ARTICLE IN PRESS
Journal of Biomechanics 43 (2010) 1806–1816
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Journal of Biomechanics
0021-92
doi:10.1
n Corr
E-m
www.JBiomech.com
Confocal microscopy-based three-dimensional cell-specific modeling forlarge deformation analyses in cellular mechanics
Noa Slomka, Amit Gefen n
Department of Biomedical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
a r t i c l e i n f o
Article history:
Accepted 9 February 2010This study introduces a new confocal microscopy-based three-dimensional cell-specific finite element
(FE) modeling methodology for simulating cellular mechanics experiments involving large cell
Keywords:
Pressure ulcer
Deep tissue injury
Finite element model
Cell compression
Cell stretching
90/$ - see front matter & 2010 Elsevier Ltd. A
016/j.jbiomech.2010.02.011
esponding author. Tel.: +972 3 640 8093; fax
ail address: [email protected] (A. Gefen).
a b s t r a c t
deformations. Three-dimensional FE models of undifferentiated skeletal muscle cells were developed
by scanning C2C12 myoblasts using a confocal microscope, and then building FE model geometries from
the z-stack images. Strain magnitudes and distributions in two cells were studied when the cells were
subjected to compression and stretching, which are used in pressure ulcer and deep tissue injury research
to induce large cell deformations. Localized plasma membrane and nuclear surface area (NSA) stretches
were observed for both the cell compression and stretching simulation configurations. It was found that
in order to induce large tensile strains (45%) in the plasma membrane and NSA, one needs to apply more
than �15% of global cell deformation in cell compression tests, or more than �3% of tensile strains in the
elastic plate substrate in cell stretching experiments. Utilization of our modeling can substantially enrich
experimental cellular mechanics studies in classic cell loading designs that typically involve large cell
deformations, such as static and cyclic stretching, cell compression, micropipette aspiration, shear flow
and hydrostatic pressure, by providing magnitudes and distributions of the localized cellular strains
specific to each setup and cell type, which could then be associated with the applied stimuli.
& 2010 Elsevier Ltd. All rights reserved.
1. Introduction
At the organ scale, image-based finite element (FE) modeling iscurrently considered a state-of-the-art biomechanical researchmethodology, where typically, computed tomography or magneticresonance imaging are employed to obtain a set of planar imagesused to reconstruct a three-dimensional (3D) organ (Pistoia et al.,2004; Linder-Ganz et al., 2007; Portnoy et al., 2008; Lenaerts andvan Lenthe, 2009). The equivalent of this at a cellular scale couldbe confocal microscopy imaging, which allows obtaining a set ofplanar images for individual cells; however, image-based model-ing has not yet been employed at a cell-scale for large deformationanalyses. The only attempt to develop a full-3D confocal-based cellmodel was made recently by Dailey et al. (2009) who recon-structed alveolar epithelial cells for FE modeling of shear floweffects, but employed the small strain theory to predict celldeformations. Another limitation of the Dailey study was theabsence of cellular organelles, e.g. nucleus and cytoskeletal fibers.There was another recent publication by Gladilin et al. (2008) whostudied topologies of nuclei reconstructed from fibroblasts usingthe FE modeling, but they also used small strain elasticity.
ll rights reserved.
: +972 3 640 5845.
Utilization of confocal microscopy-based FE cell models withdetails of intracellular structures in large deformation analyses islikely to support experimental work in cellular mechanics.Specifically, it will allow realistic modeling of strains at a scaleof individual cells in classic cell loading designs ,which typicallyinvolve large cell deformations, e.g. static and cyclic stretching, cellcompression, micropipette aspiration, shear flow, and hydrostaticpressure. Nevertheless, 3D computational modeling of cell-specificstrains in cellular structures subjected to large deformationspresents a challenging task, particularly in numerically represent-ing mechanical interactions at the cell-scale (mm-scale), coupledwith interactions at scales that are orders-of-magnitude smaller,e.g. involving the plasma membrane or cytoskeletal fibers (forwhich characteristic dimensions are at a nm-scale).
