Fourier Transform Traction Cytometry Research Articles

Enhancing robustness, precision, and speed of traction force microscopy with machine learning

Traction patterns of adherent cells provide important information on their interaction with the environment, cell migration, or tissue patterns and morphogenesis. Traction force microscopy is a method aimed at revealing these traction patterns for adherent cells on engineered substrates with known constitutive elastic properties from deformation information obtained from substrate images. Conventionally, the substrate deformation information is processed by numerical algorithms of varying complexity to give the corresponding traction field via solution of an ill-posed inverse elastic problem. We explore the capabilities of a deep convolutional neural network as a computationally more efficient and robust approach to solve this inversion problem. We develop a general purpose training process based on collections of circular force patches as synthetic training data, which can be subjected to different noise levels for additional robustness. The performance and the robustness of our approach against noise is systematically characterized for synthetic data, artificial cell models, and real cell images, which are subjected to different noise levels. A comparison with state-of-the-art Bayesian Fourier transform traction cytometry reveals the precision, robustness, and speed improvements achieved by our approach, leading to an acceleration of traction force microscopy methods in practical applications.

Control of Cell Adhesion using Hydrogel Patterning Techniques for Applications in Traction Force Microscopy

Traction force microscopy (TFM) is the main method used in mechanobiology to measure cell forces. Commonly this is being used for cells adhering to flat soft substrates that deform under cell traction (2D-TFM). TFM relies on the use of linear elastic materials, such as polydimethylsiloxane (PDMS) or polyacrylamide (PA). For 2D-TFM on PA, the difficulty in achieving high throughput results mainly from the large variability of cell shapes and tractions, calling for standardization. We present a protocol to rapidly and efficiently fabricate micropatterned PA hydrogels for 2D-TFM studies. The micropatterns are first created by maskless photolithography using near-UV light where extracellular matrix proteins bind only to the micropatterned regions, while the rest of the surface remains non-adhesive for cells. The micropatterning of extracellular matrix proteins is due to the presence of active aldehyde groups, resulting in adhesive regions of different shapes to accommodate either single cells or groups of cells. For TFM measurements, we use PA hydrogels of different elasticity by varying the amounts of acrylamide and bis-acrylamide and tracking the displacement of embedded fluorescent beads to reconstruct cell traction fields with regularized Fourier Transform Traction Cytometry (FTTC). To further achieve precise recording of cell forces, we describe the use of a controlled dose of patterned light to release cell tractions in defined regions for single cells or groups of cells. We call this method local UV illumination traction force microscopy (LUVI-TFM). With enzymatic treatment, all cells are detached from the sample simultaneously, whereas with LUVI-TFM traction forces of cells in different regions of the sample can be recorded in sequence. We demonstrate the applicability of this protocol (i) to study cell traction forces as a function of controlled adhesion to the substrate, and (ii) to achieve a greater number of experimental observations from the same sample.

Control of Cell Adhesion using Hydrogel Patterning Techniques for Applications in Traction Force Microscopy.

Traction force microscopy (TFM) is the main method used in mechanobiology to measure cell forces. Commonly this is being used for cells adhering to flat soft substrates that deform under cell traction (2D-TFM). TFM relies on the use of linear elastic materials, such as polydimethylsiloxane (PDMS) or polyacrylamide (PA). For 2D-TFM on PA, the difficulty in achieving high throughput results mainly from the large variability of cell shapes and tractions, calling for standardization. We present a protocol to rapidly and efficiently fabricate micropatterned PA hydrogels for 2D-TFM studies. The micropatterns are first created by maskless photolithography using near-UV light where extracellular matrix proteins bind only to the micropatterned regions, while the rest of the surface remains non-adhesive for cells. The micropatterning of extracellular matrix proteins is due to the presence of active aldehyde groups, resulting in adhesive regions of different shapes to accommodate either single cells or groups of cells. For TFM measurements, we use PA hydrogels of different elasticity by varying the amounts of acrylamide and bis-acrylamide and tracking the displacement of embedded fluorescent beads to reconstruct cell traction fields with regularized Fourier Transform Traction Cytometry (FTTC). To further achieve precise recording of cell forces, we describe the use of a controlled dose of patterned light to release cell tractions in defined regions for single cells or groups of cells. We call this method local UV illumination traction force microscopy (LUVI-TFM). With enzymatic treatment, all cells are detached from the sample simultaneously, whereas with LUVI-TFM traction forces of cells in different regions of the sample can be recorded in sequence. We demonstrate the applicability of this protocol (i) to study cell traction forces as a function of controlled adhesion to the substrate, and (ii) to achieve a greater number of experimental observations from the same sample.

