Force traction microscopy: An inverse problem with pointwise observations

Force traction microscopy is an inversion method that allows one to obtain the stress field applied by a living cell on the environment on the basis of a pointwise knowledge of the displacement produced by the cell itself. This classical biophysical problem, usually addressed in terms of Green functions, can be alternatively tackled using a variational framework and then a finite elements discretization. In such a case, a variation of the error functional under suitable regularization is operated in view of its minimization. This setting naturally suggests the introduction of a new equation, based on the adjoint operator of the elasticity problem. In this paper we illustrate the rigorous theory of the two-dimensional and three-dimensional problems, involving in the former case a distributed control and in the latter case a surface control. The pointwise observations require one to exploit the theory of elasticity extended to forcing terms that are Borel measures.