In this study, confocal microscopy-based FE cell-specific models,consisting of nucleus, cytosol, plasma membrane and cytoskeletalfibers are analyzed under large deformations. Cellular strainsoccurring due to cell compression and stretching are investigatedin the context of pressure ulcer research, selected here as an exampleof application. Specifically, large deformation analyses of skeletalmuscle cells are important in studying the aetiology of deep tissueinjury (DTI), a serious pressure ulcer that develops in muscle tissueoverlying weight-bearing bony prominences, and may cause life-threatening complications (Agam and Gefen, 2007; Black, 2009). In aprevious paper (Slomka et al., 2009), it was postulated that sustained
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N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–1816 1807
compressive tissue deformations near bony prominences could causemuscle cell death by a mechanism of locally stretching plasmamembranes of cells. This could increase membrane permeability, andperhaps also the permeability of the nucleus envelope, andeventually disrupt cellular homeostasis. The present models weretherefore utilized for studying tensile strains in membranes andnuclei of myoblasts compressed and stretched in large deformations.
2. Methods
2.1. Cells
Skeletal muscle cells were used in their undifferentiated, myoblast form.
Undifferentiated cells have not yet retained the morphological and functional
characteristics they will acquire in their mature form. In the case of myoblasts, this
xy zy
xz
Fig. 1. A confocal-based computational cell model: (a) One z-stack image through th
confocal projections of the cell to the xz and zy planes (side views); (b) top view of the t
corresponding projections to the xz and zy planes. The geometrical similarity between t
Fig. 2. The finite element meshes of cell A (a) and cell B (b). The meshes of the isola
refers to immature muscle cells that did not yet fuse together to form myotubes
(Yun and Wold, 1996). Murine C2C12 myoblasts (cell line #CRL1722, ATCC, VA,
USA) were maintained undifferentiated in a growth medium (GM) consisting of
the Dulbecco’s Modified Eagle’s medium (with 4.5 g/l glucose, Biological
Industries, Beit-Haemek, Israel), 13% fetal bovine serum (Biological Industries),
2 mM L-glutamine, 0.8% non-essential amino acids, 1.67% HEPES, 10,000 u/ml
penicillin and 10 mg/ml streptomycin. Cells were incubated at 37 1C in 5% CO2, and
passaged every 3–4 days. Fluorescent staining of actin filaments was used for
identifying the cell morphology. One day prior to staining, cells were trypsinized
and seeded on round 12-mm-diameter cover-glasses contained in a 24-well plate
(Corning, NY, USA) with GM. Cells were incubated at 37 1C in 5% CO2 overnight to
allow cell adhesion. Cell fixation was obtained by incubation in 4% paraformalde-
hyde for 10 min at room temperature. Membrane perforation was then induced by
incubation in 0.1% Tween 20 (Sigma-Aldrich, MO, USA) dissolved in PBS for 30 min
on a rocker (60 RPM) at room temperature. Actin stress fibers were stained with
1:1000 dilution of FITC-labeled phalloidin in PBS. Each well was then washed 3
times with PBS, and cover-glasses were mounted on microscope glass slides. Cells
were observed under a Zeiss LSM-510 laser scanning microscope. Confocal z-stack
images were acquired at 0.4 mm intervals for a total height of 6 mm (Fig. 1a).
zyxy
xz
e base of an FITC-labeled phalloidin-stained myoblast (C2C12) cell (cell A), and
hree-dimensional solid model of cell A built from the z-stack confocal images, with
he confocal images of the cell and the solid model is evident for all planes of view.
ted nuclei (with cytoskeletal fibers) are shown on the right frame for each cell.
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2.2. Geometry reconstructions
The z-stack images of two cells were imported into a Solid Modeling software
(SolidWorks 2008, SolidWorks, MA, USA) and loaded into parallel planes at
the same z-coordinate intervals obtained from the confocal scans. Contours of the
boundaries of cells and their nuclei were manually drawn per slide and lofted into
3D bodies by connecting overlying planar slide-contours using splines. Cytoske-
letal fibers with diameter of 20 nm were incorporated into the confocal-based cell
geometries to connect the nucleus with the plasma membrane (Figs. 1b, 2). These
were assumed to represent bundles of protofilaments, rather than individual
protofilaments (see explanation in the discussion). The main geometrical
characteristics of the cells are reported in Table 1.
