Portal de Periódicos da CAPES

A primer to traction force microscopy

2022; Elsevier BV; Volume: 298; Issue: 5 Linguagem: Inglês

10.1016/j.jbc.2022.101867

1083-351X

Andrea Zancla, Pamela Mozetic, Monica Orsini, Giancarlo Forte, Alberto Rainer,

Advanced Electron Microscopy Techniques and Applications

Traction force microscopy (TFM) has emerged as a versatile technique for the measurement of single-cell-generated forces. TFM has gained wide use among mechanobiology laboratories, and several variants of the original methodology have been proposed. However, issues related to the experimental setup and, most importantly, data analysis of cell traction datasets may restrain the adoption of TFM by a wider community. In this review, we summarize the state of the art in TFM-related research, with a focus on the analytical methods underlying data analysis. We aim to provide the reader with a friendly compendium underlying the potential of TFM and emphasizing the methodological framework required for a thorough understanding of experimental data. We also compile a list of data analytics tools freely available to the scientific community for the furtherance of knowledge on this powerful technique. Traction force microscopy (TFM) has emerged as a versatile technique for the measurement of single-cell-generated forces. TFM has gained wide use among mechanobiology laboratories, and several variants of the original methodology have been proposed. However, issues related to the experimental setup and, most importantly, data analysis of cell traction datasets may restrain the adoption of TFM by a wider community. In this review, we summarize the state of the art in TFM-related research, with a focus on the analytical methods underlying data analysis. We aim to provide the reader with a friendly compendium underlying the potential of TFM and emphasizing the methodological framework required for a thorough understanding of experimental data. We also compile a list of data analytics tools freely available to the scientific community for the furtherance of knowledge on this powerful technique. The premise of mechanobiology is that the mechanical properties of biological tissues can direct given cellular processes, like proliferation, migration, survival, and differentiation. Therefore, mechanobiology entails the understanding of how forces are generated, maintained, and interpreted by cells which actively respond to biophysical stimuli arising from their milieu (Fig. 1). The primary sites of cell interaction to any substrate are the multiprotein complexes which connect the extracellular matrix (ECM) to cell cytoskeleton, the focal adhesions (FAs). FAs are defined as integrin-based cell-matrix physical contacts transducing and integrating mechanical and biochemical cues arising from the surrounding microenvironment, through the assembly of intracellular multiprotein complexes connected to actin cytoskeleton. The formation of alpha–beta integrin heterodimers and their clustering within the extracellular membrane induces the recruitment of cytoskeleton-docking proteins and the rearrangement of allosterically regulated ones, which in turn start the cellular signaling inward toward the nucleus (1Iskratsch T. Wolfenson H. Sheetz M.P. Appreciating force and shape — the rise of mechanotransduction in cell biology.Nat. Rev. Mol. Cell Biol. 2014; 15: 825-833Crossref PubMed Scopus (0) Google Scholar, 2Wehrle-Haller B. Structure and function of focal adhesions.Curr. Opin. Cell Biol. 2012; 24: 116-124Crossref PubMed Scopus (152) Google Scholar, 3Schiller H.B. Fässler R. Mechanosensitivity and compositional dynamics of cell–matrix adhesions.EMBO Rep. 2013; 14: 509-519Crossref PubMed Scopus (0) Google Scholar). The proteins that participate in FA formation are distributed in layers. Connected to the integrins, docking proteins like talin, vinculin, zyxin, and tensin are part of the mechanosensing layer. The continuous remodelling of actin cytoskeleton in response to external stimuli is operated by the so-called mechanosignaling proteins, which include paxillin, focal adhesion kinase, Src, and p130Cas, as well as by actin regulators, like Ena-VASP and alpha-actinin, etc. (2Wehrle-Haller B. Structure and function of focal adhesions.Curr. Opin. Cell Biol. 2012; 24: 116-124Crossref