Page 1

High Resolution Traction Force Microscopy Based on Experimental

and Computational Advances

Benedikt Sabass,* Margaret L. Gardel,yClare M. Waterman,yand Ulrich S. Schwarz*

*University of Heidelberg, Heidelberg, Germany; andyThe Scripps Research Institute, La Jolla, California

ABSTRACT

of focal adhesion to the extracellular matrix. Here we report experimental and computational advances in improving the

resolution and reliability of traction force microscopy. First, we introduce the use of two differently colored nanobeads as fiducial

markers in polyacrylamide gels and explain how the displacement field can be computationally extracted from the fluorescence

data. Second, we present different improvements regarding standard methods for force reconstruction from the displacement

field, which are the boundary element method, Fourier-transform traction cytometry, and traction reconstruction with point

forces. Using extensive data simulation, we show that the spatial resolution of the boundary element method can be improved

considerably by splitting the elastic field into near, intermediate, and far field. Fourier-transform traction cytometry requires con-

siderably less computer time, but can achieve a comparable resolution only when combined with Wiener filtering or appropriate

regularization schemes. Both methods tend to underestimate forces, especially at small adhesion sites. Traction reconstruction

with point forces does not suffer from this limitation, but is only applicable with stationary and well-developed adhesion sites.

Third, we combine these advances and for the first time reconstruct fibroblast traction with a spatial resolution of ;1 mm.

Cell adhesion and migration crucially depend on the transmission of actomyosin-generated forces through sites

INTRODUCTION

A growing body of evidence suggests that physical force

plays a crucial role as regulator of many cellular processes,

including cell adhesion and migration (1–5). In particular,

the dynamics of establishing actomyosin-generated traction

force in an elastic environment appears to be central for the

way cells sense and react to mechanical properties of their

environment. Therefore, measuring cellular traction forces

on elastic substrates is an essential tool for studying the

regulation of cell adhesion and migration in a quantitative

way (6,7).

The method of traction force microscopy was pioneered

by the seminal work of Harris and co-workers, who were the

first to use thin flexible silicone sheets as a wrinkling assay

that gave a qualitative measure for the mechanical activity of

cells (8). For quantitative studies it is essential to suppress

wrinkling, which was achieved first for thin films under pre-

stress (9,10) and later for thick films attached to a cover slide

(11,12). Today traction force microscopy on thick elastic

substrates has become a standard procedure to reconstruct

cellular traction forces. Usually the films are prepared from

polyacrylamide (PAA) or polydimethylsiloxane (PDMS)

coated with adhesive ligands like fibronectin or collagen.

PAA has the advantage that its stiffness can be tuned easily

over the physiologically relevant range from 100 Pa to 100

kPa (13–15). PDMS, which is hard to prepare with a bulk

stiffness below 10 kPa, has the advantage that it can easily

be micropatterned (16–18). An alternative to flat elastic sub-

strates is the pillar assay—an array of microfabricated PDMS-

pillars that deform easily under cellular traction due to their

small diameter (19–22). Because each pillar is a localized

force sensor, the pillar assay allows a simple, albeit spatially

constrained readout of the forces. For cell adhesion which is

not spatially confined, however, flat elastic substrates are the

method of choice.

A setup for traction force microscopy on flat elastic

substrates has to combine different experimental and com-

putational techniques. In Fig. 1 A, we show a schematic

representation of the situation of interest. A cell is adhering

to a flat substrate and exerts force through its sites of

adhesion (traction pattern in red). The resulting deformations

in the substrate are tracked by monitoring the movement of

embedded marker beads (displacement field in blue). For

PAA, usually fluorescent microbeads are employed which

are embedded near the surface of the gel. For PDMS, micro-

patterning can be used to create a pattern which is easily

detected with phase contrast microscopy. The displacement

field has to be extracted from a pair of images, one image

showing the substrate as it has deformed under cell traction,

and one reference image showing the undeformed substrate.

In general, there are two ways to approach the image

processing task: either one directly tracks the movement of

the fiducial markers (particle tracking velocimetry, PTV), or

one makes use of a cross-correlation function to derive the

local motion statistically (particle image velocimetry, PIV).

Next one has to reconstruct the cellular traction pattern from

the displacement field. For synthetic substrates like PAA

or PDMS, one can assume an elastic behavior which is

doi: 10.1529/biophysj.107.113670

Submitted May 29, 2007, and accepted for publication August 14, 2007.

Address reprint requests to Ulrich S. Schwarz, Tel.: 49-6221-54-51254;

E-mail: ulrich.schwarz@bioquant.uni-heidelberg.de.

Margaret L. Gardel’s current address is Department of Physics, University

of Chicago, Chicago, IL.

Clare M. Waterman’s current address is National Heart, Lung and Blood

Institute, National Institutes of Health, Bethesda, MD.

Editor: Elliot L. Elson.

? 2008 by the Biophysical Society

0006-3495/08/01/207/14 $2.00

Biophysical JournalVolume 94January 2008207–220207

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homogeneous, isotropic, and linear. For cell adhesion in

tissue culture, the cell is rather flat and force in the normal

direction can be neglected. Then both the displacement field

u(x) and the traction stress field f(x) are two-dimensional in

the plane of the substrate (e.g., x ¼ (x1, x2)). They are related

by the following integral equation:

Z

Given the experimental displacement u(x) and the relevant

Green function Gij(x), one needs to invert Eq. 1 to obtain the

desired traction field f(x). For thick films, the substrate can

be approximated by an elastic half-space and thus one can

use the Boussinesq Green function (23),

uiðxÞ ¼

+

j

Gijðx ? x9Þfjðx9Þdx9:

(1)

GijðxÞ¼ð11nÞ

pE

ð1 ? nÞdij

r1nxixj

r3

??

¼ð11nÞ

pEr3

ð1 ? nÞr21nx2

nxy

nxy

ð1 ? nÞr21ny2

!

; (2)

where n and E represent Poisson ratio and Young modulus,

respectively, and r ¼ jxj. For clarity, here we have written

the Green tensor both in index notation and in full form. As it

is typical for a three-dimensional elastic Green function, it

scales as ;1/r with distance, i.e., it is long-ranged and has a

singularity at the origin. Thus, Eq. 1 is a Fredholm integral

equation of the first kind with a weakly singular kernel. The

long-ranged nature of the kernel implies that the direct

problem corresponds to a smoothing operation. Therefore,

the inverse problem might be very sensitive to small differ-

ences in the displacement field, depending on the exact na-

ture of the experimental data. Noise in the experimental data

can result from different sources, including elastic inhomo-

geneities in the substrate, insufficient coupling between

marker beads and polymer matrix, deficiencies in the optical

setup, and lack of accuracy in the tracking routines.

