Recovery of Tractions Exerted by Single Cells in Three-Dimensional Nonlinear Matrices

Abstract

<jats:title>Abstract</jats:title> <jats:p>Cell-generated tractions play an important role in various physiological and pathological processes such as stem-cell differentiation, cell migration, wound healing, and cancer metastasis. Traction force microscopy (TFM) is a technique for quantifying cellular tractions during cell–matrix interactions. Most applications of this technique have heretofore assumed that the matrix surrounding the cells is linear elastic and undergoes infinitesimal strains, but recent experiments have shown that the traction-induced strains can be large (e.g., more than 50%). In this paper, we propose a novel three-dimensional (3D) TFM approach that consistently accounts for both the geometric nonlinearity introduced by large strains in the matrix, and the material nonlinearity due to strain-stiffening of the matrix. In particular, we pose the TFM problem as a nonlinear inverse hyperelasticity problem in the stressed configuration of the matrix, with the objective of determining the cellular tractions that are consistent with the measured displacement field in the matrix. We formulate the inverse problem as a constrained minimization problem and develop an efficient adjoint-based minimization procedure to solve it. We first validate our approach using simulated data, and quantify its sensitivity to noise. We then employ the new approach to recover tractions exerted by NIH 3T3 cells fully encapsulated in hydrogel matrices of varying stiffness. We find that neglecting nonlinear effects can induce significant errors in traction reconstructions. We also find that cellular tractions roughly increase with gel stiffness, while the strain energy appears to saturate.</jats:p>

Three-dimensional traction microscopy accounting for cell-induced matrix degradation

“…Next, we model the process of ECM relaxation after cell lysis. This process typically takes ∼ 45 mins [18,34], and is commonly assumed to be quasi-static (e.g., [15,17]). We further assume that ECM degradation mainly occurs during cell culturing (∼ 3 days [18]) before cell lysis, and that ECM properties do not vary significantly during the short period of ECM relaxation.…”

“…As discussed in Section 2, we assume that ECM degradation occurs only at regions close to the cell surface, so that the ECM far from the cell is unaffected by cell remodeling and remains in its pristine state (d = 0, λ = λ 0 , µ = µ 0 ). We further assume that the displacement field in the ECM is measured only at discrete bead locations, consistent with the standard experimental setup (e.g., [18,34,32,20]).…”

“…That is, one may first compute the degradation field by solving an inverse problem using an optimization approach, and then compute the stresses and tractions using the constitutive relation. However, the first step of this approach would require displacement data on the cell interface, which is often not available in a 3DTM experiment (e.g., [18,34]). Therefore, we do not adopt this method.…”

“…We solve the optimization problem using a quasi-Newton method, L-BFGS-B [53], which requires the gradient of the objective function π with respect to the traction field t and the degradation field d. The main difficulty lies in the fact that the displacement vectors u k (see (20)) depend on t and d implicitly through the constraint of the forward problem (8). Following the work of [50,33,34], we compute the gradient efficiently using an appropriate adjoint field, which can be derived by constructing a Lagrangian given by…”

“…This method was formulated in the reference (or unstressed) configuration of the ECM, and required the knowledge of the cell geometry in the unstressed state, which is often unknown due to cell lysis. In order to remedy this shortcoming, Song et al [34] reformulated the problem in the current (or stressed) configuration of the ECM, thus only requiring the cell geometry in the stressed state. These authors also computed tractions exerted by 3T3 cells in dextran-based hydrogel matrices.…”