An Augmented Reality-Based Navigation System for Stereotactic Brain Biopsy with Multi-Objective Path Planning and Hybrid Registration

1. Introduction

2. Materials and Methods

2.1. Overview of the Proposed System

The overall workflow of our AR-based navigation system for stereotactic brain biopsy is illustrated in Figure 1. First, MRI data are acquired from the patient. Skin, tumor parenchyma, and high-risk structures are segmented using automated and semi-automated methods. Secondly, given the objective of path planning is to identify the path with the lowest risk, a multi-objective optimization problem is established based on clinical criteria. A constrained MOO model is formulated to solve for the optimal path. Finally, a hybrid registration method based on optical tracking is developed. Using the HoloLens 2 (Microsoft Corp., Redmond, WA, USA), the patient’s 3D anatomical structure model and the planned path are displayed in real time at the surgical site.

Figure 1. Workflow of the AR navigation system for stereotactic brain biopsy.

2.3. Multi-Objective Path Optimization

The primary objective of the brain biopsy path-planning problem is to find the path with the least risk. In clinical surgeries, the total length of the biopsy trajectory should be minimized to reduce the incidence of complications [15,16]. To ensure surgical feasibility, the angle between the biopsy needle and the skin incision plane should be as close to 90° as possible, while angles less than 20° may cause the needle to slip or bend [18]. Additionally, it is required that the biopsy path avoid the critical anatomical structures [19]. Based on the above clinical path planning principles, the brain biopsy path-planning problem is formulated as a multi-objective optimization model. Each path planning requirement is quantified as an objective subfunction or a constraint. Accordingly, the specific optimization objectives and constraints are defined as follows:

1.

Path Length 𝑃⁢𝐿:

Path length is defined as the distance from the skin entry point to the target point on the tumor. A maximum path length of 90 mm is imposed as a constraint [15,16]. The calculation and constraints for path length are shown in Equation (1):

{𝑃⁢𝐿⁡(𝑥)=√(𝑥𝑇−𝑥𝐸)2+(𝑦𝑇−𝑦𝐸)2+(𝑧𝑇−𝑧𝐸)2𝑃⁢𝐿⁡(𝑥)≤90⁢m⁢m,

(1)

where (𝑥𝐸,𝑦𝐸, 𝑧𝐸) denotes the world coordinate of the skin entry point, and (𝑥𝑇,𝑦𝑇, 𝑧𝑇) denotes the world coordinate of the target point.

2.

Entry Angle 𝐸⁢𝐴:

The entry angle is defined as the supplementary angle between the normal vector of the skin entry point’s skin plane and the path direction vector. The minimum entry angle must not be less than 20°.The specific calculation and constraints for entry angle are shown in Equation (2):

⎧{ {⎨{ {⎩𝐸⁢𝐴⁡(𝑥)=∣90−𝑎⁢𝑟⁢𝑐⁢𝑐⁢𝑜⁢𝑠⁢(𝑃𝐸−𝑃𝑇)⋅𝑛𝐸‖𝑃𝐸−𝑃𝑇‖⋅180𝜋 ∣𝐸⁢𝐴⁡(𝑥)≥20°,

(2)

where 𝑃𝐸 and 𝑃𝑇 respectively denote the skin entry point and the target point, and 𝑛𝐸 is the normal vector of the skin cutting plane at the skin entry point.

3.