In the past, three standard methods have been established

to calculate force from displacement. Both the boundary

element method (BEM) (10,11) and the Fourier transform

traction cytometry(FTTC) (24) approximate the integral on a

grid (discretized methods). While BEM effectively corre-

sponds to inverting a large system of linear equations in real

space, FTTC uses the fact that the relevant system of linear

equations is much smaller in Fourier space, thus facilitating

inversion. Traction reconstruction with point forces (TRPF)

(16,18,25) uses additional experimental knowledge about the

location of the adhesion sites, which can be obtained for

example by fluorescence data on proteins localizing to focal

adhesions (e.g., vinculin or paxillin) (16,25) or by reflection

interference contrast microscopy (18). Then the integral in

Eq. 1 converts into a simple sum. In the past, there has been

some dispute about advantages and disadvantages of these

different methods. In particular, a priori it is not clear which

of the two discretized methods performs better in respect to

resolution and reliability. Moreover, different solutions have

been suggested to deal with the issue of experimental noise,

but a systematic and detailed analysis of the these approaches

has been missing.

Here we present different experimental and computational

advances which together allow us to achieve a much higher

resolution and reliability in traction force microscopy than

formerly possible. Experimentally we have implemented a

new method to track the deformations of a PAA substrate by

simultaneously using two differently colored nanobeads as

fiducial markers. To extract the corresponding displacement

field, we have developed a new image processing method

combining PTV and PIV. Regarding the computational

reconstruction of the traction field, we have implemented

different variants of all three standard techniques (BEM,

FIGURE 1

flat elastic substrates. Marker beads in the substrate and the corresponding

displacement vector field are shown in blue. Sites of adhesion and the

corresponding force vector field are shown in red. (B) If force is assumed to

be strongly localized, one can use the concept of point forces, which leads to

a divergent displacement field at the site of force application. Here we plot

the magnitude of the displacement in two perpendicular directions. When

relating force to displacement, the mathematical divergence can be avoided

by using a simple cutoff rule. (C) If force is assumed to be spatially extended

(here we show constant traction over a circular site of adhesion), then

displacement is finite inside the adhesion area. Again we plot the magnitude

of the displacementin two perpendicular directions. Traction of 2 kPa (2 nN/

mm2) at an adhesion of 2 mm in diameter corresponds to a maximum

displacement of 0.3 mm on a 10 kPa substrate. At a distance larger than

roughly twice the adhesion size, the displacements resulting from the two

assumptions are identical.

(A) Schematic representation of traction force microscopy on

208Sabass et al.

Biophysical Journal 94(1) 207–220

Page 3

FTTC, and TRPF) and systematically compared their per-

formance using extensive data simulation. In particular, we

have implemented an improved version of BEM and com-

pared it to different new variants of FTTC. Analysis of sim-

ulated data revealed the importance of filtering for FTTC,

and showed that certain variants of FTTC can perform al-

most as well as BEM while being much more efficient in

terms of computer time demands. Both discretized methods

are found to be strongly biased with respect to small ad-

hesion sites. TRPF does not suffer from this limitation, but

its underlying assumption of accurate knowledge of adhe-

sion site location limits its applicability. Finally, we dem-

onstrate with fibroblasts that our overall setup results in a

spatial resolution which constitutes a 5–10-fold improve-

ment over earlier methods.

EXPERIMENTAL METHODS

Cell culture

Mouse embryo fibroblasts were cultured in a humid environment at 37?C

in 5% CO2in DMEM (cat. No. 10303; GIBCO BRL, Gaithersburg, MD)

containing 10% fetal bovine serum (cat. No. 15630; GIBCO BRL). For

experiments, cells were transfected with a plasmid encoding GFP-tagged

paxillin (kind gift of A. R. Horwitz) using FuGENE 6 Transfection Reagent

(cat. No. 11814443001; Roche, Nutley, NJ). After 12 h, cells were replated

on the PAA substrates described below. Coverslips were mounted in a

Warner Perfusion Chamber (Warner Instruments, Hamden, CT) and cells

were imaged ;12–16 h after replating. Imaging media consisted of DMEM

supplemented with 10 mM HEPES (GIBCO) pH 7.0 and Oxyrase (cat. No.

EC0050; Oxyrase, Mansfield, OH).

Preparation of PAA substrates

We modified previously published protocols for making polyacrylamide

substrates to maximize coupling of ECM and minimize variation between

preparations. The surface chemistry of 22 3 40 mm rectangular, No. 1.5 glass

coverslips (Corning, Corning, NY) was modified to facilitate a tight cou-

pling of the PAA gel to the glass surface using protocols described by Wang

and Pelham (26) and Damljanovic et al. (27). Briefly, coverslips are incubated

in 0.5% 3-aminopropyltrimethyoxysilane (Pierce Biotechnology, Rockford,

IL) for 5 min. After extensive rinsing, coverslips are then incubated in 0.5%

glutaraldehyde solution (Electron Microscopy Sciences, Fort Washington,

PA) for 30 min. After rinsing, coverslips are dried and stored for future use.

The 40% polyacrylamide and 2% bis-acrylamide solutions (Bio-Rad,

Hercules, CA) were diluted to make stock solutions of 12% acrylamide/

0.1% bis-acrylamide. Five-hundred milliliters of solution was degassed for

20 min under house vacuum and 0.75 mL of TEMED and 2.5 mL of 10%

ammonium persulfate were added to initiate polymerization and mixed

thoroughly. A volume of 20–25 mL of the polyacrylamide mixture was

immediately pipetted onto the surface of a 22 3 80 mm glass slide and the

activated coverslip was carefully placed on top. The polymerization is

complete within 10–15 min and the top coverslip and attached polyacryl-

amide sheet were slowly peeled off and immediately immersed in ddH2O.

The ratio of TEMED and APS as well as the concentrations of acrylamide

and bis were chosen to be identical to those published by Yeung et al. (15),

such that we could use the values of the shear elastic modulus that they

measured. The reported shear elastic modulus for the samples used here is

15.6 kPa. The thickness of the films varied between 20 and 30 mm.

The heterobifunctional cross-linker sulfo-SANPAH (cat. No. 22589,

Pierce) is used to crosslink extracellular matrix molecules onto the gel

surface. One-milligram aliquots of sulfo-SANPAH were stored at ?80?C in

40 mL anhydrous DMSO and diluted in 1 mL ddH20 immediately before

coupling. The coverslipswerequickly(,2 s) spunona homemadecoverslip

spinner (http://www.proweb.org/kinesin/Methods/SpinnerBox.html) to elim-

inate the bulk of the water from the surface, but not dry out the gel. Five-

hundred microliters of the sulfo-SANPAH solution was pipetted on the

surface. The PAA gel was then placed three inches under a 10 W ultraviolet

lamp and irradiated for 5 min at 4?C. It was then washed thoroughly with

ddH2O,spunandtheninvertedon50mLof1mg/mLFibronectin(ChemiCon,

Temecula, CA) for several hours at 4?C.

