A Rectangular Toroidal Current-Based Approach for Lung Biopsy Needle Tracking

Abstract

Biopsy remains the gold standard for diagnosing lung cancer, with high-quality tissue samples being critical for accurate results. To improve puncture accuracy, reduce reliance on CT imaging, and minimize procedural complications, it is essential to address the challenges of tracking the biopsy needle’s trajectory and providing real-time positional guidance to physicians. In this study, we propose a tracking model based on a rectangular toroidal current distribution to determine the biopsy needle’s relative position within the electromagnetic tracking system. A printed circuit board (PCB) is used as the platform for generating the rectangular circulating magnetic field. An alternating electromagnetic field (~70 kHz) is modeled based on the Biot–Savart law. Induced voltages from multiple transmitting coils are processed using Fourier transform algorithms to separate frequencies, enabling the independent extraction of each coil’s signal. A least squares method is employed to solve the five-degree-of-freedom electromagnetic positioning equations for the receiving coils. The objective is to establish a precise and computationally efficient electromagnetic localization model for the biopsy needle. An experimental setup simulating lung biopsy procedures is implemented, utilizing the proposed rectangular toroidal current configuration. Results demonstrate an average localization error of less than 1.76 mm, validating the effectiveness of the system in addressing the challenges of real-time biopsy needle tracking.

1. Introduction

Early diagnosis is a critical step in the treatment of lung cancer, as earlier detection provides patients with better treatment options and improved outcomes [1]. The five-year survival rate for stage I lung cancer can reach up to 80% following treatment [2]. Currently, lung cancer diagnosis is categorized into two main approaches: imaging-based diagnosis and pathological diagnosis [3]. Pathological diagnosis involves performing a biopsy on a detected lung mass to examine whether cancer cells are present in the tissue, allowing for a definitive confirmation of lung cancer [4]. Visual feedback remains an effective tabletop experimental tracking method and is still applicable in medical procedures where direct visual access to the surgical area is feasible, such as subretinal injections [5]. X-ray imaging is a widely used medical imaging modality due to its high penetration capability, enabling deep tissue visualization [6]. While X-ray imaging provides high-resolution anatomical images (~100 μm) with a sufficient refresh rate (30 ms), concerns regarding ionizing radiation exposure pose risks to both patients and physicians [7,8]. Magnetic resonance imaging (MRI) is another powerful imaging tool capable of generating detailed images of internal organs and soft tissues. However, its incompatibility with magnetic materials makes it challenging to integrate with magnetic tracking systems [9]. Ultrasound imaging is a bio-friendly, non-invasive imaging technique that has been utilized for real-time tracking of magnetic devices within the vascular system [10]. Nevertheless, ultrasound imaging is highly sensitive to interfaces, making it difficult to visualize tissues located behind rigid structures such as bones. Designing an assisted lung biopsy needle tracking system that ensures high tracking accuracy, aids physicians in puncture path planning, and minimizes the number of CT scans required for patients remains a significant research focus.

Various short-range techniques for spatial position tracking include electromagnetic, optical, ultrasonic, mechanical, and inertial navigation systems [10,11,12]. The scleral search coil technique is limited to angular position tracking within a small range and is entirely unsuitable for medical biopsy procedures due to its inability to provide precise spatial localization [12,13,14]. Among these, optical and electromagnetic tracking are the most commonly used techniques [15]. Optical tracking systems offer high tracking accuracy but are susceptible to occlusion. In contrast, electromagnetic tracking systems can track positions without occlusion, ensuring reliable tracking [16]. Electromagnetic tracking is currently the gold standard for surgical tracking and guided interventions, primarily used in bronchoscopy, urological examinations, orthopedic, and lung puncture procedures [17]. Notably, electromagnetic tracking provides an economical alternative to X-ray fluoroscopy in interventional surgical navigation systems. These methods eliminate radiation requirements and enable three-dimensional and six-degree-of-freedom (6DOF) navigation in position and orientation. To implement electromagnetic tracking and enhance its accuracy, speed, and robustness, researchers have developed various electromagnetic tracking methods [18]. Optical tracking offers high accuracy (often below 1 mm), but its performance is heavily dependent on maintaining a clear line of sight, which can be difficult in clinical settings due to occlusions and anatomical constraints. Ultrasound-based systems provide real-time imaging and are widely accessible, but they are operator-dependent and generally yield lower spatial resolution, with tracking errors typically in the range of 2–5 mm. Mechanical tracking systems, such as articulated arms or encoders, can offer good repeatability but tend to be bulky and invasive, limiting their applicability during delicate interventions. In contrast, an electromagnetic tracking system demonstrates an average localization error of less than 2 mm, without requiring a line of sight and with minimal mechanical constraints. These advantages make it particularly well suited for in situ lung biopsy procedures, where internal access and soft tissue deformation pose significant challenges to traditional techniques.

Magnetic tracking is widely adopted in medical applications due to its inherent advantages, such as minimal distortion when passing through the human body and its harmlessness to living tissues [19]. One approach to magnetic tracking involves embedding magnetic sensors within a device to measure the surrounding magnetic field and determine its position [20,21]. By comparing sensor measurements with a pre-characterized magnetic field distribution within the workspace, the target can be accurately tracked [22]. However, internal magnetic sensors require onboard transmission circuits and batteries to support the measurements, which reduces the available space inside the capsule. Another magnetic tracking method employs an external sensor array to capture the magnetic field generated by magnets embedded in the device. Sensor arrays have been developed for applications such as force sensing and in vivo muscle tension monitoring and remain a promising approach for active capsule tracking [23]. These arrays are externally powered and do not require onboard components, allowing for the integration of functional units such as biopsy tools and drug delivery chambers [24].

The electromagnetic tracking system consists of three main components: a magnetic field transmitter, a magnetic field sensor, and a computer control unit. The magnetic field transmitter generates alternating electromagnetic signals, which induce a voltage in the sensor. The computer control unit estimates the spatial position of the sensor by solving an inverse problem based on the relationship between the induced voltage and the electromagnetic field distribution [25,26]. Search coil technology has a long history, but in 2014, Song et al. proposed an electromagnetic tracking system, where signals of different frequency must be simultaneously fed to the tri-axial transmitting coils [27]. Tian et al. adopted three coils as a tri-axial magnetic dipole source, but the inducing sensor was composed of two non-orthogonal coils [28]. In [29], an analytical algorithm for magnetic tracking and orientation is presented, and orthogonal tri-axial coils are used as the transmitting coils and sensing coils. Yang presented a novel electromagnetic tracking system (EMTS), whose induction coils are separated from the control unit; hence, wireless pose tracking (six-degree-of-freedom) can be achieved [29]. The transmitting coils are simultaneously excited by nine-channel sinusoidal signals with different frequencies, instead of the switching channels in the chronological order. Inspired by the idea of frequency division, we designed our model accordingly. Zhang designed an accurate magnetic field modeling method that can attenuate the electromagnets’ hysteresis and provide a precise actuation field estimation [30]. Guo proposed a ResNet-LM algorithm, based on the fusion of the deep learning algorithm and optimization algorithm, to improve permanent magnet-based tracking (PMT) performance [31,32].