PubMed Scopus (152) Google Scholar). The transmission of the signal inward is ensured by the dynamic rearrangement of the proteins composing the FA complex in response to chemical and mechanical stimuli, thus contributing to both cell–ECM interaction and intracellular signaling (4Hytönen V.P. Wehrle-Haller B. Mechanosensing in cell–matrix adhesions – converting tension into chemical signals.Exp. Cell Res. 2016; 343: 35-41Crossref PubMed Scopus (54) Google Scholar). Most components of the focal adhesions display some degree of mechanosensitivity (i.e., their localization or conformation changes following the application of physical and biochemical stimuli generated at the ECM). The cooperative activity of these components makes it difficult to determine the specific mechanosensitivity of single FA members. An established example of an FA mechanosensitive protein is talin, a 270-kDa protein which interacts directly with both β-integrin cytoplasmic domain and F-actin. The protein acts as a force buffer by unfolding the numerous rod domains following mechanical load, thus exposing cryptic hydrophobic binding domains able to interact with vinculin (5del Rio A. Perez-Jimenez R. Liu R. Roca-Cusachs P. Fernandez J.M. Sheetz M.P. Stretching single talin rod molecules activates vinculin binding.Science. 2009; 323: 638-641Crossref PubMed Scopus (1014) Google Scholar, 6Hirata H. Tatsumi H. Lim C.T. Sokabe M. Force-dependent vinculin binding to talin in live cells: A crucial step in anchoring the actin cytoskeleton to focal adhesions.Am. J. Physiol. Cell Physiol. 2014; 306: C607-C620Crossref PubMed Scopus (59) Google Scholar, 7Rahikainen R. von Essen M. Schaefer M. Qi L. Azizi L. Kelly C. Ihalainen T.O. Wehrle-Haller B. Bastmeyer M. Huang C. Hytönen V.P. Mechanical stability of talin rod controls cell migration and substrate sensing.Sci. Rep. 2017; 7: 3571Crossref PubMed Scopus (34) Google Scholar, 8Maki K. Nakao N. Adachi T. Nano-mechanical characterization of tension-sensitive helix bundles in talin rod.Biochem. Biophys. Res. Commun. 2017; 484: 372-377Crossref PubMed Scopus (4) Google Scholar). The successful binding of vinculin to talin is considered essential to stabilize the interaction between the FA and F-actin and thus transfer the mechanical signal inward (9Humphries J.D. Wang P. Streuli C. Geiger B. Humphries M.J. Ballestrem C. Vinculin controls focal adhesion formation by direct interactions with talin and actin.J. Cell Biol. 2007; 179: 1043-1057Crossref PubMed Scopus (608) Google Scholar). FA dynamics promotes the propagation of forces to the cytoskeleton, as summarized in the study by Spill et al (10Spill F. Bakal C. Mak M. Mechanical and systems biology of cancer.Comput. Struct. Biotechnol. J. 2018; 16: 237-245Abstract Full Text Full Text PDF PubMed Scopus (19) Google Scholar). Composed of microfilaments, intermediate filaments, microtubules, and adaptor proteins, the cytoskeleton represents the scaffolding structure of the cell. Its timely rearrangement is necessary for the cell to control its mechanical properties and exert all its functions. A comprehensive description of force transfer from cell periphery toward the cytoskeleton can be found in the review by Martino et al. (11Martino F. Perestrelo A.R. Vinarský V. Pagliari S. Forte G. Cellular mechanotransduction: From tension to function.Front. Physiol. 2018; 9: 824Crossref PubMed Scopus (254) Google Scholar). Due to its ability to directly affect the genetic landscape of the cell in response to extracellular stimuli, modifications in intracellular mechanics induced by cytoskeleton remodeling are now known to also participate in shaping cell identity (12Pagliari S. Vinarsky V. Martino F. Perestrelo A.R. Oliver De La Cruz J. Caluori G. Vrbsky J. Mozetic P. Pompeiano A. Zancla A. Ranjani S.G. Skladal P. Kytyr D. Zdráhal Z. Grassi G. et al.YAP–TEAD1 control of cytoskeleton dynamics and intracellular tension guides human pluripotent stem cell mesoderm specification.Cell Death Differ. 