## 3 Results  
### Accuracy of the Insertion Force Model of W Microneedles into In Vivo Peripheral Nerves  
The Oliver-Pharr approach [64; 65] was tested to fit experimental data through Equation (1). The evolution of the interaction force between tungsten microneedles and in vivo peripheral nerves was investigated for increasing values of the normalized indentation depth \(\xi\) (see Figure 1). Equation (1) was tested for \(k=3,7,13,17\), and the results are shown in Figure 3a,d,g,j, whereas the residual plots are shown in Figure 3b,e,h,k. For \(k=3,7\) they slightly exceeded the range \(-5,0\) mN up to \(\xi=0.9\), and for \(k=13,17\) the difference between experimental values and theoretical predictions was fully within the range \(-5,+2\) mN up to \(\xi=0.9\). On the contrary, for \(\xi>0.9\) the residuals increased and for \(\xi\to 1\) exceeded the range \(0,\,+10\) mN for \(k=3,7\) and the range \(-2,+5\) mN for \(k=13,17\). As shown in Figure 3c,f,i,l, the increase of the \(m\) parameter resulted in a different degree of correlation between the maximum indentation depth \(h_{max}\) and the value of the \(a_{m}\) coefficient in Equation (1). In this case, the values of the \(R^{2}\) statistic (between \(h_{max}\) and \(a_{m}\)) ranged between \(0.499\) and \(0.543\) for increasing values of \(m\). The variation of the accuracy of the Oliver-Pharr model is shown in Figure 4 for increasing values of \(m\). The average values of the \(y\) variable (i.e., \(y=R^{2}\) was the standard \(R^{2}\) statistic computed when experimental data were fitted through Equation (1)) had a non-monotonic evolution, whereas the errors (\(\pm\)one standard deviation) tended to decrease for increasing values of \(m\). A suitable functional form for Equation (3) was found as  
\[\Phi(y,m)=y-am^{2}-bm-c=0. \tag{13}\]  
Equation (13) was able to effectively match data (\(R_{p}^{2}=0.99\)) for the values of \(a,b,c\) listed in Table 1. \begin{table}
\begin{tabular}{c c c} \hline
**Coefficient** & **Value** & **(95\% Confidence Interval)** \\ \hline \(a\) & \(-1.024\times 10^{-1}\) & \((-1.163,-0.885)\times 10^{-1}\) \\ \(b\) & \(+4.349\times 10^{-1}\) & \((+3.823,+4.876)\times 10^{-1}\) \\ \(c\) & \(+5.224\times 10^{-1}\) & \((+4.747,+5.701)\times 10^{-1}\) \\ \hline \end{tabular}
\end{table} 
Table 1: \(\Phi(y,m)\) numerical coefficients. Figure 4: Variation of the accuracy of the Oliver-Pharr model with the value of \(m\). The average accuracy to fit experimental data is plotted as a function of the value of m (black circles, error bars \(=\pm\) one standard deviation). The function \(\Phi(R^{2},m)=0\) (solid line) is able to effectively match experimental data (\(R_{p}^{2}=0.99\)) as a function of the value of the exponent \(m\). It can be used to explicitly predict the best value of \(m\) (i.e., abscissa corresponding to the vertex of the parabola). Figure 3: Investigation of the accuracy of Equation (1) to match experimental data as a function of \(m\) (top row). In the first row, experimental curves were fitted with values of \(k=3,7,13,17\) (see sub-figures **a**,**d**,**g**, respectively). In the second row (sub-figures **b**,**e**,**h**,**k**), the resulting residuals have been calculated. In the third row, the correlation between the maximum indentation depth and the value of the coefficient \(a_{m}\) has been investigated (sub-figures **c**,**f**,**i**,**l**). Then Equations (3) and (4) were used together with the condition \(\frac{\partial^{2}\Psi(y,m)}{\partial m^{2}}<0\) to find an expression for the optimal value of \(m\), which was written as  
\[m_{m}=\frac{1+\sqrt{\Omega}}{2}, \tag{14}\]  
where the optimized value \(\Omega=10.546\), which was a generalization of the \(k\) values of Equation (2) for real values, resulted in \(m_{m}=2.124\) (95% confidence interval: 1.643-2.755). The optimized value \(m_{m}\) was tested to match the experimental dataset through Equation (1).