# Deflection Modeling and Curvature Manipulation of a Variable-Stiffness Flexible Needle
## 2 Materials and Methods
### Model Analysis of Flexible Needle  
Similarly to a normal bevel-tip needle, a stiffness-tunable needle interacts with the surrounding tissue when inserted into soft tissue, as shown in Figure 2a. The needle is inserted into the soft tissue with an external insertion force \(P\). As the needle is inserted, a shearing force, or cutting force \(F_{cut}\), is applied to the needle in a direction perpendicular to the beveled tip. The transverse and axial components of \(F_{cut}\) are \(F\) and \(Q\), respectively. The soft tissues are deformed and compressed, imparting a supporting force \(F_{support}\) to the needle. Studies have shown that the deflection of a long, flexible, bevel-tip needle can be approximated by modeling it as a cantilever beam during insertion into soft tissue [17; 23]. Before analyzing the deflection, the following notes and explanations are provided:  
1. The beveled tip is designed to generate an asymmetric deflection force. In fact, it is possible to make the beveled tip very short without compromising the deflection force. Compared to the needle tip, which is several centimeters long, the beveled tip is relatively short (as short as 3 mm), and its effect on beam deflection is negligible. 2. The variable-stiffness needle bends because the extending stylet adjusts the deflection of the needle, not because the beveled tip bends directly. The bending deformation  
Figure 1: Structure of stiffness-tunable flexible needle. of the beveled tip is minimal. Consequently, the beveled tip can be ignored when modeling the physics of the variable-stiffness needle. 3. The cutting force \(F_{cut}\) is an equivalent force applied to the beveled tip. For uniform tissue, the cutting force is essentially constant during puncture, while for non-uniform tissue, this force varies. Based on the quasi-static assumption, a linear superposition can be applied to estimate the deflection estimation under a changing force. 4. From our previous puncture force studies [24] and others' puncture force simulations [25], it is clear that although bending occurs in the puncture, the cutting force is essentially constant in uniform tissue. The deformation of the beveled tip during puncture is so small that it can be neglected, and thus the partial forces \(F\) and \(Q\) are also essentially constant. The transverse force \(F\) is the key factor affecting deflection, while the axial force \(Q\), transmitted through the stylet to the base of the needle, affects the insertion force. Consequently, the needle tip can be ignored when modeling the physics of the variable-stiffness needle. The deflection model can be simplified to a cantilever beam model subjected to a transverse force \(F\), as illustrated in Figure 2b. The right part of the needle tip consists of the protruding stylet, while the left part is the supporting tube. The interior of the tube is filled with the stylet, and both parts are solid cylinders. The moments of inertia can be expressed as  
\[I_{s} =\frac{\pi R_{s}^{4}}{4} \tag{1}\] \[I_{t} =\frac{\pi R_{t}^{4}}{4} \tag{2}\]  
where \(I_{s}\) and \(I_{t}\) are the moments of inertia of the stylet and tube, and \(R_{s}\) and \(R_{t}\) are the radii of the stylet and tube. The shear stiffnesses can be expressed as  
\[K_{s} =\frac{\pi R_{s}^{2}G}{\kappa} \tag{3}\] \[K_{t} =\frac{\pi R_{t}^{2}G}{\kappa} \tag{4}\]  
Figure 2: Tip force distribution and simplified model. (a) Forces acting on the needle tip. (b) Equivalent model of a flexible needle with variable stiffness. where \(K_{s}\) and \(K_{t}\) are the shear stiffnesses of the stylelet and tube, \(G\) is the shear modulus, and \(\kappa\) is the shear correction factor. In Timoshenko beam theory, the bending deflection is caused by the bending moments. The tip length is \(L=L_{s}+L_{t}\), where \(L_{s}\) is the stylelet extension length and \(L_{t}\) is the tube length.