An iterative finite element-based method for solving inverse problems in traction force microscopy

During the last years different model solutions were proposed for solving cell forces under different conditions. The solution relies on a deformation field that is obtained under cell relaxation with a chemical cocktail. Once the deformation field of the matrix is determined, cell forces can be computed by an inverse algorithm, given the mechanical properties of the matrix. Most of the Traction Force Microscopy (TFM) methods presented so far relied on a linear stress-strain response of the matrix. However, the mechanical response of some biopolymer networks, such as collagen gels is more complex. In this work, we present a numerical method for solving cell forces on non-linear materials.

The proposed method relies on solving the inverse problem based on an iterative optimization. The objective function is defined by least-square minimization of the difference between the target and the current computed deformed configuration of the cell, and the iterative formulation is based on the solution of several direct mechanical problems. The model presents a well-posed discretized inverse elasticity problem in the absence of regularization. The algorithm can be easily implemented in any kind of Finite Element (FE) code as a sequence of different standard FE analysis.

To illustrate the proposed iterative formulation we apply the theoretical model to some illustrative examples by using real experimental data of Normal Human Dermal Fibroblast cells (NHDF) migrating inside a 2 mg/ml collagen-based gel. Different examples of application have been simulated to test the inverse numerical model proposed and to investigate the effect of introducing the correct cell properties onto the obtained cell forces. The algorithm converges after a small number of iterations, generating errors of around 5% for the tractions field in the cell contour domain. The resulting maximum traction values increased by 11% as a consequence of doubling the mechanical properties of the cell domain.

With the results generated from computations we demonstrate the application of the algorithm and explain how the mechanical properties of both, the cell and the gel, domains are important for arriving to the correct results when using inverse traction force reconstruction algorithms, however, have only a minor effect on the resulting traction values.