Live cell microscopy

Coverslips of cells were mounted in a microscope perfusion chamber

(Warner Instruments) and imaged on a multispectral spinning disk confocal

microscope similar to that described in Adams et al. (28), but with several

upgrades. Briefly, the illumination system consisted of a 2.5 W water-cooled

Innova 70c Krypton/Argon ion laser (Coherent, Santa Clara, CA) with the

488, 568, and 647 nm lines selected via a polychromatic acousto-optical

modulator (Neos Technologies, Melbourne, FL) for excitation of GFP,

x-rhodamine and cy-5, respectively. The laser illumination was delivered to

the confocal scanhead (model No. CSU-10; Yokogawa, Tokyo, Japan)

via a single mode optical fiber (Oz Optics, Ottawa, Ontario, Canada). The

scanhead was equipped with a triple dichromatic mirror (Semrock, Rochester,

NY) and bandpass emission filters (Semrock) mounted in a filterwheel ap-

paratus (Sutter Instrument, Novato, CA) toallowcaptureof GFP, x-rhodamine,

and cy-5 images in rapid succession. The scanhead was mounted on a model

No. TE-2000E (Nikon, Tokyo, Japan) automated inverted microscope

equipped with a Perfect Focus system (Nikon) to maintain focus at the PAA/

cell interface to within 620 nm over time and a linearly encoded robotic

stage (Applied Scientific Instruments, Eugene, OR) to allow sequential

analysis of cells at multiple stage positions. Temperature control was

maintained on the microscope stage using an air curtain incubator (Nevtek,

Burnsville, VA). Fluorescence images were generated with a 603 1.2 NA

Plan Apo water immersion objective lens (Nikon) using a 1.53 optovar.

Triplets of images of GFP-Paxillin, 568/580 40-nm spheres, and 647/670

40-nm latex spheres were captured at 30-second intervals using a model No.

HQ 2 camera (Roper, Sarasota, FL) equipped with an interline transfer CCD

(6.4 mm pixel size) cooled to ?30?C and operated in the 14 bit A/D mode.

The exposure time is 0.5 s and therefore much smaller than the typical

timescale on which cell traction changes (minutes). After imaging, cells

were perfused with 2 mL of 0.5% trypsin to release cells from the PAA

substrate, and an image of the unstrained substrate was taken in the 568 and

647 channels.

COMPUTATIONAL METHODS

Correlation-based particle tracking velocimetry

Extraction of a discrete displacement field describing the

deformation of PAA substrates is done by comparing images

before and after removal of the adhering cell with trypsin.

For this purpose, we use a combination of particle image

velocimetry (PIV) and particle tracking velocimetry (PTV)

which we call correlation-based PTV. We acquire images of

the fluorescent nanobeads at two different colors in two

different channels of a spinning disk confocal microscope. In

a first step, either of the images is partitioned into large

windows (?4–10 mm2) and a standard PIV routine (29) is

used to determine the deformation of the gel on a coarse

scale. The result is used as an offset for the subsequent

analysis and facilitates the detection of large deformations.

Traction Force Microscopy 209

Biophysical Journal 94(1) 207–220

Page 4

Subsequently, we use PTV and segment individual beads

from both channels to gain higher resolution. Small windows

(15 3 15 pix2? 1 mm2) are placed on top of each bead

before deformation Wfc1;c2gðx;yÞ and compared with shifted

windows after deformation Wd

ces c1, c2 indicate the different channels from which the

images are taken. Tracking is done by maximizing the cor-

relation of the mean-subtracted and normalized windows˜W:

fc1;c2gðx1x9;y1y9Þ: The indi-

˜Wciðx;yÞ ¼

Wciðx;yÞ ?1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

x;y¼1

N+

x;y¼1

N

Wciðx;yÞ

+

N

Wciðx;yÞ ?1

N+

x;y¼1

N

Wciðx;yÞ

!2

v

u

u

t

+

;

(3)

ccðx9;y9Þ¼1

2

ci¼fc1;c2g

+

x;y¼1

N

˜Wd

ciðx1x01x9;y1y01y9Þ˜Wciðx;yÞ;

(4)

where x0and y0are the offset values. The cross-correlation

formula normalizes each window to a standard brightness

and it is only the spatial contrast of the markers which is

taken into account. Therefore changes in brightness of fluo-

rescent marker beads (e.g., due to bead movement under

traction, slight focus drift, or photobleaching) are not a

problem. Note that by summing over the two channels, the

correlation in both channels is maximized simultaneously.

Mathematically, we avoid mixing terms in the cross corre-

lation by averaging the correlation coefficients and not the

windows themselves. The size of the windows, indicated by

N, is a major factor for the distinctness of the global corre-

lation peak. Therefore, we follow the suggestion of Ji and

Danuser (30) and start with a small window size which we

iteratively enlarge until a sufficient confidence level is

reached. Subpixel accuracy is achieved by finding the maxi-

mum of the correlation matrix with the three-point Gaussian

fit formula. For example, for the x-coordinate, it reads

x9subpix¼ x9

1

2fln½ccðx911;y9Þ?1ln½ccðx9 ? 1;y9Þ? ? 2ln½ccðx9;y9Þ?g:

ln½ccðx9 ? 1;y9Þ? ? ln½ccðx911;y9Þ?

(5)

This procedure yields precisions of ;0.2–0.4 pixel depend-

ing on the quality of images. Although peak locking was

usually not observed, we nonetheless implemented a con-

tinuous window shift routine (31) for usage with densely

covered and noisy images.

Boundary element method (BEM)

The BEM was the first method applied to quantitative trac-

tion force reconstruction on elastic substrates (10). The es-

sential idea here is to discretize the integral in Eq. 1 on a

computational mesh in such a way that the distance between

two nodes is sufficiently small as to justify an interpolation

between them. This permits us to carry out the integration

with a bilinear interpolation scheme with the values of the

traction at the nodes being still undetermined. First, a bound-

ary for the computational mesh is established. The minimal

choice is the cell boundary, but typically we fix the com-

putational boundary outside the cell boundary to facilitate

the discrimination of noise induced boundary effects from

real traction. Then a node is created above each measured

displacementwhich is located inside the computationalbound-

ary. Next, a triangular grid is produced with these nodes

using Delaunay triangulation. During force reconstruction,

extra nodes are added to produce a finer tessellation where

needed. It is a particular useful feature of the BEM that the

density of nodes can be increased at those locations where

the force is localized.

As shown in the Appendix, Eq. 1 can now be written in

matrix form:

uix¼ +

j

+

x9

Mijxx9fjx9:

(6)

As it is typical for boundary element techniques, M is

densely populated and ill-conditioned, thus making inver-

sion time-consuming and difficult. To make the whole pro-

cess more efficient, we have conceived a new method which

splits the calculation of M into subroutines for near, inter-

mediate and far fields. For the near field, the problem of a

divergent integrand is remedied by using polar coordinates.

Here, analytical integrals are calculated with the integration

boundaries being determined by the position of the nearest

triangles. The intermediate field, with a typical extension of

five triangles in radial direction, is treated numerically with

Gaussian quadrature in barycentric coordinates. The far field

is evaluated analytically in the framework of a multipolar

expansion. These three procedures are explained in more

detail in the Appendix.

Even if the matrix M is calculated in this way, additional

measures are required to arrive at a robust force estimate. The

usual approach in the field of boundary element techniques is

to implement a regularization scheme. This means con-

straining the solution to not only maximize its probability to

estimate the experimental data in a least-square sense, but to

also incorporate prior information about the expected trac-

tion field. We choose Tikhonov regularization and require

the discrete field fjx9to minimize the following target func-

tion:

kuix? +

j;x9

Mijxx9fjx9k21l2k+

j;x9

Lijxx9fjx9k2:

(7)

The simplest choice for L is the identity matrix (0thorder

regularization). Constraining the amplitude of the solution

with 0thorder regularization means suppressing high fre-

quency and thus potentially noisy components in fjx9(25).