Risk Cost 𝑅⁢𝐶:

To quantify the risk cost of trajectories, we construct a risk map across all brain voxels. A neurosurgeon assigns risk coefficients to each risk structure, representing the relative incidence of biopsy instruments crossing these areas [20]. Specifically, blood vessels, the M1, and the ventricles are assigned coefficients of 95%, 70%, and 45%, respectively. The risk value of any voxel is determined by its Euclidean distance from nearest-neighbor voxels within these risk structures, as defined in Equation (3):

𝑟⁢𝑖⁢𝑠⁢𝑘⁡(𝑣)=∑𝑆∈𝕊∑𝑁𝑖=1𝑟𝑆⋅⁢𝑒⁢𝑥⁢𝑝⁡(−𝑑22⁢𝜎2),

(3)

where 𝕊 is the set of all risk structures 𝑆, 𝑁 is the number of nearest-neighbor voxels selected from each risk structure, 𝑑𝑆,𝑖 denotes the physical distance between voxel 𝑣 and its 𝑖 nearest neighbor in risk structure S, 𝑟𝑆 denotes the risk coefficient assigned to structure 𝑆, and 𝜎 is the distance decay constant. In this work, 𝑁 was set to 5 and 𝜎 to 10 mm.

The biopsy trajectory is a set of contiguous voxels that begins at an entry point and ends at a target point. Subsequently, the risk cost of the trajectory is defined as the sum of the risk values of all voxels constituting that path, as shown in Equation (4):

𝑅⁢𝐶⁡(𝑥)=∑𝑣𝑖∈𝑈𝑥𝑟⁢𝑖⁢𝑠⁢𝑘⁡(𝑣𝑖),

(4)

where 𝑈𝑥 denotes the set of voxels comprising the path 𝑥, and 𝑟⁢𝑖⁢𝑠⁢𝑘⁡(𝑣𝑖) denotes the risk value of the voxel 𝑣𝑖.

4.

Restricted Areas 𝑅⁢𝐴:

We consider the following clinically restricted areas: the midline of the brain, which contains the median sagittal sinus vessels and is filled with cerebrospinal fluid [21]; blood vessels, to avoid intracranial hemorrhage [22]; ventricles, to preclude cerebrospinal fluid leakage that could lead to brain shift [23]; M1, to preserve postoperative motor function [24]; and the facial and ear areas, to minimize postoperative cosmetic impact [25]. The restricted puncture areas are defined as equation constraints for path optimization, as shown in Equation (5):

𝑈𝑥∩𝑈𝑅⁢𝐴=𝜙,

(5)

where 𝑈𝑥 denotes the set of voxels comprising the path 𝑥, and 𝑈𝑅⁢𝐴 denotes the set of voxels comprising the restricted area 𝑅⁢𝐴.

With the objective functions and constraints defined above, the MOO model for surgical path planning is established as shown in Equation (6):

𝑀⁢𝑖⁢𝑛⁢𝑖⁢𝑚⁢𝑖⁢𝑧⁢𝑒⁢𝐹⁡(𝑥)=[(𝑓𝑃⁢𝐿⁡(𝑥),1−𝑓𝐸⁢𝐴⁢(𝑥),𝑓𝑅⁢𝐶⁡(𝑥))]𝑇,  𝑥∈𝐷𝐼⁢𝐶𝑖⁡(𝑥)≤0,𝑖=1,2,…,𝑛,𝐸⁢𝐶𝑘⁡(𝑥)=0,𝑘=1,2,…,𝑚,,

(6)

where the objective function 𝐹⁡(𝑥) is defined to identify the puncture trajectory with the lowest risk. The decision variable 𝑥 is a biopsy trajectory. The search space 𝐷 is defined as the set of one-to-one mappings between candidate skin entry points and candidate target points. Inequality constraints 𝐼⁢𝐶𝑖⁡(𝑥) are imposed on path length and entry angle, while equality constraints 𝐸⁢𝐶𝑘⁡(𝑥) are used to restrict traversal through specific regions.

The optimal path is determined by minimizing the objective function 𝐹⁡(𝑥) subject to the inequality and equality constraints. However, MOO problem faces the challenge of conflicting sub-objectives, making it difficult to achieve optimal outcomes across all sub-objectives simultaneously. To circumvent this inherent issue, we employ a vector-based method that utilizes the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [26]. This algorithm treats the value of each sub-objective function as a feature of a single solution and optimizes them concurrently.

