Confocal microscopy-based three-dimensional cell-specific modeling for large deformation analyses in...

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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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N. Slomka, A. Gefen / Journal of Biomechanics 43 (2010) 1806–18161808

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


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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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


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


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


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


% 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.


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

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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


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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.


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) [%]

100

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8090Cell B Cell B

6070

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>5% >10% >15% >20% >25% >30% >35% >40%

% o

f mem

bran

e ar

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%te

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ain

% o

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bran

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%te

nsile

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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

1416

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68

24

00

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.


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,

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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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