Cellular traction as an inverse problem

The evaluation of the traction exerted by a cell on a planar substrate is here considered as an inverse problem: shear stress is calculated on the basis of the measurement of the deformation of the underlying gel layer. The adjoint problem of the direct two-dimensional plain stress operator is derived by a suitable minimization requirement. The resulting coupled systems of elliptic partial differential equations (the direct and the adjoint problem) are solved by a finite element method and tested against experimental measures of displacement induced by a fibroblast cell's traction.