# Deflection Modeling and Curvature Manipulation of a Variable-Stiffness Flexible Needle
## 2 Materials and Methods
### Simulation of the Effect of Stylet Diameter on Deflection  
In addition to the structure shown in Figure 1 (noted as Structure 1), two other structures were designed, noted as Structure 2 and Structure 3, as shown in Figure 3a-c. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Deflection** & **Extension Length** & **0 mm** & **5 mm** & **10 mm** & **15 mm** & **20 mm** \\ \hline Bending deflection (mm) & 7.1788 & 8.0402 & 10.0862 & 14.0704 & 20.6389 \\ Shear deflection (mm) & 0.0008 & 0.0013 & 0.0015 & 0.0017 & 0.0020 \\ Total deflection (mm) & 7.1796 & 8.0415 & 10.0877 & 14.0721 & 20.6409 \\ Bending deflection/Total deflection & 99.99\% & 99.98\% & 99.99\% & 99.99\% & 99.99\% \\ \hline \end{tabular}
\end{table} 
Table 1: Theoretical deflection under different stylet extension lengths. Unlike Structure 1, Structure 2 and Structure 3 have incomplete circles in the cross section of the stylet. The difference between Structure 2 and Structure 3 lies in the direction of the beveleed tip. The effect of the stylet's cross-sectional shape on the bending was investigated by comparing Structures 1 and 2. In order to eliminate the influence of the needle tip level in Structure 1 and Structure 2, Structure 3 was used to analyze whether the direction of the bevele affects the bending behavior of Structure 2. The structural models of three needles were created in ANSYS WORKBENCH 2022R2, and a constant compressive load was applied to the tip bevel to analyze the effects of stylet length and thickness (the thickness of the remaining material) on the pure bending of flexible needles. The material properties [26], geometries, and loads are detailed in Table 2.  
A structural model with a stylet diameter of 0.6 mm and a stylet length of 20 mm was created in ANSYS WORKBENCH 2022R2, as shown in Figure 4a. The automated meshing method was used, creating 1989 nodes and 846 elements. A fixed support was applied to the left side of the needle, and a vertical force of 0.3 N [24] was applied to the beveleed tip. A path from (0, 0, 0) to (49, 0, 0) was created to obtain the deformations. The deflection results of the flexible needle under the applied force are shown in Figure 4b. \begin{table}
\begin{tabular}{c c c} \hline Total Length (mm) & Tip Angle (\({}^{\circ}\)) & Tip Length (mm) \\ \hline
50 & 30 & 5 \\ \hline Tube Diameter (mm) & Styllet Diameter (mm) & Styllet Length (mm) \\ \hline
0.8 & 0.8, 0.7, 0.6, 0.5, 0.4 & 0, 5, 10, 15, 20 \\ \hline Young’s Modulus (MPa) & Poisson’s Ratio & Load (N) \\ \hline
75,000 & 0.3 & 0.3 \\ \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters of a flexible needle with variable stiffness. Figure 3: Structures and cross sections of three variable-stiffness needles. (**a**) Structure 1. (**b**) Structure 2. (**c**) Structure 3. (**d**) Cross section of Structure 1. (**e**) Cross section of Structure 2. (**f**) Cross section of Structure 3. The extension length of the stylet was fixed at 20 mm, and the diameters of the stylet were set to 0.8 mm, 0.7 mm, 0.6 mm, 0.5 mm, and 0.4 mm for the simulation. The deflections of the path were recorded and are depicted in Figure 5.  
Note that the coordinate system in Figure 5 is the same as in Figure 4a, where the X-axis represents the direction along the needle and the deflection corresponds to the displacement along the Z-direction. From Figure 5a-c, it can be seen that the bending of the three needle bodies is similar, with the differences primarily occurring at the tips. For the same stylet extension length, the change in diameter has the most significant effect on the bending of Structure 1: the maximum deflection of Structure 1 increases from 7.798 mm to 20.604 mm as the diameter decreases from 0.8 mm to 0.4 mm; the deflections of Structure 2 and Structure 3 are similar, with the maximum deflection increasing to approximately 11.70 mm. Structures 2 and 3, having identical cross-sectional shapes, exhibit nearly identical deflections  
Figure 4: Simulation models in ANSYS WORKBENCH 2022R2. (**a**) Constraints and loads.