Force Traction Microscopy: an inverse problem with pointwise observations
Seminar Room 1, Newton Institute
Force Traction Microscopy is an inversion method that allows 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 Tichonov 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. The pointwise observations require to exploit the theory of elasticity extended to forcing terms that are Borel measures. In this work we show the proof of well posedness of the above problem, borrowing technics from the field of Optimal Control. We also illustrate a numerical strategy of the inversion method that discretizes the partial differential equations associated to the optimal control problem. 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 have in the numerical results.