Mathematical framework for traction force microscopy

This paper deals with the Traction Force Microscopy (TFM) problem. It consists in obtaining stresses by solving an inverse problem in an elastic medium, from known experimentally measured displacements. In this article, the application is the determination of the stresses exerted by a living cell at the surface of an elastic gel. We propose an abstract framework which formulates this inverse problem as a constrained minimization one. The mathematical constraints express the biomechanical conditions that the stress field must satisfy. From this framework, two methods currently used can be derived, the adjoint method (AM) and the Fourier Transform Traction Cytometry (FTTC) method. An improvement of the FTTC method is also derived using this framework. The numerical results are compared and show the advantage of the AM, in particular its ability to capture details more accurately.