Determining substrate displacement and cell traction fields--a new approach

This paper presents a new approach for the traction force microscopy (TFM) method which determines traction forces exerted by adherent cells on a thin, elastic polyacrylamide gel embedded with fluorescent microbeads. In this enhanced TFM method, a pattern recognition technique is first applied to match the pair of microbead embedded images before and after deformation, which subsequently provides the displacement field of the elastic substrate. Once the displacement field is obtained, the 3-D finite element method (FEM) is used to compute cell traction forces. The new TFM has been applied to determine traction forces of human tendon fibroblasts. Compared to existing TFM methods, the present method has the following advantages: (1) its displacement field obtained is associated with microbead movements; (2) it considers the finite thickness of the thin polyacrylamide gel and is therefore free from the infinite half-space approximation adopted by existing TFM methods; and (3) its computation procedure for determining cell traction forces is fast.     

Simulation and evaluation of 3D traction force microscopy

Measuring cell-generated forces by Traction Force Microscopy (TFM) has become a standard tool in cell mechanobiology. Although widely used in two dimensional (2D) experiments, only a few methods exist to measure traction in three-dimensional (3D) cell culture, since 3D volumetric high-resolution microscopy and more demanding computational approaches are required. Although it is commonly known that the selected experimental and computational setup highly influence the quality and accuracy of the results, no existing methods can adequately assess the errors involved in this process. We present a fully integrated simulation and evaluation platform that allows one to simulate TFM images and quantify errors of an applied approach for traction stress reconstruction, in order to improve experiments that attempt to measure mechanical interaction in cellular systems. In this context, we show that a careful parameter selection can decrease the reconstructed traction error by up to 40%.     

Free Form Deformation–Based Image Registration Improves Accuracy of Traction Force Microscopy

Traction Force Microscopy (TFM) is a widespread method used to recover cellular tractions from the deformation that they cause in their surrounding substrate. Particle Image Velocimetry (PIV) is commonly used to quantify the substrate’s deformations, due to its simplicity and efficiency. However, PIV relies on a block-matching scheme that easily underestimates the deformations. This is especially relevant in the case of large, locally non-uniform deformations as those usually found in the vicinity of a cell’s adhesions to the substrate. To overcome these limitations, we formulate the calculation of the deformation of the substrate in TFM as a non-rigid image registration process that warps the image of the unstressed material to match the image of the stressed one. In particular, we propose to use a B-spline -based Free Form Deformation (FFD) algorithm that uses a connected deformable mesh to model a wide range of flexible deformations caused by cellular tractions. Our FFD approach is validated in 3D fields using synthetic (simulated) data as well as with experimental data obtained using isolated endothelial cells lying on a deformable, polyacrylamide substrate. Our results show that FFD outperforms PIV providing a deformation field that allows a better recovery of the magnitude and orientation of tractions. Together, these results demonstrate the added value of the FFD algorithm for improving the accuracy of traction recovery.     

Three-dimensional traction force microscopy: a new tool for quantifying cell-matrix interactions

The interactions between biochemical processes and mechanical signaling play important roles during various cellular processes such as wound healing, embryogenesis, metastasis, and cell migration. While traditional traction force measurements have provided quantitative information about cell matrix interactions in two dimensions, recent studies have shown significant differences in the behavior and morphology of cells when placed in three-dimensional environments. Hence new quantitative experimental techniques are needed to accurately determine cell traction forces in three dimensions. Recently, two approaches both based on laser scanning confocal microscopy have emerged to address this need. This study highlights the details, implementation and advantages of such a three-dimensional imaging methodology with the capability to compute cellular traction forces dynamically during cell migration and locomotion. An application of this newly developed three-dimensional traction force microscopy (3D TFM) technique to single cell migration studies of 3T3 fibroblasts is presented to show that this methodology offers a new quantitative vantage point to investigate the three-dimensional nature of cell-ECM interactions.     

