Determination of Green's function for three-dimensional traction force reconstruction based on geometry and boundary conditions of cell culture matrices

Cell migration plays a particular important role in the initiation and progression of many physical processes and pathological conditions such as tumor invasion and metastasis. Three-dimensional traction force microscopy (TFM) of high resolution and high accuracy is being developed in an effort to unveil the underlying mechanical process of cell migration in a vivo-like environment. Linear elasticity-based TFM (LETM) as a mainstream approach relies on the Green's function (that relates traction forces to matrix deformation), of which the inherent boundary conditions and geometry of the matrix could remarkably affect the result as suggested by previous 2D studies. In this study, we investigated this close linkage in 3D environment, via modeling of a cell sensing a close-by fixed boundary of a 3D matrix surrounding it, and comparing the reconstructed traction forces from three different solutions of the Green's function, including a fully matching solution derived using the adapted Mindlin's approach. To increase fidelity in the estimate of traction forces for extreme conditions such as a sparse sampling of deformation field or targeting small focal adhesions, we numerically solved the singularity problem of the Green's function in a non-conventional way to avoid exclusion of singular point regions that could contain representative deformation indicators for such extreme conditions. A single case experimental study was conducted for a multi-cellular structure of endothelial cells that just penetrated into the gel at the early stage of angiogenesis. Statement of Significance: This study focused on the fundamental issue regarding extension of linear elasticity-based TFM to deal with physically realistic matrices (where cells are encapsulated), which concerns determination of the Green's function matching their geometry and boundary conditions. To increase fidelity in the estimate of traction forces for extreme conditions such as a sparse sampling of deformation field or targeting small focal adhesions, we numerically solved the singularity problem of the Green's function to avoid exclusion of singular point regions that could contain representative deformation indicators for such extreme conditions. The proposed approach to adapting the Green's function for the specific 3D cell culture situation was examined in a single case experimental study of endothelial cells in sprouting angiogenesis.