2021; 28: 1193-1207Crossref PubMed Scopus (0) Google Scholar) and have been involved in several pathological processes, including cancer (13Verbruggen S.W. Mechanobiology in Health and Disease. Academic Press, London2018Google Scholar). All this accumulated evidence on the fundamental role of mechanical cues points at the increasing demand for in vitro platforms compatible with the measurement of cell–cell and cell–substrate mechanical interactions. Conventional cell culture systems are based on two-dimensional (2D) monolayer cultures routinely used to study cellular mechanisms. However, the predictivity of in vitro monolayers when compared to native tissues is known to get poorer with the increase in the system complexity. Moving to three-dimensional (3D) culture allows cells to undergo indirect mechanical stimulation by controlling the rigidity and stiffness of the ECM in which they are embedded (14Campàs O. Mammoto T. Hasso S. Sperling R.A. O'Connell D. Bischof A.G. Maas R. Weitz D.A. Mahadevan L. Ingber D.E. Quantifying cell-generated mechanical forces within living embryonic tissues.Nat. Methods. 2014; 11: 183-189Crossref PubMed Scopus (232) Google Scholar). In fact, 3D tissue models can be designed to produce and control dynamic mechanical stimuli such as fluid flow, stretch/strain, and compression (15Farooque T.M. Camp C.H. Tison C.K. Kumar G. Parekh S.H. Simon C.G. Measuring stem cell dimensionality in tissue scaffolds.Biomaterials. 2014; 35: 2558-2567Crossref PubMed Scopus (47) Google Scholar). Quantitative analysis of single-cell behavior easily extends its interpretation and results in higher-scale models such as native tissues, engineered tissue constructs, and organs-on-chip, as reviewed by Ergir et al. (16Ergir E. Bachmann B. Redl H. Forte G. Ertl P. Small force, big impact: Next generation organ-on-a-chip systems incorporating biomechanical cues.Front. Physiol. 2018; 9: 1417Crossref PubMed Scopus (37) Google Scholar). This experimental landscape is driving the future of computational models in tissue growth and remodeling cases, which are of interest due to their close relationship with the clinical landscape. The field has seen significant advances in recent times, and its development has led to significant improvements in functional tissue engineering approaches (17Butler D.L. Goldstein S.A. Guilak F. Functional tissue engineering: The role of biomechanics.J. Biomech. Eng. 2000; 122: 570-575Crossref PubMed Scopus (454) Google Scholar). Additionally, these new strategies proved to be useful for investigating the molecular basis of cell–cell signaling and contributed to unveil the transmission and regulation mechanisms driving signaling pathways in tissue environments. Special attention has been paid to understand the function and regulation of YAP/TAZ proteins, which are known to play a pivotal role downstream of mechanosensitive Hippo pathway in transducing mechanical signals to the nucleus, in order to dictate focal adhesion assembly, cytoskeleton, and ECM remodeling (18Yan L. Cai Q. Xu Y. Hypoxic conditions differentially regulate TAZ and YAP in cancer cells.Arch. Biochem. Biophys. 2014; 562: 31-36Crossref PubMed Scopus (25) Google Scholar, 19Elosegui-Artola A. Oria R. Chen Y. Kosmalska A. Pérez-González C. Castro N. Zhu C. Trepat X. Roca-Cusachs P. Mechanical regulation of a molecular clutch defines force transmission and transduction in response to matrix rigidity.Nat. Cell Biol. 2016; 18: 540-548Crossref PubMed Scopus (0) Google Scholar, 20Elosegui-Artola A. Andreu I. Beedle A.E.M. Lezamiz A. Uroz M. Kosmalska A.J. Oria R. Kechagia J.Z. Rico-Lastres P. Le Roux A.-L. Shanahan C.M. Trepat X. Navajas D. Garcia-Manyes S. Roca-Cusachs P. Force triggers YAP nuclear entry by regulating transport across nuclear pores.Cell. 2017; 171: 1397-1410Abstract Full Text Full Text PDF PubMed Scopus (501) Google Scholar, 21Nardone G. Oliver-De La Cruz J. Vrbsky J. Martini C. Pribyl J. Skládal P. Pešl M. Caluori G. Pagliari S. Martino F. Maceckova Z. Hajduch M. Sanz-Garcia A. Pugno N.M. Stokin G.B. et al.YAP regulates cell mechanics by controlling focal adhesion assembly.Nat. Commun. 2017; 8: 15321Crossref PubMed Scopus (247) Google Scholar, 22Rape A.D. Guo W. Wang Y. The regulation of traction force in relation to cell shape and focal adhesions.Biomaterials. 