2.3. Finite element modeling
To apply large compressive deformations to cells, a culture dish was further
modeled as a foundation, and a plate was added on top, both an order-of-
magnitude larger than the largest cell dimension (Table 1; Fig. 3a). To apply large
stretches, a 1-mm-diameter elastic plate was modeled as the cell’s substrate, and
cells were located at 2/3 the radius, as measured from the center of the plate
(Fig. 3b). The geometries were then imported to an FE solver (ABAQUS version 6.8,
SIMULIA, RI, USA) where a 10-nm-thick plasma membrane (Ishiko et al., 1993;
Freitas, 1999; Chen and Moy, 2000; Hine, 1999; Becker et al., 2003) was added to
envelope the cytosol.
In the present displacement-driven FE models, calculations were limited to
strain measures, because calculating strain energy density (SED) or stress requires
knowledge of mechanical properties of cell organelles, which are not available in
the literature for large deformations. The nucleus, cytosol, plasma membrane and
the elastic plate (used for stretching simulations) were assumed to be isotropic
compressible materials with Neo–Hookean SED functions (Breuls et al., 2002;
Baaijens et al., 2005; Peeters et al., 2005a):
SED¼1
2kðJ1=2
3 �1Þ2þ1
2GðJ1�3J1=3
3 Þ ð1Þ
Fig. 3. Geometry and boundary conditions used for simulating c
Table 1Geometrical properties of the studied C2C12 cells, computed from the solid
models (see Fig. 1b for cell A).
Geometrical property Cell A Cell B
Height (mm) 4.86 4.04
Maximal width at base (mm) 57.6 42.2
Cellular volume including nucleus (mm3) 3248 1535
Nuclear volume (mm3) 244 54
Surface area of plasma membrane (mm2) 4070 2084
Area of cell base (mm2) 1676 900
Surface area of nucleus (mm2) 623 235
where J1=tr(B) and J3=det(B) are the first and third invariants of the finger tensor
B, respectively, and k and G are the bulk and shear moduli specified in Table 2.
Cytoskeletal fibers were assumed to be linear–elastic (Hawkins et al., 2009) with
elastic modulus of 20 MPa (Kis et al., 2002). Despite that cells are reported to
exhibit non-linear viscoelasticity (Peeters et al., 2005b). These assumptions
consider only the non-linear instantaneous elastic response. The compressive
plate and culture dish were assumed to be rigid.
Automatic meshing was used (Fig. 2), with element types and number of
elements as specified in Table 3. Transitions between nucleus and cytosol
materials, and between cytosol and membrane were modeled as discontinuous.
A ‘‘no-slip’’ contact condition was assumed at all nucleus–cytosol and cytosol–
membrane interfaces (by employing the ‘‘tie’’ constraint in ABAQUS). Cytoskeletal
fiber elements were represented using the 3D slender, line-like elements that can
transmit only axial forces (Table 3), which were embedded in the cytosol ‘‘host’’
elements using the ’’embedded region’’ constraint in ABAQUS.
2.4. Compression simulations
For compression simulations, frictional sliding with friction coefficient of 0.2
was assumed between the cell base and culture dish to represent the effective
resistance of integrins (Slomka et al., 2009). This is an intermediate value within
the range of coefficients for frictional sliding in a wet environment (0–0.4; Booser,
1983), which allowed large cell deformations without convergence problems.
Frictionless contact, representing interaction in a fluidic medium, was assumed
between the plasma membrane and compressive plate. ‘‘Penalty normal contact’’
with a stiffness scale factor of 1 was assumed between the plasma membrane and
plate, and between the cell base and culture dish, for imposing the contact
constraints. The culture dish was fixed for all translations and rotations, and the
cell base was fixed for translations along an axis vertical to the culture dish
(Fig. 3a). Global cell deformation (GCD) is defined as
GCD %½ � ¼Undeformed cell height�Compressed cell height
Undeforemed cell height� 100; ð2Þ
onfigurations of: (a) cell compression, and (b) cell stretching.
Table 2Mechanical properties used in the analyses for nucleus, cytosol, plasma membrane
and the elastic plate (used for stretching simulations), corresponding to Eq. (1).
