Approval for this research was granted by the Ethics Committee at Charité - Universitätsmedizin Berlin (EA2/214/24). Anonymized preoperative CT scans were retrospectively collected at the Department of Oral and Maxillofacial Surgery, Charité-Universitätsmedizin Berlin, Campus Virchow-Klinikum. The scans were taken as part of routine preoperative assessments in patients presenting with mandibular fractures. Image acquisition followed a standardized clinical protocol and was performed on General Electric CT systems (LightSpeed VCT and Pro 16, Fairfield, CT, USA). Digital Imaging and Communications in Medicine (DICOM) data sets of CT scans from 88 individuals were analyzed and served as the basis for VFR algorithm development. Scans were acquired with a slice thickness of 0.625 mm and reconstructed using a bone kernel with a reconstruction interval of 0.5-1.0 mm. Acquisition parameters included 120 kVp, automated mA modulation, a matrix size of 512 × 512, and a field of view (FOV) adapted to the craniofacial region (~180-220 mm). All scans were performed without contrast enhancement and in axial acquisition mode.

Segmentation of the mandibular bone and associated fracture fragments was performed using Materialise Mimics 23.0 (Materialise NV, Leuven, Belgium). The software provides intensity-based segmentation directly from DICOM-formatted CT data using predefined Hounsfield unit (HU) thresholds for bone tissue. An automatic thresholding procedure was applied to isolate cortical and trabecular bone structures, typically within the HU range of 300-2,000. This mask was refined using region-growing and morphological editing tools to separate individual fracture fragments and eliminate noise from surrounding soft tissues. After segmentation, the resulting masks were converted into 3D surface models. Each fragment was exported in stereolithography (STL) format for further processing and algorithmic repositioning.

The algorithm was initially developed and tested using a synthetic mandibular model based on two anonymized full-body CT scans. Both volumes were manually cropped to the head and neck region and forwarded to our artificial intelligence (AI)-segmentation pipeline for automatic annotation of relevant skull structures^[38]. The mandible mean-shape model, including teeth, was created by averaging vertex coordinates from their meshes while retaining the main mesh topology. The resulting model preserves the original vertex connectivity and topology, representing an averaged mandible geometry without any individual anatomical information [Figure 2].

To simulate a fracture scenario, the virtual mandible was sectioned with random rotational displacements [Figure 3]. This synthetic fracture model provided a controlled environment for iterative parameter tuning, performance evaluation, and optimization of the VFR algorithm.

A dedicated in-house-developed Landmark Editor was used, with Qt Quick/QML for the graphical user interface and Visualization Toolkit (VTK) for interactive three-dimensional visualization. Landmarks were manually placed on STL surface models and could subsequently be selected, renamed, renumbered, or deleted. Landmark coordinates were exported for each surface model as three-dimensional (x), (y), and (z) values with four decimal places in a JSON-based file for further processing.

The algorithm utilizes three types of anatomical landmarks to guide fracture repositioning: (i) Dental Landmarks: These are specific reference points on the teeth critical for achieving proper occlusion. Key landmarks include the mesiopalatinal cusp of the maxillary molars, the central fossa of the mandibular molars, and the midpoints of the maxillary and mandibular arches (MidU and MidS, respectively), which ensure symmetry and functional harmony in dental articulation [Figures 4 and 5]; (ii) Fracture Landmarks: These are defined points characterizing the sectional features of fractured surfaces. Each landmark on one fragment corresponds to a counterpart on the opposing fragment, enabling precise alignment and analysis of fracture patterns [Figure 6]; (iii) Condylar Head Landmarks: These are anatomical reference points on the mandibular condyle heads (CHL, CHLT for the left side; CHR, CHRT for the right side), critical for positioning the condyle within the mandibular fossa to ensure TMJ functionality [Figure 7].

The algorithm employs a weighted objective function to optimize the alignment of mandibular segments, incorporating multiple constraints to address dental occlusion, condylar positioning, and fracture gap minimization:

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The objective function is defined as a weighted sum of nine terms: (i) Terms 1-3 align mandibular and maxillary dental landmarks (LS1, LS2, LUS1, LUS2, MidS, MidU) to ensure proper occlusion; (ii) Terms 4-5 regulate condylar head positioning (CHL, CHLT, CHR, CHRT) for TMJ articulation; (iii) Term 6 minimizes the gap between fracture landmarks (JS1, JS2S1) to enhance bone healing; and (iv) Terms 7-9 prevent interpenetration and volume overlaps between 3D segments to ensure anatomical feasibility. The objective function quantifies alignment error as the weighted sum of squared differences between transformed and target landmark positions, with weights reflecting clinical priorities. A genetic algorithm (GA) facilitated by the DEAP library framework was employed to optimize the transformation parameters (rotation and translation) for each mandibular segment, with each GA individual encoding 12 parameters: six for rotation (three angles per segment, constrained to ± 0.349 radians) and six for translation (three values per segment, constrained to ± 10 mm)^[39].

