averaged continuum model accompanied by
appropriate testing protocols if the overall cell mechanics is
of interest. One should also treat structure-based models
and the continuum models as complementary to each
other. Various factors (e.g. cell morphology, substrate, tip
geometry and relative indentation penetration with regard
to cell size) that affect quantifying the mechanical properties
of cells have been evaluated. When researchers compare the
measured cell mechanical properties across the literature,
they should be aware that interpretation of the results can
be significantly affected by the mechanical models adopted
at given test conditions. Accordingly, this paper presents
strategies to select tip geometries and the associated
Interface Focus 4: 20130055
Figure 14. Flowchart of finite-element analysis to determine the mechanical properties of cells during nanoindentation.
rsfs.royalsocietypublishing.org
deformed cell
shape by
measurement
match
load–displacement
curve
three-dimensional reconstruction
of cell model
Downloaded from http://rsfs.royalsocietypublishing.org/ on October 2, 2016
0.35
effective strain
0.30
13
y = 0.1433x2 + 0.205x + 0.0191
R2 = 0.9862
rsfs.royalsocietypublishing.org
mechanical models at given test conditions. When the irregular shape of a cell plays an important role in cell mechanics,
it violates the boundary conditions of all the analytical
and semi-analytical mechanical models. In that case, FEA
incorporating the true cell shape is essential.
The way that cells sense and respond to the substrate is
expected to be temporal in nature [126]. In addition, passive
stresses may be generated in the cell during indentation. Therefore, the outlook for modelling cell mechanics should be to
incorporate these dynamic effects to the mechanical modelling
and FEA.
0.25
0.20
0.15
0.10
0.05
0
0.2
0.4
0.6
0.8
1.0
Acknowledgements. The author acknowledges Dr Bill Chaudhry for his
contribution of figure 10. The author also acknowledges Prof.
Van. C. Mow and Prof. Farshid Guilak for useful discussion on
viscoelastic, poroelastic (biphasic) and triphasic models. In addition,
the author acknowledges the anonymous reviewers for very constructive comments.
Funding statement. This work is partially supported by ‘A new frontier in
design: the simulation of open engineered biological systems’, EP/
K039083/1 and EPSRC-Newcastle University sandpit workshop award.
Figure 16. The effective strain changes with the semi-apical angle of the
conical tip, with the raw data taken from [41].
Appendix A
Figure 16 presents how the effective strain changes with the
semi-apical angle of the conical tip, with the raw data taken
from [41].
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