Mechanical property Shear modulus (G)
(kPa)
Bulk modulus (k)
Nucleusa (Jean et al., 2005) 1.7 16.4 kPa
Cytosola (Breuls et al., 2002) 1 9.7 kPa
Plasma membranea
(Hochmuth et al., 1973)
2.5 24.2 kPa
Elastic plateb (Bernal et al., 2007) 671 33.3 MPa
a The Poisson’s ratios for all cellular components were assumed to be 0.45.b The Poisson’s ratio for the elastic plate was assumed to be 0.49.
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Tensile strain [%]
80Radial displacement [µµm]
6156
CellCell
746760
50464054
47 363040
34 252027
2014
1510
70
50
Boundary effects
NNon-uniform tensile strainsinduced in the cell
Uniform strain field on the elastic plate
Fig. 4. Cell stretching simulations: (a) Radial displacement of the elastic plate with the attached cell (marked); and (b) tensile strains in the elastic plate with the attached
cell, demonstrating uniformity of tensile strains in the plate, excluding an approximately 40-mm-wide marginal zone associated with boundary effects of the radial
displacement applied along the plate’s perimeter. It is demonstrated that the cell resides far enough from the boundary effects (approximately 170 mm away), consistent
with the Saint Venant’s principle, which recommends that in such a problem, the region of interest should be located at a minimal distance of 1/3 the plate’s radius with
respect to the edge. The cell and its immediate surroundings are magnified for clarity at the bottom frames.
Table 3Element types and number of elements per each model component.
Component Element type in ABAQUS Number of elements Cell A Number of elements Cell B
Nucleus C3D10M 3348 6967
Cytosol C3D10M 20,592 30,883
Plasma membrane M3D6 7820 8260
Cytoskeletal fibers T3D2 139 (for 19 fibers) 271 (for 33 fibers)
Culture dish R3D4, R3D3 9308, 250 9313, 234
Plate R3D4, R3D3 9149, 234 9338, 244
Elastic plate C3D15 16,090 16,156
2 Tensile strains in nucleus elements directly connected to cytoskeletal fibers
were not considered while calculating the outcome measures, because mechanical
N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–1816 1809
was induced quasi-statically lowering the compressive plate parallel to the culture
dish (Fig. 3a), until reaching GCD of 65% and 45% for cells A and B, respectively.1
2.5. Stretching simulations
For stretching simulations, a ‘‘tie’’ constraint was assumed between the cell base
and elastic plate, to represent that focal adhesions (integrins) effectively resist
tension applied to cells, as manifested in cell stretching experiments where cells did
not detach from the elastic plate at stretches o18% (Stroetz et al., 2001). The base of
the elastic plate was fixed for translations along the vertical axis, and for all
rotations (Fig. 3b). Cell stretching was applied quasi-statically by radially displacing
the perimeter of the elastic plate outwards (Fig. 3b), thereby inducing a
displacement field in the elastic plate that gradually decays towards its center,
and a uniform field of tensile strains in the elastic plate (TSEP) up to 15% tensile
strain (Fig. 4). Saint Venant’s principle was used to verify that cells were located far
enough from the edge of the elastic plate, allowing for boundary effects to be
neglected (Fig. 4b).
2.6. Data analysis
The cell compression and stretching models were analyzed for maximal local
principal strains in the plasma membrane and nuclear surface area (NSA) as
function of the GCD and TSEP, respectively. Strain data are reported as engineering
1 Cell B was smaller in size (Table 1), which numerically limited the maximal
GCD of this cell to be lower than that of cell A.
(Cauchy) strains, calculated by ABAQUS from the Green–LaGrange strains.
Outcome measures included: (i) peak tensile strains in the plasma membrane
and NSA,2 (ii) average tensile strains in the membrane and NSA, (iii) percentage of
membrane area subjected to tensile strains over x%, with x ranging between 5%
and 40% at 5% intervals, and (iv) tensile strains in cytoskeletal fibers. In addition,
the global nuclear deformation (GND) is defined as
GND %½ � ¼Undeformed nuclear height�Compressed nuclear height
Undeforemed nuclear height� 100; ð3Þ
was plotted as a function of the GCD and TSEP for the compressed and stretched
cell models, respectively.
