3D migration of cells solving an inverse problem

Traction Force Microscopy (TFM) is an inverse 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 during its migration. This biophysical problem, usually addressed in terms of Green functions, can be alternatively tackled in a variational framework. In such a case, a suitable penalty functional has to be minimized. The resulting Euler-Lagrange equations include both the direct problem based on the linear elasticity operator as well as an equation built on its adjoint. Results from a two-dimensional model, i.e. where living cancer cells are migrating on a plane substrate, are briefly presented. While the mathematics is well established also in the three-dimensional case, i.e. where cells are completely embedded in the gel matrix, the experimental data needed are more difficult to obtain than the two-dimensional counterpart. First steps towards the complete three-dimensional traction reconstruction are reported.