Is it mathematically correct to fit AFM data (obtained on biological materials) to equations arising from Hertzian mechanics?

When testing soft biological samples using the Atomic Force Microscopy (AFM) nanoindentation method, the force-indentation data is usually fitted to the equations provided by Hertzian mechanics. Nevertheless, a significant question remains up to date; is this a correct approach from a mathematical perspective? Biological materials are heterogeneous, so 'what do we calculate' when using a classic fitting approach? In this paper, conclusive answers to the abovementioned questions are provided. In addition, a new tool for the nanomechanical characterization of biological samples, the depth-dependent mechanical properties maps, is introduced.

Is It Possible to Directly Determine the Radius of a Spherical Indenter Using Force Indentation Data on Soft Samples?

An important factor affecting the accuracy of Young's modulus calculation in Atomic Force Microscopy (AFM) indentation experiments is the determination of the dimensions of the indenter. This procedure is usually performed using AFM calibration gratings or Scanning Electron Microscopy (SEM) imaging. However, the aforementioned procedure is frequently omitted because it requires additional equipment. In this paper, a new approach is presented that focused on the calibration of spherical indenters without the need of special equipment but instead using force indentation data on soft samples. Firstly, the question whether it is mathematically possible to simultaneously calculate the indenter's radius and the Young's modulus of the tested sample (under the restriction that the sample presents a linear elastic response) using the same force indentation data is discussed. Using a simple mathematical approach, it was proved that the aforementioned procedure is theoretically valid. In addition, to test this method in real indentation experiments agarose gels were used. Multiple measurements on different agarose gels showed that the calibration of a spherical indenter is possible and can be accurately performed. Thus, the indenter's radius and the soft sample's Young's modulus can be determined using the same force indentation data. It is also important to note that the provided accuracy is similar to the accuracy obtained when using AFM calibration gratings. The major advantage of this paper is that it provides a method for the simultaneous determination of the indenter's radius and the sample's Young's modulus without requiring any additional equipment.

The Hertzian theory in AFM nanoindentation experiments regarding biological samples: Overcoming limitations in data processing.

Atomic Force Microscopy (AFM) nanoindentation is a powerful tool for the mechanical nano-characterization of biological samples. However, the range of Young's modulus values for the same type of samples usually varies significantly in the literature. This fact is partly related to the inhomogeneity of biological samples at the nanoscale and partly to significant mistakes during data processing. This review depicts that common errors related to (i) the real shape of the AFM tip, (ii) the range of data for which the sample presents an approximate linear elastic response, (iii) the sample's viscoelasticity, (iv) the sample's shape and (v) the substrate effects can be easily avoided without increasing the complexity of data processing. Thus, the present review paper focuses on the procedures that should be followed for the accurate processing of force-indentation curves regarding experiments on biological samples.

A New Approach for the AFM-Based Mechanical Characterization of Biological Samples.

The AFM nanoindentation technique is a powerful tool for the mechanical characterization of biological samples at the nanoscale. The data analysis of the experimentally obtained results is usually performed using the Hertzian contact mechanics. However, the aforementioned theory can be applied only in cases that the sample is homogeneous and isotropic and presents a linear elastic response. However, biological samples often present depth-dependent mechanical properties, and the Hertzian analysis cannot be used. Thus, in this paper, a different approach is presented, based on a new physical quantity used for the determination of the mechanical properties at the nanoscale. The aforementioned physical quantity is the work done by the indenter per unit volume. The advantages of the presented analysis are significant since the abovementioned magnitude can be used to examine if a sample can be approximated to an elastic half-space. If this approximation is valid, then the new proposed method enables the accurate calculation of Young's modulus. Additionally, it can be used to explore the mechanical properties of samples that are characterized by a depth-dependent mechanical behavior. In conclusion, the proposed analysis presents an accurate yet simple technique for the determination of the mechanical properties of biological samples at the nanoscale that can be also used beyond the Hertzian limit.

A simplified approach for the determination of fitting constants in Oliver-Pharr method regarding biological samples.

The atomic force microscopy (AFM) nanoindentation regarding biological samples is a challenging procedure. Biological samples at the nanoscale can be considered as purely elastic materials under the condition that the indentation depth is very small and the indenter is smooth. However, the indenters that are commonly used are pyramidal and in several cases the indentation depths are big comparing to the dimensions of the tip apex. Hence, pyramidal indenters usually cause a permanent damage to the sample. In this case, the best model that can be applied for the data processing is the Oliver-Pharr model which takes into account the elastic-plastic behavior of the sample. The Oliver-Pharr model is based on the fitting of the unloading load-indentation data to a power law equation. In this paper a simplified procedure which ensures the accurate fitting of the unloading load-indentation data to the Oliver-Pharr model is presented and validated on experimental data obtained from a human glioma cell line. It should be noted that the proposed method can be also applied for the data fitting in the case of purely elastic response.