Fig. 2. Assessment of the impact of the Poisson’s ratio on the accuracy of traction force microscopy (TFM) through finite element (FE) simulations.

a-d Steps to compute error in the traction force estimation. a In 2D and 2.5D-TFM, shear and normal stresses are applied on a circular region located at the top surface of a cubic region with zero displacement boundary condition at the bottom surface. For 3D-TFM, a spherical traction region is defined within a cubic region whose deformation is again constrained at the bottom surface. Considering the linear constitutive law, the deformations resulting from the applied stress fields were calculated at all nodes within the FE mesh using the so-called forward Poisson’s ratio ν = ν _forward, which represents the true Poisson’s ratio of the material. b The displacements were selected at random nodes (representing the finite sampling imposed by a given BD in a TFM experiment) and interpolated on the whole domain to obtain the displacement field. For the 2D case only lateral components of the displacement are extracted, while the axial component is discarded, replicating the lack of axial sensitivity in 2D-TFM. For 2.5D, both lateral and axial components of displacements are sampled from nodes at the top surface of the elastic substrate representing the typical experimental condition in 2.5D-TFM. To replicate 3D-TFM experiments, all components of the displacement from nodes throughout the cubic region were sampled. c The interpolated displacements were used to solve the inverse problem considering a ν = ν _inverse, which is a Poisson’s ratio that is hypothetically assumed to reconstruct the forces in typical TFM analysis. d The initially applied 3D traction forces were compared with the reconstructed ones to find the error. e Even considering the ideal case of ν _forward = ν _inverse, some intrinsic errors are generated which are highly dependent on the sampling density. 2D and 2.5D results are represented with dash and solid curves, respectively (n = 10 for 2D/2.5D and n= 5 for 3D simulations). f The mismatch error was estimated considering ν _inverse = 0.5 and ν _forward to vary from 0 to 0.5 generation Poisson’s ratio mismatch (ν _inverse – ν _forward) ranging from 0.5 to 0 (n = 8 for 2D/2.5D and n = 5 for 3D simulations). g Contour maps showing the errors generated as the result of considering all combinations of ν _forward (possible true value) and ν _inverse (used in TFM analysis) varying from 0 to 0.5 for the 2.5D (left) and 3D (right) scenarios. h Bilateral tolerance in the Poisson’s ratio for three allowable levels of traction error for the 2.5D (left) and 3D (right) scenarios. i Effects of force anisotropy on the error. To generate different force anisotropy, the ratio of normal to shear stress was varied from 0 to 1 for 2D and 2.5D cases and from 0 to 2 for the 3D case. (n = 5 simulations). Error bars represent standard deviation in all panels.