The prior information here is that the traction field is mostly

located in small patches and zero elsewhere. Hence, sharp

210Sabass et al.

Biophysical Journal 94(1) 207–220

Page 5

peaks in the estimated solution are possible and will occur

locally. Another choice for L is a discrete difference between

the nodes (first-order regularization) (10,11). This choice is

based on the assumption that the traction field is smooth

rather than localized. Positive effects in regard to noise are

similar to those with 0thorder regularization while traction

plateaus are emphasized. For the numerical minimization of

Eq. 7 we used the MatLab package Regularization Tools

from Hansen (32). The regularization parameters were chosen

using the L-curve criterion and a cross-validation Ansatz.

Fourier-transform traction cytometry (FTTC)

Fourier methods have been introduced for traction force

reconstruction(24)becausetheconvolutionofEq.1becomes

a simple product in Fourier space. In detail, one has

(

k

˜ uik¼

+

j

˜Gij˜fj

)

;fik¼

+

j

˜G?1

ij˜ uj

()

k

:

(8)

The Green function can be calculated from Eq. 2:

?

¼2ð11nÞ

Ek3

?nkxky

˜Gijk¼2ð11nÞ

E

dij

k?nkikj

ð1 ? nÞk21nk2

k3

?

y

?nkxky

ð1 ? nÞk21nk2

x

!

:

(9)

Again we give both index notation and the full form. Note

that this tensor differs by an important minus sign in the off-

diagonal elements from the formula printed in Butler et al.

(24). The method proposed in that work proceeds as follows.

The regular displacement field uix is smoothed using a

frequency cutoff rule. Then it is transformed into Fourier

space, multiplied with the inverse Green function, and the

result is transformed back to real space. The transformations

can be done with standard techniques for fast Fourier

transform (FFT). The use of the FFT-algorithm to compute

the Fourier transform of the displacement field ˜ uikrequires a

regular, rectangular grid covering the whole image, which is

obtained from our irregular fields by biquadratic interpola-

tion. We found that the adaptive Gaussian window technique

performs rather badly in this context, because it is sensitive

to the geometry of the given field. Note that the use of a

regular rectangular grid is an important difference to the

BEM-procedure, which uses irregular grids, which in turn

allows locally adapting the node density.

In our work, we modified the FTTC suggested by Butler

et al. (24) by testing different smoothing procedures for the

displacement field, including an adaptive Wiener filter and a

Gaussian filter. As an alternative approach, we removed the

effects of noise directly from the calculated traction field.

Here we choose, in analogy to the BEM, a regularization

scheme and derive the corresponding variational equation:

Z Z

+l;jGliðx;x$ÞGljðx$;x9Þfjðx9Þ?+

1l2Z

j

Gjiðx;x$Þujðx$Þ

"#

dx$dx9

+

j

Hjiðx;x9Þfjðx9Þdx9¼0:

(10)

H is the square of L introduced above. This equation can now

be transformed to Fourier space and solved there:

?

For the regularization kernel Hij(x, x9) we choose the identity

matrix (0thorder regularization) or the square of an approxi-

mation for the Laplace operator (second-order regulariza-

tion). The whole expression on the right-hand side of Eq. 11

can, like in the former method, be calculated at once, making

the regularized method only marginally slower. The final

step is, like above, the inverse Fourier transformation.

˜fik¼

+

l;j

+

m

˜Gml˜Gmi1l2˜Hil

??1

˜Gjl˜ uj

()

k

:

(11)

Traction reconstruction with point forces (TRPF)

Here we follow the procedure introduced by Schwarz et al.

(25) for a traction microscopy study in which focal adhe-

sions had been tagged with a fluorescent label. In brief, one

assumes that all the traction is localized at discrete and

known positions, i.e., the focal adhesions. Then the traction

field is described by a sum of delta functions fiðxÞ ¼

+x9Fix9dðx ? x9Þ: This amounts to keeping only the first

order term in a force multipolar expansion. The integral in

the forward problem Eq. 1 is thus turned into a sum con-

necting each measured displacement to the set of force lo-

cations. The mathematical singularity has to be avoided by

introducing a cutoff in which displacements closer to a point-

force than approximately twice the typical size of a site of

adhesion are ignored. One thus arrives, similar to the BEM,

at a discrete, inverse problem where G is the Boussinesq

solution in real space from Eq. 2:

uix¼+

j;x9

Gijxx9Fjx9:

(12)

This formula can be easily inverted by means of 0thorder

Tikhonov regularization as described above for the BEM.

Simulated data: pointlike adhesions

To test the different variants of traction force microscopy

described above, we used extensive data simulation. In gen-

eral, we simulated both pointlike and spatially extended sites

of adhesion. The difference in the resulting displacement

fields is shown schematically in Fig. 1, B and C, respectively.

For pointlike adhesions, we randomly distributed point

forces over a finite region (the cell) and biased them to point

to the center of this field (the cell body). The force

Traction Force Microscopy 211

Biophysical Journal 94(1) 207–220

Page 6

magnitudes were taken to fluctuate randomly in a given

interval. For point forces, calculating the displacement field

simply corresponds to the direct problem defined in Eq. 12.

Displacement data was sampled from a random set of loca-

tions. Local displacement averaging was performed to mimic

the effect of presmoothing by the correlation tracker on real

data (window size 2 mesh sizes). Gaussian noise with a fixed

standard deviation of 0–10% of the maximum displacement

was added and then the traction force reconstruction was

performed. Note that even for 0% noise, the reconstructed

traction pattern will not reproduce the original traction pat-

tern due to the different discretization steps involved.

Simulated data: finite-sized adhesions

We calculated an analytical solution for the strain field

induced by a single circular adhesion with traction which is

constant over the whole adhesion site, compare Fig. 1 C.

Fourier transformation of the traction profile described by a

two-dimensional step function yields a Bessel function of the

first kind: FT½f0

placement field was calculated by back-transforming the

convolution with the Green function (9) into real space:

uiðxÞ ¼ ð1=2pÞ2Rð+j˜Gij˜fjÞðkÞe?ikxdk: The resulting field is

solutions for the inside and the outside of the adhesion. Use

of the superposition principle permitted assembling artificial

cells similar to the procedure described above for pointlike

adhesions. In contrast to this case, however, now the indi-

vidual adhesions not only differ in location and force magni-

tude, but also in size. The corresponding displacement fields

were obtained as above for the pointlike adhesions by choos-

ing random sites, averaging locally and addition of noise.

jQðr ? RÞ? ¼ f0

jð2pR=kÞJ1ðkRÞ: The dis-

composed of hypergeometric functions and splits up into

Dimensionless scores for quality of

force reconstruction

All presented traction force routines were used on the test

data. Optimization of filter parameters was done using a

parameter scan and were then held constant for all samples at

a given noise level. Three dimensionless scores were defined

and measured to quantitatively assess the relative perfor-

mance of the different methods.