To enhance both accuracy and system simplicity, we propose a rectangular circulating current-based magnetic field tracking system. In this study, we develop a model of the rectangular circulating emitted magnetic field to determine the relative position of the biopsy needle within the tracking system. The main focus of this research is on establishing a model based on a rectangular circulating magnetic field that accurately represents the position of the biopsy needle while minimizing computational load. We also present an experimental setup for a lung puncture biopsy needle motion tracking system, which is based on electromagnetic tracking using a rectangular toroidal current. The XYZ coordinates of the biopsy needle relative to the emitted magnetic field, along with the pitch and pendulum angles, were calculated using the least squares method. Unlike traditional electromagnetic tracking methods, our approach utilizes a rectangular toroidal current distribution to improve spatial accuracy while maintaining computational efficiency. This novel configuration, combined with optimized signal processing algorithms, enables high-precision tracking of biopsy needles, making it a viable alternative for real-time medical guidance.

2. Materials and Methods

2.1. Rectangular Toroidal Needle Tracking System

In medical puncture procedures, physicians rely on CT imaging data to perform high-precision three-dimensional reconstruction, accurately restoring the patient’s internal anatomical structures, including organs, major blood vessels, and trachea. Based on the reconstructed model, doctors can precisely plan the puncture path, ensuring that the target tissue is reached while avoiding critical blood vessels and airways, thereby reducing the risk of intraoperative complications.

Traditionally, puncture procedures depend largely on the surgeon’s experience and guidance from two-dimensional imaging, which can result in significant errors and risks. However, with the aid of electromagnetic tracking technology, physicians can monitor the real-time spatial position and orientation of the biopsy needle and compare it with the preplanned trajectory, allowing for adjustments during the procedure. This technology employs an external electromagnetic field generator and integrates magnetic sensors at the needle tip to track the needle’s movement in three-dimensional space in a non-contact manner. This enables surgeons to dynamically adjust the insertion angle and depth, significantly enhancing precision, as Figure 1 shows.

This approach, which integrates CT-based reconstruction, trajectory planning, and electromagnetic tracking for guidance, not only enhances the safety and success rate of puncture procedures but also minimizes the likelihood of severe medical incidents caused by accidental puncture of critical structures. As medical imaging processing and electromagnetic navigation technologies continue to advance, this method is expected to play a greater role in precision medicine, providing more reliable technical support for minimally invasive surgeries and interventional treatments.

According to the Biot–Savart law $d\overrightarrow{B}={\displaystyle \frac{{\mu }_0}{4\pi }}{\displaystyle \frac{Id\overrightarrow{l}\times \overrightarrow{R^{\prime }}}{{R^{\prime }}^3}}$, by establishing a coordinate system at the location of the current element and measuring the magnetic induction at a point in space, the relative position of that point with respect to the current element can be determined.

The choice of a rectangular toroidal current distribution in this study is driven by both practical and theoretical considerations. From a practical standpoint, this geometry facilitates efficient implementation using printed circuit board (PCB) technology, allowing for precise coil configuration and uniform magnetic field generation within a confined three-dimensional space. Theoretically, the rectangular toroidal distribution enables a simplified application of the Biot–Savart law, ensuring accurate magnetic field calculations and reliable positioning of the biopsy needle. While alternative coil geometries, such as circular coils and solenoids, were considered, they were ultimately excluded. Circular coils tend to generate less uniform magnetic fields and require more complex modeling approaches to achieve the same level of spatial accuracy. Solenoids, although providing a uniform magnetic field along the center axis, present challenges in terms of boundary effects and are less suitable for compact, PCB-based implementations. Consequently, the rectangular toroidal configuration offers an optimal balance of uniformity, computational efficiency, and practical implementation.

The circular current model complicates the integration process when calculating the toroidal integral model. In contrast, the calculation of the magnetic field is significantly simpler when using the long straight wire current model. The strength of the magnetic field is directional and vectorial. Since the magnetic field is vectorial, when calculating the rectangular current-based magnetic field, only the magnetic fields generated by the currents in the four sides of the rectangle need to be added vectorially. From the Biot–Savart law, we can derive the spatial magnetic field distribution model for the current-carrying long straight wires. As shown in Figure 2a, the spatial coordinate system is established with the current-carrying long straight wire CD aligned along the z-axis, and the spatial point P is located on the x-axis at a distance $r_0$ from the origin. The magnetic field expression of the long straight wire $\mathrm{C}\mathrm{D}$ at point P is as shown in Equation (1). The differentiation of the z-axis can be expressed as Equation (3).

From Equation (5), it follows that when the current size of the current-carrying long straight wire, the positions of the two endpoints, and the positions of point $P$ in space are known, the induction of the magnetic field generated by the long straight wire at point $P$ can be known. As shown in Figure 2b, the center of the rectangular annulus $ABCD$ is located at the origin of the coordinates, the vector of the four edges is expressed as $\overrightarrow{a_i}$, the magnetic field generated by the four edges at point $P$ in space is $\overrightarrow{B_p}$, and the vector of the line connecting the termination position and the starting position of point $P$ and $\overrightarrow{a_i}$ is $\overrightarrow{b_i}$ and $\overrightarrow{c_i}$, where $i=\mathrm{1,2},\mathrm{3,4}$. The magnetic field of the rectangular circulation is given by Equation (7).

$$B_P={\int }_{CD}dB={\displaystyle \frac{{\mu }_{0}I}{4\pi }}{\int }_{CD}{\displaystyle \frac{sin\theta }{r^2}}dz$$

(1)

$$dz={\displaystyle \frac{r_0}{{\mathit{sin}}^2\theta }}dz$$

(2)

$$B_P={\displaystyle \frac{{\mu }_{0}I}{4\pi r_0}}\left(cos{\theta }_1-cos{\theta }_2\right)$$

(3)

$$\begin{gathered} {\displaystyle \frac{1}{r_0}}·\left(cos{\theta }_1-cos{\theta }_2\right)={\displaystyle \frac{1}{\left|\overrightarrow{c}\right|sin{\theta }_1}}·\left({\displaystyle \frac{\overrightarrow{a}·\overrightarrow{c}}{\left|\overrightarrow{a}\right|·\left|\overrightarrow{c}\right|}}-{\displaystyle \frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{a}\right|·\left|\overrightarrow{b}\right|}}\right) \\ ={\displaystyle \frac{1}{\left|\overrightarrow{a}\right|·\left|\overrightarrow{c}\right|·sin{\theta }_1}}·\left({\displaystyle \frac{\overrightarrow{a}·\overrightarrow{c}}{\left|\overrightarrow{c}\right|}}-{\displaystyle \frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{b}\right|}}\right) \\ ={\displaystyle \frac{1}{\left|\overrightarrow{c}\times \overrightarrow{a}\right|}}·\left({\displaystyle \frac{\overrightarrow{a}·\overrightarrow{c}}{\left|\overrightarrow{c}\right|}}-{\displaystyle \frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{b}\right|}}\right) \end{gathered}$$