High-Resolution, Highly-Integrated Traction Force Microscopy Software.

Accurate measurement of cellular traction force is critical for understanding physical interaction between cells and the extracellular matrix. Traction force microscopy (TFM) has become the most widely used tool for this purpose. While TFM has made continual progress in terms of resolution and accuracy, there have been challenges regarding obtaining user-friendly software and choosing the right values for parameters and sub-processes associated with the software. Here we provide step-by-step instructions for a MATLAB-based TFM software application equipped with multiple methods for image deformation quantification and force reconstruction, along with clarification on the computational meaning of the parameters within the software. We outline how to choose the optimal sub-methods and values for parameters for each process, depending on the characteristics of images and purpose of the analyses. The software's runtime is 20, 4, and 0.05 min by Fast BEM L1 (Boundary Element Method L1-regularization), Fast BEM L2 (L2-regularization), and FTTC (Fourier Transform Traction Cytometry), respectively, in addition to 7 min of particle-tracking velocimetry-based deformation tracking, for a single image (1280 × 960 pixel) on a standard workstation. Finally, the colocalization accuracies, in reference to a paxillin-GFP image, are compared between the three force reconstruction methods. © 2021 Wiley Periodicals LLC. Basic Protocol 1: Setting up the TFM package in MATLAB Basic Protocol 2: Running the TFM package Alternate Protocol 1: Stage drift correction: Efficient subpixel registration Alternate Protocol 2: Force field calculation: FastBEM.

A Bayesian traction force microscopy method with automated denoising in a user-friendly software package

Adherent biological cells generate traction forces on a substrate that play a central role for migration, mechanosensing, differentiation, and collective behavior. The established method for quantifying this cell–substrate interaction is traction force microscopy (TFM). In spite of recent advancements, inference of the traction forces from measurements remains very sensitive to noise. However, suppression of the noise reduces the measurement accuracy and the spatial resolution, which makes it crucial to select an optimal level of noise reduction. Here, we present a fully automated method for noise reduction and robust, standardized traction-force reconstruction. The method, termed Bayesian Fourier transform traction cytometry, combines the robustness of Bayesian L2 regularization with the computation speed of Fourier transform traction cytometry. We validate the performance of the method with synthetic and real data. The method is made freely available as a software package with a graphical user-interface for intuitive usage. Program summaryProgram Title: Easy-to-use TFM softwareProgram Files doi:http://dx.doi.org/10.17632/229bnpp8rb.1Licensing provisions: GNU General Public License v3.0Programming language: Matlab version R2010b or higherSupplementary material: A user manual for the software and a test data set.Nature of problem: Calculation of the traction forces on the surface of an elastic material from observed displacements.Solution method: Traction forces are efficiently calculated by combining L2 regularization in Fourier space with Bayesian inference of the regularization parameter.

Non-intrusive measurement of wall shear stress in flow channels

Flow shear stress measurement plays an important role in the characterization of macro and microfluidic systems. Many currently used wall shear sensors quantify local shear stress with the use of fluid-disruptive probes, unless installed accurately flush to the channel surface. Non-intrusive shear stress measurement systems capable of quantifying the shear stress vector field in larger areas are highly desirable. The present study reports on non-intrusive direct measurement of wall shear stress under pressure-driven fluid flows with the use of particle imaging velocimetry and Fourier transform traction cytometry. This method uses the known mechanical properties of a soft substrate strained under the flow to quantify the shear stress field. Under fully developed pressure-driven laminar flows of different flow rates in a rectangular channel, the average magnitude of wall shear stress thus obtained matched with the theoretical results obtained for Poiseuille flow. The major advantage of this method is the direct experimental characterization of wall shear stress vector field without disruption of the flow itself. The method shows promise in the characterization of shear flow in diverse areas such as aerospace and bioengineering.

