Novel inverse finite-element formulation for reconstruction of relative local stiffness in heterogeneous extra-cellular matrix and traction forces on active cells.

Accurately resolving the traction forces on active cells in 3D extra-cellular matrix (ECM) is crucial to understanding stress homeostasis in cellularized ECM systems and the resulting collective cellular behavior. The majority of 3D traction force microscopy techniques, which compute the stress distribution in ECM as well as cellular traction forces from experimentally measured deformation field in the ECM using dispersed tracing particles or fluorescently-tagged matrix proteins, have assumed a spatially homogeneous ECM with constant material properties at every location in the system. Recent studies have shown that ECM can exhibit significant heterogeneity due to the disordered nature of collagen network as well as cell remodeling. In this paper, we develop a novel inverse finite-element formulation for accurately resolving the cellular traction forces by explicitly reconstructing the relative local elastic modulus values of the heterogeneous ECM containing an arbitrary shaped cell from a measured displacement field in the ECM. Our formulation does not require any a priori knowledge of the boundary conditions, and simultaneously results in the distribution of the heterogeneous modulus values and stress field in the ECM, as well as the traction forces on the cell, given experimentally measured average modulus of the ECM. We first validate our procedure in artifical model cell-ECM systems, and then employ the procedure to compute the distribution of elastic modulus in a heterogeneous type-I collagen gel as well as the traction force on a rounded breast cancer cell in the gel, based on the deformation field data obtained via 3D reflectance force microscopy. Our results indicate that the majority part of the cell is in a tensile state, while a local region on the cell is in a tri-axial compressive state, indicating the possible development of a local protrusion in this region. This is further verified by tracking the subsequent evolution of the cell morphology.