(4)

$$\overrightarrow{B_P}={\displaystyle \frac{{\mu }_{0}I}{4\pi }}·{\displaystyle \frac{\overrightarrow{c}\times \overrightarrow{a}}{{\left|\overrightarrow{c}\times \overrightarrow{a}\right|}^2}}·\left({\displaystyle \frac{\overrightarrow{a}·\overrightarrow{c}}{\left|\overrightarrow{c}\right|}}-{\displaystyle \frac{\overrightarrow{a}·\overrightarrow{b}}{\left|\overrightarrow{b}\right|}}\right)$$

(5)

$$\overrightarrow{B_P}={\displaystyle \frac{{\mu }_{0}I}{4\pi }}{\sum }_{i=1}^4{\displaystyle \frac{\overrightarrow{c_i}\times \overrightarrow{a_i}}{{\left|\overrightarrow{c_i}\times \overrightarrow{a_i}\right|}^2}}·\left({\displaystyle \frac{\overrightarrow{a_i}·\overrightarrow{c_i}}{\left|\overrightarrow{c_i}\right|}}-{\displaystyle \frac{\overrightarrow{a_i}·\overrightarrow{b_i}}{\left|\overrightarrow{b_i}\right|}}\right)$$

(6)

where $B_P$ is the magnitude of the magnetic induction generated by the current carrying a long straight wire at point $P$, whose direction is compounded by the right-hand rule; $dz$ is the length of long straight wire in microns, whose distance to the origin of coordinates is $z$; the angle between $d\overrightarrow{z}$ and $\overrightarrow{r}$ is $\theta$.; and ${\theta }_1$ and ${\theta }_2$ are the angles between point $P$ and points $C,D$.

Let the coordinates of the space point $P$ be $\left(\mathit{x},\mathit{y},\mathit{z}\right)$, the length of the $AB$ side be ${2l}_1$, and the length of the $BC$ side be $2l_2$; then, the coordinates of the four vertices $ABCD$ of the rectangular annulus can be represented by $l_1$ and $l_2$, $A\left(-l_1,l_2,0\right)$, $B\left(l_1,l_2,0\right)$, $C\left(l_1,-l_2,0\right)$, $D\left(-l_1,-l_2,0\right)$, which are given by Equation (7).

$$\left\{\begin{array}{c} \overrightarrow{a_1}=2l_1\overrightarrow{i} \\ \overrightarrow{a_2}=-2l_2\overrightarrow{j} \\ \overrightarrow{a_3}=-2l_1\overrightarrow{i} \\ \overrightarrow{a_4}=2l_2\overrightarrow{j} \\ \overrightarrow{b_1}=\overrightarrow{c_2}=\left(l_1-x\right)\overrightarrow{i}+\left(l_2-y\right)\overrightarrow{j}-z\overrightarrow{k} \\ \overrightarrow{b_2}=\overrightarrow{c_3}=\left(l_1-x\right)\overrightarrow{i}+\left(-l_2-y\right)\overrightarrow{j}-z\overrightarrow{k} \\ \overrightarrow{b_3}=\overrightarrow{c_4}=\left(-l_1-x\right)\overrightarrow{i}+\left(-l_2-y\right)\overrightarrow{j}-z\overrightarrow{k}\\ \overrightarrow{b_4}=\overrightarrow{c_1}=\left(-l_1-x\right)\overrightarrow{i}+\left(l_2-y\right)\overrightarrow{j}-z\overrightarrow{k} \end{array}\right.$$

(7)

Substituting Equation (9) into Equation (8) yields an expression for the components of the magnetic field in the three directions of the rectangular circulation:

$$\begin{gathered} B_{Px}={\displaystyle \frac{{\mu }_{0}Iz}{4\pi \left[{\left(l_1-x\right)}^2+z^2\right]}}\left({\displaystyle \frac{l_2+y}{\sqrt{{\left(l_1-x\right)}^2+{\left(l_2+y\right)}^2+z^2}}}+{\displaystyle \frac{l_2-y}{\sqrt{{\left(l_1-x\right)}^2+{\left(l_2-y\right)}^2+z^2}}}\right) \\ -{\displaystyle \frac{{\mu }_{0}Iz}{4\pi \left[{\left(l_1+x\right)}^2+z^2\right]}}\left({\displaystyle \frac{l_2-y}{\sqrt{{\left(l_1+x\right)}^2+{\left(l_2-y\right)}^2+z^2}}}+{\displaystyle \frac{l_2+y}{\sqrt{{\left(l_1+x\right)}^2+{\left(l_2+y\right)}^2+z^2}}}\right) \end{gathered}$$

(8)

$$\begin{gathered} B_{Py}={\displaystyle \frac{{\mu }_{0}Iz}{4\pi \left[{\left(l_2-y\right)}^2+z^2\right]}}\left({\displaystyle \frac{l_1-x}{\sqrt{{\left(l_1-x\right)}^2+{\left(l_2-y\right)}^2+z^2}}}+{\displaystyle \frac{l_1+x}{\sqrt{{\left(l_1+x\right)}^2+{\left(l_2-y\right)}^2+z^2}}}\right) \\ -{\displaystyle \frac{{\mu }_{0}Iz}{4\pi \left[{\left(l_2+y\right)}^2+z^2\right]}}\left({\displaystyle \frac{l_1+x}{\sqrt{{\left(l_1+x\right)}^2+{\left(l_2+y\right)}^2+z^2}}}+{\displaystyle \frac{l_1-x}{\sqrt{{\left(l_1-x\right)}^2+{\left(l_1+y\right)}^2+z^2}}}\right) \end{gathered}$$