Forward and Inverse Approaches to Characterizing Cellular Traction Forces

Adherent cells exert tractional stresses on substrates through transmission of intracellular contractile forces. Traction forces influence processes including cell migration, crucial in cancer metastasis and wound healing. Quantification of these forces is hence essential to our understanding of cell-ECM interactions in normal and pathological conditions. In this study, we adapted the finite element (FE) method to quantify cellular tractions and used strain energy, stress and reaction forces as measures to characterize cell-substrate interactions using fibroblasts transfected with GFP-paxillin cultured on polyacrylamide substrates. FE analysis showed rapid convergence in strain energy and net reaction force with mesh refinement; stresses however converged slowly. We compared these results with Fourier Transform Traction Cytometry (FTTC) and regularised FTTC (Reg-FTTC) method which overcomes problems in inversion occurring due to noise. Mesh convergence for high stress nodes in FE was comparable to Reg-FTTC; FTTC stresses had poor convergence in comparison. Strain energy from FE method was lower than that obtained from both Fourier methods. Further, maximum stress was highest for FTTC, followed by FE and Reg-FTTC methods. Overall, the FE method performed significantly better than FTTC based on these metrics. The Reg-FTTC provides an improvement over slow stress convergence in FE but depends crucially on choice of regularization parameter which necessitates optimization for individual cell types. We used reaction force output, unique to FE, to quantify localized adhesion forces acting on discrete regions of cell periphery that may provide insights into spatial organization of cell-ECM interactions. Together, we hope these studies will help in creation of robust tools to characterize the mechanical interplay between cell and microenvironment which will further our understanding of disease progression and cell behaviour.

Mathematical framework for traction force microscopy

This paper deals with the Traction Force Microscopy (TFM) problem. It consists in obtaining stresses by solving an inverse problem in an elastic medium, from known experimentally measured displacements. In this article, the application is the determination of the stresses exerted by a living cell at the surface of an elastic gel. We propose an abstract framework which formulates this inverse problem as a constrained minimization problem. The mathematical constraints express the biomechanical conditions that the cell must satisfy. From this framework, two methods currently used can be derived, the adjoint method (AM) and the Fourier Transform Traction Cytometry (FTTC) method. An improvement of the FTTC method is also derived using this framework. The numerical results are compared and show the advantage of the AM, in particular it can capture details more accurately.

Inverse problems for the determination of traction forces by cells on a substrate: a comparison of two methods

Traction forces exerted by cells on soft elastic substrates are important for characterizing the types of mechanisms used for cell migration. Classical tools for the determination of traction forces include the knowledge of the displacement field thanks to fluorescent beads embedded into the substrate. Then, from the discrete beads displacements, an inverse problem is solved to obtain the stress field. Two currently used methods in the literature are the well-known Fourier Transform Traction Cytometry (FTTC) method and the adjoint method (AM), which are compared here. A real case is presented where the displacement field is known from cancer cell migration study. The two methods allow the recovery of the traction stresses and their results are compared. Similar results are seen as long as an adequate projection technique is used (zero force imposed outside the cell). It is found that the AM allows a finer resolution of the traction forces, in particular at the cell edge. This is a strong incentive to use this method for the investigation of cancer cell migration on soft substrates.

Cellular Traction Force Reconstruction Based on a Self-adaptive Filtering Scheme

As an emerging measurement technique in cell mechanics, cellular traction force microscopy has begun to be applied to monitor spatiotemporal dynamics of cell-substrate interactions, in which Fourier transform traction cytometry (FTTC) can be utilized to reconstruct cellular traction fields from substrate displacement knowledge in a computationally cheap way. Owing to the intrinsic ill-posed property, cellular traction recovery founded on FTTC is extremely susceptible to measurement noise so that it is very likely to produce unreliable traction results. This paper investigates the nature of noise amplification during the process of deconvolution and accordingly proposes a set of filtering algorithms to evaluate cellular tractions. By self-adaptively filtering out the noise components in the two-dimensional Fourier space, high-resolution traction fields can thus be recovered in a relatively efficient manner. The feasibility and effectiveness of the set of algorithms are verified by both systematic simulation analyses and actual cellular traction reconstructions.

Optimization of traction force microscopy for micron-sized focal adhesions

To understand how adherent cells regulate traction forces on their surroundingextracellular matrix (ECM), quantitative techniques are needed to measure forces atthe cell–ECM interface. Microcontact printing is used to create a substrate of1 µm diameter circles of ECM ligand to experimentally study the reconstruction oftraction stresses at constrained, point-like focal adhesions. Traction reconstructionwith point forces (TRPF) and Fourier transform traction cytometry (FTTC)are used to calculate the traction forces and stress field, respectively, at isolatedadhesions. We find that the stress field calculated with FTTC peaks near thecenter of individual adhesions but propagates several microns beyond the adhesionlocation. We find the optimal set of FTTC parameters that yield the highest stressmagnitude, minimizing information lost from over-smoothing and sampling of thedisplacement or stress field. A positive correlation between the TRPF and FTTCmeasurements exists, but integrating the FTTC stress field over the adhesion area yieldsonly a small fraction of the force calculated by TRPF. An effective area similarto that defined by the width of the stress distribution measured with FTTC isrequired to reconcile these measurements. These measurements set bounds on thespatial resolution and precision of FTTC measurements on micron-sized adhesions.

