Full L₁-regularized Traction Force Microscopy over whole cells

Traction Force Microscopy (TFM) is a widespread technique to estimate the tractions that cells exert on the surrounding substrate. To recover the tractions, it is necessary to solve an inverse problem, which is ill-posed and needs regularization to make the solution stable. The typical regularization scheme is given by the minimization of a cost functional, which is divided in two terms: the error present in the data or data fidelity term; and the regularization or penalty term. The classical approach is to use zero-order Tikhonov or L₂-regularization, which uses the L₂-norm for both terms in the cost function. Recently, some studies have demonstrated an improved performance using L₁-regularization (L₁-norm in the penalty term) related to an increase in the spatial resolution and sensitivity of the recovered traction field. In this manuscript, we present a comparison between the previous two regularization schemes (relying in the L₂-norm for the data fidelity term) and the full L₁-regularization (using the L₁-norm for both terms in the cost function) for synthetic and real data.

Our results reveal that L₁-regularizations give an improved spatial resolution (more important for full L₁-regularization) and a reduction in the background noise with respect to the classical zero-order Tikhonov regularization. In addition, we present an approximation, which makes feasible the recovery of cellular tractions over whole cells on typical full-size microscope images when working in the spatial domain.

The proposed full L₁-regularization improves the sensitivity to recover small stress footprints. Moreover, the proposed method has been validated to work on full-field microscopy images of real cells, what certainly demonstrates it is a promising tool for biological applications.