(9)

$$\begin{gathered} B_{Pz}={\displaystyle \frac{{\mu }_{0}I\left(l_2+y\right)}{4\pi \left[\left(l_2+y\right)+z^2\right]}}\left({\displaystyle \frac{l_1+x}{\sqrt{{\left(l_1+x\right)}^2+{\left(l_2+y\right)}^2+z^2}}}+{\displaystyle \frac{l_1-x}{\sqrt{{\left(l_1-x\right)}^2+{\left(l_2+y\right)}^2+z^2}}}\right)+{\displaystyle \frac{{\mu }_{0}I\left(l_2-y\right)}{4\pi \left[{\left(l_2-y\right)}^2+z^2\right]}}\left({\displaystyle \frac{l_1-x}{\sqrt{{\left(l_1-x\right)}^2+{\left(l_2-y\right)}^2+z^2}}}+{\displaystyle \frac{l_1+x}{\sqrt{{\left(l_1+x\right)}^2+{\left(l_2-y\right)}^2+z^2}}}\right)+ \\ {\displaystyle \frac{{\mu }_{0}I\left(l_1-x\right)}{4\pi \left[{\left(l_1-x\right)}^2+z^2\right]}}\left({\displaystyle \frac{l_2+y}{\sqrt{{\left(l_1-x\right)}^2+{\left(l_2+y\right)}^2+z^2}}}+{\displaystyle \frac{l_2-y}{\sqrt{{\left(l_1-x\right)}^2+{\left(l_2-y\right)}^2+z^2}}}\right)+ {\displaystyle \frac{{\mu }_{0}I\left(l_1+x\right)}{4\pi \left[{\left(l_1+x\right)}^2+z^2\right]}}\left({\displaystyle \frac{l_2-y}{\sqrt{{\left(l_1+x\right)}^2+{\left(l^2-y\right)}^2+z^2}}}+{\displaystyle \frac{l_2+y}{\sqrt{{\left(l_1+x\right)}^2+{\left(l_2+y\right)}^2+z^2}}}\right) \end{gathered}$$

(10)

From Equations (8)–(10), if the positions of the four vertices $A,B,C,D$ and the point t in space $P(x,y,z)$ are known, the magnitude of the magnetic field generated by the rectangular circulation at the point $B_{Px},B_{Py},B_{Pz}$ can be determined based on its current.

2.2. Simulation of the Rectangular Circulating Current Emission Model

To validate the rectangular circulating current magnetic field model, simulations were conducted to compare the simulated magnetic flux density with the calculated values from the proposed model, ensuring its accuracy. COMSOL software (version 6.1) is commonly used for simulating the magnetic field generated by the emission board. This software is a general-purpose simulation tool applicable across various engineering, manufacturing, and research fields. Its AC/DC Module is a powerful and flexible tool for analyzing static and low-frequency electromagnetic problems, providing extensive modeling capabilities and numerical methods to solve Maxwell’s equations for in-depth electromagnetic field and EMI/EMC studies. For the simulation of the electromagnetic tracking system, this module supports detailed 3D modeling of coils, allowing for precise calculations of current distribution within each wire. It also provides an equivalent approximation approach to simplify modeling by analyzing current distributions in complex coil structures, which is particularly effective for multi-turn coils.

The simulation process includes the following steps, as shown in Figure 3a:

(1)

Modeling the transmitting coil: A single transmitting coil is modeled with coordinates identical to those of the actual magnetic field emission board. A rectangular simulation domain is created to enclose the transmitting coil, with a total volume of (insert volume value here).

(2)

Assigning material properties: The transmitting coil is assigned copper as its material, while the surrounding simulation domain is set as air.

(3)

Steady-state simulation: The transmitting coil is excited with a constant current of 1A.

(4)

Magnetic field analysis: The simulation results provide the magnetic flux density at each point within the simulation domain.

The simulation results are exported, including the coordinates, 1200 simulation points, and their corresponding magnetic flux density values. By substituting these coordinates into the mathematical model of the rectangular circulating magnetic field, the computed magnetic flux density values are obtained.

The ratio between the computed magnetic flux density and the corresponding simulated magnetic flux density is calculated for each point. The distribution of these ratio values across all simulation points is shown in Figure 3b. A ratio closer to 1 indicates higher consistency between the simulation and the theoretical model. Figure 3 demonstrates that most ratio values fall within the range of 1 to 1.3, indicating that the simulated rectangular circulating magnetic field closely aligns with the theoretical model. This confirms that the simulation approach can be effectively used for electromagnetic tracking calculations.

2.3. Magnetic Field Sensor Model

In the Cartesian coordinate system, assuming that the coordinates of the sensor’s centroid is $\left(x,y,z\right)$, the coordinates of the center of mass $\left(x,y,z\right)$ need to be solved for in a lung puncture biopsy needle tracking system. The angle of its mid-axis and the z-axis is $\theta$, and the angle of its mid-axis and the x-axis is $\phi$, as shown in Figure 4. Thus, the unit vector in the direction of the mid-axis of the inductor coil is given in Equation (11).

$$\overrightarrow{n_A}=(sin\theta \times cos\phi ,sin\theta \times sin\phi ,cos\theta )$$

(11)

A time-varying magnetic field induces an induced electromotive force in the induction coil. Depending on the core material, the induced voltage can be increased by up to 500 times, as shown by Equation (12).

$$v\left(t\right)=-N{\displaystyle \frac{{\mu }_0{\mu }_{r}Ad\overrightarrow{B}·\overrightarrow{n_A}}{dt}}$$

(12)

where ${\mu }_r$ is the permeability of iron oxide magnetism and ${\mu }_0$ is the absolute permeability of the iron oxide core. $A$ refers to the cross-sectional area of a circular induction coil, and $\overrightarrow{n_A}$ is a unit vector along its central axis.

The induced voltage in the induction coil depends on its attitude parameter, while the position parameter is contained in $B_0\overrightarrow{n_A}$, and the attitude parameter is contained in $\overrightarrow{n_A}$. By determining the induced voltage in the induction coil, the attitude information of the induction coil can be solved.

2.4. Position Calculation Algorithm

To determine the pose parameters of the receiving coil, a mathematical equation is formulated that establishes the relationship between the induced voltage and the coil’s spatial position and orientation. Since this equation contains five unknowns, at least five independent equations are required to solve for the pose parameters. Each equation can be derived from the voltage induced by a distinct transmitting coil, meaning that a minimum of five transmitting coils is necessary to provide sufficient data for accurate localization.

However, relying solely on five transmitting coils poses limitations due to the rapid attenuation of the magnetic field with distance, resulting in a constrained operational range that may be insufficient for lung biopsy procedures. To address this, the system employs a total of eight transmitting coils, each operating at a distinct frequency. This expanded coil configuration enhances the spatial coverage and signal robustness while mitigating localization errors caused by field inhomogeneities. Nevertheless, increasing the number of coils introduces additional complexities in signal separation, hardware calibration, and pose computation, potentially affecting system latency and real-time performance. Thus, the selection of eight coils represents a balance between accuracy, computational efficiency, and practical feasibility in medical applications.

Considering these factors, this study employs a magnetic field emission board incorporating a total of eight transmitting coils, each operating at distinct frequencies. Four of these coils are positioned in a standard orientation, while the remaining four are rotated 90 degrees about the central axis. This strategic configuration not only optimizes the utilization of the available layout space but also enhances the uniformity and strength of the generated magnetic field within the designated working area, thereby improving the system’s overall localization accuracy and stability.