Improved-Throughput Traction Microscopy Based on Fluorescence Micropattern for Manual Microscopy

Traction force microscopy (TFM) is a quantitative technique for measuring cellular traction force, which is important in understanding cellular mechanotransduction processes. Traditional TFM has a significant limitation in that it has a low measurement throughput, commonly one per TFM dish, due to a lack of cell position information. To obtain enough cellular traction force data, an onerous workload is required including numerous TFM dish preparations and heavy cell-seeding activities, creating further difficulty in achieving identical experimental conditions among batches. In this paper, we present an improved-throughput TFM method using the well-developed microcontact printing technique and chemical modifications of linking microbeads to the gel surface to address these limitations. Chemically linking the microbeads to the gel surface has no significant influence on cell proliferation, morphology, cytoskeleton, and adhesion. Multiple pairs of force loaded and null force fluorescence images can be easily acquired by means of manual microscope with the aid of a fluorescence micropattern made by microcontact printing. Furthermore, keeping the micropattern separate from cells by using gels effectively eliminates the potential negative effect of the micropattern on the cells. This novel design greatly improves the analysis throughput of traditional TFM from one to at least twenty cells per petri dish without losing unique advantages, including a high spatial resolution of traction measurements. This newly developed method will boost the investigation of cell-matrix mechanical interactions.     

Couple-stress moduli of a trabecular bone idealized as a 3D periodic cellular network

Trabecular bone is modeled as a cellular material with an idealized periodic structure made of open cubic cells, which is effectively orthotropic. We evaluate apparent couple-stress moduli of such a periodic material; apparent moduli refer to the moduli obtained using a domain smaller than a Representative Volume Element and they depend on boundary conditions. We conduct this analysis computationally (using ANSYS) by subjecting a unit cell of this periodic cellular material to either displacement or traction boundary conditions. Cell walls, representing bone tissue, and void space, representing bone marrow, are both modeled and they are assumed to be linear elastic. The applied loadings include a uniaxial extension (or uniaxial stress), a hydrostatic deformation (or hydrostatic stress) and a shear deformation (or shear stress) to evaluate the first stiffness (or compliance) tensor, and an applied curvature (or bending moment), a uniaxial twist (or torsion), and a triaxial twist (or triaxial torsion) to evaluate the second couple-stress stiffness (or compliance) tensor. Apparent couple-stress moduli are computed by equating the total strain energy stored in the unit cell with the energy of an equivalent homogeneous orthotropic couple-stress material for each applied loading. The moduli computed using displacement boundary conditions give upper bound, while those obtained using traction boundary conditions give lower bound on effective couple-stress moduli. These bounds are very wide due to a large mismatch in elastic moduli of bone tissue and bone marrow. These results are in agreement with our studies on composite materials with very stiff or very compliant inclusions.     

Validation tool for traction force microscopy

Traction force microscopy (TFM) is commonly used to estimate cells' traction forces from the deformation that they cause on their substrate. The accuracy of TFM highly depends on the computational methods used to measure the deformation of the substrate and estimate the forces, and also on the specifics of the experimental set-up. Computer simulations can be used to evaluate the effect of both the computational methods and the experimental set-up without the need to perform numerous experiments. Here, we present one such TFM simulator that addresses several limitations of the existing ones. As a proof of principle, we recreate a TFM experimental set-up, and apply a classic 2D TFM algorithm to recover the forces. In summary, our simulator provides a valuable tool to study the performance, refine experimentally, and guide the extraction of biological conclusions from TFM experiments.     

Traction force microscopy by deep learning

Cells interact mechanically with their surroundings by exerting and sensing forces. Traction force microscopy (TFM), purported to map cell-generated forces or stresses, represents an important tool that has powered the rapid advances in mechanobiology. However, to solve the ill-posed mathematical problem, conventional TFM involved compromises in accuracy and/or resolution. Here, we applied neural network-based deep learning as an alternative approach for TFM. We modified a neural network designed for image processing to predict the vector field of stress from displacements. Furthermore, we adapted a mathematical model for cell migration to generate large sets of simulated stresses and displacements for training and testing the neural network. We found that deep learning-based TFM yielded results that resemble those using conventional TFM but at a higher accuracy than several conventional implementations tested. In addition, a trained neural network is appliable to a wide range of conditions, including cell size, shape, substrate stiffness, and traction output. The performance of deep learning-based TFM makes it an appealing alternative to conventional methods for characterizing mechanical interactions between adherent cells and the environment.     

