Soft Matter

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3D Viscoelastic t

aSchool of Engineering, Brown University, 1

E-mail: franck@brown.edubThe Wallace H. Coulter Department of Bio

Technology and Emory University, and

Bioengineering and Bioscience, Georgia InsticDepartment of Molecular Pharmacology, P

Biomedical Engineering, Department of Ort

RI, USA

† Electronic supplementary informa10.1039/c4sm01271b

Cite this: Soft Matter, 2014, 10, 8095

Received 12th June 2014Accepted 14th August 2014

DOI: 10.1039/c4sm01271b

www.rsc.org/softmatter

This journal is © The Royal Society of C

raction force microscopy†

Jennet Toyjanova,a Erin Hannen,b Eyal Bar-Kochba,a Eric M. Darling,ac

David L. Henanna and Christian Franck*a

Native cell–material interactions occur onmaterials differing in their structural composition, chemistry, and

physical compliance. While the last two decades have shown the importance of traction forces during cell–

material interactions, they have been almost exclusively presented on purely elastic in vitro materials. Yet,

most bodily tissue materials exhibit some level of viscoelasticity, which could play an important role in how

cells sense and transduce tractions. To expand the realm of cell traction measurements and to encompass

all materials from elastic to viscoelastic, this paper presents a general, and comprehensive approach for

quantifying 3D cell tractions in viscoelastic materials. This methodology includes the experimental

characterization of the time-dependent material properties for any viscoelastic material with the

subsequent mathematical implementation of the determined material model into a 3D traction force

microscopy (3D TFM) framework. Utilizing this new 3D viscoelastic TFM (3D VTFM) approach, we

quantify the influence of viscosity on the overall material traction calculations and quantify the error

associated with omitting time-dependent material effects, as is the case for all other TFM formulations.

We anticipate that the 3D VTFM technique will open up new avenues of cell–material investigations on

even more physiologically relevant time-dependent materials including collagen and fibrin gels.

1 Introduction

Over the past two decades Traction Force Microscopy (TFM) hasemerged as a powerful, quantitative approach for characterizingthe physical interactions between cells and their surround-ings.1–4 TFM has provided a deeper understanding on the role ofphysical forces during cellular homeostasis, disease progres-sion, and many other cellular processes. For example in 2005,Paszek et al. showed, using TFM that tensional homeostasis,i.e., the regulation of intracellular tension, plays a signicantrole in determining the ultimate outcome in cancer cellmalignancy.5

With the availability of improved spatial imaging tech-niques, TFM has evolved to deliver two-dimensional and three-dimensional (3D) descriptions of cell-surface tractionmeasurements of single cells and cell sheets.6–11 It has beenshown that most of the tissues in the body are characterized astime-dependent viscoelastic materials.12–14 Yet, until recently,almost every TFM approach featured a linear elastic continuum

82 Hope St. Box D, Providence, RI, USA.

medical Engineering, Georgia Institute of

The Parker H. Petit Institute for

tute of Technology, Atlanta, Georgia, USA

hysiology, and Biotechnology, Center for

hopaedics, Brown University, Providence,

tion (ESI) available. See DOI:

hemistry 2014

mechanics framework.15–17 Through a recent signicantadvancement in our displacement detection scheme we showedthat cells on so gels are capable of generating large materialdeformations exceeding the traditional linear formulationlimits.18 We addressed these observations by presenting a new,reformulated hyperelastic 3D TFM approach, capable of accu-rately capturing nite elastic material deformations.18

Still, there is currently no documented approach of incor-porating material viscosity into a TFM framework. Such aformulation may be of signicant importance given that theliterature has shown that most tissues, including moresophisticated in vitro culture systems, possess some level ofviscosity as part of their microstructure.19–22 Since cells possessthe ability to sense and probe their microenvironment dynam-ically, incorporating viscoelastic material properties into a TFMframework is important to provide the most accuratemeasurements of cell–material interactions.23–26

In this paper, we address this challenge by presenting anintegrated material characterization and 3D TFM approach toperform cell traction force measurements on materials withtime-dependent physical properties. This technique, named 3DViscoelastic Traction Force Microscopy (3D VTFM) is generalenough to allow the incorporation of any linear or non-linearelastic and viscous material properties. Specically, we presenta combined experimental and numerical approach on deter-mining the viscoelastic material properties of so agarosesubstrates, and their subsequent use in determining cell-induced material tractions. To examine differences in thespatial distribution and magnitude of the time-dependent

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surface traction elds due to the viscosity of the substratematerial, we construct an analytical test case. Finally, wepresent actual experimental cell surface traction data using our3D VTFM technique of breast cancer cells deforming collagen-functionalized agarose substrates.

2 Materials and MethodsGlass coverslips and microscope slides surface modication

Circular glass coverslips (25 mm, diameter, Fisher Scientic,Waltham, MA) were chemically modied to allow covalentattachment of polyacrylamide substrates using previouslydeveloped protocols.1,6,7 Briey, glass coverslips and slides wererinsed with ethanol and placed in a Petri dish containing asolution of 0.5% (v/v) 3-aminopropyltrimethoxysilane (Sigma-Aldrich, St. Louis, MO) in ethanol for 5minutes. Next, coverslipswere washed with ethanol and submersed in a solution of 0.5%glutaraldehyde (Polysciences, Inc., Warrington, PA) in deion-ized (DI) water for 30 minutes. Activated coverslips were washedwith DI water and le to dry.