Since the sensor coil generates eight induced voltages caused by the transmitting coil currents when operating in an electromagnetic positioning system, the signal collected by the sensor is the sum of the eight induced voltages, as shown in Equation (13).

$$v\left(t\right)={\sum }_{i=1}^8{V}_i\mathrm{s}\mathrm{i}\mathrm{n}({\omega }_{i}t+{\phi }_{V_i})$$

(13)

where $V_i$ is the amplitude of each induced voltage component and ${\phi }_{V_i}$ is the phase of each induced voltage at the start of timing.

At a given moment, when calculating the pose of the receiving coil based on the induced voltage, the acquired voltage signal is the superposition of the induced voltages generated by eight transmitting coils on the receiving coil. However, the required induced voltage corresponds to each transmitting coil individually. Therefore, it is necessary to extract the induced voltage corresponding to each transmitting coil. A commonly used method is to apply the Fourier transform algorithm to convert the acquired signal from the time domain to the frequency domain.

In this study, the electromagnetic tracking system utilizes sinusoidal excitation signals, resulting in induced voltages that are also sinusoidal. Consequently, Fourier transform can be effectively employed to accurately obtain the frequency domain information of the sinusoidal components within the acquired voltage signal. By analyzing the frequency spectrum, the voltage amplitude corresponding to each specific frequency can then be determined.

Both the transmitting coil current and the sensing coil are demodulated using the Fourier transform, as described in the synchronous demodulation method described in the previous section. The amplitude and phase of the transmitter coil current and sensor coil-induced voltage can be obtained as follows:

$$\left\{\begin{array}{c}V=[V_1,V_2,\dots ,V_8]\\ I=[I_1,I_2,\dots ,I_8]\\ {\phi }_V=[{\phi }_{V_1},{\phi }_{V_2},\dots ,{\phi }_{V_8}]\\ {\phi }_I=[{\phi }_{I_1},{\phi }_{I_2},\dots ,{\phi }_{I_8}]\end{array}\right.$$

(14)

where ${\mathrm{V}}_i$ is the induced voltage of the 8 coils, ${\mathrm{I}}_{\mathrm{i}}$ is the induced current of the 8 coils to the sensor, ${\phi }_V$ is voltage phase angle, and ${\phi }_I$ is current phase angle.

2.5. System Calibration Algorithm

The magnetic induction strength of the space magnetic field is positively correlated with the size of the excitation current of the electromagnetic transmitter plate and the size of the magnetic permeability in the vacuum. Due to an error in the hardware, in the process of practical application, the size of the signal current is not perfect with the theoretical value; the magnetic permeability of the air and the magnetic permeability of the vacuum are not identical, and there exists a certain proportionality relationship; at the same time, the error in the other hardware in the circuit is also transferred to the bit gesture solving process. At 70 kHz, the effect of eddy currents in biological tissues is negligible due to their relatively high resistivity compared to metal conductors. The magnitude of eddy currents depends on the material’s conductivity, permeability, and the frequency of the applied field. While metals with high conductivity generate significant eddy current losses, biological tissues, such as muscle and fat, have conductivities ranging from 0.01 to 1 S/m, which are several orders of magnitude lower. At 70 kHz, the skin depth in biological tissues is several centimeters, allowing for the magnetic field to penetrate effectively with minimal attenuation or distortion.

As the frequency increases, factors such as eddy currents and other losses become more significant. On the other hand, lower frequencies are susceptible to environmental interference and result in longer sampling times, which can compromise system efficiency. Therefore, the selection of 70 kHz optimizes the trade-off between minimizing eddy current effects and ensuring a reasonable sampling time, thus improving the overall computational efficiency of the electromagnetic tracking system.

At the same time, other hardware errors in the circuit will also be transferred to the bit pose solution process, and these errors are uncontrollable and cannot be accurately obtained through analysis. To solve this problem, it is necessary to calibrate the system before the operation to obtain the calibration coefficients $k_i$ for each transmitting coil of the electromagnetic transmitting board.

The calibration process needs to be carried out individually for each transmitting coil by setting up some test points at known locations, using the nonlinear least squares method. It assumes that the measured value of the magnetic flux of the $i$ transmitting coil at the $j$ test point in the electromagnetic transmitting plate is $B_{meas}^i$, and at the same time, the theoretical flux of the $i$ transmitting coil at the same point, which is derived according to the magnetic field model, is $B_{calc}^i$. Then, the solution for the objective function to minimize the error between the two can be expressed as

$$F_i\left(k_i\right)={\sum }_{j=i}^m{\left(V_{meas}^i\left(x_j,y_j,z_j,{\theta }_j,{\phi }_j\right)-k_i{V}_{calc}^i\left(x_j,y_j,z_j,{\theta }_j,{\phi }_j\right)\right)}^2$$

(15)

where $k_i$ is the calibration coefficient of the $i$ coil, $F_i\left(k_i\right)$ is the sum of errors corresponding to the $i$ coil, $meas$ is the number of test points, and $\left(x_j,y_j,z_j,{\theta }_j,{\phi }_j\right)$ is the position parameter of the inductive coil at the point, which is minimized by iteratively iterating and correcting the value of $k_i$ to make $F_i\left(k_i\right)$ reach the minimum value.

Since the sensor cannot be completely fixed along the axial direction, there are a few errors in the z-axis direction. To reduce this error, it is necessary to introduce the z-axis compensation constant $r_i^{offset}$, and thus, the error objective function can be written as follows:

$$F_i\left(k_i,r_i^{offset}\right)={{\sum }_{j=i}^m\left[V_{meas}^i\left(r_j-r_i^{offset},{\theta }_j,{\phi }_j\right)-k_i{V}_{calc}^i\left(r_j-r_i^{offset},{\theta }_j,{\phi }_j\right)\right]}^2$$

(16)

To solve the electromagnetic positioning equations derived from the induced voltages, a standard least squares (LS) method is employed. This approach involves matrix inversion, with a computational complexity of $O\left(N^3\right)$, where n is the number of variables in the system. While this complexity is acceptable for offline computation, it can become a limiting factor in real-time applications. The convergence of the LS solution is generally stable when the system is well conditioned and the initial estimates are reasonably close to the true values. However, poor initial conditions may lead to suboptimal convergence or sensitivity to measurement noise. In practice, this issue is mitigated through a pre-calibration process and by using a physically constrained initial guess based on prior sensor positions. To further enhance the computational efficiency and robustness in future iterations of this system, we propose adopting recursive least squares (RLS) methods, which can incrementally update the solution with significantly reduced computational overhead, making them more suitable for real-time tracking scenarios.

The least squares method is employed to obtain the true displacement by finding the optimal point of an approximate (quadratic) function of the objective function, within a region centered at the initial point with a given radius. The specific procedure of the five-degree-of-freedom (DOF) solution algorithm is as follows:

(1)

The voltage information corresponding to the eight transmitting coils is input as the parameters of the objective function model, and the parameters for the least squares function are set.

