Force traction microscopy: An inverse problem with pointwise observations

Force Traction Microscopy is an inversion method that allows to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself. This classical biophysical problem, usually addressed in terms of Green functions, can be alternatively tackled using a variational framework and then a finite elements discretization. In such a case, a variation of the error functional under suitable regularization is operated in view of its minimization. This setting naturally suggests the introduction of a new equation, based on the adjoint operator of the elasticity problem. In this paper we illustrate the rigorous theory of the two-dimensional and three dimensional problem, involving in the former case a distributed control and in the latter case a surface control. The pointwise observations require to exploit the theory of elasticity extended to forcing terms that are Borel measure

Non-regularised inverse finite element analysis for 3D traction force microscopy

The tractions that cells exert on a gel substrate from the observed displacements is an increasingly attractive and valuable information in biomedical experiments. The computation of these tractions requires in general the solution of an inverse problem. Here, we resort to the discretisation with finite elements of the associated direct variational formulation, and solve the inverse analysis using a least square approach. This strategy requires the minimisation of an error functional, which is usually regularised in order to obtain a stable system of equations with a unique solution. In this paper we show that for many common threedimensional geometries, meshes and loading conditions, this regularisation is unnecessary. In these cases, the computational cost of the inverse problem becomes equivalent to a direct finite element problem. For the non-regularised functional, we deduce the necessary and sufficient conditions that the dimensions of the interpolated displacement and traction fields must preserve in order to exactly satisfy or yield a unique solution of the discrete equilibrium equations. We apply the theoretical results to some illustrative examples and to real experimental data. Due to the relevance of the results for biologists and modellers, the article concludes with some practical rules that the finite element discretisation must satisfy.Peer ReviewedPostprint (author's final draft

3D migration of cells solving an inverse problem

International audienceTraction Force Microscopy (TFM) is an inverse method that allows to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself during its migration. This biophysical problem, usually addressed in terms of Green functions, can be alternatively tackled in a variational framework. In such a case, a suitable penalty functional has to be minimized. The resulting Euler-Lagrange equations include both the direct problem based on the linear elasticity operator as well as an equation built on its adjoint. Results from a two-dimensional model, i.e. where living cancer cells are migrating on a plane substrate, are briefly presented. While the mathematics is well established also in the three-dimensional case, i.e. where cells are completely embedded in the gel matrix, the experimental data needed are more difficult to obtain than the two-dimensional counterpart. First steps towards the complete three-dimensional traction reconstruction are reported

Computation of forces from deformed visco-elastic biological tissues

We present a least-squares based inverse analysis of visco-elastic biological tissues. The proposed method computes the set of contractile forces (dipoles) at the cell boundaries that induce the observed and quantified deformations. We show that the computation of these forces requires the regularisation of the problem functional for some load configurations that we study here. The functional measures the error of the dynamic problem being discretised in time with a second-order implicit time-stepping and in space with standard finite elements. We analyse the uniqueness of the inverse problem and estimate the regularisation parameter by means of an L-curved criterion. We apply the methodology to a simple toy problem and to an in vivo set of morphogenetic deformations of the Drosophila embryo.Peer ReviewedPostprint (author's final draft

Modeling cell movement in anisotropic and heterogeneous network tissues

Cell motion and interaction with the extracellular matrix is studied deriving a kinetic model and considering its diffusive limit. The model takes into account of chemotactic and haptotactic effects, and obtains friction as a result of the interactions between cells and between cells and the fibrous environment. The evolution depends on the fibre distribution, as cells preferentially move along the fibre direction and tend to cleave and remodel the extracellular matrix when their direction of motion is not aligned with the fibre direction. Simulations are performed to describe the behavior of ensemble of cells under the action of a chemotactic field and in presence of heterogeneous and anisotropic fibre networks

Traction force microscopy on soft elastic substrates: a guide to recent computational advances