To allow easier gel detachment, rectangular glass micro-scope slides (75 � 25 mm) were chemically modied to createhydrophobic surface. Briey, glass coverslips were placed in aPetri dish containing 97% (v/v) hexane (Fisher Scientic, Wal-tham, MA), 2.5% (v/v) (tridecauoro-1,1,2,2-tetrahydrooctyl)-triethoxysilane (SIT) (Gelest, Morrisville, PA), and 0.5% (v/v)acetic acid (Sigma-Aldrich, St. Louis, MO) for approximately 1minute. Coverslips were then removed and le to dry.

Preparation of polyacrylamide-covered coverslips

Polyacrylamide (PA) gels were prepared from acrylamide (40%w/v, Bio-Rad Laboratories, Hercules, CA) and N,N-methylene-bis-acrylamide (BIS, 2% w/v, Bio-Rad Laboratories, Hercules,CA) stock solutions as described previously.1,6,7 The concentra-tions of acrylamide and BIS were chosen to be 8%/0.1%.Crosslinking was initiated through the addition of ammoniumpersulfate (Sigma-Aldrich, St. Louis, MO) and N,N,N,N-tetra-methylethylenediamine (Life Technologies, Grand Island, NY).The PA solution was vortexed for about 30 seconds, and 12 mL ofthe PA solution were pipetted on the surface of an activatedcoverslip and sandwiched with a hydrophobic glass slide. PA gelthickness was measured to be �5 mm. PA gel substrates werethen submerged in distilled water and allowed to polymerizeand hydrate for approximately an hour. Once the coverslip wasremoved, PA gel substrates were le uncovered at roomtemperature to dry out. Exploiting the strong cohesion betweenagarose and PA, the thin PA layer was used to rmly attach eachagarose substrate to the underlying glass coverslip (Fig. 1(a)).

Preparation of agarose substrates

For agarose substrate preparation, 0.5% (w/v) agarose powder(Benchmark Scientic, Edison, NJ) was dissolved in 1� PBS(Invitrogen, Carlsbad, CA). Each agarose solution was heated inan oven to approximately 90 �C for complete dissolution. Itshould be noted, that the vial with the solution was looselycovered, yet boiling was avoided at all times. While heating, the

8096 | Soft Matter, 2014, 10, 8095–8106

solution was stirred occasionally, until the agarose powder hadcompletely dissolved. Next, the agarose solution was combinedwith 10% (v/v) of 0.5 mm yellow-green uorescent microspheres(Invitrogen, Carlsbad, CA). All agarose samples were thoroughlyvortexed, and 35 mL of the nal agarose solution were sand-wiched between a dried PA gel-covered coverslip and a plainglass slide (75� 25mm). Upon complete gelation of the agarosesolution, the assembly was immediately immersed in distilledwater for approximately one hour. Finally, agarose coverslipswere carefully peeled off the plain glass cover slide (Fig. 1(a)).The nal thickness of agarose gel layer aer swelling wasmeasured to be �40 mm.

Collagen-functionalization of agarose substrates

To promote cell attachment, agarose substrates were function-alized with type I collagen using the bifunctional crosslinker,sulfo-SANPAH (Thermo Fisher Scientic, Waltham, MA).2,6,7

Fig. 1 presents a complete overview of the functionalizedagarose substrate preparation setup. Excess water was removedprior to deposition of 100 mL of sulfo-SANPAH (1 mgmL�1) ontothe surface of each lm, followed by a 15 minute exposure to UVlight. The darkened sulfo-SANPAH solution was aspirated andthe procedure was repeated. The samples were thoroughlywashed with DI water, and covered with a solution of 0.2 mgmL�1 bovine type I collagen (BD Bioscience, Franklin Lakes,New Jersey) and le undisturbed at 4 �C overnight. Followingovernight incubation, the substrates were rinsed three timeswith 1� PBS and sterilized with UV irradiation before depos-iting cells.

Cell culture and seeding

MDA-MB-231 metastatic breast adenocarcinoma cells werechosen as a representative contractile cell phenotype. Cells werecultured in medium containing 10% fetal bovine serum and 1%penicillin–streptomycin in Dulbecco's Modied Eagle Medium(DMEM, Invitrogen, Carlsbad, CA). Cells were maintained in theincubator at 37 �C and 5% CO2. Seeding density was 30 cells permm2. Once seeded, cells were allowed to spread for approxi-mately 5 hours in the incubator prior to imaging.

Live cell imaging

Three-dimensional image stacks were acquired using a Nikon A-1 confocal system mounted on a TI Eclipse inverted opticalmicroscope controlled by NI-Elements Nikon Soware. A 40�plan uor air objective mounted on a piezo objective positionerwas used for all the experiments, which allowed imaging atspeeds up to 30 frames per second. Green 0.5 mm uorescentmicrospheres were embedded into the substrate and excitedwith an Argon (488 nm) laser. 512 � 512 � Z voxels (102 mm �102 mm � Z) confocal volume stacks were recorded every 100seconds with Z � 128 voxels (38 mm), illustrated in Fig. 1(b). Toensure physiological imaging conditions within the imagingchamber, temperature and pH were strictly maintained at 37 �Cand�7.4 during time lapse recording as previously described.6,7

The outline of the cell was estimated from phase contrastmicroscopy images.

This journal is © The Royal Society of Chemistry 2014

Fig. 1 Schematic of (a) the agarose substrate preparation and (b) a representative volumetric image of the agarose substrate embedded withgreen fluorescent microspheres recorded by laser scanning confocal microscopy.

