# Deep Learning Regressors of Surface Properties from Atomic Force Microscopy Nanoindentations
## 2 Methods
### Synthetic Nanindentation Curves  
Before starting the development of a machine learning model, a suitable synthetic dataset must be defined, representing the underlying patterns of a real dataset obtained from experimental AFM nanoindentations in soft biological samples, for both approach and retraction stages. Hertzian theory-based Equation (3) was the foundation to compute the force for the approach stage, while Equation (6) was used for the withdraw phase. It is thus important to define what the fundamental variables are that will affect the forces for each step (\(F_{\text{Hertz}}\) and \(F_{\text{JKR}}\)), so it can be written that  
\[F_{\text{Hertz}}=F_{\text{Hertz}}(\delta,E,\
u,R) \tag{9}\]  
and  
\[F_{\text{JKR}}=F_{\text{JKR}}(\delta,E,\gamma,\
u,R). \tag{10}\]  
Therefore, all these variables must be defined to reach the final curves. The Poisson ratio and the tip radius will be the same for all instances, while the other parameters are unique for a specific curve. Thus, each indentation vector will have material parameters assigned to it, and jointly with \(R\) and \(\
u\) will allow tone to compute the force for all indentation values in the vector. Knowing the force and indentation for each point, the curves can be obtained. Starting with the Poisson ratio, as the targets of this model are soft tissues, they can be approximated as incompressible due to their high water content [39], so a Poisson ratio of 0.5 was assigned. The tip radius value, presented in Table 1, was chosen according to the probe that was used to produce the experimental nanoindentations [38] that will be later used to test the developed models, and it is within a common range of values for spherical tip radii. Figure 2: Experimental approach curves used to test the ML models. Only the contact region (\(\delta>0\)) is relevant for the analysis. (**a**) [PERSON]’s modulus distribution, mean and standard deviation values. (**b**) Examples of experimental approach curves in the available AFM nanoindentation data. Regarding indentation depth vectors, their length and the maximum indentation value must be defined, so that the inputs to the ML models are consistent. Only the contact region (\(\delta>0\)) was considered, since negative indentation values do not yield any information on surface properties according to the contact models chosen. In order for the indentation vectors to resemble real data more closely and to have a greater impact on the ML models, they cannot all be composed of the same evenly spaced indentation values, so different instances are associated to different indentation values, always keeping the maximum indentation at 150 nm. After specifying the Poisson ratio, tip radius and a set of displacement vectors, the only remaining parameters are the material properties \(E\) and \(\gamma\), which must also be comprised in a typical range of values for soft tissues. This work focuses on studying a specific subset of soft tissues, with surface properties compatible with the available experimental data. However, if a broader range of materials was being analysed, standardizing the units of the physical quantities could potentially be helpful to improve the model's behaviour. For the [PERSON]'s modulus, its values commonly fluctuate between the orders of magnitude of \(10^{-1}\) to \(10^{1}\) kPa, reaching the hundreds of kPa in more specific scenarios [27; 40; 41; 42]. Taking also into account the [PERSON]'s moduli of the samples in the experimental data available, a minimum \(E\) of 0.2 kPa and a maximum of 10 kPa were fixed, using a triangular distribution function with the mode as 1.8 kPa, so that the ML models could be trained more thoroughly on the most frequent range of values for the [PERSON]'s modulus of biological cells or hydrogels. On the other hand, the same dataset does not contain any information about adhesion energy, since the available retraction curves are scarce and the corresponding samples did not necessarily exhibit strong adhesion properties.