Deviation of traction magnitude (DTM)

Relative difference of reconstructed and real traction mag-

nitude summed over all adhesions:

DTM ¼1

N+

i

kReconstructedtractionk?kRealtractionk

kRealtractionk

:

(13)

N represents the number of adhesions and i runs over all

adhesions. A negative DTM indicates that the measured

tractions are too small, a positive DTM indicates that they are

too large. The data on each adhesion is averaged before

calculating the indicated norms.

Deviation of traction magnitude in the surrounding (DTMS)

Traction magnitude in a circular ring around the adhesions

relative to the measured magnitude at the respective site:

DTMS¼1

N+

i

kReconstructedtractioninsurroundingk

kReconstructedtractiononadhesionk:

(14)

Width of the ring was fixed to 1.5 mesh sizes. The DTMS

quantifies the ability of different methods to reconstruct

edges and contours, because for a successful reconstruction,

the traction should decay rapidly outside the adhesion

boundary. DTMS is expected to lie between 0 (optimal) and

1 (worst).

Deviation of traction angle (DTA)

Correlation of simulated traction and measured mean trac-

tion vector on adhesion:

DTA¼arccos1

N+

i

Reconstructedtraction?Realtraction

kReconstructedtractionkkRealtractionk:

(15)

This definition corresponds to the difference in degree be-

tween the two (averaged) traction vectors.

RESULTS

Usage of two distinct kinds of fluorescent marker

beads improves measured displacement field

The spatial resolution of the reconstructed traction field is

directly related to the spatial resolution with which the dis-

placement field is sampled. The use of a multispectral con-

focal spinning disk microscope permits acquisition of images

from different fluorescence channels at high resolution (Fig.

2 A). To achieve an information content as high as possible,

we used two differently colored beads densely embedded in

the PAA gel. Fig. 2 B shows that the number of features is

much higher if a combination of both channels is used. Fig.

2 C shows the tracking result for one channel. Open and solid

circles indicate positions where beads could be only tracked

by enlarging the correlation window (effectively lowering

the spatial resolution) or not at all, respectively. Because the

quality of images usually differs from channel to channel, it

is impossible to interlace both extracted vector fields when

tracking each channel by its own. Fig. 2 D shows the cor-

responding result for two channels obtained with the newly

developed method of correlation-based PTV described above.

Because the density of tracked displacements is now ap-

proximately twice as large, the mesh size determining spatial

resolution is decreased by a factor of

ffiffiffi

2

p

; from ;700 nm for

212 Sabass et al.

Biophysical Journal 94(1) 207–220

Page 7

one channel to 500 nm for two channels. Thus our new

method using two colors results in a much denser and more

precise sampling of the displacement field than the tradi-

tional approach with one color only.

Spatial resolution of discretized methods strongly

depends on traction magnitude and chosen filter

According to the Nyquist-Shannon sampling theorem, the

optimal spatial resolution which can be achieved is deter-

mined by half the sampling frequency. For one- and two-

channel tracking, this should be ;1.5 and 1 mm, respectively.

To check whether this limit can be reached, we next

investigated the capability of the different computational

methods to distinguish very small traction sources. Resolu-

tion is defined by the Rayleigh criterion, i.e., as the minimum

distance between two point forces at which they can still be

separated. Fig. 3 A shows the chosen pattern of point forces

(red) and the resulting displacement field (blue). Fig. 3, B

and C, shows the results of the boundary element method

(BEM) with 0thand first-order regularization, respectively.

Fig. 3, D–G, shows the results of Fourier-transform traction

cytometry (FTTC) with four different treatments of noise

(Gaussian filtering, Wiener filtering, 0th, and second-order

regularization, respectively). Visual inspection of this ex-

ample demonstrates that very weak traction is not repro-

duced at all (marked with solid arrows in the original pattern,

Fig. 3 A). The reconstructions from Fig. 3, B–G, also show

that in several cases, close-by forces cannot be separated

anymore (marked with open arrows in the reconstructions,

Fig. 3, B–G). Yet all methods give a clear representation of

the overall traction pattern, and strong point forces can

always be distinguished clearly. Differences in resolution

between BEM and FTTC were mainly due to mesh geometry

and this gave the BEM a slight advantage. Moreover the

choice of the filter procedure strongly influences point-force

resolution. Qualitatively, those methods seem to work best

for which no open arrows had to be added, that is, BEM with

0thorder regularization (Fig. 3 B), FTTC with Wiener

filtering (Fig. 3 E), and FTTC with 0thorder regularization

(Fig. 3 F). As indicated by the space bar (5 mm) in Fig. 3, in

these cases we indeed reach a spatial resolution of 1 mm for

displacement data corresponding to two-channel tracking.

To quantitatively determine the detection limit of the

traction reconstruction, we generated 10 point force traction

patterns and measured, for a substrate stiffness of E ¼ 10

kPa, the reconstructed traction well outside the adhesion.

These values are interpreted to represent a background in the

traction pattern which potentially masks small forces. In Fig.

3 H we plot two times the standard deviation of this traction

background as a function of the noise level (measured in

absolute values, that is, pixel). This shows that increasing the

noise level only leads to a slight increase in the detection

limit.However,increasingtheabsolutelevelofthedisplacement

FIGURE 2

ment of two types of nanobeads with correlation-based

particle tracking velocimetry. (A) The two different nano-

beads (each with diameter 40 nm) are shown in red and

blue. A fibroblast fluorescently marked with GFP-paxillin

is shown in green. (B) Combination of both channels

(bottom) shows more, but less distinct features than one

channel alone (top). (C) Displacement field extracted from

one channel only. Open circles at the base of an arrow

indicate that this bead could only be tracked by enlarging

the correlation window, effectively lowering the spatial

resolution. Solid circles indicate beads which could not be

tracked. The resulting mesh size is 700 nm. (D) Displace-

ment field extracted from both channels simultaneously.

Despite the displacement density being twice as high as in

panel C, the number of enlarged windows and discarded

beads is significantly lower. The resulting mesh size is

500 nm. Space bar 5 mm.

Extractingthedisplacementfieldfrommove-

Traction Force Microscopy213

Biophysical Journal 94(1) 207–220

Page 8

field (represented by the median and maximum values in

pixel) by increasing the original traction leads to a strong

increase in the traction background. For example, for a

maximum displacement of approximately eight pixels, the

detection limit is ;200 Pa, corresponding to a force of 0.2

nN per mm2. For the same displacement, but for a doubled

substrate stiffness of E ¼ 20 kPa, the traction background

goes up by a factor of two, that is, to 400 Pa, corresponding

to a force of 0.4 nN per mm2. Therefore small forces cannot

be resolved if the overall magnitude of traction is high,

irrespective of the noise level.