Cellular traction force recovery: An optimal filtering approach in two-dimensional Fourier space

Quantitative estimation of cellular traction has significant physiological and clinical implications. As an inverse problem, traction force recovery is essentially susceptible to noise in the measured displacement data. For traditional procedure of Fourier transform traction cytometry (FTTC), noise amplification is accompanied in the force reconstruction and small tractions cannot be recovered from the displacement field with low signal–noise ratio (SNR). To improve the FTTC process, we develop an optimal filtering scheme to suppress the noise in the force reconstruction procedure. In the framework of the Wiener filtering theory, four filtering parameters are introduced in two-dimensional Fourier space and their analytical expressions are derived in terms of the minimum-mean-squared-error (MMSE) optimization criterion. The optimal filtering approach is validated with simulations and experimental data associated with the adhesion of single cardiac myocyte to elastic substrate. The results indicate that the proposed method can highly enhance SNR of the recovered forces to reveal tiny tractions in cell–substrate interaction.

Determination of Cellular Tractions on Elastic Substrate Based on an Integral Boussinesq Solution

Cell-substrate interaction is implicated in many physiological processes. Dynamical monitoring of cellular tractions on substrate is critical in investigating a variety of cell functions such as contraction, migration, and invasion. On account of the inherent ill-posed property as an inverse problem, cellular traction recovery is essentially sensitive to substrate displacement noise and thus likely produces unstable results. Therefore, some additional constraints must be applied to obtain a reliable traction estimate. By integrating the classical Boussinesq solution over a small rectangular area element, we obtain a new analytical solution to express the relation between tangential tractions and induced substrate displacements, and then form an alternative discrete Green's function matrix to set up a new framework of cellular force reconstruction. Deformation images of flexible substrate actuated by a single cardiac myocyte are processed by digital image correlation technique and the displacement data are sampled with a regular mesh to obtain cellular tractions by the proposed solution. Numerical simulations indicate that the 2-norm condition number of the improved coefficient matrix typically does not exceed the order of 100 for actual computation of traction recovery, and that the traction reconstruction is less sensitive to the shift or subdivision of the data sampling grid. The noise amplification arising from ill-posed inverse problem can be restrained and the stability of inverse solution is improved so that regularization operations become less relevant to the present force reconstruction with economical sampling density. The traction recovery for a single cardiac myocyte, which is in good agreement with that obtained by the Fourier transform traction cytometry, demonstrates the feasibility of the proposed method. We have developed a simple and efficient method to recover cellular traction field from substrate deformation. Unlike previous force reconstructions that numerically employ some regularization schemes, the present approach stabilizes the traction recovery by analytically improving the Green's function such that the intricate regularizations can be avoided under proper conditions. The method has potential application to a real-time traction force microscopy in combination with a high-efficiency displacement acquisition technique.

High Resolution Traction Force Microscopy Based on Experimental and Computational Advances

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

Spatial and temporal traction response in human airway smooth muscle cells.

Tractions that cells exert on their substrates are essential in cell spreading, migration, and contraction. These tractions can be determined by plating the cells on a flexible gel and measuring the deformation of the gel by using fluorescent beads embedded just below the surface of the gel. In this article we describe the image correlation method (ICM) optimized for determining the displacement field of the gel under a contracting cell. For the calculation of the traction field from the displacement field we use the recently developed method of Fourier transform traction cytometry (FTTC). The ICM and FTTC methods are applied to human airway smooth muscle cells during stimulation with the contractile agonist histamine or the relaxing agonist isoproterenol. The overall intensity of the cell contraction (the median traction magnitude, the energy transferred from the cell to the gel, and the net contractile moment) increased after activation with histamine, and decreased after treatment with isoproterenol. Cells exhibited regional differences in the time course of traction during the treatment. Both temporal evolution and magnitude of traction increase induced by histamine varied markedly among different cell protrusions, whereas the nuclear region showed the smallest response. These results suggest that intracellular mediators of cell adhesion and contraction respond to contractile stimuli with different rates and intensities in different regions of the cell.