2011; 32: 2043-2051Crossref PubMed Scopus (215) Google Scholar). All these events are crucial to ensure the tight control of cell adhesion, migration, proliferation, and differentiation, which in turn underlies the correct orchestration of vital processes like angiogenesis and immune response, among the others (23Reinhart-King C.A. Dembo M. Hammer D.A. The dynamics and mechanics of endothelial cell spreading.Biophys. J. 2005; 89: 676-689Abstract Full Text Full Text PDF PubMed Scopus (282) Google Scholar, 24Fang Y. Lai K.W.C. Modeling the mechanics of cells in the cell-spreading process driven by traction forces.Phys. Rev. E. 2016; 93: 42404Crossref PubMed Scopus (0) Google Scholar). The paradigm shift between the macroscale observation of forces in biomechanics to the single-cell micrometric scale of mechanobiology reasonably began with the work of Harris et al. in 1980 (25Harris A.K. Wild P. Stopak D. Silicone rubber substrata: A new wrinkle in the study of cell locomotion.Science. 1980; 208: 177-179Crossref PubMed Google Scholar), in which the first known indirect estimation of cellular traction forces was performed through microscope images. The method was based on cell cultures on distortable sheets of silicone rubber. Although only in a qualitative way, the technique enabled examination of single-cell tractions. Starting from this seminal study, two main approaches have been pursued to study forces at the cellular level: (i) active stimulation methods, which measure cell response to mechanical force application, and (ii) passive methods, which sense mechanical forces generated by cells without applying any external stimulus. Hereafter, we will focus on passive stimulation methods, with particular regard to traction force microscopy (TFM). A more detailed overview on active versus passive platforms for single-cell biomechanical characterization can be found in the review by Basoli et al. (26Basoli F. Giannitelli S.M. Gori M. Mozetic P. Bonfanti A. Trombetta M. Rainer A. Biomechanical characterization at the cell scale: Present and prospects.Front. Physiol. 2018; 9: 1449Crossref PubMed Scopus (33) Google Scholar). Microfabricated platforms have been investigated to measure cellular tractions in controlled mechanical environments, including both hard silicon-based devices and soft polymer/gel devices. In particular, soft polymer and gel microsystems obtained through soft lithography techniques are characterized by biocompatibility, optical transparency, and the possibility to functionalize the surface as well as to tune its mechanical properties to match those of the in vivo environment. Soft lithography structures are realized by replica molding of a patterned silicon master. Several research groups have highlighted the use of elastomeric microfabricated pillars (microfabricated post-array-detectors) as engineered tools to measure single-cell adhesion forces (27Yang M.T. Sniadecki N.J. Chen C.S. Geometric considerations of micro- to nanoscale elastomeric post arrays to study cellular traction forces.Adv. Mater. 2007; 19: 3119-3123Crossref Scopus (135) Google Scholar, 28Kaylan K.B. Kourouklis A.P. Underhill G.H. A high-throughput cell microarray platform for correlative analysis of cell differentiation and traction forces.J. Vis. Exp. 2017; 121e55362Google Scholar, 29Ghassemi S. Meacci G. Liu S. Gondarenko A.A. Mathur A. Roca-Cusachs P. Sheetz M.P. Hone J. Cells test substrate rigidity by local contractions on submicrometer pillars.Proc. Natl. Acad. Sci. U. S. A. 2012; 109: 5328-5333Crossref PubMed Scopus (0) Google Scholar, 30Razafiarison T. Holenstein C.N. Stauber T. Jovic M. Vertudes E. Loparic M. Kawecki M. Bernard L. Silvan U. Snedeker J.G. Biomaterial surface energy-driven ligand assembly strongly regulates stem cell mechanosensitivity and fate on very soft substrates.Proc. Natl. Acad. Sci. U. S. A. 2018; 115: 4631-4636Crossref PubMed Scopus (31) Google Scholar). The analysis of pillar displacement is performed by particle tracking software to detect and label the deflection of each post over the temporal series of images. Tracking can be performed by means of either bright field or fluorescent microscopy (the latter following coating of the pillar tips with fluorescent probes). The lattice arrangement of pillars also offers a means for the calculation of rest (zero-stress) position, which can be obtained by linear fitting starting from the position of the posts not covered by cells belonging to the same row (31Saez A. Anon E. Ghibaudo M. du Roure O. Di Meglio J.-M. Hersen P. Silberzan P. Buguin A. Ladoux B. Traction forces exerted by epithelial cell sheets.J. Phys. Condens. Matter. 