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We develop a method to reconstruct, from measured displacements of anunderlying elastic substrate, the spatially dependent forces that cells ortissues impart on it. Given newly available high-resolution images of substratedisplacements, it is desirable to be able to reconstruct small scale, compactlysupported focal adhesions which are often localized and exist only within thefootprint of a cell. In addition to the standard quadratic data mismatch termsthat define least-squares fitting, we motivate a regularization term in theobjective function that penalizes vectorial invariants of the reconstructedsurface stress while preserving boundaries. We solve this inverse problem byproviding a numerical method for setting up a discretized inverse problem thatis solvable by standard convex optimization techniques. By minimizing theobjective function subject to a number of important physically motivatedconstraints, we are able to efficiently reconstruct stress fields withlocalized structure from simulated and experimental substrate displacements.Our method incorporates the exact solution for the stress tensor accurate tofirst-order finite-differences and motivates the use of distance-based cut-offsfor data inclusion and problem sparsification.

High-resolution reconstruction of cellular traction-force distributions: the role of physically motivated constraints and compressive regularization