(2)

The least squares algorithm is used for iterative computation of the five DOF of the sensing coil corresponding to the eight voltage signals. In each iteration, the electromagnetic tracking coupling equations are first used to compute the eight sensing voltages at the current iteration point. Then, the difference between these calculated voltages and the input voltage parameters is taken as the residual. Using the electromagnetic tracking coupling equation as the model, the algorithm is applied to compute the next iteration point until the residual is smaller than the iteration accuracy.

(3)

Once the algorithm iteration is completed, the pose of the sensing coil from the final iteration is provided, which corresponds to the result of the five DOF solution algorithm.

Due to the ability to compute the position of the next point based on the previous point, the algorithm iterates quickly by employing the least squares method. The objective function is then solved to determine the displacement of the target function. After obtaining the displacement, the value of the objective function is calculated. If the decrease in the objective function satisfies a given condition, the displacement is considered reliable, and iterative calculations continue according to this rule. If the decrease in the objective function does not satisfy the given condition, the range of the domain should be reduced, and the problem should be resolved accordingly.

2.6. Positioning of System Measurements

In obtaining the coil calibration coefficients $k_i$ and z-axis compensation constants $r_i^{offset}$ for electromagnetic positioning, the coils corresponding to different voltages $V=[V_1,V_2,\dots ,V_8]$ obtained from sampling can be obtained by the signal acquisition card, and can be used according to the $F_i\left(k_i\right)={\sum }_{j=i}^m{\left(V_{meas}^i\left(x_j,y_j,z_j,{\theta }_j,{\phi }_j\right)-k_i{V}_{calc}^i\left(x_j,y_j,z_j,{\theta }_j,{\phi }_j\right)\right)}^2$; in order to minimize $F_i\left(k_i\right)$, the least squares algorithm is used to calculate $\left(x_j,y_j,z_j,{\theta }_j,{\phi }_j\right)$.

3. Results

3.1. System Building

After establishing the rectangular circulating magnetic field model and deriving the electromagnetic tracking coupling equations, the design of the transmitting coils can be carried out according to the requirements of lung puncture procedures. Based on standard human body dimensions, the 50th percentile chest width is 280 mm, and the 99th percentile is 331 mm, while the 50th percentile chest depth is 212 mm, and the 99th percentile is 216 mm. Since the abdominal volume is comparable to the thoracic volume, the working area of the transmitting coils is set to 300 mm × 300 mm × 300 mm directly above the transmitting coils, ensuring complete coverage of the patient’s abdomen.

Traditional magnetic field generation devices typically use wound coils, where enameled copper wires are tightly wound around an ABS plastic frame using a winding machine. The coil ends are connected to a signal generation and amplification system. The entire magnetic field generation device is box-shaped and suspended above the surgical area. To avoid obstructing the surgeon, a certain distance must be maintained between the device and the patient, which prevents the full utilization of the device’s working area. Since the generated magnetic flux density is inversely proportional to the square of the distance, as the receiving coil moves away from the working area, the induced voltage drops rapidly, making signal acquisition and computation difficult. To expand the effective working area, a stronger magnetic field is required, which increases hardware complexity and significantly raises costs.

To address these limitations, PCB coils are used as transmitting coils in the design of the magnetic field emission board. Compared to wound coils, PCB coils offer advantages such as simpler fabrication, lower cost, and higher precision. With a line width of 0.5 mm, PCB manufacturing can achieve a precision of 0.02 mm, and PCB coils are significantly thinner, with a thickness of only 1.6 mm. This allows for the magnetic field emission board to be placed directly beneath the patient, in close contact with the body, thereby fully utilizing the working area while reducing hardware requirements for the transmitting coils.

We selected planar coils on a single PCB for manufacturability, repeatability, and integration; chose 25 turns to balance signal strength, power consumption, and manufacturability; optimized resistance, inductance, and current for efficiency and stability; and used a single-layer PCB to simplify fabrication, reduce cost, and maintain mechanical stability.

The PCB-based magnetic field emission board is designed as shown in Figure 4. The board has an overall size of 300 mm × 300 mm with a thickness of 1.6 mm, and the generated magnetic field effectively covers a volume of at least 300 mm × 300 mm × 300 mm. During lung biopsy procedures, the board can be placed underneath the patient without occupying additional space on the surgical table or interfering with the surgeon’s workflow. The outer coils are aligned parallel to the X-axis, while the middle coils are rotated 45 degrees around the Z-axis.

PCB coils are employed as transmitting coils in the design and fabrication of electromagnetic transmitting boards. These coils offer advantages such as simpler fabrication and lower cost compared to traditional single-layer coils wound with enameled wire. The design parameters for these coils are detailed in

Table 1. PCB rectangular coil design parameters.

Parameter	Value
Initial length	70 mm
Turns	25 turns
Line width	0.5 mm
Line distance	0.25 mm
PCB Thickness	1.6 mm

.

Each coil operates with a distinct frequency AC excitation current, with the frequency range controlled between 68 kHz and 72 kHz. Additionally, to prevent signal degradation due to the filtering process, the signal frequency must be slightly higher than the filter cutoff frequency (by approximately 500 Hz). To mitigate the potential interference from power line noise (50/60 Hz), an analog high-pass filter is implemented using operational amplifiers (op-amps). The filter is designed with a cutoff frequency of 500 Hz to effectively remove unwanted high-frequency noise while maintaining the integrity of the signal of interest. The filter allows for immediate noise reduction at the signal acquisition stage, ensuring that the electromagnetic tracking system remains responsive and accurate. The operational amplifier used in the filter is a TL081, chosen for its low noise and fast response characteristics.

The receiving coil in the electromagnetic tracking system utilizes a ferrite core, which offers several advantages, including a high magnetic permeability (~500), low eddy current loss, and favorable magnetic saturation characteristics with a saturation flux density of approximately 0.3–0.5 T. Compared to metal cores, ferrite cores have high resistivity, which effectively suppresses eddy current losses at 70 kHz, making them more suitable for electromagnetic tracking systems. The coil structure adopts a helical winding design, using high-purity enameled copper wire as the winding material. The winding is arranged in a single winding configuration to optimize distributed capacitance and reduce high-frequency losses. To enhance the signal strength, a ferrite core is used in the tracking system sensor. The design parameters for the sensor are given in

Table 2. Sensor coil design parameters.

Parameter Name	Value
Diameter	7.2 mm
Length	25 mm
Turns	100 turns
Wire diameter	0.3 mm
Diameter	25 mm
Permeability	10 K
Magnetic permeability	435 μH
Resistance	0.79 Ω

, and the design results are shown in Figure 5.

3.2. System Calibration Error Analysis and Optimization

The system requires calibration before operation, and several calibration position schemes are set up, including 50 mm 4 × 4, 40 mm 5 × 5, 30 mm 6 × 6, and 30 mm 7 × 7 calibration grids. These calibration positions are used to solve the system and calculate the modification coefficients, as shown in

Table 3. Calculation of system parameters under different calibrations.