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Calculation of cell-generated displacements

Cell-generated full-eld displacements were determinedfollowing a similar methodology described in Tojanova et al.18

First, three-dimensional time-lapse volumetric images of uo-rescent beads embedded in agarose substrates were recordedusing laser scanning confocal microscopy (LSCM). Then, themotion of embedded uorescent beads was tracked in all threedimensions using a new fast iterative digital volume correlation(FIDVC) algorithm.27 This cross correlation-based algorithm hasthe capability to capture large material deformations byutilizing a built-in iterative deformation method (IDM). Byutilizing the IDM, our FIDVC technique is capable of reducingits volumetric subset size to 32 voxels, or lower depending onthe signal content within each subset, without introducingsignicant correlation error.27 The nal results feature signi-cantly higher spatial resolution, signal-to-noise, and fastercomputation times than our previous DVC method.28

Nanoindentation measurements

Force–indentation measurements of agarose gels were collectedusing an Asylum MFP-3D-BIO atomic force microscope (AFM)(Asylum Research, Santa Barbara, CA) following a similarprocedure as previously described.29,30 The indenter geometryconsisted of individual borosilicate glass spheres (5 mm diam-eter) attached to the tip of AFM cantilevers (Novascan Tech-nologies, Inc., Ames, IA). Fig. 2(a) provides a schematic overviewof the indentation experiments. The cantilever spring constant

Fig. 2 Schematic of (a) the experimental setup of AFM indentation of thModeling setup mimicking the indentation experiment. The reference poSchematic of the finite-deformation viscoelastic agarose constitutive m

This journal is © The Royal Society of Chemistry 2014

was determined to be 0.02 N m�1 using the power spectraldensity of the inherent thermal noise uctuations of thecantilever.31

To probe the viscoelastic material properties of our agarosesubstrates, loading–unloading indentation experiments wereperformed at three different cantilever approach velocities: 0.1mm s�1, 1 mm s�1, and 10 mm s�1. Additionally, 30 second forcerelaxation tests were performed using an initial loading velocityof 10 mm s�1 and an indentation depth of 450 nm.

Data were sampled at 5 kHz for loading–unloadingmeasurements and at 200 Hz for 30 s for viscoelastic relaxationmeasurements. A 2–3 nN force trigger, resulting in 400–600 nmindentation depths, was used to prescribe the point at which thecantilever approach was stopped and either retracted forloading–unloading tests, or held constant for relaxation tests.To determine the initial point of contact, we used a contactpoint extrapolation method.32 In brief, the upper 50% of forceand indentation data were t with a linearized version of theHertz model, and the contact point was extrapolated by backcalculation with the slope and intercept. All data past thecontact point were used in our least-squares analysis with ournite element simulation results. To account for any day-to-dayvariability among the agarose substrates, all AFM indentationtests were performed on the same batch of agarose, and on thesame day as the cell experiments. All experiments were carriedout at room temperature in a uid environment. The AFM wasallowed to equilibrate before each measurement period tominimize deection laser and/or piezo dri.

e agarose substrate with a spherical cantilever tip radius of 2.5 mm. (b)int (RF) to control the indentation is set at the center of the sphere. (c)odel.

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Finite element simulation

Simulations of the nanoindentation process were conductedusing the commercial nite element soware, ABAQUS 6.12(Dassault Systemes Simulia Corporation, Providence, RI). Thegeometry of the agarose substrate was discretized by using 2Daxisymmetric elements (CAX4H). The indenting spherical tipwas modeled as analytical rigid solid with a radius of R ¼ 2.5mm. The agarose substrate boundary conditions were set tomimic those of the indentation experiments with frictionlessinteractions between the indenter tip and sample surface(Fig. 2(b)). The indentation simulation was executed using thesame displacement control loading condition as in the AFMexperiments with the same loading rates. The material behaviorof agarose used for tting the experimental force–indentationcurves is a Neo-Hookean rubber elasticity model with a nite-deformation Maxwell element in parallel, shown schematicallyin Fig. 2(c) and detailed mathematically in the Appendix. Ourprocess of material parameter determination is described in thefollowing section.

Table 1 Table of the viscoelastic material constants for agarosesubstrate

m Shear modulus of hyperelastic spring 1600 PaK Bulk modulus of hyperlastic spring 3700 Pamneq Non-equilibrium shear modulus 700 Pah Maxwell element viscosity 800 Pa s

3 Agarose material characterization

The objective of this investigation is to present the reader with acomplete approach of determining the viscoelastic materialproperties of a given substrate material, and provide astraightforward approach of implementing the determinedmaterial model into a 3D TFM framework for cell tractionmeasurements. To illustrate this process in detail, we chose anagarose gel as a model viscoelastic substrate material.33,34

Typical material characterization techniques for determiningthe material properties of so hydrogels and tissue-mimickingmaterials have involved uniaxial compression, tension, AFMindentation and shear rheometry experiments. AFM indenta-tion characterizations are attractive for determining TFM-substrate properties, since they have the ability to measure thelocal mechanical properties directly at the surface where the cellis applying its tractions.

While elastic material AFM-based characterizationapproaches are relatively straightforward, the analysis of time-dependent material properties usually requires additionalexperimental measurements, such as creep or stress (force)relaxation tests.30 If a large strain constitutive relation is sought,numerical simulation of the experimental procedure, followedby least-square tting of the numerical to the experimental dataprovides a robust overall material characterization approach.