The flowchart of the MOO algorithm is illustrated in Figure 2. Given the MRI-based segmentations of critical anatomical structures and the risk map, the algorithm generates Pareto-optimal entry and target point pairs for biopsy path planning.

Figure 2. Flowchart of the MOO algorithm.

First, preprocessing is performed on the tumor and skin masks. According to expert consensus, a sufficient volume of lesion tissue is required in stereotactic biopsy to ensure reliable histopathological diagnosis. Specimens smaller than approximately 1 mm3 often result in diagnostic uncertainty [27]. Therefore, to ensure sufficient pathological tissue acquisition, we extracted voxels with a boundary thickness greater than 2 mm from the original tumor mask as candidate target points. To avoid generating paths through facial and auricular restricted areas, voxels outside the facial and auricular areas on the skin mask are extracted to form the candidate skin entry point set.

The initial population for NSGA-II is generated by randomly pairing candidate skin entry points with candidate target points. Any trajectory violating hard constraints is pre-filtered before being incorporated into the population. Each individual in the population represents a biopsy trajectory from a skin entry point to a target point, encoded as a 6-dimensional vector containing the coordinates of the entry and target voxels. Each individual corresponds to three sub-objective function values. Individuals are selected using non-dominated sorting and crowding distance to retain those with higher fitness. These high-fitness individuals, along with new individuals generated through crossover and mutation, form a new population to continue iterative evolution. In this work, the crossover process is performed by exchanging the skin entry points or target points between two parent individuals. Mutation updates the entry or target points of the parent individuals using a feasible neighbor identified by a K-nearest-neighbor search within the segmentation mask. After the final generation, the Pareto-optimal solution set is extracted and used for subsequent decision-making. The quality of these solutions is then evaluated based on an expert weight matrix and visualized through color mapping as shown in Figure 3.

Figure 3. Pareto front with color-mapped quality assessments.

Finally, the optimal paths are selected from the Pareto front using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method, incorporating expert weights. The expert weight vector 𝑤 =[𝑤1,𝑤2,…,𝑤𝑛]𝑇 is derived from the expert priority matrix. The raw decision matrix is normalized and weighted to construct the weighted normalized matrix, as shown in Equation (7):

[𝑣𝑖⁢𝑗]𝑚×𝑛=⎡⎢ ⎢ ⎢⎣𝑤𝑗⁢𝑥𝑖⁢𝑗√∑𝑚𝑖=1𝑥2𝑖⁢𝑗⎤⎥ ⎥ ⎥⎦𝑚×𝑛,∀𝑖∈{1,…,𝑚},𝑗∈{1,…,𝑛},

(7)

where 𝑥𝑖⁢𝑗 represents the 𝑗 −𝑡⁢h objective value of the 𝑖 −𝑡⁢h candidate path in the Pareto set, and 𝑚 and 𝑛 denote the number of candidate solutions and evaluation criteria, respectively.

Based on the decision matrix, the Euclidean distances of each candidate path to the positive and negative ideal solutions were subsequently computed. The comprehensive performance score 𝐶*𝑖 for each candidate path is defined by a unified expression of relative closeness to the ideal solution, as shown in Equation (8):

𝐶*𝑖=√∑𝑛𝑗=1(𝑣𝑖⁢𝑗−𝑣−𝑗)2√∑𝑛𝑗=1(𝑣𝑖⁢𝑗−𝑣+𝑗)2+√∑𝑛𝑗=1(𝑣𝑖⁢𝑗−𝑣−𝑗)2,

(8)

where 𝑣+𝑗 and 𝑣−𝑗 denote the weighted normalized values of the positive-ideal and negative-ideal solutions, respectively.

The candidate path with the highest value 𝐶*𝑖 was ultimately selected as the optimal trajectory for surgical planning.

