# Discrepancies stemming from the Hertz contact mechanics theory

The Hertz contact mechanics [47, 48] is widely applied in quantification of the AFM data despite the required assumptions that are only partially fulfilled:

- (a) A cell is treated as a isotropic and purely elastic material;
- (b) A cell can be approximated by an infinitely thick half space.
- (c) There are no adhesion forces within the contact area between the probing tip and the cell surface.

These assumptions simplify Young's modulus determination but make it difficult to obtain the absolute modulus value and, simultaneously, introduce additional ambiguity in the estimated elastic properties. Some consequences of this issue are discussed below.

The theoretical model used for Young's modulus determination assumes a particular shape of the indenter. The shape of a typical AFM tip is a four-sided pyramid with a height varying from 2.5 gm to even 15 gm and a distinct tip apex ranging from 2 nm to 50 nm. The most popular approximations of the AFM tip shape relate the load force *F* and the indentation depth *z* as follows:

- (1)
*F ~ z*(cone and pyramid)^{2} - (2)
*F*~ z^{3/2}(paraboloid)

Due to heterogenic nature of cells, experimental data do not always follow the chosen theoretical model. In such case, one must bear in mind that imperfections in the fit are unavoidable and not much can be done about uncertainty in the final results.

Figure 4.23 shows fits obtained for indentation depths of 200 nm, 800 nm, and 1400 nm, carried out using two most common AFM tip geometries: paraboloid and cone.

Figure 4.23 The function used to fit the experimental data is *y = ax ^{b }*where

*b*denotes the assumed shape of the AFM tip (2 for cone and 3/2 for paraboloid) and

*a*is the coefficient proportional to Young's modulus. Violet, orange and brown lines correspond to the fits performed for the indentations of 200, 800 and 1400 nm, respectively while gray points denote the experimental data. Reprinted with permission from [51].

Most of AFM experimental data show a kind of relationship contained in the area between two model shapes, i.e., cone and paraboloid, where sometimes the former sometimes the latter assumption fits better. Usually, the form of the fitted function is *y = a-x ^{b}* where

*b*is usually set to be either 2 either 3/2. To estimate more precisely how the performed fit affects the determination of Young’s modulus, the following method can be applied. The experimental data may be represented by the following relation:

where *n* is the number of point in the force-indentation curve, *y _{n}* is the force value corresponding to the indentation of

*x*is the fitted parameter that is proportional to elastic modulus value,

_{n}, c*b*includes the information on the shape of the AFM tip, and e denotes the low value expressing how much the fitted line deviates from the theoretical model. By comparing the fitted function (Eq. 4.28) with the theoretical model

*y = ax*one can estimate that

^{b},

For e close to zero the fitted *c* parameter approximates better the coefficient *a* proportional to Young’s modulus.

Geometrical properties of the AFM cantilever also influence modulus values estimation. The theoretical relation presented in Fig. 4.24 illustrates Young's modulus relation on load force calculated for three types of indenting probes: pyramidal, flat- ended, and spherical ones [61].

Figure 4.24 Elastic modulus variations as a function of load force calculated for two types of living cells: (a) NIH3T3 fibroblasts and (b) 7-4 (Ha-Ras^{V12} oncogene transformed) cells. Images prepared using data from [61].

From the presented comparison, one can clearly see that the use of pyramidal tip delivers larger elastic modulus values while values obtained for flat-ended and spherical (a bead with diameter of 5 gm) indenters are lower for a given load force. The increase of the indenting force manifests in gradual modulus increase, which is better visible for NIH3T3 fibroblasts than for 7-4 cells [61].