# The Development of Piezo-Driven Tools for Cellular Piercing
## Materials and Methods
### Design of the Piezo-Drill  
The vibration of a conventional injector is approximately 37 \(\upmu\)m and 270 \(\upmu\)m in the deionized water and air, respectively, even with the assistance of mercury [19]. In the conventional injector, the piezoelectric actuator is placed behind the holder (Figure 2A). Taking into consideration the inner diameter, outer diameter, and the extent of the pulled section, the pulled section of the injection pipette can be modeled as a flexible beam [19]. This section is connected to the injection pipette's shoulder, which has a high rigidity. Thus, the conventional injector, except the drawn section, is rigid, and the pulled section is connected to the rigid base. Installation errors from the front of the piezoelectric actuator generate a slight deviation, \(\theta\). Axial force \(F\) is produced by the piezo-pulse, as shown in Figure 2B. We analyze the conventional injector in terms of the lateral vibration and kinetic energy. Here, a periodic driving signal is applied to the piezoelectric actuator, and the value of signal is positive. Displacement of the injection pipette generated from piezoelectric actuators is also periodic. When the pipette is placed against an oocyte, the indention force to the oocyte is linear to the displacement of the piezoelectric actuator [20], and the proportionality coefficient is defined as \(k\). In addition, axial output force (\(F\)) of the piezoelectric actuator is equal to the indention force, which indicates that the axial output force is also periodic. Figure 1: Mouse oocytes injection system. (**A**) Schematic diagram of the injection system architecture. (**B**) Part of the injection system: 1 is the microrobot-1, which is a three-degrees-of-freedom micromanipulator with a travel distance of 25 mm and a positioning accuracy of 0.04 \(\upmu\)m along each axis; 2 is the holding pipette which is used to immobilize the oocytes; 3 is the medium for placing the oocytes; 4 is the injection pipette for cellular piercing; 5 is the injector for cellular piercing; 6 is the microrobot-2, which is a three-degrees-of-freedom micromanipulator with a stroke of 20 mm and a step resolution of 0.04 \(\upmu\)m along each axis; 7 is the microscope; 8 is the Charged Coupled Device (CCD). Visual feedback is gained by the microscope and CCD. Due to the existence of the installation error, a periodic point force \(F_{y}\), inducing lateral vibrations of the injection pipette, is derived from the axial output force \(F\). \[F_{y}=F\cdot\sin\theta=\frac{k\cdot d_{\max}}{V_{\max}}\cdot g(t)\cdot\sin\theta, \tag{1}\]  
where \(d_{\max}\) represents the maximum displacement of the piezoelectric actuator when the maximum voltage \(V_{\max}\) is applied to the piezoelectric actuator, and \(g(t)\) is a periodic signal applied to the piezoelectric actuator. Figure 3 shows a coordinate system for the pulled section of the injection pipette. The bending vibration equation of the cantilever is  
\[\rho A\frac{\partial^{2}w(x,t)}{\partial^{2}t}+EI\frac{\partial^{4}w(x,t)}{ \partial^{2}x}=f(x,t)+\frac{\partial m(x,t)}{\partial t}, \tag{2}\]  
where \(\rho\) is mass per unit length, \(A\) is the cross section area of the pulled section, \(E\) is the Young's modulus of the structure of the drawn section, \(I\) is the bending moment of inertia, \(F\) is the axial force produced by the piezo-pulse, \(\theta\) is the deviation generated by the error of installation, \(w\) is the deflection of injection pipette tip, and \(f(x,t)\) and \(m(x,t)\) are the external force and moment both derived from the \(F_{y}\). Modal coordinate transform is given below:  
\[w(x,t)=\sum_{r=1}^{+\infty}W_{r}(x)q_{r}(t), \tag{3}\]  
where \(W_{r}(x)\) describes the form of vibration, and \(q_{r}(t)\) is a function of the time used to describe the motion of the pulled section.