Spatial Parameter	Correction Factor	Coil 1	Coil 2	Coil 3	Coil 4	Coil 5	Coil 6	Coil 7	Coil 8
4 × 4	$r_i^{offset}$	0.0172	0.0231	0.0166	0.0184	0.016	0.0141	0.0173	0.0155
4 × 4	$k_i$	17.923	20.282	20.102	23.309	23.952	24.711	29.677	31.952
5 × 5	$r_i^{offset}$	0.0191	0.0148	0.0187	0.0182	0.0155	0.0135	0.0157	0.0165
5 × 5	$k_i$	19.55	17.622	20.952	23.144	24.661	25.268	29.163	32.976
6 × 6	$r_i^{offset}$	0.018	0.0172	0.0203	0.0088	0.0136	0.013	0.0183	0.0166
6 × 6	$k_i$	18.168	17.150	21.846	16.850	22.429	22.532	29.284	32.456
7 × 7	$r_i^{offset}$	0.0113	0.0172	0.0184	0.0175	0.0145	0.0151	0.0161	0.0144
7 × 7	$k_i$	18.172	17.480	20.899	22.876	22.494	24.261	28.674	29.303

. 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 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.

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. Among the different calibration position schemes, the 30 mm calibration scheme results in the lowest calibration error. Using this scheme, the system’s calibration parameters are obtained in

Table 4. Static sample variance values for different locations.

Location	(0, 0, 80)	(40, −40, 80)	(−40, 40, 80)	(40, 40, 130)	(80, −80, 80)	(−80, 80, 80)
$x(m{m}^2$)	4.27	1.58	0.94	4.11	2.32	7.84
$y(m{m}^2$)	4.72	0.11	0.87	0.36	2.90	5.11
$z(m{m}^2$)	3.19	6.81	1.65	3.48	3.56	5.51

.

3.3. System Positioning Stability Analysis

The static stability of the positioning system is crucial for the overall measurement accuracy. To assess the system’s stability, the electromagnetic sensor coil was fixed at specific positions, and its jitter error relative to the system was measured. The sensor was fixed at the following coordinates: (0, 0, 80), (40, −40, 80), (−40, 40, 80), (40, 40, 130), (80, −80, 80), and (−80, −80, 80).

For precise calculations, the sample variance is used to evaluate the static stability of the system. A higher sample variance indicates greater fluctuation in the sample data. The sample variance $X_s$ is employed to assess the continuous tracking stability of the induction coil at each position.

The sample variance values at each of the six points are recorded separately. As shown in Table 4, the static error is generally within the range of 3–4 mm², which meets the system’s stability requirements.

3.4. System Accuracy Analysis

To perform an overall precision analysis of the system, the sensor probes were placed within the system to measure the error. The measurements were conducted on the horizontal plane by moving the probes 50 mm to different positions, and the position of the sensors was averaged over three consecutive measurements at 1 mm intervals. To ensure consistent placement of the sensing coil, the operator underwent multiple training sessions before the experiment. During each test, the predetermined position of the induction coil was recorded first, followed by the recording of the system’s output position.

3.4.1. Deviation Between True and Measured Values of Horizontal Movement

The electromagnetic sensor probe was placed at the origin of the electromagnetic tracking system and subjected to 1 mm movements in the front–back and left–right directions. The displacement values were then collected and calculated by the electromagnetic tracking system, as shown in Figure 7. As observed in the figure, the error between the measured and theoretical values remains within 2 mm, indicating that the performance is influenced by random error.

3.4.2. Deviation Between the Real and Measured Values at Positive and Negative Half-Axis of y

To measure the error across the measured area, the electromagnetic tracking system probe is positioned along the positive and negative half-axes of the y-direction and moved along the x-direction. The test results are shown in Figure 8. From the figure, it can be observed that the error between the measured and theoretical values remains within 2 mm, indicating random error in the performance. Additionally, compared to the central area, the error at the edges is slightly larger.

3.4.3. Measurement Deviation When Moving up and Down

To measure the sensor in space up and down to move the measurement accuracy, the electromagnetic sensor probe was placed in the electromagnetic tracking system at the origin, for the up and down movement, respectively, to measure its z-axis position, and compared with the theoretical value and the theoretical deviation, and the actual measured value is shown in Figure 9. In the figure, it can be seen that the z-axis direction, the theoretical value, and the real basic overlap.

3.4.4. Fixed Angle and Real Deviation

To measure the accuracy when the sensor is moved at specific angles, the electromagnetic sensor probe is placed at the origin of the electromagnetic tracking system and tilted at angles of 30°, 45°, and 60° for measurement. The probe then undergoes front, back, left, and right movements, respectively. The theoretical deviation and the actual measured values are shown in Figure 10. As seen in the figure, compared to the vertically placed sensor, the measurement accuracy remains within 2 mm, and the accuracy is not significantly affected by the angular changes.

Root Mean Square Error (RMSE) is used to evaluate the average accuracy of both position and orientation. A smaller RMSE indicates higher accuracy. By combining all the above measurements, the RMSE values of the electromagnetic (EM) tracking system can be calculated, as shown in

Table 5. RMSE values for different movement directions and positions.

Orientation	Shift Left	Right Shift	Upward Shift	Move Down	Left Edge Move	Right Edge Move	Up	Down	30 Degrees	60 Degrees	45 Degrees
$x$ (mm)	1.61	1.42	0.57	1.17	2.12	1.05	/	/	1.01	0.89	0.77
$y$ (mm)	1.52	1.25	1.29	1.34	1.46	1.34	/	/	0.36	0.53	0.69
$z$ (mm)	2.22	1.89	1.41	1.78	2.58	1.71	2.46	1.26	1.07	1.04	1.02

. It can be observed that the z-axis exhibits the lowest accuracy compared to the x- and y-axes. This is primarily because the z-axis is farther from the measured electromagnetic field area, while the x- and y-axes remain within the core region. Additionally, when the sensor is offset by a certain angle, the displacement accuracy improves. This is mainly due to the smaller rate of change in the magnetic field caused by the displacement, which allows for the least squares method to more effectively solve for the specific position.

3.5. System Test and Response

A robotic arm was used to test the system’s stability and performance. The experiment included position variation tests and angle variation tests, aimed at evaluating the system’s positioning accuracy under changes in both position and angle. The robotic arm was employed for control, with the arm’s pose used as the true value of the simulated biopsy needle’s pose. The deviation between the true value and the system’s calculated value was then computed to assess the positioning accuracy of the inertial-assisted electromagnetic positioning system.

In the experiment, the robotic arm controlled the movement of the simulated biopsy needle. The robotic arm used was the FR-3 model, shown in Figure 11, and it was fixed on a metal frame to ensure the base of the arm remained stationary. The key parameters of the robotic arm are listed in

Table 6. Parameters of the robotic arm.