Fig. 2(a) presents a schematic of the AFM indentationexperiments that were utilized to collect force–indentationcurves of agarose substrates at different loading rates, as well asforce relaxation tests (see technical details in Materials andMethods). Next, we constructed a simple, axisymmetric FEMmodel (Fig. 2(b)) mimicking the experimental indentationprocess using the non-linear viscoelastic material model shownschematically in Fig. 2(c). For a full mathematical description ofthe model, the reader is referred to the appendix. Briey, theconstitutive model consists of two contributions, represented asbranches in Fig. 2(c). The equilibrium, or time-independent,

8098 | Soft Matter, 2014, 10, 8095–8106

response is modeled through a Neo-Hookean spring (repre-sented schematically as the le branch in Fig. 2(c)) with its freeenergy given mathematically by eqn (1). The material parame-ters associated with the equilibrium response are the equilib-rium ground-state shear modulus m and bulk modulus K. Theright branch of Fig. 2(c) represents the non-equilibrium, ortime-dependent, contribution and is modeled through a nite-deformation Maxwell element. We note that the Maxwellelement only affects the deviatoric, or shear, response, leavingthe volumetric response time-independent. The Maxwellelement includes a Hencky spring with free energy given by eqn(17) – referred to in this way due to the use of the Hencky, orlogarithmic, strain in eqn (17) – in series with a Mises dashpotwhose ow rule is given by eqns (20) and (21). The non-equi-librium materials parameters are the non-equilibrium shearmodulus mneq and viscosity h. Here, the use of the word non-equilibrium is for purely descriptive purposes to denote allmaterial properties associated with the time-dependentMaxwell element.

The emphasis behind the model was to use a minimal set oftting parameters to appropriately describe the viscoelasticbehavior of agarose under nite deformations. The character-istic size of our indentation process – as well as the deformationprocesses induced by cells – is comparable or smaller than theaverage pore size of typical agarose gels.35,36 Therefore, it isreasonable to assume that solvent diffusion relative to the gelnetwork equilibrates quickly compared to deformation time-scales, so that the diffusion process is always in equilibrium. Toinform our choice of bulk modulus, we utilize the observationsof Zhao et al.37 and set the equilibrium Poisson's ratio in thelinear (small) deformation regime to be 0.3. That is to say, usingthe familiar relations between elastic constants in linear elas-ticity, we have that the bulk modulus is given by K¼ 2m(1 + n)/(3� 6n) with m the equilibrium shear modulus and n taken to be0.3. Once the bulk modulus K has been xed in this manner, theremaining three material parameters, m, mneq, and h weredetermined by iteratively adjusting the numerical FEM simu-lation to t the experimental force relaxation data in a least-squares sense. The resulting best-t material parametersdescribing the nite viscoelastic behavior of agarose are pre-sented in Table 1.

Fig. 3(a) and (b) present the experimental and best-tnumerical indentation results for a typical force-relaxationexperiment. The agarose sample is rst indented to a depth ofapproximately 450 nm using a constant loading rate of 10 mms�1 (Fig. 3(a)). Then, while maintaining the current indentationdepth, the force is allowed to relax over a total period of 30seconds (Fig. 3(b)). Comparison between the experimental and

This journal is © The Royal Society of Chemistry 2014

Fig. 3 Representative experimental indentation results for agarosesubstrates (dots) and corresponding numerical results (solid line) ascomputed from fitted material constants. (a) Force (nN) vs. indentationdepth, d (mm) at a loading rate of 10 mm s�1. (b) A representative forcerelaxation plot for agarose substrates.

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numerical force–indentation curves shows good agreement.Given the relatively simple nature of the viscoelastic materialmodel presented in Fig. 3(b) (see simple shear example in ESIFig. 1 and 2†), some differences especially at larger indentationdepths (Fig. 3(a)) can be expected.

Fig. 4 compares the experimental indentation loading–unloading force curves to the numerical indentation resultsbased on the same material parameter set (given in Table 1)used to describe the force relaxation curves in Fig. 3. As Fig. 4shows, the FEM simulations are in good agreement withthe experimental results for all three different loadingrates. Indenting the agarose gel at different loading ratesallows for the examination of the time-dependent, viscousmaterial behavior, which the model seems to captureadequately. When comparing Fig. 4(a)–(c), only the loadingrate of 0.1 mm s�1 leads to signicant loading–unloadinghysteresis, which is indicative of viscoelastic relaxationduring the indentation cycle. This is expected as the intrinsicmaterial relaxation time, s, is on the same order of magnitudeas the time duration for the loading–unloading cycle for the0.1 mm s�1 loading rate. The indentation force curves shownin Fig. 4 are representative of 5 indentation measurements ateach loading rate.

‡ Notational conventions: The symbol V denotes the gradient with respect to thematerial point x in the reference conguration; a superposed dot denotes thematerial time-derivative. We write tr A and det A for the trace and determinateof A, and dev A ¼ A � (1/3)(tr A)I denotes the deviatoric part of A. The productof tensors A and B is denoted by AB (or AikBkj in component form), and theinner product is A : B (or AijBij). The magnitude of A is |A| ¼ ffiffiffiffiffiffiffiffiffiffi

A : Ap

.

4 3D Viscoelastic Traction ForceMicroscopy (3D VTFM)

Cell-generated full-eld displacements and tractions can bedetermined in a variety of different ways, with two of the mostcommon approaches being image correlation-based measure-ments and single particle tracking algorithms. We previouslypresented a large-deformation 3D image correlation-basedapproach for determining cellular displacement elds with highresolution.18,27 Regardless of the chosen displacement detectionscheme, once the 3D displacement eld is obtained, its spatialderivative, i.e. the deformation gradient, is used within theframework of the previously described viscoelastic materialmodel (Fig. 2(c)) to determine the cell surface tractions. Giventhat most tissue materials and more complex hydrogels

This journal is © The Royal Society of Chemistry 2014

exhibit some time-dependent and viscous behavior,19–21 and theobservation that cells can generate nite deformations,18 accu-rate cell traction analysis requires the utilization of a nite-deformation viscoelastic material model. Although our exampleis specic to agarose, we purposefully set up the methodologysuch that the same steps can be utilized to incorporate other,experiment-specic, viscoelastic material models into the sameoverall 3D VTFM framework.