2.4. Hybrid Registration

3D registration is the foundation, key, and difficulty of building an AR system [28]. To achieve accurate alignment between the virtual 3D anatomical model and the patient, we propose a hybrid registration framework that combines quick response (QR) codes with anatomical landmarks. The coordinate transformation for the hybrid registration is illustrated in Figure 4. The HoloLens 2 uses a spatial world anchor to define the Holenlens2 coordinate system {H}. After anatomy reconstruction and preoperative planning, all the medical models and path data are registered to {H}. We define the patient coordinate system {P} within the anatomical space of the patient’s head, while {A} and {B} denote the local coordinate systems associated with Marker A and Marker B, respectively.

Figure 4. Coordinate systems and transformation matrices in the hybrid registration.

Vuforia SDK is a development toolkit for detecting and tracking image targets with high accuracy, supporting the development of AR applications within the Unity platform [29]. The system uses two marker tools developed by Vuforia SDK. Marker A and Marker B are used for intraoperative localization and spatial drift compensation, respectively. Upon detection, the Vuforia Engine automatically identifies image features and estimates the poses of Marker A and Marker B relative to the {H}, denoted as 𝐻𝐴𝑇 and 𝐻𝐵𝑇, respectively.

The hybrid registration process consists of initial registration and fine registration. The initial registration employs a QR code-based alignment method. Preoperatively, Marker A is manually fixed at a stable location on the patient or operating table, serving as an external reference marker within the surgical environment. The HoloLens 2 HMD identifies and tracks the real-time pose of the QR code, which determines the initial pose of the virtual 3D model in {H} via 𝐻𝐴𝑇. This step establishes a preliminary spatial correspondence between the virtual 3D model and the real surgical scene.

Prior to the procedure, anatomical landmarks (e.g., inner/outer canthus and nasal tip) were manually labelled on the reconstructed 3D mesh. In regions lacking distinct biological features, supplementary reference markers were strategically defined as virtual fiducials. To compensate for the inaccuracies of the initial registration, Marker B facilitates fine registration by localizing these predefined landmarks on the patient through multiple image target tracking. Marker B is mounted on a handheld probe and defines a local coordinate system {B} centered at its geometric origin. The transformation matrix 𝐵𝑃𝑇 is derived from the real-time pose of the probe’s needle tip relative to the coordinate system {B}. When the operator aligns the probe’s needle tip with a landmark and maintains the position beyond a predefined duration, the system automatically records the needle tip’s coordinates in the coordinate system {B}. A red sphere is then generated at the sampled position to provide visual feedback, as shown in Figure 5.

Figure 5. Spatial coordinate acquisition of reference points using Marker B: (a) Localization of anatomical landmarks on the phantom; (b) Localization of corner points from a high-precision checkerboard calibration board.

Subsequently, these coordinates centered on Marker B are transformed into the coordinate system {H} via 𝐻𝐵𝑇. The coordinates of the acquired patient anatomical landmarks in the coordinate system {H} are given by Equation (9):

𝑃𝐻=𝐻𝐵𝑇⁢𝐵𝑃𝑇⁢𝑃𝑃,

(9)

where 𝑃𝐻 denotes the coordinates of the anatomical landmarks in the coordinate system {H}, and 𝑃𝑃 denotes the coordinates of the anatomical landmarks in the patient coordinate system {P}.

The 3D coordinates of feature points acquired from the patient are defined as the target point set 𝑆⁢𝑒⁢𝑡𝑃 ={𝑃𝐻,𝑖}. Correspondingly, the feature point set on the virtual model constitutes the source point set 𝑆⁢𝑒⁢𝑡𝐴 ={𝐴𝐻,𝑖}. The Trimmed Iterative Closest Point (Tr-ICP) algorithm is utilized to minimize the sum of squared distances between the target point set and the source point set. The obtained optimal transformation is shown in Equation (10):

𝑃𝐴𝑇=𝑚⁢𝑖⁢𝑛⁢∑𝑁𝑖=1∥𝑅⁢𝐴𝐻,𝑖+𝑡−𝑃𝐻,𝑖∥2,

(10)

where 𝑅 is the rotation matrix, and 𝑡 is the translation vector.