BEM and FTTC produce comparable results

To arrive at a more general quantitative comparison between

the different variants of traction force microscopy, we mea-

sured the three different scores defined in Computational

Methods for circular adhesions with different noise levels in

the displacement field. Our results are summarized in Table

1. For all three scores used (DTM, DTMS, and DTA), the

closer its value to zero, the better the performance. In the

absence of noise, filtering or regularization do not improve

the results of FTTC. Here one sees that FTTC works

comparably well to BEM with 0thorder regularization, but

that BEM with first-order regularization is much worse. This

result persists in the presence of noise, so in general BEM is

best with 0thorder regularization. For FTTC, Table 1 shows

that Wiener filtering is always better than Gaussian filtering

and that 0thorder regularization is always better than second-

order regularization. Moreover, Wiener filtering and 0th

order regularization perform equally well, and similar to

BEM with 0thorder regularization. Thus these three methods

indeed perform equally well, as suggested qualitatively by

Fig. 3. However, there are dramatic difference in the required

time for computation. While on a standard desktop computer

BEM runs for several hours and requires large storage re-

sources, FTTC only requires seconds. In addition, the pro-

gramming effort is also highly reduced for FTTC, which can

be encoded in a few pages whereas the BEM is, in our hands,

spread out into more than a dozen subroutines. Thus FTTC

with Wiener filtering or 0thorder regularization is both

reliable and the most efficient choice in terms of resources.

FIGURE 3

tion at pointlike adhesions. (A) Random

configuration of point forces (red) and

resulting displacement field (blue).

Space bar 10 mesh sizes (?5 mm). (B

and C) Reconstructed traction magni-

tude using the boundary element method

(BEM). (B) BEM, 0thorder regulariza-

tion. (C) BEM, first-order regularization.

(D–G) Reconstructed traction magni-

tude using Fourier-transform traction

cytometry (FTTC). (D) FTTC, Gaussian

filter.(E)FTTC,Wiener filter.(F)FTTC,

0thorder regularization. (G) FTTC,

second-order regularization. Solid ar-

rows in panel A mark weak forces

which cannot be detected at all. Open

arrows mark positions where the recon-

struction cannot distinguish between

close-by forces in the original pattern.

The performance is best for panels B, E,

andF, witha spatial resolutionof 1mm.

(H) Traction background is a statistical

measure (n ¼ 10) for the detection limit

and is plotted here as a function of

absolute noise (in pixel) for different

absolute magnitudes of displacement

(median and maximum value given in

pixel). Traction background increases

stronger with displacement magnitude

than with noise.

Reconstruction of trac-

214Sabass et al.

Biophysical Journal 94(1) 207–220

Page 9

Traction magnitude of small adhesion sites is

underestimated with discretized methods

Table 1 also shows that the deviation of traction magnitude

(DTM) is always negative for both BEM and FTTC, thus

significant underestimation of traction is a persistent chal-

lenge with the discretized methods. The causes of systematic

and random underestimation are manifold but depend

strongly on the sampling density of the displacement field,

as suggested by the Nyquist criterion. Fig. 4 confirms this

view and shows that the analysis of adhesion sites which are

smaller than two mesh sizes is problematic. Fig. 4 A shows

the original traction pattern with nine randomly placed

circular adhesions (in red) and the resulting displacement

field (in blue). Fig. 4, B and C, show traction force recon-

struction with BEM with 0thorder regularization and FTTC

with Wiener filtering, respectively. The difference between

reconstructing traction on an irregular versus regular grid is

clearly visible. In both cases, the overall traction pattern is

nicely reproduced. In Fig. 4, D and E, the DTM is plotted as

a function of adhesion size (measured in units of mesh size)

for noise levels of 0 and 10%. Below a critical size, there

is an inverse relation between DTM and adhesion size,

meaning that force reconstruction is only reliable for large

adhesions. For 0% noise, the critical value is approximately

two mesh sizes, but shifts to larger values for higher noise

levels. In the presence of 10% noise, only adhesions with a

size approximately four times larger than the mesh size are

reconstructed in a reliable way. For smaller adhesions, the

DTM becomes strongly negative, thus the traction at these

adhesions (and, concomitantly, the overall strain energies)

are severely underestimated. In conclusion, it is critical to

start with a dense sampling of the displacement field,

because this results in small mesh sizes and thus in reliable

force reconstruction for small adhesions and better estimates

for strain energy.

Traction reconstruction with point forces is

precise, but highly depends on correct

localization of focal adhesions

To improve the force reconstruction at small adhesions, we

employed traction reconstruction with point forces (TRPF),

which explicitly starts from the assumption of highly lo-

calized force (25). Fig. 5 A shows a typical force recon-

struction with TRPF (green vectors) from a displacement

field with 10% noise (blue vectors). Fig. 5 B demonstrates

that now the DTM depends much less on the size of the

adhesion, resulting in a reliable force reconstruction over a

large range of adhesion sizes. However, in this method it is

critical that the localization of the adhesions is very precise.

Fig. 5 C shows the results of the force reconstruction (orange

vectors) if the sites of adhesion used for reconstruction have

been deliberately misplaced in regard to the original sites of

adhesion. The quantitative analysis using DTM and DTA in

Fig. 5 D shows that directional errors and strong magnitude

fluctuations occur if the adhesions are not placed well. This

finding limits the applicability of the method to cells with

few and distinct focal adhesions.

High resolution reconstruction for

fibroblast traction

We finally demonstrate how the whole setup can be used to

analyze fibroblast traction. Fig. 6 A shows an extension of a

stationary mouse embryo fibroblast with discrete focal

adhesions (marked by GFP-paxillin). To obtain the required

TABLE 1

element method (BEM), and Fourier-transform traction cytometry (FTTC)

Quantitative performance of different variants of two standard methods in traction force microscopy, namely boundary

BEM with regularizationFTTC with filteringFTTC with regularization

NoiseScore0thorder First-orderGaussian Wiener0thorderSecond-order

0% DTM

DTMS

DTA [?]

?0.24 6 0.19

0.18 6 0.06

8.8

?0.24 6 0.20

0.24 6 0.10

19.6

?0.28 6 0.19

0.23 6 0.02

7.92

5% DTM

DTMS

DTA [?]

?0.47 6 0.2

0.36 6 0.09

12.3

?0.59 6 0.10

0.56 6 0.07

32.8

?0.55 6 0.27

0.65 6 0.16

25.8

?0.43 6 0.21

0.37 6 0.06

12.3

?0.48 6 0.23

0.35 6 0.04

11.7

?0.47 6 0.25

0.42 6 0.09

14.1

10%DTM

DTMS

DTA [?]

?0.58 6 0.14

0.50 6 0.09

17.4

?0.62 6 0.16

0.64 6 0.17

33.0

?0.61 6 0.25

0.86 6 0.39

32.2

?0.46 6 0.22

0.48 6 0.12

19.9

?0.55 6 0.2

0.48 6 0.1

17.6

?0.48 6 0.25

0.55 6 0.18

23.2

Computation time 2–10 h

,2 s

The three scores used are deviation of traction magnitude (DTM), deviation of traction magnitude in the surrounding (DTMS), and deviation of traction angle

(DTA) (n ¼ 10). In each case, the methods performs the better the closer the score is to zero. For BEM, 0thorder regularization works better than first-order

regularization. For FTTC without noise, filtering or regularization does not improve the results. In the presence of noise, Wiener filtering works better than

Gaussian filtering and 0thorder regularization works better than second-order regularization. Both Wiener filtering and 0thorder regularization perform

similarly well, and comparable to BEM with 0thorder regularization. However, FTTC requires much less computation time. All computations were done on a

single 2 GHz Pentium processor with 4 GB memory.

