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 estabilished 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.