Finally, the optimal transformation is applied to the initial pose of the virtual model. Through a secondary pose transformation, precise alignment between the virtual model and the patient’s anatomical structures is achieved. Since Marker A remains fixed relative to the patient, the system can continuously track intraoperative patient or operating table movements. The system dynamically updates the pose of the registered virtual model, which effectively compensates for navigation inaccuracies caused by patient motion during surgery.

After the registration process, the operator can visualize the 3D anatomical structures and the planned path overlaid on the patient through the HoloLens 2, facilitating an intuitive surgical navigation.

2.5. Experiment

2.5.1. Expert Study

We utilized the IEEE Visualization Contest 2010 Case 2 dataset [30], which comprises multi-modal MRI data, including T1WI, T2, FLAIR, and fMRI. The T1WI volume serves as the spatial reference for this study, featuring a matrix size of 512 × 512 with 176 slices and a voxel spacing of 0.49 × 0.49 × 1.0 mm3. All other multimodal data were linearly co-registered to this T1WI coordinate space. Based on neurosurgical consensus, four typical brain tumor locations were established. Target 1 represents the actual tumor location in the patient, while the remaining tumors were artificially placed within the scene, as shown in Figure 6. Target 1 is situated in the precentral gyrus, in close proximity to major blood vessels. Target 2 is located in the left temporal lobe. Target 3 lies adjacent to the cerebral ventricle within the right parietal lobe, and Target 4 is positioned within the left prefrontal cortex.

Figure 6. Four common locations (in green) of brain tumors used during evaluation.

For each target, the proposed MOO method was executed 20 times to automatically generate the optimal path. The MOO algorithm was implemented in Python 3.11. The NSGA-II was configured for use with a population of 200 individuals in a total of 1000 generations. The probability of crossover and mutation was configured to 100% and 30%, respectively. The algorithm terminated after reaching the maximum generation limit. To identify the optimal paths, TOPSIS was employed with a fixed weight vector of [0.35, 0.2, 0.45]. All computational tasks were conducted on a workstation equipped with an Intel Core i7-12700H CPU (2.3 GHz) and 16 GB RAM.

To evaluate the quality of paths generated by the MOO model, an experienced neurosurgeon and three novice surgeons were recruited to perform manual path planning and AR-based interactive path planning, as shown in Figure 7. For each target, the experienced neurosurgeon repeated the planning three times, whereas each novice surgeon performed one planning trial. Manual path planning requires the surgeon to manually select skin entry points and target points according to the standard of care. We developed an AR-based interactive path planning system [31]. Users wearing the HoloLens 2 HMD could clearly visualize the patient’s 3D anatomical model. When the operator manually selects a target location, the system automatically generates a path based on the skin entry point closest to the target. Subsequently, users can interactively adjust the path via gesture input, enabling rotation operations centered on the target. During this process, path color feedback is triggered if the trajectory intersects restricted areas or violates the length or angle constraints.

Figure 7. Two manual path planning methods: (a) Manual path planning based on 2D images (in pink); (b) Interactive path planning using AR (safe mode in green and risk mode in red).

During the experiment, four indicators were used to evaluate paths. Specifically, path length, entry angle, and risk cost are defined in Section 2.3. The minimum Euclidean distance from the path to the risk structure serves as the safe margin metric, as shown in Equation (11):

𝑑𝑚𝑖𝑛=𝑚𝑖𝑛(𝑥,𝑦,𝑧)∈𝑃,(𝑢,𝑣,𝑤)∈𝑆⁡√(Δ𝑥⁢(𝑥−𝑢))2+(Δ𝑦⁢(𝑦−𝑣))2+(Δ𝑧⁢(𝑧−𝑤))2 ,

(11)

where (𝑥,𝑦,𝑧) denotes the voxel coordinates of an arbitrary voxel within along the path 𝑃. (𝑢,𝑣,𝑤) denotes the voxel coordinates of an arbitrary voxel within the risk structure 𝑆. Δ𝑥, Δ𝑦 and Δ𝑧 denote the voxel spacing along the three spatial directions of the MRI volume.

