The average Young's modulus as a physical quantity for describing the depth-dependent mechanical properties of cells

Abstract

Hertzian mechanics is a useful theoretical tool for data processing regarding Atomic Force Microscopy (AFM) nanoindentation experiments on cells and other biological samples. A common approach that is usually followed in the literature is to fit the experimentally obtained data to the equations provided by Hertzian analysis. However, there are many different approaches regarding the selection of the maximum indentation depth even for the same biological sample and many of them depend on the experience of the AFM-user. In addition, the results as provided by different research groups on the same sample (e.g. the same cell type) usually vary significantly. Thus, it is crucial to develop novel user-independent methods for assessing the mechanical properties of cells. In this paper a new approach, based on the work done by the indenter, is proposed for data processing. At first, a hypothetical nanoindentation experiment on an elastic half space is assumed in which the maximum indentation depth and the work done by the indenter are equal to the actual experiment. The Young's modulus of the hypothetical ideal material is the actual sample's average Young's modulus. In addition, the function of the average Young's modulus with respect to the indentation depth is introduced to reveal the depth dependent mechanical properties of biological samples. Secondly, the proposed methodology was applied on experimental data from fibroblasts and H4, A172 human glioma cells. The results showed a ‘softening’ of cells as the indentation depth increases. However, for big indentation depths the average Young's modulus always reaches an asymptotic value (the overall average Young's modulus). Thus, the overall average Young's modulus can be used for comparing the results of AFM nanoindentation experiments on cells between different research groups. In addition, the average Young's modulus function can also reveal the maximum value of the indentation depth that should not be surpassed in order to avoid a substrate effect. The analysis proposed by this paper can be applied to any biological sample using conical or spherical indenters. The great advantage of the proposed by this paper technique is that it does not require a linear elastic response of the sample, thus it is appropriate for the mechanical nano-characterization of highly heterogeneous biological materials.