Traction Force Measurements of Human Aortic Smooth Muscle Cells Reveal a Motor-Clutch Behavior

The contractile behavior of smooth muscle cells (SMCs) in the aorta is an important determinant of growth, remodeling, and homeostasis. However, quantitative values of SMC basal tone have never been characterized precisely on individual SMCs. Therefore, to address this lack, we developed an in vitro technique based on Traction Force Microscopy (TFM). Aortic SMCs from a human lineage at low passages (4-7) were cultured 2 days in conditions promoting the development of their contractile apparatus and seeded on hydrogels of varying elastic modulus (1, 4, 12 and 25 kPa) with embedded fluorescent microspheres. After complete adhesion, SMCs were artificially detached from the gel by trypsin treatment. The microbeads movement was tracked and the deformation fields were processed with a mechanical model, assuming linear elasticity, isotropic material, plane strain, to extract the traction forces formerly applied by individual SMCs on the gel. Two major interesting and original observations about SMC traction forces were deduced from the obtained results: 1. they are variable but driven by cell dynamics and show an exponential distribution, with 40% to 80% of traction forces in the range 0-10 μN. 2. They depend on the substrate stiffness: the fraction of adhesion forces below 10 μN tend to decrease when the substrate stiffness increases, whereas the fraction of higher adhesion forces increases. As these two aspects of cell adhesion (variability and stiffness dependence) and the distribution of their traction forces can be predicted by the probabilistic motor-clutch model, we conclude that this model could be applied to SMCs. Further studies will consider stimulated contractility and primary culture of cells extracted from aneurysmal human aortic tissue.     

Cellular contractility and substrate elasticity: a numerical investigation of the actin cytoskeleton and cell adhesion

Numerous experimental studies have established that cells can sense the stiffness of underlying substrates and have quantified the effect of substrate stiffness on stress fibre formation, focal adhesion area, cell traction, and cell shape. In order to capture such behaviour, the current study couples a mixed mode thermodynamic and mechanical framework that predicts focal adhesion formation and growth with a material model that predicts stress fibre formation, contractility, and dissociation in a fully 3D implementation. Simulations reveal that SF contractility plays a critical role in the substrate-dependent response of cells. Compliant substrates do not provide sufficient tension for stress fibre persistence, causing dissociation of stress fibres and lower focal adhesion formation. In contrast, cells on stiffer substrates are predicted to contain large amounts of dominant stress fibres. Different levels of cellular contractility representative of different cell phenotypes are found to alter the range of substrate stiffness that cause the most significant changes in stress fibre and focal adhesion formation. Furthermore, stress fibre and focal adhesion formation evolve as a cell spreads on a substrate and leading to the formation of bands of fibres leading from the cell periphery over the nucleus. Inhibiting the formation of FAs during cell spreading is found to limit stress fibre formation. The predictions of this mutually dependent material-interface framework are strongly supported by experimental observations of cells adhered to elastic substrates and offer insight into the inter-dependent biomechanical processes regulating stress fibre and focal adhesion formation.     

Intercellular Stress Reconstitution from Traction Force Data

Cells migrate collectively during development, wound healing, and cancer metastasis. Recently, a method has been developed to recover intercellular stress in monolayers from measured traction forces upon the substrate. To calculate stress maps in two dimensions, the cell sheet was assumed to behave like an elastic material, and it remains unclear to what extent this assumption is valid. In this study, we simulate our recently developed model for collective cell migration, and compute intercellular stress maps using the method employed in the experiments. We also compute these maps using a method that does not depend on the traction forces or material properties. The two independently obtained stress patterns agree well for the parameters we have probed and provide a verification of the validity of the experimental method.     