Finite deformations are described by a displacementfunction, y ¼ x + u(x, t), which represents a one-to-onemapping of points x in the reference conguration to points yin the deformed conguration at each time t.‡ The deforma-tion gradient is then the gradient of the displacement functionwith respect to x, that is F ¼ Vy ¼ I + Vu(x), with the volumeratio given by the Jacobian of F, J ¼ det F > 0. Experimentally,the deformation gradient tensor, F, can be directly calculatedfrom the full-eld displacements using various discretedifferentiation kernels. Farid et al., and Bar-Kochba et al.provide a description of several robust differentiationkernels.27,38

To specify the mechanical response of the equilibriumbranch of the material model in Fig. 2(c), we assume that thespring is described by a Neo-Hookean free energy of the form

j eqðFÞ ¼ m

2

�ðtr BÞJ 2=3

�3

�þ1

2KðJ�1Þ2; (1)

where m and K are the ground-state, equilibrium shear and bulkmoduli, respectively, and B ¼ FFu is the le Cauchy–Greentensor. The rst term represents the free energy due to distor-tional (constant volume) deformations, and the second termrepresents purely volumetric deformations. The associatedCauchy stress is then

s eq ¼ J�1vj eqðFÞ

vFFu

¼ m

J5=3

�B� 1

3

�tr B

�I

�þ K

�J �1

�I:

(2)

The time-dependent, viscous material response is describedby theMaxwell element (Fig. 2(c), right branch). For this branch,the deformation gradient, F, is decomposed into its elastic andviscous parts: F ¼ F eF v, where F e is the elastic deformationgradient, representing the deformation of the spring, and Fv isthe viscous deformation gradient, representing the deformationof the dashpot. To specify the mechanical response of the non-equilibrium branch, we take the spring to be described by aHencky spring with shear modulus mneq and the dashpot to bein a linearly viscous Mises form with viscosity h. The details ofthe constitutive equations describing this branch are given in

Soft Matter, 2014, 10, 8095–8106 | 8099

Fig. 4 Representative loading–unloading indentation results for loading rates of (a) 0.1 mm s�1, (b) 1 mm s�1, and (c) 10 mm s�1. Dotted curvedenotes experimental results, continuous solid lines are best fits based on the fitted agarose material constants.

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the Appendix. Here, we give the recipe for handling the Maxwellelement in time-discrete form.

The approach is summarized in Fig. 5. At a given time t¼ tn,the deformation gradient Fn and viscous deformation gradientF vn (and hence the non-equilibrium elastic deformation

gradient F en) are known (note that if tn corresponds to the

initial time, Fn ¼ F vn ¼ I). Then at time t ¼ tn+1 ¼ tn + Dt, the

deformation gradient Fn+1 is given. In an experiment, this issimply the deformation gradient measured at the subsequenttime point. To proceed forward in time, we need to calculatethe Cauchy stress due to the Maxwell element, sneq

n+1, andviscous deformation gradient, F v

n+1, at time tn+1. The steps aregiven below:

(1) First, we consider a trial state, in which viscous ow isfrozen, see Fig. 5, and calculate a trial non-equilibrium elasticdeformation gradient, F e

tr, based on the old value of the viscousdeformation gradient F v

n:

F etr ¼ Fn+1(F

vn)�1. (3)

In this intermediate state, all deformation between times tn andtn+1 is assumed to be accommodated by the Hencky spring. Wethen perform the polar decomposition of Fetr and calculate thetrial strain:

Fig. 5 Schematic of the time-discrete algorithm for the non-equilibrium

8100 | Soft Matter, 2014, 10, 8095–8106

Fetr ¼ Re

trUetr, E

etr ¼ ln U e

tr, (4)

where Retr is the trial rotation, U e

tr the trial right stretch tensor,and Ee

tr the trial Hencky (logarithmic) strain.(2) Next we use the trial strain to calculate a trial stress.

Metr ¼ 2mneq dev(E

etr), (5)

where mneq is the non-equilibrium shear modulus. This partic-ular stress measure is work conjugate to the Hencky strain andis referred to as the Mandel stress, and the specic stress–strainrelation reected in eqn (5) arises due to a free energy functionjneq associated with the non-equilibrium spring (see theAppendix for details). Note that we only consider the contribu-tion to the stress due to the deviatoric, or shear, strain. In thismanner, the non-equilibrium Maxwell element only addsviscoelasticity to the shear deformation and leaves the volu-metric response of eqn (2) unaffected.

(3) Next, we calculate the viscous stretching – that is, thestrain rate in the dashpot – at time tn+1:

Dvnþ1 ¼

M etr

2�hþ Dtmneq

� ; (6)

where h is the constant viscosity of the dashpot. See theAppendix for the derivation of this important step.

, Maxwell branch of the constitutive model.

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Fig. 6 Plot of the ratio of the total traction magnitude to the purelyelastic traction contribution vs. relaxation time, s ¼ h/mneq. The

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(4) Calculation ofDvn+1 allows for the update of F

v by means ofthe exponential map:

Fvn+1 ¼ exp(DtDv

n+1)Fvn. (7)

This brings us to the corrected, updated state in Fig. 5.(5) Finally, we may update the Cauchy stress due to the

Maxwell element sneq:

Men+1 ¼ 2hDv

n+1, sneqn+1 ¼ (det Fe

tr)�1Re

trMen+1R

eutr , (8)

where Men+1 is the Mandel stress at time tn+1.

The total Cauchy stress due to both the equilibrium and non-equilibrium branches (Fig. 2(c), le and right branch) may thenbe calculated by

sn+1 ¼ seqn+1 + sneq

n+1, (9)

where seqn+1 is calculated by eqn (2) using Fn+1. One may then

repeat these steps, marching forward in time, to calculate thestress history corresponding to a measured deformationhistory.39,40 We encourage readers who are interested in thedetails of themodel as well as the derivation of the time-discreteversion to refer to the Appendix.