3. Results
3.1. Compression simulations
Compressing the cells induced localized tensile strains in theirmembrane and NSA (see Fig. 5 for cell A) with peak and average
interactions at the nucleus–cytoskeleton interface are not characterized experi-
mentally (see also in the discussion), e.g. there could be some fiber sliding in the
real-world case, and so, we were careful not to derive outcome measures directly
at the connection sites.
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N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–18161810
tensile strains that increased in a power–law relationship withthe GCD (Table 4; Fig. 6). Average tensile strains in the NSA werehigher than in the membrane for both cells (Fig. 6). Peak andaverage membrane and NSA strains for the maximally reachedGCD of 65% and 45% for cells A and B, respectively are detailed in
Plasma membrane Tenstra
GCD = 15%
GCD = 27%
GCD = 35%
GCD = 45%
GCD = 56%
GCD = 65%
Fig. 5. Development of tensile strains in the plasma membrane (left column) and nucleu
(GCD).
Table 4Curve fitting of peak and average tensile strains (e) in the plasma membrane and nuclea
where x is the global cell deformation (GCD) (%), a, b, g are the curve-fitted constants,
Constant Membrane
Peak Cell A Peak Cell B Average Cell A Average Cell
R 0.999 0.999 0.999 0.998
a 0.0998 0.2607 0.0022 0.0055
b 1.6895 1.2801 2.1039 1.9077
g 0 0 0 0
Table 5. Plots of percentage membrane area and percentage NSAover a certain strain level indicated that membrane and NSA ofboth cells were subjected to large tensile strains (45%) only forGCD values exceeding approximately 15% (Fig. 7). The GND werenegligible for GCDo5% and started increasing linealy at a similar
Nucleus Tensile sile strain [%]
80
in [%]40 80
7367
403733 67
6053
333027 53
4740
72320 40
3327
1713 27
2013
107 13
10
30 0
s (right column) of compressed cell A for increasing levels of global cell deformation
r surface area (NSA) of compressed cells A and B to a power–law function: e=axb+gand R is the Pearson correlation coefficient. The curve fitting is valid for GCD45%.
Nucleus
B Peak Cell A Peak Cell B Average Cell A Average Cell B
0.990 0.998 0.999 0.999
1.2362 0.2428 0.0045 0.0387
1 1.3604 2.127 1.6397
�2.7706 0 0 0
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Plasma membrane Nuclear surface area 120
120100
10080
60
8060
40
20
40
20
Peak
tens
ile s
trai
n [%
]
Peak
tens
ile s
trai
n [%
]
Ave
rage
tens
ile s
trai
n [%
]
Ave
rage
tens
ile s
trai
n [%
]
000
% Global cell deformation (GCD)
3535
25
3030
2020
25
10
1515
55
10
000 5 10 15 20 25 30 35 40 45 50 55 60 65 70
% Global cell deformation (GCD)0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
% Global cell deformation (GCD)
Cell A Cell B
5 10 15 20 25 30 35 40 45 50 55 60 65 70 0% Global cell deformation (GCD)
5 10 15 20 25 30 35 40 45 50 55 60 65 70
Fig. 6. Peak (top row) and average (bottom row) tensile strains in the plasma membrane (left column) and nuclear surface area (NSA) (right column) for the compressed
cells versus the global cell deformation (GCD). Solid lines are power–law curves fitted to the discrete finite element computations, for which parameter values and the
Pearson correlation coefficients are provided in Table 4.
Table 5Peak and average tensile strains in the plasma membrane and nuclear surface area
(NSA) for the maximally reached global cell deformation (GCD), for cells A
(maximal GCD=65%) and B (maximal GCD=45%).
Cell Membrane Nucleus
N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–1816 1811
slope for GCD=5% and 15% for cells A and B, respectively, whereapproximately 0.9% GND was added for every 1% increase in GCD(Fig. 8). In cell A, which reached a higher final GCD (GCD=65%), aplateau in GND (GND=47%) was observed for GCD463% (Fig. 8).Tensile strains in cytoskeletal fibers were in the order of 1% forboth cells.
Peak Average Peak Average
A 113%a 15% 71% 28%
B 32%a 8% 47% 19%
a At the cell base.