Heterogeneous drying stresses in stratum corneum

We study the drying of stratum corneum, the skin's outermost layer and an essential barrier to mechanical and chemical stresses from the environment. Even though stratum corneum exhibits structural features across multiple length-scales, contemporary understanding of the mechanical properties of stratum corneum is based on the assumption that its thickness and composition are homogeneous. We quantify spatially resolved in-plane traction stress and deformation at the interface between a macroscopic sample of porcine stratum corneum and an adherent deformable elastomer substrate. At length-scales greater than a millimeter, the skin behaves as a homogeneous elastic material. At this scale, a linear elastic model captures the spatial distribution of traction stresses and the dependence of drying behavior on the elastic modulus of the substrate. At smaller scales, the traction stresses are strikingly heterogeneous and dominated by the heterogeneous structure of the stratum corneum.     

The biochemical composition of the actomyosin network sets the magnitude of cellular traction forces

The regulation of cellular force production relies on the complex interplay between a well-conserved set of proteins of the cytoskeleton: actin, myosin, and α-actinin. Despite our deep knowledge of the role of these proteins in force production at the molecular scale, our understanding of the biochemical regulation of the magnitude of traction forces generated at the entire-cell level has been limited, notably by the technical challenge of measuring traction forces and the endogenous biochemical composition in the same cell. In this study, we developed an alternative Traction-Force Microscopy (TFM) assay, which used a combination of hydrogel micropatterning to define cell adhesion and shape and an intermediate fixation/immunolabeling step to characterize strain energies and the endogenous protein contents in single epithelial cells. Our results demonstrated that both the signal intensity and the area of the Focal Adhesion (FA)–associated protein vinculin showed a strong positive correlation with strain energy in mature FAs. Individual contents from actin filament and phospho-myosin displayed broader deviation in their linear relationship to strain energies. Instead, our quantitative analyzes demonstrated that their relative amount exhibited an optimum ratio of phospho-myosin to actin, allowing maximum force production by cells. By contrast, although no correlation was identified between individual α-actinin content and strain energy, the ratio of α-actinin to actin filaments was inversely related to strain energy. Hence, our results suggest that, in the cellular model studied, traction-force magnitude is dictated by the relative numbers of molecular motors and cross-linkers per actin filament, rather than the amounts of an individual component in the cytoskeletal network. This assay offers new perspectives to study in more detail the complex interplay between the endogenous biochemical composition of individual cells and the force they produce.     

A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure

A 3D anisotropic micropolar continuum model of vertebral trabecular bone is presently developed accounting for the influence of microstructure-related scale effects on the macroscopic effective properties. Vertebral trabecular bone is modeled as a cellular material with an idealized periodic structure made of open 3D cells. The micromechanical approach relies on the discrete homogenization technique considering lattice microrotations as additional degrees of freedom at the microscale. The effective elastic properties of 3D lattices made of articulated beams taking into account axial, transverse shearing, flexural, and torsional deformations of the cell struts are derived as closed form expressions of the geometrical and mechanical microparameters. The scaling laws of the effective moduli versus density are determined in situations of low and high effective densities to assess the impact of the transverse shear deformation. The classical and micropolar effective moduli and the internal flexural and torsional lengths are identified versus the same microparameters. A finite element model of the local architecture of the trabeculae gives values of the effective moduli that are in satisfactory agreement with the homogenized moduli.     

Active Prestress Leads to an Apparent Stiffening of Cells through Geometrical Effects

Tuning of active prestress, e.g., through activity of molecular motors, constitutes a powerful cellular tool to adjust cellular stiffness through nonlinear material properties. Understanding this tool is an important prerequisite for our comprehension of cellular force response, cell shape dynamics, and tissue organization. Experimental data obtained from cell-mechanical measurements often show a simple linear dependence between mechanical prestress and measured differential elastic moduli. Although these experimental findings could point to stress-induced structural changes in the material, we propose a surprisingly simple alternative explanation in a theoretical study. We show how geometrical effects can give rise to increased cellular force response of cells in the presence of active prestress. The associated effective stress-stiffening is disconnected from actual stress-induced changes of the elastic modulus, and should therefore be regarded as an apparent stiffening of the material. We argue that new approaches in experimental design are necessary to separate this apparent stress-stiffening due to geometrical effects from actual nonlinearities of the elastic modulus in prestressed cellular material.     