Once the Cauchy stress is calculated, the cell surface trac-tions can be found using the well-known Cauchy relation.

t ¼ sn, (10)

where n is the surface normal, and we have dropped the (n + 1)subscript for simplicity of notation. The surface normal vectorscan be determined directly from the laser scanning confocalimages. Toyjanova et al. previously described such a method in-detail.18 Themagnitude of the three-dimensional traction vectoris then calculated as

|t|¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitx2 þ ty2 þ tz2

q; (11)

where tx and ty are the in-plane traction force componentsunder the cell and tz corresponded to the out-of-planecomponent. The total cell force, a common metric usedin cell TFM studies, is calculated by integrating the magni-tude of the substrate surface tractions over the total cellarea S, i.e.,

F¼ðS

|t|dS: (12)

Another metric that can be calculated from measured tractions– the root mean-squared tractions, tRMS is dened as

tRMS ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

N

XNi¼1

ti2;

vuut (13)

where ti is the traction vector located at every ith point along thesurface of the cell boundary, enclosing total of N points. Furtherdetails on how to calculate cell forces from 3D traction data canbe found elsewhere.18

This journal is © The Royal Society of Chemistry 2014

5 Analytical and experimental 3DVTFM examples – influence of materialviscosity on surface tractions

Given the slightly higher complexity of a viscoelastic over apurely elastic 3D TFM formulation, our intent is to provide thereader with a quantitative metric to determine the penaltyassociated with ignoring the viscous effects for a given substratematerial. Fig. 6 plots the ratio between the total tractionmagnitude that includes the time-dependent contribution, |t|,and the purely elastic traction magnitude, |te|, in the absence ofthe viscous dashpot in Fig. 2(c) as a function of the viscoelasticmaterial properties of the substrate. Depending on the intrinsicrelaxation time, s ¼ h/mneq, and the viscosity, h, of the materialat hand, the error associated with neglecting the viscoelasticnature of the substrate can be signicant. Since any 3D and 2DTFM framework requires careful material characterization of itssubstrate material, Fig. 6 can be utilized to determine whether aviscoelastic 3D TFM framework is necessary or whether asimpler, purely elastic formulation will suffice to estimate thecell-generated surface tractions.

Analytically simulated viscoelastic cell deformation elds

While Fig. 6 provides a quantitative estimate on the penaltyassociate with neglecting the viscous stresses in a material as afunction of the characteristic relaxation time, s, it is importantto understand the relationship between s and the actualimaging intervals utilized during live cell imaging. If s is muchless than the minimal time-lapse imaging interval, i.e., s � Dt,then only the elastic contributions of the material behavior canbe estimated. This concept is illustrated in Fig. 7 using ananalytically generated Gaussian displacement dipole on aviscoelastic material obeying the constitutive model shown inFig. 2(c). The material constants were chosen such that itmatches the properties of embryonic liver and heart tissues.19

The characteristic relaxation time, s, is set to 10 seconds. Theanalytical displacement dipole is applied along the free surfaceof the x1 � x2 plane in the form of

calculations are shown for (h)/(ms) of 0.6 (triangle), 2 (diamond), 5(square) and 10 (circle).

Soft Matter, 2014, 10, 8095–8106 | 8101

Fig. 7 Analytical traction example of (a) prescribedGaussian displacement dipole and its (b) associated displacement gradientmagnitude. (c) Plotof the magnitude of the maximum total surface tractions, |tmax| (Pa), vs. imaging time interval, Dt (seconds). (d) Contour maps of the calculatedsurface tractions at imaging time intervals of Dt ¼ 0.1 seconds, Dt ¼ 10 seconds, and Dt ¼ 1000 seconds.

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uðxÞ¼ A exp

� ðx1 � b1Þ2 þ ðx2 � b2Þ2 þ ðx3 � b3Þ2

2s2

!(14)

Fig. 7(a) and (b) depict the magnitude of the appliedGaussian displacement dipole and its associated displacementgradient. Using the displacement data, the viscoelastic cellsurface tractions are calculated and are plotted in Fig. 7(d) fordifferent time-lapse imaging intervals. Fig. 7(d) presents a panelseries of the magnitude of the 3D surface traction vector for theanalytical displacement dipole shown in Fig. 7(a) for differentimaging intervals, Dt. Given a material relaxation time, s of 10seconds, Fig. 7(c) quantitatively describes the effect differenttime-lapse imaging intervals have on accurately estimating the

Fig. 8 Experimental example of a migrating MDA-MB-231 cell on the spresents the outline of the cell, followed by contour plots of surface tra

8102 | Soft Matter, 2014, 10, 8095–8106

applied tractionmagnitude. For fast acquisition times, i.e.,Dt#1 second, most of the viscoelastic traction magnitudes aresufficiently captured. If the imaging interval is increased to �10seconds, the peak tractions will be underestimated by approx-imately 17%, and with imaging times on the order of 100seconds most of the viscoelastic material response has reachedthe equilibrium state, resulting in an traction estimation errorof �30%. The shape of the curve and the absolute numbers willvary depending on the particular viscoelastic material modelemployed, but can be generated via means of eqn (1)–(12).

While Fig. 7 shows that the particular time-lapse imaginginterval, Dt, can have a profound effect on accurately repre-senting the spatial distribution and magnitude of the tractionpattern in a viscoelastic medium, practical considerations, such

urface of an agarose substrate. The phase contrast image (left panel)ctions, t1, t2, and t3 in Pa.

