# A Rectangular Toroidal Current-Based Approach for Lung Biopsy Needle Tracking
## 3 Results
### System Test and Response
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**This Article** & **Wireless 5-d [14]** & **5-d Large Work Area [15]** &
\begin{tabular}{c} **Residual Network-Based** \\ **Electromagnetic [16]** \\ \end{tabular} \\ \hline Average error (mm) & 1.75 & 1.8633 & 2.02 mm & 1.25 \\ Maximum error (mm) & 3.13 & 2.3 & 2.56 & 2.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of RMSE with the other literature. each position change, the inertial-assisted electromagnetic positioning system performed a calculation of induced voltage and pose. The data and errors calculated from 2000 experiments can be found in Supplementary Materials. To display the system's stability, data within the ranges of 0-20 mm for the \(x\)-axis, 0-20 mm for the \(y\)-axis, and 100-120 mm for the \(z\)-axis were selected for display in Figure 12. From the error bars, it can be observed that the system error is consistently within 1.75 mm. Additionally, the spatial distance errors of the 2000 data sets for x, y, and z are calculated, and Figure 12d shows that the distance error is less than 3 mm after 2000 large-scale experiments. To verify the instrument's response performance, we conducted systematic time response data collection across 2000 experiments. In each experiment, the data acquisition card began collecting the current signal and transmitted the data to the computing system, where it was processed to yield results. The time of signal collection and result output were precisely recorded in each experiment to analyze the response time of the instrument throughout the process. Through statistical analysis of the 2000 experimental data, we focused on the system's response time, including the total delay from signal acquisition to result processing, system stability, and time fluctuations in each experiment, as shown in Figure 13a. The experimental data indicated that the majority of response times remained within the expected range, and across the 2000 trials, the average response time consistently stayed within the target time range (50 ms), demonstrating good repeatability (20 hz). Figure 12: Overall error analysis of the system: (**a**) The measurement error of the \(x\)-axis between 0 and 20 mm. (**b**) The measurement error of the \(y\)-axis between 0 and 20 mm. (**c**) The measurement error of the \(z\)-axis between 100 and 120 mm. (**d**) Overall error of 2000 test points. Figure 13: System response time and system dynamic response test: (**a**) Statistical histogram of system response time after 2000 tests. (**b**) Screenshot of system dynamic response test video. For assessing the accuracy of the system, we conducted 30 repeated measurements of the positioning error. The average error was calculated as 1.76 mm, and the standard deviation of the measurements was 0.6 mm. Using a 95% confidence level, we calculated the 95% confidence interval for the error to be [1.12 mm, 2.4 mm], indicating that the positioning error falls within this range with 95% confidence. We compared and validated our method with several algorithms from previous studies in the literature. As shown in Table 7, the accuracy reported in this paper is comparable to that in other studies, but our method is simpler and better suited to meet the fine requirements of puncture tracking. In comparison to the other three studies, our approach offers several distinct advantages: Firstly, while previous methods often rely on complex optimization algorithms or machine learning techniques for pose tracking, which can increase computational time and reduce accuracy, our method uses a simple yet effective least squares algorithm to solve the five-degree-of-freedom positioning equations, ensuring both precision and computational efficiency. Moreover, unlike traditional methods that may be limited by sensor arrays or use of fewer transmitting coils, we employ a rectangular toroidal current distribution model, which provides more accurate and reliable tracking of the biopsy needle, especially in real-time tracking scenarios.