# Deflection Modeling and Curvature Manipulation of a Variable-Stiffness Flexible Needle
## 4 Conclusions  
In this paper, a deflection model of a variable-stiffness flexible needle is established, and the influence of parameters such as the stylet diameter and extension length on bending is analyzed and optimized by simulation. The puncture experiments show that the deflection of the flexible needle is independent of the puncture speed. When inserted to a depth of 100 mm, the variable-stiffness needle with a diameter of 0.4 mm and an extension length of 8 mm exhibits 8.841 mm more deflection than the conventional flexible needle (diameter: 0.8 mm, which is equivalent to the variable-stiffness needle at 0 mm extension length). There is a cubic relationship between the bending deflection \(w\) and the length of the stylet extending out of the tube \(L_{s}:w\propto L_{s}^{3}\).  
The advantages of the proposed technology can be summarized as follows: (1) The developed deflection model effectively identifies key factors influencing the bending behavior of the variable-stiffness flexible needle; (2) the simplified structural design eliminates the need for complex actuation mechanisms (e.g., tendon wires or shape memory alloys), reducing manufacturing costs and operational complexity; (3) dynamic curvature control achieved by adjusting the stylet extension length enhances the flexibility of the puncture path.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**PunctureLength** & **0 mm** & **2 mm** & **4 mm** & **6 mm** & **8 mm** \\
**Speed** & & & & & \\ \hline At 2 mm/s (mm) & 18.232 & 18.576 & 19.608 & 21.156 & 27.004 \\ At 4 mm/s (mm) & 17.888 & 18.404 & 19.436 & 21.672 & 26.488 \\ At 6 mm/s (mm) & 18.06 & 18.576 & 19.436 & 21.844 & 26.66 \\ At 8 mm/s (mm) & 18.232 & 18.748 & 19.78 & 21.672 & 26.66 \\ Average (mm) & 18.103 & 18.576 & 19.565 & 21.586 & 26.703 \\ SD (mm) & 0.165 & 0.140 & 0.165 & 0.298 & 0.216 \\ \hline \end{tabular}
\end{table}
Table 6: Deflections of stiffness-tunable flexible needles.  
Figure 11: Influence of puncture speed on flexible needle deflection.  
This study qualitatively analyzes the relationship between needle deflection and stylelet extension length. In the future, we will implant three-dimensional force sensors to analyze the lateral force \(F\) and axial force \(Q\) at the beveled tip and study their contribution to deflection. A kinetic model that accounts for tissue interaction forces, such as friction and anisotropy, will be developed to predict deformation during puncture in real time.  
Conception, design, acquisition of the data, and drafting of the article: G.W., Z.L. and K.L.; acquisition and interpretation of the data: H.Y. and Z.Z.; writing and editing: G.W. All authors have read and agreed to the published version of the manuscript.  
This work was supported by Natural Science Foundation of Shandong Province under Grant No. ZR20240F082; Doctoral Research Fund Project of Shandong Jianzhu University under Grant No. X24034; Youth Advisor Visiting Program of Shandong Provincial Department of Education; and Shandong Provincial Key R&D Program under Grant No. 2019 GGX104056.  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
The authors declare that they have no conflicts of interest.