## 3 Results
### Validation of the Simulated Model  
[PERSON] [31] used the AFM technique to consider the effect of strain rate on the mechanical properties of three different cell types. He studied a single osteocyte, osteoblast, and chondrocyte separately under AFM with varying rates of strain. He used the obtained data to validate a finite element model for seeking the cells' hyper-elastic and viscoelastic properties. The chondrocyte data was chosen to compare these three cells because the diameter size (\(15\pm 3\) um) was similar to the cell height (\(11.7\) um) of the simulated cell. As noted above, these results were achieved with distinct strain rates. However, the current research only utilized the elastic properties of all the modeled subcellular elements. It is obvious that linear materials behavior is time-independent, so the data from the largest strain rate was chosen in order to validate the model.  
Figure 4: Stress distribution in the cell components: (**a**) principal stress in the cytoskeleton (CSK) and nucleoskeleton (NSK), the bar elements can be seen in dark color. (**b**) Principal stress in the cytoplasm. (**c**) Principal stresses on the substrate at the contact points between the CSK and substrate.  
Figure 5: Force-indentation data obtaining from finite element model (FEM) (solid dots) and fitted curve (solid line).  
Figure 6 shows the F-8 curve for the experimental and simulated data. The dotted curve is fitted to the experimental data from the literature, and the solid curve was obtained from the finite element analysis (Figure 5). Both curves show an exponential relation between the AFM indentation and the measured force. The difference between the plots is due to the differences in the radius of the micro spherical tip (Equation (4)). [PERSON] et al. [30] have shown that for a sample with a height \(h\), the smallest radius of the bead must be equal to \(\frac{1}{12.8}\) in order to prevent the strains exceed the material linear behavior. They also concluded that using microspheres with a 2 to 5 \(\upmu\)m radius provides more accurate results. For validating the model, a tip with a 2.5 \(\upmu\)m diameter, as in the reported experimental test, was modeled. The dashed line represents the results obtained with a 2.5 \(\upmu\)m tip. The calculated data is very similar to the experimental report. However, the current model is based on a linear elastic analysis of the cell. Thus, to ensure linearity and better accuracy, the model was developed using a rigid sphere with a radius of 4.5 \(\upmu\)m for further observation. It is evident that a bigger tip produces a higher reaction force.  
According to the Hertz equation [32], the reaction force for the spherical tip on the elastic matter is a power-law relationship:  
\[F\ =\ \frac{3}{4}\frac{E}{1-\
u^{2}}\sqrt{R}\delta^{1.5} \tag{4}\]  
where \(F\) is the reaction force, \(R\) is the radius of the spherical indenter, \(\delta\) is the applied indentation, \(E\) and \(\
u\) are the Young's modulus and Poisson's ratio of the elastic material, respectively. The power-law fit for the F-\(\delta\) relationship (Figure 5) obtained in this study was 1.68. This value is very close to the 1.5 that is provided by the theory.