## 3 Results
### Analysis of the Post Insertion Shape  
After defining the theoretical shape of the intact microneedle in Equation (17), all the geometrical changes due to the insertion procedure were quantified with respect to this standard configuration (see Figure 2). In particular, the track of the geometrical deformation was obtained as the difference between the post insertion and the theoretical microneed tip shape, resulting in the function \(\Gamma(x)\), described through Equation (10) (see Figure 9a-e). This function was further split in two different parts: a low frequency contribution \(R_{L}(x)\), which was modelled through smooth 15-th degree polynomials in the range \(-\)30, 30 \(\upmu\)m (\(R^{2}=0.9976,0.9957,0.9958,0.9944,0.9981\), Figure 9f-j), and a high frequency jagged and irregular contribution \(R_{H}(x)\), as shown in Figures 9k-o. In addition, in Figure 9f-j the values of the metric \(D(R_{L})\), defined in Equation (11), were indicated through the length of the yellow arrows, whereas in Figure 9k-o the values of the metric \(D(R_{H})\), defined in Equation (12), were indicated through the length of the blue arrows. Nevertheless, to provide a physical meaning to the previous metrics, the experimental images were presented together with the comparison between the post insertion and the intact shapes of the microneedle tips. Indeed, in Figure 10a1-e1, the five experimental images of the W microneedle tips were shown with a metric bar, and in Figure 10a2-e2, the comparison between the intact reference profile of the tip (in red) and the post insertion profiles (in black) was presented. The values of both metrics \(D(R_{L})\) and \(D(R_{H})\) were also inserted in the plots in order to visualize their physical relationship with deformed shape of post  
Figure 8: Comparison between puncture force into in vivo and in vitro nerves and predictions of the size of the microneedle shaft. (**a**) Comparison between insertion forces of W microneedles with a tip angle of 18 degrees into in vivo [49] nerves and in vitro specimens [48]: the two groups do not show any significant statistical difference (Wilcoxon–Mann–Whitney test \(p=0.58\)). (**b**) Experimental correlation between the mean value of the insertion force and the microneedle tip angle for in vitro specimens [48]. (**c**) Theoretical relationships between the diameter of 15-mm-long microneedles and their tip angle. Different boundary conditions are investigated to assess the minimum diameter needed to enter in vivo nerves without failure of the main shaft for buckling: pinned–pinned ends (circles), pinned–fixed ends (triangles), and fixed–fixed ends (squares). (**d**) Assessment of the diameter size of a 15-mm-long microneedle with a flat tip to puncture the perineuilrial and epineuil sheets with different boundary conditions (pinned–pinned, pinned–fixed, fixed–fixed ends), starting from literature data [67]. insertion tips. More specifically, in Figure 10f the values of the \(D(R_{L})\) metric were compared in a bar plot (all the values are reported in Appendix A, Table A3 first column), as well as in Figure 10g the values of the \(D(R_{H})\) metric (all the values are in Appendix A, Table A3 second column). The length of the bars was related to the amount of folding, and their signum was related to the direction of the folding of the extreme section of the post insertion microneedle tips: the signum \"\(-\)\" indicated that the tip folded up toward the left, and the signum \"+\" indicated a folding toward the right. The reference (0 value) was the position of the symmetry axis of the theoretical intact tip. In particular, this metric was able to measure how the post insertion tip folded up toward the right, up to 11.63 \(\upmu\)m in the first specimen, toward the left up to 18.46 \(\upmu\)m in the second specimen. Similarly, this metric measured that the deformed tip folded up toward the left, up to 9.47 \(\upmu\)m, in the third specimen, whereas it folded up toward the right up to 18.59 and 16.26 \(\upmu\)m in the fourth and fifth specimens (see Figure 10g).