Traction Force Microscopy for Viscoelastic Substrates: A Semi-Analytical Method.

Traction force microscopy (TFM) quantifies cellular forces at the cell-extracellular matrix interface, yet elastic formulations neglect viscous dissipation and can misinterpret cellular forces on viscoelastic substrates. We introduce a semi-analytical 2D viscoelastic TFM (veTFM) that generalizes the Boussinesq framework of elastic TFM to Generalized Maxwell (GMX) substrates with one or two components. By combining Fourier and Laplace transforms, veTFM quantifies time-resolved tractions in finite-thickness substrates and resolves stress-free reference and substrate pre-stress. We derive criteria for when elastic regimes remain valid in this framework. This positions veTFM as a scalable extension of standard 2D TFM (eTFM) to viscoelastic substrates, identifying when eTFM remains sufficient, which elastic limit applies, and when full viscoelastic quantification is required. Applied to beating cardiomyocytes, epithelial cells, and dermal fibroblasts cultured on linear-polyacrylamide and alginate viscoelastic hydrogels, veTFM shows that the elastic or viscoelastic regime engaged by the cell depends on timescale matching between the loading rate and the substrate's relaxation times. Notably, for the Generalized Maxwell substrates analyzed here, viscoelastic traction magnitudes scale with the substrate's total dissipation rather than individual relaxation times, with total dissipation setting traction magnitude and timescale matching determining whether the cell engages the substrate in an elastic or viscoelastic regime.