The measurement of cellular traction forces on soft elastic substrates has become a standard tool for many labs working on mechanobiology. Here we review the basic principles and different variants of this approach. In general, the extraction of the substrate displacement field from image data and the reconstruction procedure for the forces are closely linked to each other and limited by the presence of experimental noise. We discuss different strategies to reconstruct cellular forces as they follow from the foundations of elasticity theory, including two- versus three-dimensional, inverse versus direct and linear versus non-linear approaches. We also discuss how biophysical models can improve force reconstruction and comment on practical issues like substrate preparation, image processing and the availability of software for traction force microscopy.Comment: Revtex, 29 pages, 3 PDF figures, 2 tables. BBA - Molecular Cell Research, online since 27 May 2015, special issue on mechanobiolog

Mechanical cell-matrix feedback explains pairwise and collective endothelial cell behavior in vitro

In vitro cultures of endothelial cells are a widely used model system of the collective behavior of endothelial cells during vasculogenesis and angiogenesis. When seeded in an extracellular matrix, endothelial cells can form blood vessel-like structures, including vascular networks and sprouts. Endothelial morphogenesis depends on a large number of chemical and mechanical factors, including the compliancy of the extracellular matrix, the available growth factors, the adhesion of cells to the extracellular matrix, cell-cell signaling, etc. Although various computational models have been proposed to explain the role of each of these biochemical and biomechanical effects, the understanding of the mechanisms underlying in vitro angiogenesis is still incomplete. Most explanations focus on predicting the whole vascular network or sprout from the underlying cell behavior, and do not check if the same model also correctly captures the intermediate scale: the pairwise cell-cell interactions or single cell responses to ECM mechanics. Here we show, using a hybrid cellular Potts and finite element computational model, that a single set of biologically plausible rules describing (a) the contractile forces that endothelial cells exert on the ECM, (b) the resulting strains in the extracellular matrix, and (c) the cellular response to the strains, suffices for reproducing the behavior of individual endothelial cells and the interactions of endothelial cell pairs in compliant matrices. With the same set of rules, the model also reproduces network formation from scattered cells, and sprouting from endothelial spheroids. Combining the present mechanical model with aspects of previously proposed mechanical and chemical models may lead to a more complete understanding of in vitro angiogenesis.Comment: 25 pages, 6 figures, accepted for publication in PLoS Computational Biolog

Mechanical cell-substrate feedback explains pairwise and collective endothelial cell behavior in vitro

In vitro cultures of endothelial cells are a widely used model system of the collective behavior of endothelial cells during vasculogenesis and angiogenesis. When seeded in a extracellular matrix, endothelial cells can form blood vessel-like structures, including vascular networks and sprouts. Endothelial morphogenesis depends on a large number of chemical and mechanical factors, including the compliancy of the extracellular matrix, the available growth factors, the adhesion of cells to the extracellular matrix, cell-cell signaling, etc. Although various computational models have been proposed to explain the role of each of these biochemical and biomechanical effects, the mechanisms underlying in vitro angiogenesis are still poorly understood. Most explanations focus on predicting the whole vascular network or sprout from the underlying cell behavior, and ignore the intermediate organizational levels of the system. Here we show, using a hybrid Cellular Potts and finite-element computational model, that a single set of biologically plausible rules describing (a) the contractile forces that endothelial cells exert on the ECM, (b) the resulting strains in the extracellular matrix, and (c) the cellular response to the strains, suffices for reproducing the behavior of individual endothelial cells and the interactions of endothelial cell pairs in compliant matrices. With the same set of rules, the model also reproduces network formation and sprouting from epithelial spheroids. Combining the present, mechanical model with aspects of previously proposed mechanical and chemical models may lead to a more complete understanding of in vitro angiogenesi

Modeling cell movement in anisotropic and heterogeneous network tissues

Cell motion and interaction with the extracellular matrix is studied deriving a kinetic model and considering its diffusive limit. The model takes into account of chemotactic and haptotactic effects, and obtains friction as a result of the interactions between cells and between cells and the fibrous environment. The evolution depends on the fibre distribution, as cells preferentially move along the fibre direction and tend to cleave and remodel the extracellular matrix when their direction of motion is not aligned with the fibre direction. Simulations are performed to describe the behavior of ensemble of cells under the action of a chemotactic field and in presence of heterogeneous and anisotropic fibre networks

Force transmission in migrating cells

Analysis of the relationship between actin network velocity and traction forces at the substrate shows that force transmission mechanisms vary with distinct regions of the cell