3.2. Stretching simulations
Stretching the cells caused localized tensile strains in theirmembrane and NSA (particularly around attachment sites ofcytoskeletal fibers) (see Fig. 9 for cell A), but unlike thecompression simulations, peak and average tensile strainsincreased linearly with the TSEP (Table 6; Fig. 10). Averagetensile strains were higher in the NSA compared to the membranefor both cells (Fig. 10). Peak and average membrane and NSAstrains for the maximally reached TSEP=15% are providedin Table 7. Values of cell elongation along the radial axis ofthe plate were 9.3 and 6.8 mm for cells A and B, respectively.Plots of percentage membrane area and percentage NSA overa certain strain level demonstrated that the membrane andNSA of both cells were subjected to large tensile strains forTSEP values exceeding approximately 3% (Fig. 11). The GNDincreased linearly with TSEP for both cells until reaching amaximal value of approximately 22% for TSEP=15% (Fig. 12).Consistent with the compressed cell models, tensile strainsexperienced by cytoskeletal fibers were in the order of 1% forboth cells.
4. Discussion
This study introduced a new confocal microscopy-based 3Dcell-specific modeling methodology for simulating cellular me-chanics experiments involving large cell deformations. Thismethodology was tested by simulating two specific setups, cellcompression and cell stretching, which are physiological loadingregimes, and are particularly used in pressure ulcer and DTIresearches (Peeters et al., 2005a; Gawlitta et al., 2007; Gefen et al.,2008). Localized tensile strains in plasma membrane and NSAwere induced in cell stretching simulations (Fig. 10), which is wellexpected, but also when compressing the cells (Fig. 6), consistentwith the Slomka et al. (2009) two-dimensional (2D) cell modelstudy. It was found that in order to induce large tensile membraneand NSA strains (45%), one needs to apply more than �15% GCDin cell compression tests (Fig. 7), or more than �3% TSEP in cellstretching experiments (Fig. 11). Comparing tensional strain data
ARTICLE IN PRESS
Plasma Membrane Nuclear surface area
70100
60 8090Cell A
50 70
40 5060
303040
10
20
1020
0 00 5 10 15 20 25 30 35 40 45 50 55 60
% Global cell deformation (GCD)0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
% Global cell deformation (GCD)
70 100
60 8090
40
506070
4050
20
30
30
10 1020
00 5 10 15 20 25 30 35 40 45
00 5 10 15 20 25 30 35 40 45
% Global cell deformation (GCD) % Global cell deformation (GCD)
>5%
% o
ff m
embr
ane
area
ove
r x %
tens
ile s
trai
n
% o
ff m
embr
ane
area
ove
r x %
tens
ile s
trai
n
% o
ff m
embr
ane
area
ove
r x %
tens
ile s
trai
n
% o
ff m
embr
ane
area
ove
r x %
tens
ile s
trai
n
>10% >15% >20% >25% >30% >35% >40%
Cell B Cell B
Cell A
Fig. 7. Percentage of plasma membrane area (left column) and nuclear surface area (NSA) (right column) subjected to tensile strain values exceeding a certain strain level
(in the range 5–40%, at 5%-strain steps) for the compressed cells versus the global cell deformation (GCD).
50
45 Cell A Cell B
40
30
35
25
20
15
10
5
00 5 10 15 20 25 30 35 40 45 50 55 60 65 70
% Global cell deformation (GCD)
% G
loba
l Nuc
lear
dee
form
atio
n (G
ND
)
Fig. 8. Global nuclear deformation (GND) versus the global cell deformation (GCD) for the compressed cells.
N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–18161812
obtained from the two cells show that cell-to-cell data were moresimilar for the stretching configuration (Fig. 10) with respect tocompression (Fig. 6), which is likely associated with the directtensional loads applied to cells during stretching (Fig. 4), asopposed to when compressing the cells.
Though there are no available experimental methods tomeasure localized intracellular strains at the resolution providedby the present FE analyses (Figs. 5,9), which makes it difficult tofully validate the modeling, the stretching simulation dataprovide some verification. Specifically, the strain coordinate of
ARTICLE IN PRESS
Plasma membrane NucleusTensile Tensile strain [%]
40
strain [%]80
3734
7467
3027
6054TSEP = 4%
2420
4842
181411
3529
85
2216
2
10
TSEP = 15%
TSEP = 12%
TSEP = 10%
TSEP = 8%
TSEP = 6%
Fig. 9. Development of tensile strains in the plasma membrane (left column) and nucleus (right column) of stretched cell A for increasing levels of tensile strain in the elastic
plate (TSEP).