2010; 22: 194119Crossref PubMed Scopus (101) Google Scholar). Forces can be calculated with single-pillar resolution from measured deflections, assuming a (quasi) linear relationship between the two entities. A comprehensive discussion on fabrication route, imaging, and evaluation of traction forces can be found in the works of Polacheck & Chen (32Polacheck W.J. Chen C.S. Measuring cell-generated forces: A guide to the available tools.Nat. Methods. 2016; 13: 415-423Crossref PubMed Scopus (258) Google Scholar) and Gupta et al. (33Gupta M. Kocgozlu L. Sarangi B.R. Margadant F. Ashraf M. Ladoux B. Micropillar substrates: A tool for studying cell mechanobiology.Biophys. Methods Cell Biol. 2015; 125: 289-308Crossref PubMed Scopus (0) Google Scholar). As a notable advancement in the field, Xiao et al. (34Xiao F. Wen X. Tan X.H.M. Chiou P.-Y. Plasmonic micropillars for precision cell force measurement across a large field-of-view.Appl. Phys. Lett. 2018; 112033701Crossref PubMed Scopus (10) Google Scholar) designed a plasmonic micropillar platform with self-organized gold nanospheres, precisely resolving cell tractions across a large field of view. In their work, micropillars were modified with gold nanospheres, which were precisely allocated at the center of each micropillar tip via laser annealing process. Gold served as a point source–like light scattering marker, allowing every micropillar to be tracked even under low-magnification objective lenses. TFM represents the most widely used technique for measuring cell forces. The core strength of the method is that it can generate quantitative stress maps, resuming the stress of an elastically deformed substrate at the level of the cell adhesion plane. The foundation of the technique is that when a cell is adherent to a soft substrate, it exerts a contractile force causing a strain, which is measurable. TFM commonly relies on thin hydrogel films, endowed with nanoscopic fluorescent beads, which are either embedded in the substrate or attached to its surface to be used as fiducial markers for optical tracking in space and time (35Holenstein C.N. Silvan U. Snedeker J.G. High-resolution traction force microscopy on small focal adhesions - improved accuracy through optimal marker distribution and optical flow tracking.Sci. Rep. 2017; 7: 41633Crossref PubMed Scopus (27) Google Scholar). A typical TFM experiment consists of two subsequent image acquisition phases. During the first phase, the bead positions are recorded in the stressed state when cells are contracting the elastic substrate they have been seeded onto (cell-loaded image). Then, cells are detached by trypsinization, releasing the gel to its unstressed state, where a new image is captured (reference image). The vector displacement field for the substrate at each cell position is computed into a displacement map resuming the deviation (in pixel) of each bead from its reference position as a consequence of the force exerted by the cell (31Saez A. Anon E. Ghibaudo M. du Roure O. Di Meglio J.-M. Hersen P. Silberzan P. Buguin A. Ladoux B. Traction forces exerted by epithelial cell sheets.J. Phys. Condens. Matter. 2010; 22: 194119Crossref PubMed Scopus (101) Google Scholar, 36Style R.W. Boltyanskiy R. German G.K. Hyland C. MacMinn C.W. Mertz A.F. Wilen L.A. Xu Y. Dufresne E.R. Traction force microscopy in physics and biology.Soft Matter. 2014; 10: 4047-4055Crossref PubMed Scopus (178) Google Scholar). Polyacrylamide (PA) or silicon-based gels are common substrates for TFM. Both types of gels exhibit a linear elastic behavior under deformations produced by cell traction, and their stiffness can be varied over a range of several orders of magnitude. Interestingly, mechanical properties of those gels have been proven not to change under the action of biochemical factors that may occur during a TFM measurement, including cell proteases (37Rocha M.S. Extracting physical chemistry from mechanics: A new approach to investigate DNA interactions with drugs and proteins in single molecule experiments.Integr. Biol. 2015; 7: 967-986Crossref Google Scholar). The above-described setup for TFM made possible for the technique to reach a high level of diffusion and replicability among different laboratories. Partially