A numerical method for the inverse problem of cell traction in 3D

Force traction microscopy is an inversion method that allows us 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's functions, can be alternatively tackled in a variational framework. 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 a numerical strategy of the inversion method that discretizes the partial differential equations associated with the optimal control problem by finite elements. A detailed discussion of the numerical approximation of a test problem (with known solution) that contains most of the mathematical difficulties of the real one allows a precise evaluation of the degree of confidence that one can achieve in the numerical results.