Table 6
Curve fitting of peak and average tensile strains (e) in the membrane and nucleus of stretched cell A and compressed cell B, to a linear function: e=dz+f where z is the
tensile strains in the elastic plate (TSEP) (%), d, f are the curve-fitted constants, and R is the Pearson correlation coefficient. The curve fitting is valid for TSEP43%.
Constant Membrane Nucleus
Peak cell A Peak cell B Average cell A Average cell B Peak cell A Peak cell B Average Cell A Average Cell B
R 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999
d 2.4729 2.0298 1.0281 1.0056 5.1817 5.8317 1.4472 1.5617
f 1.001 0.9773 0.0031 1.0012 3.5431 0.9808 0.2145 0.9927
N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–1816 1813
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Plasma membrane Nuclear surface area
90
708090
7080
5060
5060
3040
3040
1020
1020
000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Tensile strains in the elastic plate (TSEP) [%]Tensile strains in the elastic plate (TSEP) [%]
202222
1618
161820
1214
101214
810
68
246
24
000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Tensile strains in the elastic plate (TSEP) [%]Tensile strains in the elastic plate (TSEP) [%]
Cell A Cell B
Peak
tens
ile s
trai
n [%
]
Peak
tens
ile s
trai
n [%
]
Aver
age
tens
ile s
trai
n [%
]
Aver
age
tens
ile s
trai
n [%
]
Fig. 10. Peak (top row) and average (bottom row) tensile strains in the plasma membrane (left column) and nuclear surface area (NSA) (right column) for the stretched cells
versus the tensile strain in the elastic plate (TSEP). Solid lines are power–law curves fitted to the discrete finite element computations, for which parameter values and the
Pearson correlation coefficients are provided in Table 6.
Table 7Peak and average tensile strains in the plasma membrane and nuclear surface area
(NSA) for the maximally reached tensile strain in the elastic plate (TSEP=15%), for
cells A and B.
Cell Membrane Nucleus
Peak Average Peak Average
A 38%a 15% 80% 22%
B 28%b 15% 81% 23%
a At the superior cell surface.b At the cell base.
N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–18161814
each curve shoulder in Fig. 11 approximately equals its corre-sponding TSEP, e.g., the percentage membrane area over 5%tensile strain increases sharply from less than 10% to nearly 100%when the TSEP crosses the 5% level (Fig. 11). Analogue trends existfor NSA (Fig. 11).
Cellular scale FE modeling is a rapidly growing field ofresearch, and progress is evident over the last decade in termsof representing both the geometry and constitutive modeling. Inorder to gain basic understanding of deformations, strains andstresses in and around cells, and in order to perform reverseengineering analyses of cell mechanical properties, some authorsdeveloped 2D cell models (Guilak and Mow, 2000; Jean et al.,2005; Ofek et al., 2009; Peeters et al., 2005a; Slomka et al., 2009),which, due to their 2D nature, require some idealization ofcellular geometry. Other authors attempted to develop morecomplicated models, extending the geometry to 3D, but stillemploying idealized cell-shape representations such as spheres orellipsoids (McGarry and Prendergast, 2004; De Santis et al., 2009).Some of these models, either 2D or 3D, also incorporated nucleus
and/or cytoskeletal fibers (McGarry and Prendergast, 2004; Jeanet al., 2005; De Santis et al., 2009; Ofek et al., 2009). Just twoattempts were made to develop 3D cell models whose geometry isbased on imaging individual cells rather than on assumingidealized cell shapes, and these include the study by Ferko et al.(2007) who built an endothelial cell model from multimodalfluorescence images, and the study of Dailey et al. (2009) whodeveloped alveolar epithelial cell models from confocal micro-scopy scans. However, both the Ferko and the Dailey studiesutilized the small deformation theory. Accordingly, our presentstudy is the first to introduce a modeling methodology which(i) provides realistic 3D representation of individual cells based onconfocal microscopy imaging, (ii) incorporates cellular compo-nents (nucleus, cytosol, plasma membrane and cytoskeletalfibers), and (iii) employs the large deformation theory to analyzecellular strains.
