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 vs. experimental measures of displacement induced by a fibroblast cell traction. Introduction. The study of the basic mechanisms of cell migration has received a tremendous increment in the last few years. Cell locomotion occurs through a very complex interaction that involves, among others, actin polymerization, matrix degradation, chemical signaling, adhesion and pulling on substrate and fibers [12]. All these ingredients concur not only in single cell migration but also in collective morphogenetic behaviors [15]. When focusing on mechanical aspects only, a major issue is the determination of the dynamical action of the cells on the environment during migration: the cells adhere, pull the surrounding matrix and move. As a cell can have more than one hundred of focal adhesion sites, each one with thousand of integrins, it is quite difficult to obtain a pointwise description of the forces exerted by moving cells on a direct basis. Nevertheless, the striking improvement of nanotechnology has very recently lead to direct measurements of the cell traction: cells are deposed on a bed of microneedles and, on the basis of the Young modulus, the moment of inertia, length and displacement of the microneedles, one can directly obtain the exerted deflective force [4,14]. However, these very recent experimental achievements still provide partial information on a very special configuration only: the resolution of the displacement field is at most the distance between two microneedles (2 microns) and, more important, this setting is far from being a natural migration environment. This kind of considerations suggests that the dynamics of cell locomotion can be fruitfully studied as an inverse problem, an idea that dates back to the seminal paper of Harris and coworkers [8]. A thin elastic film over a fluid is deformed under cell traction in a wrinkled pattern and the size of the crimps is correlated to the shear load. Unfortunately, buckling of thin film is an essentially nonlinear phenomena and a quantitative reconstruction of the exerted traction would call for a non–trivial stability analysis in nonlinear elasticity. A quantitative methodology has been proposed in 1996 by Dembo et al [3], using prestressed silicone rubber, an approach further improved by Dembo & Wang in 1999 [2]. They deduce the traction exerted by a fibroblast on a polyacrilamide substrate from the measured displacement of several fluorescent beads merged in the upper layer of the gel. The gel is soft enough to remain in a linear elasticity regime and no wrinkles form. The cellular traction is then computed by maximizing the total Bayesian likelihood of the markers displacement predicted on the basis of the Boussinesq solution for the linear elastic half–plane with pointwise traction. The same approach (in two spatial dimensions) is followed by Schwarz and others [11] who invert numerically the Boussinesq integral operator, thus expressing the displace∗ Dipartimento di Matematica, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy 1