2.5.2. Phantom Experiment

To measure the registration errors of the AR navigation system, we conducted 20 phantom experiments. The phantom used in the experiments was obtained through 3D reconstruction and 3D printing, preserving the precise relative positions of anatomical structures. Four yellow spheres were placed on the head model to serve as reference points for accuracy testing, as shown in Figure 8a. Micron Tracker (ClaroNav, North York, ON, Canada) served as the ground truth spatial coordinate acquisition device, achieving an accuracy of 0.2 mm after calibration. The spatial coordinates of the point were obtained by maintaining the optical marker within the trinocular camera’s detection range and aligning the probe tip with the target point, as shown in Figure 8b.

Figure 8. Materials of phantom experiment: (a) Reference points (yellow spheres) for registration accuracy testing; (b) Subjects recording spatial coordinate information using Micron Tracker.

First, the operator wore the HoloLens 2 HMD and initiated the AR navigation system. Initial registration was performed by gazing at Marker A. The appearance of a virtual 3D model near Marker A confirms successful initial alignment. The Micron Tracker was used to obtain the spatial coordinates of reference points on both the phantom and the virtual model. Subsequently, fine registration was performed using Marker B. The Micron Tracker was used again to acquire updated coordinates of the reference points. The Fiducial Registration Error (FRE) of both registrations was calculated to evaluate the accuracy of the hybrid registration. FRE is defined as the Euclidean distance between the reference points of the virtual model and the phantom, as shown in Equation (12):

𝐹⁡𝑅⁢𝐸=√(𝑥1−𝑥2)2+(𝑦1−𝑦2)2+(𝑧1−𝑧2)2,

(12)

where (𝑥1,𝑦1,𝑧1) denotes the coordinates of the reference point on the virtual model, and (𝑥2,𝑦2,𝑧2) denotes the coordinates of the corresponding reference point on the phantom.

After registration, the Target Registration Error (TRE) was additionally measured at the planned biopsy target. TRE is defined as the Euclidean distance between the target point of the virtual model and the phantom, as given in Equation (13):

𝑇⁢𝑅⁢𝐸=√(𝑥1−𝑥2)2+(𝑦1−𝑦2)2+(𝑧1−𝑧2)2,

(13)

where (𝑥1,𝑦1,𝑧1) denotes the coordinates of the target point on the virtual model, and (𝑥2,𝑦2,𝑧2) denotes the coordinates of the corresponding target point on the phantom.

3. Results

3.1. Path Evaluation

A comparative analysis was conducted across manual path planning, AR-based path planning and the MOO algorithm. Representative path planning results were visualized in three dimensions, as shown in Figure 9. The top-ranked path obtained by the MOO algorithm is considered the optimal path, marked with purple lines in each image. Manually planned paths and those derived from AR-based path planning are labeled with pink and blue lines, respectively. The path planning process considers the feasibility of a cylindrical puncture zone centered at the tumor target point with a 2 mm radius. Path planning results are presented as line segments within the 3D head reconstruction, aiming to clearly visualize the entry and target points, along with the spatial extent of key anatomical structures for the surgeons. Each image in Figure 9 is derived from the 3D reconstruction results. Within the 3D reconstruction, certain vital organs are rendered using pseudo-color processing techniques. The anatomical structures are visualized with varying transparency and color intensity, including the tumor (light green), primary motor cortex (yellow), blood vessels (deep red), and ventricles (deep blue).

Figure 9. 3D visualization of the path planning results.