Despite the progress made in cellular modeling in the presentstudy, there are still some limitations that should be borne inmind when interpreting the data, mainly as related to representa-tion of the cytoskeleton, constitutive modeling of cellularcomponents, and component-to-component mechanical interac-tions. Individual actin filaments (microfilaments), intermediatefilaments or microtubules in the cytoskeleton were not modeledbecause these are numerous in a single cell, and their cross-sectional area is orders-of-magnitude smaller than a cell size(Fig. 1a). Accordingly, when analyzing consecutive z-stack images,it is technically not feasible to track contours and 3D orientationspecific to each filament or microtubule. In order to still be able torepresent the mechanical contribution of the cytoskeleton inresisting large deformations, at least to some extent, fewercytoskeletal components than in reality were incorporated inthe models (Figs. 1 and 2), and each component was assumed to
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Plasma Membrane Nuclear surface area
100
8090
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8090 Cell A
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Tensile strain in the elatic plate (TSEP) [%]0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Tensile strain in the elastic plate (TSEP) [%]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Tensile strain in the elastic plate (TSEP) [%]
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Tensile strain in the elastic plate (TSEP) [%]
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8090Cell B Cell B
6070
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>5% >10% >15% >20% >25% >30% >35% >40%
% o
f mem
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%te
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ain
% o
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Cell A
Fig. 11. Percentage of plasma membrane area (left column) and nuclear surface area (NSA) (right column) subjected to tensile strain values exceeding a certain strain level
(in the range of 5–40%, at 5%-strain steps) for the stretched cells versus the tensile strain in the elastic plate (TSEP).
2422 Cell A Cell B
2018
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Tensile strains in the elastic plate (TSEP) [%]
% G
loba
l nuc
lear
def
orm
atio
n (G
ND
)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Fig. 12. Global nuclear deformation (GND) versus the tensile strain in the elastic plate (TSEP) for the stretched cells.
N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–1816 1815
effectively represent a bundle of real protofilaments. Anothermodeling problem is the general lack of experimental dataregarding constitutive laws and properties of cellular compo-nents, due to technological limitations in performing cell-scaleexperiments. The quality of cell stress analyses depends on theassumed mechanical properties. Considering this issue, it wasdecided to limit the present analyses to strains only (given thatdeformation boundary conditions applied), and avoid extrapola-tion to stress. However, our modeling is perfectly capable ofsolving stress states of cells, if, in the future, relevant constitutiveproperties can be measured. Emerging tools that can provide suchexperimental data include atomic force microscopy, optical andmagnetic tweezers (Kamgoue et al., 2007; Carl and Schillers,2008). A similar problem is how to characterize component-to-
component mechanical interactions in modeled cells and betweencells and their simulated surroundings, such as the frictionalproperties assumed between cells and substrates, or the mechan-ical nature of connections between cytoskeletal fibers and thenucleus or membrane. Again, new experimental data in thisregard can be implemented directly in the present modeling.
In conclusion, a new methodology for the confocal micro-scopy-based cell-specific modeling of strain states in individualcells subjected to large deformations was introduced. Applica-tions in modeling cell compression and stretching were demon-strated. Utilization of our modeling can substantially enrichexperimental cellular mechanics studies in classic cell loadingdesigns that typically involve large cell deformations, e.g. staticand cyclic stretching, cell compression, micropipette aspiration,
ARTICLE IN PRESS
N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–18161816
shear flow and hydrostatic pressure, by providing magnitudes anddistributions of localized cellular strains specific to each setup andcell type, which could then be associated with the applied stimuli.
Conflict of interest
The authors of the above paper state that they have no conflictof interest.
Acknowledgements
The authors appreciate the help of Dr. Eran Linder-Ganz andMs. Sigal Portnoy from the Musculoskeletal BiomechanicsLaboratory at the Department of Biomedical Engineering of theFaculty of Engineering at Tel Aviv University, for their technicalassistance in using SolidWorks and ABAQUS. We are also thankfulto Ms. Shira Or-Tzadikario from the same lab, for helping with thecell culturing. We appreciate the help of Mr. Shmulik Keidar(ADCOM Ltd., Givatayim, Israel) with technical issues related tothe ABAQUS analyses.
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