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

Highlights

• Formulation of a new 3-D traction force microscopy inverse method by using a recursive algorithm based on finite elements.

• Explore cellular forces in physiologically relevant 3-D matrices with highly non-linear mechanical properties, such as collagen type-I hydrogels.

• Analysis of the effect of the mechanical properties of both cell and matrix domains in the accuracy of the predictions in traction force microscopy.

Abstract

Background and Objective

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.

Methods

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.

Results

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.

Conclusions

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.

Introduction

Cell migration through a three-dimensional (3-D) matrix depends strongly on the ability of cells to generate traction forces. In contrast to cell migration on a two-dimensional (2-D) matrix, when cells migrate through a 3-D matrix these must overcome not only the adhesion forces, but also the resisting forces imposed by the surrounding matrix [1], [2], [3]. Resisting forces mainly arise from steric effects against the movement of the cell. Thus, to overcome the steric hindrance imposed by the matrix the cell needs to generate traction forces that depend on the matrix properties as well as cell properties and shape. Studying the dependency of these forces on the mechanical properties of cells and the surrounding matrix is therefore important for a mechanistic understanding of many physiological and pathological cell functions in health and disease that involve cell adhesion, shape changes and migration, such as tissue formation during embryogenesis or tumor cell invasion [4], [5], [6], [7], [8].

Traction forces can be computed from cell-induced deformation of the matrix, using different mathematical frameworks. Basically, the displacement field of the hydrogel caused by the cell tractions is quantified by tracking the positions of embedded beads before and after cell tractions are relaxed with a drug cocktail containing trypsin and/or high concentrations of actin-disrupting drugs such as cytochalasin-D. Once the matrix displacements are known, the matrix strain field can be computed [9].

In order to quantitatively measure the traction field generated during cell migration, several techniques have been developed. These include surface wrinkle analysis, deflection of micropillars and Traction Force Microscopy (TFM) [10], [11], [12], [13], [14], [15]. The most popular technique relies on the Boussinesq solution in the Fourier space, the so called Fourier Transform Traction Cytometry (FTTC) [13], [16], but this solution is limited for 2-D substrates undergoing small strains (with a linear elastic behavior of the substrate). However, more recent studies have demonstrated important differences when comparing cell behavior in 2-D vs. 3-D environments. Accordingly, 3-D TFM techniques have increased in sophistication and now feature high-spatial displacement resolution and advanced computation formalisms to connect the displacement information to complex material constitutive laws [14], [17], [18].

To measure the 3-D forces several research groups, such as Legant et al. have proposed the use of Finite Element Methods (FEM) to solve the inverse problem [15]. However, previously published approaches mostly ignore the non-linear behavior of 3-D biopolymer networks, such as reconstituted collagen gels. There have been only a few attempts at solving 3-D non-linear TFM problem [19], [20], [21], by using non-linear constitutive models and finite element analysis. Some of the previous models assumed idealized shapes for cells and intracellular structures and just a few of the studies used microscopy-based realistic cell geometries for cell forces reconstruction [15], [20], [22]. The precise knowledge of the cell surface domain contributes to reduce the computational cost for solving the inverse method, while at the same time provides the possibility to understand the effect of internal cell properties and other intracellular structures on the cell forces reconstruction.

In this work, we describe a method for quantifying cell forces during cell migration in physiologically relevant 3-D matrices with highly non-linear mechanical properties, such as collagen type-I hydrogels. To that end, we use Laser Scanning Confocal Microscopy (LSCM) images of Normal Human Dermal Fibroblast (NHDF) to develop 3-D finite element models of in vitro cell geometries. Subsequently, these models are used to explore the influence of both cell and matrix mechanical properties on the cell traction force reconstruction method. The imaging of cells was performed inside a microfluidic-based chip, to ensure live cell conditions and to recreate the physiological microenvironment of real tissues [23]. Moreover, microfluidic devices allow for a better control of the boundary conditions of the problem to solve.

We propose a numerical method to solve the inverse problem based on an iterative optimization algorithm. The objective function is defined by least-square minimization of the difference between the target/measured and the computed deformed configuration of the cell by means of an iterative formulation, which is based on the solution of several direct mechanical problems. First, from the measured deformed configuration the known displacements serve as boundary conditions for the first direct mechanical problem to be solved. Subsequently, the reaction forces obtained in the previous direct calculation serve as boundary conditions for the second direct mechanical problem to be solved. The so obtained deformed configuration is compared with the measured deformed configuration. If this difference is smaller than a critical value, the iterative algorithm has converged to a solution. If not, this process is repeated until convergence is achieved. Moreover, this method presents a well-posed discretizing inverse elasticity problem in the absence of regularization, as demonstrated in other previous works [24].

To illustrate the proposed iterative method we apply the theoretical model to some examples where cell forces are computed on real geometries of NHDF cells migrating inside a 2 mg/ml collagen-based gel. Moreover, the effect of incorporating the real mechanical properties of both domains (cell and hydrogel) onto the cell traction force solution obtained with the FEM method is investigated.