Traction Force Microscopy 215

Biophysical Journal 94(1) 207–220

Page 10

spatial resolution for the displacement field, it is imperative

to image only a limited region of the cell. Therefore we have

chosen here a region of interest which does not have any

additional close-by adhesions that might change the traction

pattern inside the region of interest. Fig. 6 B shows an over-

lay with the displacement field extracted with two differently

colored nanobeads. As explained above, the mesh size of

;500 nm results in a spatial resolution of ;1 mm because

the computational reconstruction is combined with adequate

filtering. For the substrate stiffness E ¼ 15.6 kPa and the

given fibroblast strength, the traction background is ;500

Pa, corresponding to a force of 0.5 nN per mm2. In the recon-

struction we often find forces of up to 10 kPa, corresponding

to a force of 10 nN per mm2, that is, 20-fold above the

detection limit. Fig. 6, C and D, show the force vector fields

obtained with BEM and FTTC, respectively, both with 0th

order regularization. Note that for the BEM the computa-

tional mesh is a irregular grid restricted to the wedgelike

region including the cell contour. For FTTC, force vectors

are reconstructed on a square lattice covering the whole

image. In Fig. 6, E and F, we show color plots for the traction

magnitudes for BEM and FTTC, respectively. Both methods

give very similar results. In Fig. 6 G we show our results

obtained with TRPF. In this case, first one point has been

selected for each adhesion (for very large adhesions, two

points have been selected). The resulting forces then show

visual agreement with the results from BEM and FTTC. The

direct comparison between the three different methods

shown in Fig. 6 H for the small region marked in Fig. 6 A

nicely summarizes the difference between the three methods:

one force vector for TRPF, an irregular pattern for BEM, and

a regular pattern for FTTC. In particular, it shows that the

spatial resolution of BEM and FTTC is indeed ;1 mm.

DISCUSSION

This work aims at a thorough discussion of established and

newly developed procedures to reconstruct cellular traction

force on flat elastic substrates. In particular, we asked which

approach will guarantee an optimal spatial resolution and

magnitude reconstruction for different situations of interest.

As an initial step, we extended the known experimental

traction force protocols to the use of very small and differ-

ently colored fluorescent beads and a confocal microscope.

FIGURE 4

at finite-sized adhesion sites. (A) Circu-

lar adhesion sites with constant traction

(red) and resulting displacement field

(blue). Space bar 10 mesh sizes

(?5 mm). (B) Traction reconstruction

with BEM and 0thorder regularization

for 5% noise. (C) Traction reconstruc-

tion with FTTC and Wiener filtering for

5% noise. (D) Deviation of traction

magnitude(DTM)asafunctionofadhe-

sion size (measured in units of mesh

size) for 0% noise. DTM is optimal at

zero and worst for ?1 (complete under-

estimation) or 11 (complete overesti-

mation). Here DTM is negative, i.e., the

traction is systematically underesti-

mated. The inverse scaling (dotted line)

of magnitude deviation as a function of

adhesion size indicates sampling prob-

lems for adhesion sites which are

smaller than two mesh-sizes (;1 mm).

(E) Same plot but with 10% noise.

Necessary filtering shifts the effective

sampling frequency and only adhesion

siteslargerthanfourmeshsizes(;2mm)

can be properly examined (n ¼ 10).

Reconstruction of traction

216Sabass et al.

Biophysical Journal 94(1) 207–220

Page 11

The usage of two kinds of distinctly colored beads densely

embedded in the gel permits the extraction of displacement

fields with an average mesh size of 500 nm, setting the basic

scale for the spatial resolution of the force reconstruction to

1 mm. This resolution is an improvement by a factor of ;5–

10 compared with earlier work.

Cellular adhesion sizes vary considerably, ranging from a

few hundred nanometers in nascent adhesions of locomoting

cells to tens of microns for supermature focal adhesions in

myofibroblasts. For small adhesions, the mesh size of the

displacement data is larger than the adhesion feature size and

thus the Nyquist frequency determining the theoretical upper

limit for the resolution of traction forces is typically too low

to capture all the details of the exerted traction field. Still one

can ask which of the different computational methods per-

forms best in reconstructing cellular traction fields. The main

methods investigated here were BEM (10,11) and FTTC

(24). For both methods, here we suggested different im-

provements, including analytical integration procedures and

adaptive mesh generation for BEM and different filtering and

regularization schemes for both BEM and FTTC.

An intrinsic feature of both methods is that solving the

inverse stress-strain problem is equivalent to multiplying

each Fourier component of the displacement field by its

respective wave number. High frequency noise will thus be

amplified and has to be removed while leaving as much as

possible of the signal untouched. One can follow different

philosophies here: physically, it seems reasonable to impose

a smoothness constraint on the displacement field and filter it

before the calculation of traction stress is performed. This

technique only works with the Fourier transform methods.

On the other hand, one can argue that it is better to choose a

good solution a posteriori by inspection of its properties.

Regularization constrains the resulting traction field and one

can fine-tune it such as to compensate the above-mentioned

effect of noise amplification to a desired level in the solution.

FTTC is seen to work best with Wiener filtering and 0th

order regularization in simulated data while experience with

real data suggested that regularization is a more robust

approach. BEM mostly work well with 0thorder regulariza-

tion. An overall comparison of both approaches leads to the

conclusion that FTTC, when combined with a proper fil-

tering procedure, is in large parts comparable with BEM

in regard to resolution. Initial advantages of the boundary

element approach, resulting from the exact incorporation of

irregular data, are lost in the presence of noise. Aliasing and

boundary effects are not very prominent with boundary ele-

ment methods, but can limit the performance of FTTC. The

big advantage of FTTC is the very small run time compared

with the BEM.

Both discretized methods suffer from a systematic under-

estimation of traction at small adhesion sites. It is thus inter-

esting to ask whether traction force reconstruction with point

forces (TRPF) can properly measure force magnitudes inde-

pendent of adhesion size. The involved concept of a priori

localized traction sources does, in fact, serve to avoid above

bias. However, it is only applicable if the main assumption of

this method, a reasonable localization of all traction sources,

can be ensured. In general, traction measurement systems

which do not permit the presumably force-mediated, yet tem-

porary assembly of very small adhesion sites do seem to be a

slightly ill-fated choice if the whole spectrum of forces and

FIGURE 5

with point forces (TRPF). (A) Displace-

ment field includes 10% noise. Point

forces as calculated with TRPF (green)

are localized in the center of adhesion

sites (red). Space bar 10 mesh sizes (?5

mm). (B) Deviation of traction magni-

tude (DTM) as a function of adhesion

size as in Fig. 4, D and E. The under-

estimation of traction magnitude is

small and does not depend strongly on

adhesion size with TRPF. (C) Recon-

struction of traction when point forces

(orange) are not localized exactly in the

center of adhesions, as it may occur

with small and ill-defined adhesion

sites. (D) Standard deviation of traction

magnitude and directional error are much

higher if point forces are not localized

properly (orange) compared to correct

localization (green) (n ¼ 10).