Parm Name	Value
Effective payload	3 kg
Work radius	$622 \mathrm{m}\mathrm{m}$
Repeatability accuracy	$\pm 0.02 \mathrm{m}\mathrm{m}$

.

As shown in the

Table 7. Comparison of RMSE with the other literature.

	This Article	Wireless 5-d [14]	5-d Large Work Area [15]	Residual Network-Based Electromagnetic [16]
Average error (mm)	1.75	1.8633	2.02 mm	1.25
Maximum error (mm)	3.13	2.3	2.56	2.8

, the robotic arm’s effective payload is capable of bearing the weight of the simulated biopsy needle. Its working radius covers the operational area of the electromagnetic positioning system, and its repeatable positioning accuracy significantly outperforms that of electromagnetic positioning products available on the market. Therefore, this robotic arm can be used to test the positioning accuracy of the inertial-assisted electromagnetic positioning system.

The initial position of the simulated biopsy needle was determined, and the specific location of the starting point was obtained using the robotic arm’s position information. Induced voltage and angle signals were collected, and the position of the simulated biopsy needle was calculated. The electromagnetic positioning system was used for the positioning of the simulated biopsy needle, and the position error was calculated. The position of the simulated biopsy needle was changed, and the experiment was repeated 2000 times to obtain the computed data and the actual true positions. The robotic arm was used to control the simulated biopsy needle’s movement within the working area, with the arm’s position information serving as the true position value of the simulated biopsy needle. After 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).

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. Additionally, the magnetic field model we develop, with an average error of less than 1.75 mm, surpasses the tracking accuracy demonstrated in the other studies. Furthermore, the use of a PCB as the carrier for the magnetic field emitter allows for a compact, cost-effective, and scalable solution, while the 70 kHz operating frequency minimizes eddy current interference in biological tissues, enhancing system performance. In summary, our system’s improved accuracy, computational efficiency, and reduced system complexity make it a strong candidate for medical applications such as biopsy needle tracking.

In addition, to comprehensively validate the instrument’s response performance under different motion modes, we conducted a continuous positioning motion experiment, which included both square trajectory motion and a random motion trajectory. The square trajectory motion simulated the continuous operation of the positioning system along a standard path, while the random motion trajectory tested the system’s dynamic response capability on nonlinear and irregular paths. The entire experimental process was recorded using high-frame-rate video (https://zenodo.org/records/14636130 (accessed on 13, January 2025)), ensuring the precise capture of every movement and response change of the positioning system. The video screenshot is shown in Figure 13b. This experiment provided comprehensive data support for the practical application of the instrument, confirming the system’s reliability and response capability under various operating conditions.

3.6. Equivalent Material Simulation of Biological Tissues

In this study, equivalent materials were used to simulate biological tissues instead of conducting real biological experiments. This approach was chosen due to the complexity of real experiments, ethical considerations, and the challenges associated with in vivo testing. The selected materials simulate muscle (conductivity: 0.4 S/m; relative permittivity: 260; permeability: 1) and bone (conductivity: 0.05 S/m; relative permittivity: 15; permeability: 1), as their electromagnetic properties closely resemble those of human tissues, making them suitable for experimental validation. These equivalent materials effectively replicate the electromagnetic characteristics of biological tissues and can be applied in the validation of electromagnetic tracking systems.

In the electromagnetic tracking system, calibration was performed based on the procedures outlined in previous sections, and the system was established accordingly, as shown in Figure 14. The equivalent muscle and bone materials were then introduced into the system, and random positioning experiments were conducted to obtain absolute positioning error data, as illustrated in Figure 14c. The results indicate that the inclusion of human-equivalent tissues and organs in the tracking system does not affect the overall positioning performance and accuracy. Experimental results confirm that skin-depth effects do not significantly weaken the induced signal, ensuring accurate localization. The field strength remains within acceptable limits, and any signal loss in biological tissues does not impact the system’s overall performance.

Since the magnetic permeability of biological tissues is close to that of a vacuum ($\mathsf{\mu }\approx {\mathsf{\mu }}_0$), the magnetic field can penetrate the human body with minimal loss, particularly in the case of low- and mid-frequency electromagnetic fields, without inducing significant eddy current losses as seen in metal shielding. While this approach provides a controlled and repeatable testing environment, we acknowledge that the heterogeneity in real biological tissues—such as variations in conductivity across different organs—may introduce discrepancies between experimental results and real-world conditions. In future research, we plan to conduct in vivo experiments to further validate the applicability of this system in real medical scenarios.

4. Discussion

The diameter of the magnetic core used for the sensor probe in Figure 6d is 7.2 mm, which is relatively large compared to those typically used in classical electromagnetic tracking systems, resulting in decreased accuracy. The least squares method employed in this study is used to solve for the position. However, due to a pole problem during the solving process, the system may incorrectly identify the pole as the minimum value point, resulting in calculation errors. To address this, further modifications to the least squares method are proposed to minimize the error caused by this issue.

In this study, several potential sources of error could impact the accuracy and performance of the electromagnetic tracking system. Environmental interference, such as external electromagnetic fields from nearby electronic devices or power lines, could distort the system’s magnetic field and lead to incorrect position estimates. Additionally, sensor noise, originating from either the internal electronics of the sensors or external factors, can degrade the signal-to-noise ratio, reducing the precision of measurements. Calibration inaccuracies also represent a significant potential error source; misalignments or errors during the calibration process, such as incorrect alignment of the transmitter coils and sensors, can introduce considerable discrepancies between theoretical models and real-world measurements. Sensor positioning and alignment issues, even small deviations, can further affect the system’s accuracy. Furthermore, variability in the electrical conductivity and magnetic permeability of biological tissues may influence the magnetic field distribution, especially given that tissue properties can differ depending on composition or temperature. These factors must be carefully considered, and future research should focus on enhancing calibration methods, minimizing sensor noise, and improving system sensitivity to environmental variables in order to optimize tracking accuracy in practical applications.

5. Conclusions

This study introduces a novel electromagnetic localization system for lung biopsy needle tracking, aimed at improving real-time procedural guidance and localization accuracy. The system utilizes a rectangular toroidal current distribution, with a printed circuit board (PCB) serving as the magnetic field emitter. An alternating electromagnetic field at approximately 70 kHz was modeled using the Biot–Savart law. Induced voltages from multiple transmitting coils were separated through Fourier transform algorithms, enabling independent signal extraction. A five-degree-of-freedom positioning model was solved using the least squares method, ensuring both computational efficiency and spatial precision. Experimental validation demonstrated that the proposed system achieved an average localization error of less than 1.76 mm, effectively addressing core challenges in biopsy needle tracking. The results highlight the system’s potential to enhance accuracy while reducing dependency on imaging techniques such as CT, thereby contributing to safer and more efficient clinical procedures. Future work will focus on enhancing calibration strategies, minimizing susceptibility to environmental interference, and evaluating the system’s performance in biological tissues to support its translation into clinical settings.