# A Rectangular Toroidal Current-Based Approach for Lung Biopsy Needle Tracking
## 3 Results
### System Calibration Error Analysis and Optimization  
The system requires calibration before operation, and several calibration position schemes are set up, including 50 mm 4 \(\times\) 4, 40 mm 5 \(\times\) 5, 30 mm 6 \(\times\) 6, and 30 mm 7 \(\times\) 7 calibration grids. These calibration positions are used to solve the system and calculate the modification coefficients, as shown in Table 3. Table 3 shows that the vertical height offset values of the eight coils vary, with the electromagnetic positioning of the plate at approximately 15 mm. The calibration coefficients differ for each coil due to varying \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Initial length & 70 mm \\ Turns & 25 turns \\ Line width & 0.5 mm \\ Line distance & 0.25 mm \\ PCB Thickness & 1.6 mm \\ \hline \end{tabular}
\end{table} 
Table 1: PCB rectangular coil design parameters. \begin{table}
\begin{tabular}{c c} \hline
**Parameter Name** & **Value** \\ \hline Diameter & 7.2 mm \\ Length & 25 mm \\ Turns & 100 turns \\ Wire diameter & 0.3 mm \\ Diameter & 25 mm \\ Permeability & 10 K \\ Magnetic permeability & 435 uH \\ Resistance & 0.79 \(\Omega\) \\ \hline \end{tabular}
\end{table} 
Table 2: Sensor coil design parameters. drive positions of the plates and the different filtering effects on the overall signal. However, the coil calibration coefficients remain consistent across different calibration schemes. Figure 5: Electromagnetic transmitter board: (**a**) schematic diagram of the electromagnetic transmitter board; (**b**) electromagnetic field transmitter board; (**c**) schematic diagram of the sensor coil; (**d**) sensor; (**e**) system architecture; (**f**) photographed image of electromagnetic tracking. The receiving coil is fixed inside a simulated biopsy needle, with its bottom positioned 80 mm above the bottom of the simulated needle. The bottom of the receiving coil serves as the reference position for induced voltage calculation, ensuring that its height matches the calibration point height. The simulated biopsy needle is then placed vertically at the calibration point for voltage acquisition. The collected voltage data are processed by the calibration algorithm to determine the calibration coefficients and the \(Z\)-axis compensation constant. Once the hardware and software setup for calibration is complete, the calibration process is performed as follows:  
1. Induced voltage acquisition at the calibration point: The induced voltage is measured, and Fourier transform is applied to extract the individual induced voltages corresponding to each of the eight transmitting coils. 2. Nonlinear least squares optimization: The induced voltages and the known pose of the calibration point are used as function parameters, while the calibration coefficients for the eight transmitting coils and the \(Z\)-axis compensation constant of the receiving coil are treated as unknowns, initialized to zero. Nonlinear least squares optimization is employed, iteratively refining the calibration coefficients and \(Z\)-axis compensation constant to minimize the difference between the computed induced voltage (based on the calibration point pose) and the experimentally acquired voltage from step (1). 3. Output of optimized calibration parameters: The final iteration results provide the optimal calibration coefficients for each transmitting coil and the optimal \(Z\)-axis compensation constant for the receiving coil. To evaluate the performance of different system calibration methods, a mutual calibration of the coils is conducted, which allows for the calculation of the self-calibration error for each coil, as shown in Figure 6. As depicted in the figure, when different calibration position schemes are chosen, there are some variations in the calibration error. However, the overall differences are not significant. It can also be observed that the highest signal-to-noise ratio is typically found in the center of the acquisition area, with the maximum calibration error occurring more frequently at the edges.