An updated model including the deflection history of microcantilever for characterizing cellular viscoelastic properties by AFM indentation-relaxation test

Factors such as the memory effect of viscoelastic materials and the influence of microcantilever deflection still pose challenges in the mechanical modeling, mathematical solution, and material parameter identification for AFM indentation-relaxation experiments to characterize the viscoelastic properties of cells. This paper aims to provide a rational mechanical model for interpreting the detection signals in AFM indentation-relaxation experiments. Considering the contribution of microcantilever deflection under the Euler beam assumption to the viscoelastic indentation force, the Lee-Radok’s model was updated. Combining a piecewise integration method and a time-domain differential method, we derived the implicit/explicit governing equations of the probe-cell viscoelastic indentation forces under ramp-hold protocol. After building the finite element (FE) model, we examined the effect of microcantilever deflection on the acquired data in FE simulation results and the updated model predictions, and compared our updated model with relevant experiments. Then, we proposed a new two-stage (TS) approach for parameter extraction and compared the differences between this new approach and the classic single-stage (SS) approach, thus the strategy for suppressing parameter extraction error was presented. The results validate that the transient modulus extracting by the classical SS approach is essentially an equivalent transient modulus dependent on the ramp loading history, which incurs a divergence in identification of cellular viscoelastic parameter; whereas the new TS approach is valid at a broader range of loading conditions due to explicitly reflecting the dependence of viscoelastic material parameters on ramp loading history. These conclusions provide a theoretical foundation and reference for the advancement of AFM-based detection techniques for cellular mechanical properties.