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as low signal to imaging noise, phototoxicity, and photo-bleaching might limit the range of available imaging intervals.Thus, once the material of interest has been characterized andevaluated, the error associated with practical imaging intervalscan be evaluated by generating the type of plot shown inFig. 7(c). This should allow the user to directly evaluate whethera viscoelastic TFM framework is necessary for the desired TFMmeasurements or whether a simpler, purely elastic formulationsuffices. Furthermore, for small linear elastic deformations, thereader has several TFM choices with recent developmentsprovided by del Alamo in 3D,41 and the well-established Fouriertransform traction cytometry by Butler et al. in 2D15 as meansfor accurate estimation of cellular traction elds.

Experimentally observed viscoelastic cell deformation elds

To illustrate the resolution and applicability of the 3D VTFMtechnique to capture actual cell-induced traction data, Fig. 8shows snapshots of a MDA-MB-231 cell deforming an agarosesubstrate at two different time points. The le panel showsphase contrast images of the cell on top of the agarosesubstrate, followed by the in-plane and out-of-plane tractions.The traction resolution sensitivity was 50 Pa for in-plane and 80Pa for out-of-plane tractions. The observed traction patterns inFig. 8 are similar in shape and magnitude to previouslyobserved traction patterns for mesenchymal cells.6,15,42–44 As canbe seen from Fig. 8, the utilization of the 3D VTFMmethodologyproduces high-resolution traction maps with similar resolutionand sensitivity capabilities to our previously published 3D largedeformation elastic TFM approach.18

It should be noted that the characteristic relaxation time forthe agarose gels used in this study was �1.1 seconds, and thattime-lapse images were acquired every 100 seconds. Any shortertime-lapse intervals yielded negligible material deformationsindistinguishable from noise.

6 Conclusion

This paper presents a 3D Viscoelastic Traction Force Micros-copy (3D VTFM) technique that allows the computation of cell-generated material tractions on viscoelastic materials. Using acombined experimental and numerical AFM indentationapproach, our study details how to determine the materialproperties of the viscoelastic substrate material of interest,which in our simple example is an agarose gel. We would like tonote, that the extraction of the viscoelastic material propertiesof the substrate material at hand can also be achieved by variousother mechanical testing approaches besides indentation,including stress relaxation, creep and frequency-dependenttests in rheometer or tension/compression setups. As long asthe material stress–strain state is recorded as a function of time,the viscoelastic material properties of the particular substratematerial can be determined. Next, utilizing the obtainedmaterial properties we provide a general, nite deformationframework for integrating the viscoelasticity of the substratematerial into a 3D TFM methodology. The same generalframework we described can be simply modied by the reader

This journal is © The Royal Society of Chemistry 2014

to allow the incorporation of his/her own substrate specicviscoelastic material properties into our 3D VTFM code struc-ture to perform 3D traction measurements. Finally, we providethe reader with a quantitative estimate on the penalty associ-ated with neglecting the viscous effects for the given viscoelasticmaterial, and what the minimum TFM time-lapse interval isrequired to adequately capture the viscous contribution of thematerial response. We conclude by showing traction contourplots of breast cancer cells migrating on collagen-functionalizedagarose substrates. The 3D VTFM code package can be down-loaded from our website (http://franck.engin.brown.edu).

In conclusion, this study provides a new 3D TFM techniquethat is capable of incorporating time-dependent materialcharacteristics for studies that involve more realistic tissue-mimicking substrate materials. While the last two decades haveprovided a wealth of cell–material interactions on elasticsubstrates, we anticipate that the 3D VTFM technique will openup new avenues of cell–material investigations on even morephysiologically relevant time-dependent materials includingnative tissue derivatives such as collagen and brin gels.

AppendixViscoelastic model details

In this appendix, we give a detailed account of the nite-deformation viscoelastic model used to describe experimentaldata, as well as a derivation of the time-discrete versionsummarized in Section 4. Mathematically, the viscoelasticmodel involves the following denitions: u(x, t), displacement; y¼ x + u(x, t), motion; F¼ Vy, J¼ det F > 0, deformation gradient;B ¼ FFu, le Cauchy–Green tensor. We take the free energy dueto the equilibrium branch to be given in Neo-Hookean form

jeqðFÞ¼ 1

2m

�ðtr BÞJ2=3

� 3

�þ 1

2KðJ �1Þ2; (15)

where m and K are ground-state, equilibrium shear and bulkmoduli, respectively. The rst term represents free energy dueto distortional (constant volume) deformation, and the secondterm arises due to purely volumetric deformation. The associ-ated Cauchy stress is then

s eq ¼ J�1vj eqðFÞ

vFFu

¼ mJ�5=3 devðBÞ þ KðJ�1ÞI:(16)

The Maxwell element in parallel with the Neo-Hookeanspring then allows for modeling non-equilibrium time-depen-dent behavior. To describe deformation in this element, thedeformation gradient, F, is multiplicatively decomposed intoelastic and viscous parts: F ¼ FeFv; with Fv, Jv ¼ det Fv ¼ 1,denoting the (constant-volume) viscous distortion and Fe, Je ¼det Fe > 0, denoting the non-equilibrium elastic distortion. Thefree energy due to the non-equilibrium spring is based on theelastic Hencky (logarithmic) strain, utilizing the followingdenitions: Fe ¼ ReUe, polar decomposition of Fe;

Ue ¼X3a¼1

learea5r ea, spectral decomposition of Ue; and

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Ee ¼X3a¼1

�lnlea

�r ea5r ea, the elastic Hencky (logarithmic) strain.