accounting for the low throughput of the technique, some groups have introduced dedicated setups, as in the case of Yoshie et al., who designed a polydimethylsiloxane (PDMS) contractile force screening platform featuring 96 monolithic independent wells (38Yoshie H. Koushki N. Kaviani R. Tabatabaei M. Rajendran K. Dang Q. Husain A. Yao S. Li C. Sullivan J.K. Saint-Geniez M. Krishnan R. Ehrlicher A.J. Traction force screening enabled by compliant PDMS elastomers.Biophys. J. 2018; 114: 2194-2199Abstract Full Text Full Text PDF PubMed Scopus (25) Google Scholar). We will not bring further examples of conventional TFM setups (the reader is referred to the comprehensive review by Roca-Cusachs et al. for the current state of the art (39Roca-Cusachs P. Conte V. Trepat X. Quantifying forces in cell biology.Nat. Cell Biol. 2017; 19: 742-751Crossref PubMed Scopus (234) Google Scholar)); conversely, we aim to introduce some of its most innovative variants, as outlined in the following subsections and summarized in Figure 2. Interesting exploitations of surface micropatterning were used to demonstrate alternatives to the more common beads-in-a-gel practice. A novel structure has been realized by Pasqualini et al., (40Pasqualini F.S. Agarwal A. O'Connor B.B. Liu Q. Sheehy S.P. Parker K.K. Traction force microscopy of engineered cardiac tissues.PLoS One. 2018; 13e0194706Crossref PubMed Scopus (31) Google Scholar) who applied microcontact printing to the patterned deposition of cell adhesion molecules (fibronectin) on PA gel to direct cell cluster organization. Other scientists produced micropatterned elastomeric substrates by soft lithography by engineering the surface topography with a lattice of either embossed or fluorescent markers (41Balaban N.Q. Schwarz U.S. Riveline D. Goichberg P. Tzur G. Sabanay I. Mahalu D. Safran S. Bershadsky A. Addadi L. Geiger B. Force and focal adhesion assembly: A close relationship studied using elastic micropatterned substrates.Nat. Cell Biol. 2001; 3: 466-472Crossref PubMed Scopus (1708) Google Scholar). The latter approach was achieved by fabricating the pad array with a fluorescent photoresist and by achieving a controlled peel-off of the resist which remained embedded on the PDMS surface. Overall, micropatterning is an interesting approach which adds an additional degree of control over cell arrangement and which can be used to externally influence cytoskeleton architecture and cellular polarization by tailoring focal adhesion distribution, thus impacting cell migration, growth, and differentiation (42Thery M. Micropatterning as a tool to decipher cell morphogenesis and functions.J. Cell Sci. 2010; 123: 4201-4213Crossref PubMed Scopus (481) Google Scholar). Location and origin of the normal tractions with respect to the adhesive and cytoskeletal elements of cells can be further modelled to consider the 3D nature of cellular forces acting on planar 2D surfaces (hereby the notation '2.5D'). It is worth noting that under elongated focal adhesions, upward and downward normal tractions are more likely to appear on distal (toward the cell edge) and proximal (toward the cell body) ends of adhesions. The resulting rotational moments affect focal adhesions by either protruding or retracting peripheral regions. To measure this, Legant et al. (43Legant W.R. Choi C.K. Miller J.S. Shao L. Gao L. Betzig E. Chen C.S. Multidimensional traction force microscopy reveals out-of-plane rotational moments about focal adhesions.Proc. Natl. Acad. Sci. U. S. A. 2013; 110: 881-886Crossref PubMed Scopus (0) Google Scholar) developed a 2.5D expansion of the TFM protocol. Full 3D TFM was designed to measure the traction field of cells that, instead being seeded on top of the substrate, are embedded within an ECM-like 3D environment (44Hall M.S. Long R. Feng X. Huang Y. Hui C.