## 2 Materials and Methods
### Modelling of the Underformed Shape of W Microneedles  
Four experimental pictures of W microneedles (\(\varnothing\)75 \(\upmu\)m) were chosen before both contact and insertion within peripheral nerves and used to model the mean shape of the undeformed tip. More specifically, the ending part of each microneedle was considered for  
Figure 1: Evolution of the indentation force with respect to the normalized indentation depth \(\xi\): The reaction force \(\mathbf{F}\) (black arrow) of the tissue to the action of the microneedles tip (with opening angle \(\gamma\)) at the current indentation depth \(h\) (subfigure 1). Similarly, the tissue responds with a reaction force \(\mathbf{F_{max}}\) (black arrow) to the action of the microneedle tip (with opening angle \(\gamma\)) at the maximum indentation depth \(h_{max}\) (subfigure 2 ). In this work, the reaction force to the action of the microneedle tip is defined as ”interaction force”. The ratio \(\xi=h/h_{max}\) is defined as normalized indentation depth. The plot shows that increasing the normalized indentation depth also increases the interaction force up to a maximum value, when the peripheral nervous tissue is punctured by the microneedle tip. The indentation phase is defined for \(0\leq\xi\leq 1\).  
a length of about 100 \(\upmu\)m from the tip end and for a width of 30 \(\upmu\)m from the symmetry axis. ImageJ [66], through the plug-in NeuronJ, was used to provide a semiautomatic track of the microneedle profile and to extract the numerical coordinate of the microneedle shape from the experimental images. To standardize all images, they were further elaborated through a Python custom program to align the microneedles' symmetry axis with the vertical one and to center each profile with respect to the theoretical contact point (that is the lowest point of the experimental profile was superimposed to the origin of the axes). All the four images were used to achieve a single cluster of points, which was fitted to provide the mean shape of the intact tip of microneedles.  
In particular, a closed and limited set \(A=[x_{i},x_{f}]\) was considered, where \(x_{i}\) and \(x_{f}\) were the initial and final abscissas whose the experimental shape was the image. Polynomial candidate functions of \(n\) degree \(P_{th}^{n}:A\to B\subset R^{+}\), were tested to fit the cluster of experimental points for each \(x\in A\), and their suitability was assessed through the value of the standard statistic \(R^{2}\), for increasing value of the maximum power \(n\). In addition, to combine the features of polynomials of different powers, piecewise polynomial candidates were considered and compared to simple polynomial candidates in order to assess the best way to match the undeformed shape of the microneedle tips. In this case, the limited and closed sets \(A_{1}=[x_{i},x_{m}]\), \(A_{2}=[x_{m1},x_{m2}]\), \(A_{3}=[x_{m2},x_{f}]\), where, \(x_{f}>x_{m2}>x_{m1}>x_{i}\), were considered together with the functions \(P_{th}^{n_{1}}:A_{1}\to B_{1}\subset R^{+}\), \(P_{th2}^{n_{2}}:A_{2}\to B_{2}\subset R^{+}\), \(P_{th3}^{n_{3}}:A_{3}\to B_{3}\subset R^{+}\), where \(B_{1},B_{2},B_{3}\subset R^{+}\) were the images of \(A_{1},A_{2},A_{3}\), while \(A_{1}\bigcup A_{2}\bigcup A_{3}=A\), and \(n_{1},n_{2},n_{3}\in N\) were the maximum powers of the fitting functions, respectively. In the following, a theoretical candidate function matching the mean experimental microneedle shape was written as  
\[P_{th}^{n_{1},n_{2},n_{3}}(x)=\begin{cases}P_{th}^{n_{1}}\ for\ x_{i}\leq x<x_{ m1}\\ P_{th}^{n_{2}}\ for\ x_{m1}\leq x\leq x_{m2}\\ P_{th}^{n_{3}}\ for\ x_{m2}<x\leq x_{f}\end{cases} \tag{5}\]  
to specify the power of the three branches of the piecewise function, whereas, when a single polynomial candidate function of power \(n\) was used to reproduce the experimental mean shape of the microneedle, the condition \(n_{1}=n_{2}=n_{3}=n\) was imposed.