A Practical Approach for Determining Depth-Dependent Mechanical Properties of Soft Materials in AFM Indentation via Polynomial Fitting and a New Model for Cellular Mechanics

In most AFM nanoindentation experiments on soft biological samples, classical contact mechanics models, such as Hertz or Sneddon’s equations, are commonly employed to determine the Young’s modulus. However, biological materials are inherently heterogeneous, and their mechanical properties often depend on the indentation depth. In this work, we present a novel and simple approach to quantify how the apparent modulus varies with increasing indentation depth. The method is based on the general indentation equation for axisymmetric indenters combined with a straightforward polynomial fitting of the force–indentation data. The proposed approach offers significant advantages, as it greatly simplifies the fitting process without requiring any advanced algorithms, while maintaining high accuracy. In addition, it is shown that the depth-dependent mechanical properties of cells can be described by a simple law, E(h)=Cd/h+El , where El is the limiting value of the apparent modulus at large indentations, and Cd/h represents the depth-dependent contribution dominant at the initial stages of the indentation process. Here, Cd is a positive stiffness coefficient, and h is the indentation depth. This is a very important result, indicating that by using the pair of coefficients Cd and El, we can fully describe the mechanical properties of cells, capturing their depth-dependent mechanical behavior. Experiments on fibroblasts and H4 human glioma cells confirm the accuracy of this equation. The proposed methods provide an accessible and reliable framework for nanoscale mechanical characterization, offering insights into the depth-dependent elasticity of heterogeneous soft materials and revealing mechanical patterns in biological samples.