Effects of gel thickness on microscopic indentation measurements of gel modulus

In vitro, animal cells are mostly cultured on a gel substrate. It was recently shown that substrate stiffness affects cellular behaviors in a significant way, including adhesion, differentiation, and migration. Therefore, an accurate method is needed to characterize the modulus of the substrate. In situ microscopic measurements of the gel substrate modulus are based on Hertz contact mechanics, where Young's modulus is derived from the indentation force and displacement measurements. In Hertz theory, the substrate is modeled as a linear elastic half-space with an infinite depth, whereas in practice, the thickness of the substrate, h, can be comparable to the contact radius and other relevant dimensions such as the radius of the indenter or steel ball, R. As a result, measurements based on Hertz theory overestimate the Young's modulus. In this work, we discuss the limitations of Hertz theory and then modify it, taking into consideration the nonlinearity of the material and large deformation using a finite-element method. We present our results in a simple correction factor, ψ, the ratio of the corrected Young's modulus and the Hertz modulus in the parameter regime of δ/h ≤ min (0.6, R/h) and 0.3 ≤R/h ≤ 12.7. The ψ factor depends on two dimensionless parameters, R/h and δ/h (where δ is the indentation depth), both of which are easily accessible to experiments. This correction factor agrees with experimental observations obtained with the use of polyacrylamide gel and a microsphere indentation method in the parameter range of 0.1 ≤δ/h ≤ 0.4 and 0.3 ≤R/h ≤ 6.2. The effect of adhesion on the use of Hertz theory for small indentation depth is also discussed.