## 3 Results
### Quantification of the Microneedle Tip Angle and Its Relationship to the Insertion Force  
The tip angle of the microneedles was calculated starting from the mean profile of intact microneedles through Equations (17) and (6), where \(\lambda_{1}=lim_{x\to x_{m_{1}}^{+}}\partial_{x}P_{th}^{2,8,2}(x)\) and \(\lambda_{2}=lim_{x\to x_{m_{2}}^{-}}\partial_{x}P_{th}^{2,8,2}(x)\). From these equations, we obtain results showing that \(<\gamma>\simeq 18\) degrees. In addition, the experimental value of the maximum insertion force for in vivo nerves (and for \(<\gamma>\simeq 18\) degrees) resulted in \(24.573\pm 10.761\) mN (mean value \(\pm\) standard deviation). The previous set of insertion forces was compared to the literature values [48] related to the insertion of similar microneedles into in vitro specimens (see Figure 8a), showing that the difference between the two set of forces was not statistically relevant (Wilcoxon-Mann-Whitney test \(p=0.58\)). As a consequence, assuming the same phenomenon for all the tip angles ranging from 15 to 24 degrees [48], Equation (7) resulted in \(<K_{F}>=6.307\) mN/degree and \(<q>=-86.888\) mN (\(R^{2}=0.93\), see Figure 8b). This experimental correlation between the force of insertion and the tip angle was used to provide, through Equation (9), the minimum diameter needed to avoid buckling following [PERSON]'s theory. In particular, a safety coefficient of 2 was used to provide structures able to withstand a maximum force which is twice the maximum mean force of insertion [48]. To assess the effect of the boundary conditions, the combinations pinned-pinned, pinned-fixed, and fixed-fixed ends, were studied to provide the minimum diameter size needed to avoid the failure of microneedles for critical buckling load (Figure 8c). Finally, starting from Equation (8) together with experimental data of puncture force in epineurial and perineurial sheets [67], an assessment of the theoretical diameter size of 15 mm long W microneedles with a flat tip was provided in Figure 8d. In particular, a combination between minimum and maximum puncture forces and different boundary conditions (pinned-pinned, pinned-fixed, fixed-fixed ends) is shown both for epineurial and perineurial layers. More specifically, to enter the epineurial layer the median values of the microneedle diameters were about 120 \(\upmu\)m, 135 \(\upmu\)m, and 165 \(\upmu\)m for pinned-pinned, pinned-fixed, and fixed-fixed ends, respectively (Figure 8d, top). Similarly, to effectively puncture the perineurial layer  
Figure 7: Evolution of polynomial candidate functions. (**a**) The evolution of both \(P_{th}^{2}\) (2-th polynomial candidate function, blue solid line) and \(P_{th}^{8}\) (8-th polynomial candidate function, red solid line) is compared for distances greater than 20 microns from the symmetry axis of the microneedles. The \(P_{th}^{2}\) candidate is able to follow the experimental shape, and the \(P_{th}^{8}\) candidate function is affected by numeric instability for distances greater than 20 microns. (**b**) The piecewise candidate function \(P^{2,8,2}\) is able to reproduce the experimental shape of intact microneedles both for distances lower than 20 \(\upmu\)m and greater than 30 \(\upmu\)m from the symmetry axis without numeric instability phenomena (blue solid line).  
the assessed median values of the diameter size were 155 \(\upmu\)m, 185 \(\upmu\)m, and 220 \(\upmu\)m for pinned-pinned, pinned-fixed, and fixed-fixed ends, respectively (Figure 8d, bottom).