We take the free energy due to the non-equilibrium spring as

jneq(Ee) ¼ mneq|dev(Ee)|2, (17)

where mneq is the non-equilibrium shear modulus. The stressconjugate to the elastic strain, Ee, is referred to the Mandelstress:

M e ¼ vjneqðEeÞvEe ¼2mneq devðEeÞ; (18)

and the contribution to the Cauchy stress due to the Maxwellelement is then given through

sneq ¼ Je�1ReMeReu. (19)

Finally, the evolution of Fv is given by

_Fv ¼ DvFv, Fv(x, t ¼ 0) ¼ I, (20)

with the viscous stretching, Dv, given by

Dv¼ 1

2hM e; (21)

where h is the Maxwell element viscosity (we refer to thedashpot as a Mises dashpot since this choice of constitutiveequation asserts that the viscous stretching and Mandel stresstensors are codirectional, a hallmark of Mises plasticity). Notethat since jneq is taken to not depend upon Je, the contributionof the Maxwell element to the stress is purely deviatoric. In thisway, we have only added viscoelasticity to the shear deformationand le the volumetric response of eqn (16) unaffected. Thefour material parameters appearing in the model are the equi-librium shear modulus m, the non-equilibrium shear modulusmneq, the Maxwell element viscosity h, and the bulk modulus K.The three parameters {m, mneq, h} are selected to best t theexperimental behavior, while the K bulk modulus is chosen sothat the equilibrium Poisson's ratio in the linear (small)deformation regime, i.e., n ¼ (3K � 2m)/(6K + 2m), is 0.3,following ref. 37.

Since experimental measurements are made at discrete time-points, we next derive the time-discrete version of the modelused both to analyze the experimental data as well as the oneimplemented in the Abaqus UMAT used in the simulations. At atime tn, the deformation gradient, Fn, and the viscous defor-mation gradient, Fvn, are known. Then, given the deformationgradient, Fn+1, at a time tn+1 ¼ tn + Dt, we are to determine theCauchy stress, sn+1, and viscous deformation gradient, Fvn+1, attime tn+1. In the experiments, Dt is the time increment betweenthe two successive images. The evolution equation for Fv isimplicitly integrated by means of an exponential map:45,46

F vnþ1 ¼ exp

�DtD v

nþ1

�F v

n with D vnþ1¼

1

2hM e

nþ1: (22)

Using eqn (22)1, we have that the elastic deformation gradient atthe end of the step is

8104 | Soft Matter, 2014, 10, 8095–8106

F enþ1 ¼ Fnþ1F

v�1nþ1 ¼ Fnþ1F

v�1n|fflfflfflfflfflffl{zfflfflfflfflfflffl}

¼F etr

exp�� DtDv

nþ1

�¼ F e

tr exp��DtDv

nþ1

�;

(23)

where Fetr is the trial value of the elastic deformation gradient. Atrial quantity represents the value of that quantity evaluatedwhen viscous ow is frozen. With (i) the polar decompositionsFen+1 ¼ Re

n+1Uen+1 and Fetr ¼ Re

trUetr; (ii) the knowledge that Dv

n+1,Me

n+1, and Uen+1 share principal directions and therefore

commute; and (iii) the uniqueness of the polar decomposition;we have that

Ren+1 ¼ Re

tr and Uen+1 ¼ Ue

tr exp(�DtDvn+1). (24)

Taking the logarithm of (24)2, we have

En+1 ¼ Etr � DtDvn+1 with Etr ¼ ln Utr. (25)

Multiplying the deviatoric part of eqn (25) by 2mneq, by the stressrelation eqn (18), we have

Men+1 ¼ Me

tr � 2mneqDtDvn+1 with Mtr ¼ 2mneq dev(Etr), (26)

which upon combining with (22)2 yields

Dvnþ1 ¼

M etr

2�hþ Dtmneq

� : (27)

Finally, the viscous deformation gradient is updated by eqn(22)1, and the Mandel and Cauchy stresses due to the Maxwellelement are updated by

Men+1 ¼ 2hDv

n+1, sneq ¼ (det Fe

tr)�1Re

trMen+1R

eutr . (28)

The sequence of time-integration steps for theMaxwell element,with reference to the steps shown in Fig. 5, is outlined below.

(1) Compute the trial elastic deformation gradient, Fetr, andits associated kinematic quantities using the viscous deforma-tion gradient from the previous time increment:

Fetr ¼ Fn+1F

v�1n , Fe

tr ¼ RetrU

etr, E

etr ¼ ln(Ue

tr). (29)

(2) Compute the trial Mandel stress:

Metr ¼ 2mneq dev(E

etr). (30)

(3) Calculate the viscous stretching

Dvnþ1 ¼

M etr

2�hþ Dtmneq

� : (31)

(4) Update the viscous deformation gradient:

Fvn+1 ¼ exp(DtDv

n+1)Fvn. (32)

(5) Update the Mandel stress and Cauchy stress due to theMaxwell element:

Men+1 ¼ 2hDv

n+1, sneqn+1 ¼ (det Fe

tr)�1Re

trMen+1R

eutr . (33)

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This completes the calculation of the contribution to theCauchy stress due to theMaxwell element. The contribution dueto the equilibrium Neo-Hookean spring is simply computedfrom the deformation gradient at the end of the step Fn+1:

Bn+1 ¼ Fn+1Fun+1,

seqn+1 ¼ m(det Fn+1)

�5/3 dev(Bn+1) + K[(det Fn+1)�1]I. (34)

allowing for the calculation of the total Cauchy stress:

sn+1 ¼ seqn+1 + sneq

n+1. (35)

Acknowledgements

CF and DLH acknowledge Brown Start-up funds. EMDacknowledges NIH grants P20GM104937 and R00AR054673.

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