The performance of the three planning methods was evaluated across six metrics defined in Section 2.5.1. Table 1 shows the quantitative outcomes of the path evaluation derived from the multi-objective optimization functions, while Table 2 summarizes the safety distance results. The comparative results across all six metrics are illustrated in Figure 10. In the radar chart, diminished values signify superior performance for Path Length and Risk Cost, while elevated values suggest enhanced performance for the other metrics.

Table 1. Results of the path evaluation on multi-objective optimization functions.

Table 2. Results of the path evaluation on safety distances.

Figure 10. Radar chart of the path evaluation.

Subsequently, a linear mixed-effects model was applied, with Method as a fixed effect and Target as a random effect. Table 3 shows the estimated marginal means and standard error, with p-values adjusted using Bonferroni correction for multiple comparisons.

Table 3. Linear mixed-effects model analysis of three planning methods.

The results indicated that the MOO algorithm yielded significant improvements across nearly all metrics. Specifically, it reduced the mean path length by 13.6% compared to manual planning and by 13.7% compared to AR-based planning. Similarly, the MOO algorithm dramatically reduced the mean risk cost by 54.3% and 55.4% compared to manual and AR-based methods, respectively. Simultaneously, the mean entry angle of the paths generated by this method was significantly improved by 12.4% and 11.1% compared to manual planning and AR-based planning, respectively. Furthermore, the MOO method significantly increased the minimum distance to the ventricles by 18.2% and 24.3% compared to manual and AR-based methods, respectively. The minimum distance to blood vessels also significantly increased by 20.6% and 15.8%. Regarding the minimum distance to M1, the MOO method performed comparably to the AR-based planning without a significant difference, though it exhibited a slight 5.3% decrease compared to the manual planning. Compared with manual planning, the AR-based method significantly reduced the minimum distances to blood vessels and M1 by 4.96% and 5.44%, respectively, while no significant differences were observed in the remaining metrics.

Beyond the quantitative evaluation of path quality, we further assessed planning efficiency. The mean time for manual planning was 7.91 ± 1.04 min, compared with 6.65 ± 1.59 min for AR-based planning. Notably, the MOO method reduced the mean time to 2.98 ± 0.08 min.

3.2. Phantom Experiment Results

Phantom experiment results are shown in Figure 11. Using only QR code-based registration for initial alignment yielded a mean FRE of 4.19 ± 1.11 mm, with a maximum value of 6.70 mm and a minimum value of 2.12 mm. The FRE for Point1 was 4.64 ± 1.03 mm, for Point2 it was 4.24 ± 1.19 mm, for Point3 it was 3.57 ± 0.95 mm, and for Point4 it was 4.31 ± 1.03 mm.

Figure 11. Phantom experiment results: (a) FRE; (b) TRE. In the box plots, the box boundaries represent the 25th and 75th percentiles, while the horizontal lines, squares, and diamonds represent the median, mean, and individual data points, respectively.

After spatial drift compensation using anatomical landmarks, the mean FRE was 2.37 ± 0.91 mm, with a maximum of 4.50 mm and a minimum of 0.66 mm. The FRE for Point1 was 2.79 ± 0.94 mm, for Point2 it was 2.31 ± 0.58 mm, for Point3 it was 2.12 ± 0.92 mm, and for Point4 it was 2.27 ± 1.05 mm.

The proposed hybrid registration significantly reduced registration error compared to the uncompensated condition. The mean FRE decreased from 4.19 ± 1.11 mm to 2.37 ± 0.91 mm (mean difference: 1.82 ± 0.91 mm). A linear mixed-effects model demonstrated a significant reduction in FRE (β = −1.818, z = −16.23, p < 0.001), with Trial specified as a random effect.

As shown in Figure 11b, the mean TRE was 2.34 ± 0.71 mm, with a maximum of 3.72 mm and a minimum of 1.48 mm. Meanwhile, the mean hybrid registration time was 2.63 ± 0.36 min.

4. Discussion

5. Conclusions

References

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