-Y. Wu M. Towards single cell traction microscopy within 3D collagen matrices.Exp. Cell Res. 2013; 319: 2396-2408Crossref PubMed Scopus (0) Google Scholar). The measurement steps leading to the displacement field are the same that we already met for standard TFM. From there on, the discrete set of displacement data is converted into a continuous displacement field by means of interpolation. The strain field is then evaluated through a numerical evaluation of the spatial gradient of the above-calculated displacement field. Since the mechanical properties of the hydrogel are known and its constitutive model is defined, the stress field can be calculated without any a priori assumption of stress state or ECM geometry. Interestingly, in this case we are not bound to infinite substrate requirements typical of the Boussinesq theory (45Franck C. Hong S. Maskarinec S.A. Tirrell D.A. Ravichandran G. Three-dimensional full-field measurements of large deformations in soft materials using confocal microscopy and digital volume correlation.Exp. Mech. 2007; 47: 427-438Crossref Scopus (180) Google Scholar, 46Franck C. Maskarinec S.A. Tirrell D.A. Ravichandran G. Three-dimensional traction force microscopy: A new tool for quantifying cell-matrix interactions.PLoS One. 2011; 6e17833Crossref Scopus (170) Google Scholar) (this topic will be further explained in Box 1). Several significant works have been published on this subject (47Koch T.M. Münster S. Bonakdar N. Butler J.P. Fabry B. 3D traction forces in cancer cell invasion.PLoS One. 2012; 7e33476Crossref Scopus (210) Google Scholar, 48Munoz J.J. Non-regularised inverse finite element analysis for 3D traction force microscopy.Int. J. Numer. Anal. Mod. 2016; 13: 763-781Google Scholar, 49Toyjanova J. Bar-Kochba E. Hoffman-Kim D. Franck C. High resolution, large deformation 3D traction force microscopy.PLoS One. 2014; 9e90976Crossref PubMed Scopus (55) Google Scholar, 50Jorge-Peñas A. Izquierdo-Alvarez A. Aguilar-Cuenca R. Vicente-Manzanares M. Garcia-Aznar J.M. Van Oosterwyck H. de-Juan- Pardo E.M. Ortiz-de-Solorzano C. Muñoz-Barrutia A. Free form deformation–based image registration improves accuracy of Traction Force Microscopy.PLoS One. 2015; 10e0144184Crossref PubMed Scopus (15) Google Scholar), also detailing methods for the numerical solution of the problem (51Gjorevski N. Nelson C.M. Endogenous patterns of mechanical stress are required for branching morphogenesis.Integr. Biol. 2010; 2: 424-434Crossref Scopus (106) Google Scholar).BOX 1Mathematical frameworkThe math underneath the reconstruction of traction forces in TFM relies on the theory of linear elasticity (96Landau L.D. Lifshitz E.M. Theory of Elasticity.3rd Ed. Butterworth-Heinemann, Oxford1986Google Scholar). Substrates are tunable in their characteristics and can be assumed as isotropic, homogeneous, and linear. As such, they are defined by two parameters, namely the linear elastic modulus (Young's modulus) E and the Poisson's ratio ν (97Takigawa T. Morino Y. Urayama K. Masuda T. Poisson's ratio of polyacrylamide (PAAm) gels.Polym. Gels Networks. 1996; 4: 1-5Crossref Scopus (75) Google Scholar).TFM problems can be solved directly (direct TFM) by calculating the strain field ε from the measured displacement field u, according to the linearized expression, which holds for small strains:εi,j=12(∂ui∂xj+∂uj∂xi)(1) with u=(u1,u2,u3) and x=(x1,x2,x3). The stress field σ can then be derived by the constitutive law for linear elasticity (Hooke's law):σ=cε(2) where c is the stiffness tensor, which describes the substrate properties (can be expressed in terms of E and v).Direct TFM is a relatively recent approach since its implementation demands for high-resolution and high-accuracy measurement of the displacement field, which is mandatory for accurate strain reconstruction.Most commonly, an inverse TFM approach is followed, where cell tractions t can be described in terms of the displacement field u using a convolution approach, described by the following Fredholm integral (98Liangguo W. Xiao L. On the general expression of Fredholm integral equations method in elasticity.Acta Mech. Sin. 1988; 4: 138-145Crossref Scopus (1) Google Scholar), which makes use of the Green's function G:u(r)=∫dr′G(r−r′)t(r′).(3) G describes the impulse response (i.e., the output of a linear system with zero initial conditions and a unit impulse function as the input) of the system to a point load. The integral in Equation 3 can be interpreted as a summation, so that each displacement u located at r=(x,y,z) is the net effect of all the traction forces t acting in r′=(x′,y′,z′).In typical TFM settings, where cell-induced displacements are ca. two orders of magnitude smaller than the substrate thickness (usually in the 50–80 μm range), the Boussinesq approximation