## 4 Discussion
The post insertion tip shape was, then, the \"measured surface\", and the reconstructed intact tip shape was the \"form\" of the tip. The track of the geometrical changes of the tip was achieved by subtracting the measured post insertion shape from the intact shape. This track was assumed as the superimposition of a low frequency component, accounting for the \"waviness\" of the surface, and a high frequency component, accounting for the \"roughness\" of the tip surface. The first contribution, which was quantitatively assessed through the \(R_{L}(x)\) function and modelled through smooth polynomials, was related to the bulging phenomenon. This apparent phenomenon was likely physically related to the compression of the tip due to the action of the interaction force; that is, to the action of the reaction of the tissue on the microneedle tip. The second contribution, which was quantitatively assessed through the \(R_{H}(x)\) function, was characterized by the presence of a main dominant spike. The position of this spike, with respect to the reference (0 value) was measured through the metric \(D(R_{H})\) and physically related to the amount of folding of the extreme sections of the microneedle tip; that is, this metric measured the length of the bent part of the tip. In this work, a phenomenological characterization of the geometry of the post insertion microneedle tip was performed through the description of the bulging and folding phenomena. However, this geometrical description is likely related to the coupled effects of compression and bending due to the effects of the interaction force on the microneedle tip. The first phenomenon had a characteristic length, measured through the \(D(R_{L})\) of about 100 \(\upmu\)m, whereas the second one has a characteristic length, measured by the \(D(R_{H})\) metric, within 20 \(\upmu\)m. Therefore, the compression phenomenon apparently dominated on the bending of the tip due to the action of the interaction force on the extreme sections of the microneedles in contact with the nerves. The reported values of \(D(R_{L})\) and \(D(R_{H})\) supported this hypothesis, because \(|D(R_{L})|>|D(R_{H})|\) for all the analyzed W microneedles. In addition, the signum of the \(D(R_{H})\) metric was able to identify the direction of the folding of the microneedle tip (i.e., toward left or right). Furthermore, the small value of the standard statistic \(R^{2}\) in the correlation between \(D(R_{L})\) and \(D(R_{H})\) supported the hypothesis that both are needed to fully characterize the damage of the post insertion microneedle tips. This work underlined that conformational changes of the tip geometry still happen although W microneedles are designed to effectively penetrate in vivo peripheral nerves without fracture or buckling of their main shaft. These changes, which are likely related to the sharp geometry of the tip, were already related to the lack of precision in insertion procedures and to patients' discomfort [44]. As a consequence, to avoid bulging and folding effects, W microneedles with a flat tip could be used. Nevertheless, in this case, an increase of the size of their main shaft was predicted (see Figure 8d). In particular, the size of their main shaft should range between 110 \(\upmu\)m (fixed-fixed ends) and 180 \(\upmu\)m (pinned-pinned ends) to puncture the epineurium, and it should range between 150 \(\upmu\)m (fixed-fixed ends) and 230 \(\upmu\)m (pinned-pinned ends) to penetrate the perineurium. Because the available data [67] are referred to single epineurial and perineurial sheets, the size of the diameter of the main shaft of the W microneedles should likely be in the range 150-230 \(\upmu\)m to penetrate a whole nerve, where epineurial and perineurial layers are superimposed. Then, a good tradeoff should be identified between sharp microneedles with a smaller diameter but prone to tip damage, and microneedles with a flat tip, but with a bigger size of the main shaft to puncture the nerves.