Traction reconstruction

Traction Force Microscopy 217

Biophysical Journal 94(1) 207–220

Page 12

morphologies of cellular adhesion sites are to be studied in

migrating cells.

Bearing in mind that several mechanisms may contribute

to the force development at focal adhesions, a high spatial

resolution is the sine qua non condition for a quantitative

understanding of traction forces. From the work presented

here, it is clear that dense displacement fields are the most

important strategy to achieve this goal. Then a major

drawback seems to be the concern that incorporating many

marker beads in the gel contradicts the assumption of linear

elasticity in the substrate. However, inclusions in an elastic

material are known to perturb Young modulus and Poisson

ratio only in a surrounding shell of the scale of the bead

diameter (33). Therefore this effect might only be a problem

when using large beads (e.g., microbeads).

In summary, our work shows that high resolution traction

force microscopy relies on the combination of advances in

substrate preparation, image processing, and computational

force reconstruction. The systematic and quantitative com-

parison presented here shows that depending on the specific

situation of interest and the resources at hand, different ap-

proaches are useful. For example, BEM with 0thorder regu-

larization is the best choice to obtain high resolution traction

patterns for migrating cells with many small adhesions and

for small noise level in the displacement data. If computer

time is a limiting factor, FTTC with 0thorder regularization

is an almost equivalent alternative, especially at higher noise

levels. For stationary cells in which focal adhesions can be

well localized, TRPF is both computationally cheap and

reliable.

FIGURE 6

fibroblast. (A) Image section of an extension of a fibroblast

marked by GFP-paxillin (green). (B) Overlay with dis-

placement field extracted with two differently colored

nanobeads. (C) Traction vector reconstruction using BEM

and 0thorder regularization. The computational mesh

inside the wedge-shaped region is chosen such that the cell

contour is well included. (D) Traction vector reconstruc-

tion using FTTC with 0thorder regularization. The

computational mesh is a simple square lattice required

for the FFTs. (E) Traction magnitude for the BEM-result.

(F) Traction magnitude for the FTTC-result. Units of color

bar given in Pascal. Both methods give similar results with

a spatial resolution at ;1 mm and a lower bound for

traction detection of ;500 Pa. (G) Traction vector

reconstruction with TRPF. For each adhesion, one point

has been selected (for very large adhesions, two points

have been selected). (H) Comparison of TRPF, BEM, and

FTTC for the region of interest marked in panel A around

one large adhesion.

Traction forces at adhesions of a stationary

218 Sabass et al.

Biophysical Journal 94(1) 207–220

Page 13

APPENDIX: DETAILED DESCRIPTION OF THE

BOUNDARY ELEMENT METHOD

The task is here to sequentially treat all possible interactions of tractions at

nodes x9 and displacements at x. For brevity, in the following we use the

Einstein sum convention. Traction interpolation between the nodes is done

using triangular shape functions Px9

tinuous coordinates inside the triangle and x9i, i 2 {1, 2, 3} are the locations

of its corners (nodes). Shape functions are defined in the usual way as the

normalized area spanned by vectors connecting two of the corners x9 with

the point xtriand can thus be expressed as determinants:

????

Px9

scheme, incorporates information about all the triangles surrounding the

node x9

iin one matrix element:

Z

The computation time needed to loop through the combination of all nodes

with all displacements can become intolerably long if high resolution is to be

achieved with quadrature of Eq. 17. The exclusive use of numerical inte-

gration will thus demand for supercomputers or related facilities. Hence, we

resort to a partial implementation of analytical integral results and also

benefit from the high accuracy of this approach. The idea is here to compute

the displacement field emanating from each triangular element, in analogy to

a multipole field, depending on the distance. Separate subroutines were

devised for three distinct cases.

iðxtriÞ; where xtrisymbolizes the con-

Px9iðxtriÞ¼eijk

xtri x9j x9k

111

????

?????

x91 x2 x93

111

????:

(16)

Interpolation of traction is written in the following form: fjðxtriÞ ¼

iðxtriÞfjx9

i: The discretized operator M, integral over the interpolation

Mijxx9i¼ +

triangles

D

Gijðx;xtriÞPx9iðxtriÞdx2

tri:

(17)

Near field

Displacement is on top of the node. The Greens function diverges for

r ¼ kx ? x9k/0: This difficulty is circumvented by usage of polar

coordinates and the location of a node on top of each displacement. Any

dependence of the Greens function from the distance is thus removed by the

functional determinant:

Z

DGijðx;xtriÞPx9iðxtriÞdx2

tri¼

Zu

0

ZRðuÞ

0

ðGijrÞðuÞPx9iðr;fÞdrdu:

(18)

R(u) is an elementary trigonometric expression which, however, diverges

for certain u. This necessitates a coordinate rotation for each triangle in a

preparatory step. The implementation of a precomputed, analytical integral

solution makes the following evaluation very fast. The such calculated

integral in the near-field is exact to the accuracy of the approximation by the

triangulation.

Intermediate field

Displacement is in the surrounding of the node. Lack of better possibilities

necessitates the use of Gaussian quadrature in this region. However,

rewriting the integral to barycentric coordinates ðPx9

shape functions as coordinates, facilitates the implementation:

Z

¼D

0

0

1; Px9

2Þ; i.e., usage of the

D

Gijðx;xtriÞPx9iðxtriÞdx2

Z1

D is the functional determinant and equals the area of the parallelogram

spanned by two of the triangle sides.

tri

Z1?Px92

Gijðx;Px91;Px92ÞPx9idPx91dPx92:

(19)

Far field

Displacement is far from the node. The integrandin Eq. 19 is expanded up to

third order around the center of mass of the triangle (DPi¼ Px9i?1

Gijðx;Px91;Px92ÞPx9i

? +

n¼0

3):

3

@

@Px91

DP11

@

@Px92

DP2

??nGijðx;Px91;Px92ÞPx9i

n!

????

Px9¼1

3

(20)

:

This procedureleads to lengthy analyticalterms whichcan be evaluated very

quickly in a vectorized routine. The accuracy of the approximation for the

integral mainly depends on the ratio of the dimensions of the triangle to the

distancetothe displacements. We foundthat ratiosof 1:3–1:4alreadyleadto

results which differ only by a very small percentage from the numerical

values obtained with Gauss quadrature.

Profound gratitude is expressed by B.S. to Barbara Sabass for inspiration,

encouragement, and support! The authors thank Timo Betz, Ke Hu, and

Rudolf Merkel for helpful discussions.

U.S.S. was supported by the Emmy Noether Program of the German

Research Foundation. B.S. and U.S.S. are supported by the Center for

Modeling and Simulation in the Biosciences (BIOMS) at Heidelberg.

M.L.G. is supported by the Jane Coffin Childs fund and a Burroughs

Wellcome Career Awards at the Scientific Interface award. C.M.W. is

supported by National Institutes of Health Director’s Pioneer Award

DP10D435 and American Heart Association Established Investigator

0640086N.

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