We introduce a novel three-dimensional (3D) traction force microscopy (TFM)method motivated by the recent discovery that cells adhering on plane surfacesexert both in-plane and out-of-plane traction stresses. We measure the 3Ddeformation of the substratum on a thin layer near its surface, and input thisinformation into an exact analytical solution of the elastic equilibriumequation. These operations are performed in the Fourier domain with highcomputational efficiency, allowing to obtain the 3D traction stresses from rawmicroscopy images virtually in real time. We also characterize the error ofprevious two-dimensional (2D) TFM methods that neglect the out-of-planecomponent of the traction stresses. This analysis reveals that, under certaincombinations of experimental parameters (\ie cell size, substratums' thicknessand Poisson's ratio), the accuracy of 2D TFM methods is minimally affected byneglecting the out-of-plane component of the traction stresses. Finally, weconsider the cell's mechanosensing of substratum thickness by 3D tractionstresses, finding that, when cells adhere on thin substrata, their out-of-planetraction stresses can reach four times deeper into the substratum than theirin-plane traction stresses. It is also found that the substratum stiffnesssensed by applying out-of-plane traction stresses may be up to 10 times largerthan the stiffness sensed by applying in-plane traction stresses.
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