Towards a biomechanics-based technique for assessing myocardial contractility: an inverse problem approach

This work presents the initial development and implementation of a novel 3D biomechanics-based approach to measure the mechanical activity of myocardial tissue, as a potential non-invasive tool to assess myocardial function. This technique quantifies the myocardial contraction forces developed within the ventricular myofibers in response to electro-physiological stimuli. We provide a 3D finite element formulation of a contraction force reconstruction algorithm, along with its implementation using magnetic resonance (MR) data. Our algorithm is based on an inverse problem solution governed by the fundamental continuum mechanics principle of conservation of linear momentum, under a first-order approximation of elastic and isotropic material conditions. We implemented our technique using a subject-specific ventricle model obtained by extracting the left ventricular anatomical features from a set of high-resolution cardiac MR images acquired throughout the cardiac cycle using prospective electrocardiographic gating. Cardiac motion information was extracted by non-rigid registration of the mid-diastole reference image to the remaining images of a 4D dataset. Using our technique, we reconstructed dynamic maps that show the contraction force distribution superimposed onto the deformed ventricle model at each acquired frame in the cardiac cycle. Our next objective will consist of validating this technique by showing the correlation between the presence of low contraction force patterns and poor myocardial functionality.     

An array of microfabricated pillars to study cell migration

Mechanical forces play an important role in various cellular functions, such as tumor metastasis, embryonic development or tissue formation. Cell migration involves dynamics of adhesive processes and cytoskeleton remodelling, leading to traction forces between the cells and their surrounding extracellular medium. To study these mechanical forces, a number of methods have been developed to calculate tractions at the interface between the cell and the substrate by tracking the displacements of beads or microfabricated markers embedded in continuous deformable gels. These studies have provided the first reliable estimation of the traction forces under individual migrating cells. We have developed a new force sensor made of a dense array of soft micron-size pillars microfabricated using microelectronics techniques. This approach uses elastomeric substrates that are micropatterned by using a combination of hard and soft lithography. Traction forces are determined in real time by analyzing the deflections of each micropillar with an optical microscope. Indeed, the deflection is directly proportional to the force in the linear regime of small deformations. Epithelial cells are cultured on our substrates coated with extracellular matrix protein. First, we have characterized temporal and spatial distributions of traction forces of a cellular assembly. Forces are found to depend on their relative position in the monolayer : the strongest deformations are always localized at the edge of the islands of cells in the active areas of cell protrusions. Consequently, these forces are quantified and correlated with the adhesion/scattering processes of the cells.     

Eigenstrain as a mechanical set-point of cells

Cell contraction regulates how cells sense their mechanical environment. We sought to identify the set-point of cell contraction, also referred to as tensional homeostasis. In this work, bovine aortic endothelial cells (BAECs), cultured on substrates with different stiffness, were characterized using traction force microscopy (TFM). Numerical models were developed to provide insights into the mechanics of cell–substrate interactions. Cell contraction was modeled as eigenstrain which could induce isometric cell contraction without external forces. The predicted traction stresses matched well with TFM measurements. Furthermore, our numerical model provided cell stress and displacement maps for inspecting the fundamental regulating mechanism of cell mechanosensing. We showed that cell spread area, traction force on a substrate, as well as the average stress of a cell were increased in response to a stiffer substrate. However, the cell average strain, which is cell type-specific, was kept at the same level regardless of the substrate stiffness. This indicated that the cell average strain is the tensional homeostasis that each type of cell tries to maintain. Furthermore, cell contraction in terms of eigenstrain was found to be the same for both BAECs and fibroblast cells in different mechanical environments. This implied a potential mechanical set-point across different cell types. Our results suggest that additional measurements of contractility might be useful for monitoring cell mechanosensing as well as dynamic remodeling of the extracellular matrix (ECM). This work could help to advance the understanding of the cell-ECM relationship, leading to better regenerative strategies.