The GA was configured with the following parameters: (i) Population size of 1,000 individuals, initialized with zero parameters - this neutral starting point allows the algorithm to systematically explore the solution space and identify optimal transformations without an initial bias; (ii) Generations set to 6,000 iterations; (iii) Genetic operators including crossover using blend crossover (cxBlend) with a blending factor (alpha) of 0.5 and a crossover probability (cxpb) of 0.9, mutation using Gaussian mutation (mutGaussian) with a mutation strength (sigma) of 0.2 and an individual mutation probability (indpb) of 0.2, and selection using tournament selection with a tournament size of 3. Fitness was evaluated based on the total alignment error, and statistical metrics (average, standard deviation, minimum, and maximum fitness) were recorded per generation to monitor convergence. The optimal transformation parameters were then used to construct 4 × 4 transformation matrices, which were applied to the 3D mesh representations for precise segment realignment.

To ensure reproducibility, a fixed random seed (seed = 42) was set for both NumPy and Python’s random module prior to population initialization. Given identical input data and a fixed seed, the GA’s stochastic operators (Gaussian mutation, tournament selection) produce deterministic and reproducible results across repeated executions.

Weights for the objective function were determined through 14 experimental iterations by varying weights while maintaining a total sum of 1. The following considerations guided the iterations.

The highest weight is assigned to w6, as aligning the fracture sections is the most fundamental and critical aspect of the surgery. Proper alignment of fracture segments ensures the restoration of the mandible’s structural integrity and functional capacity, which are primary objectives in mandibular reconstruction. The smallest weight is assigned to w3, reflecting the clinical observation that patients often do not have perfectly aligned dental midlines prior to fracture. Additionally, in jawbone reduction, aesthetic considerations such as dental midline alignment are of secondary importance compared to functional restoration. Thus, this term is given the lowest priority in the objective function.

Larger weights are assigned to w1 and w2 to minimize the distance between the dental landmarks of the maxilla and mandible. These terms ensure proper dental occlusion, which is essential for functional recovery and patient comfort. The alignment of dental landmarks directly impacts the patient’s ability to chew, speak, and maintain oral health.

The weights for the condyle head term (w4 and w5) are assigned to anchor the condyle heads in their target positions. Proper alignment of the condyles is crucial for restoring mandibular articulation and ensuring the stability of the TMJ. These terms contribute to the overall alignment of the mandible by providing a stable reference point for repositioning. The weights w7, w8, and w9 are assigned to mitigate interpenetration and prevent volume overlap among the 3D reconstructed segments. These terms ensure anatomical feasibility and avoid unrealistic configurations that could compromise the surgical outcome and have been given the lowest weights and kept constant along with w3 throughout the experiments.

The experiments are summarized in Table 1. Experiments 7 and 10 are representative cases. In experiment 10, minimizing the distance between the fractures is given the highest priority to determine whether this alone can solve the problem, omitting the occlusion criterion and positioning of the condyle inside the mandibular fossa. Experiment 7 omitted the positioning of the condyle as a factor.

Minimization of the sum of the distances between the fracture landmarks and the dental landmarks to achieve occlusion is the clinical criterion; as such, experiment 13 yields the best performance. The best results are as follows: (i) w6 (fracture alignment) is assigned the highest weight (0.4); (ii) w1 and w2 (dental occlusion) are assigned moderate weights (0.15 each); (iii) w4 and w5 (condylar positioning) are assigned weights of 0.05 each to ensure TMJ stability; (iv) w3 (dental midline alignment) is assigned the lowest weight (0.05); and v) w7, w8, and w9 (interpenetration prevention) are assigned weights of 0.05 each to ensure anatomical feasibility.

The mean Euclidean distance for fracture landmarks was 1.39 mm, with additional metrics for dental occlusion (left: 2.53 mm, right: 3.00 mm) and condylar landmarks (left: 2.32 mm, right: 1.79 mm), yielding a combined dental and fracture alignment error (A + B + C) of 6.92 mm.

To address overlaps observed between the bone fragments, the SimpleICP - a simplified implementation of the ICP algorithm was employed for precise alignment of fracture fragments. The 3D models of repositioned segments were converted to point clouds, with one designated as the “moving” point cloud and the other as the “fixed” point cloud. The ICP algorithm iteratively minimized discrepancies in the overlap region through the following steps: (i) Matching - correspondences were established between points in the moving and fixed point clouds using a nearest-neighbor search within a threshold defined by the mean Euclidean distance from the GA (1.39 mm), with points lacking correspondences within this threshold being rejected; (ii) Minimization - a point-to-plane error metric was minimized using least-squares optimization to estimate rigid-body transformation parameters (rotation and translation), producing the transformation matrix (H) that aligned the moving point cloud with the fixed one; and (iii) Transformation - the computed transformation matrix was applied to the 3D mesh of the moving segment, and the aligned mesh was exported as an STL file.

A final rigid registration step was performed to ensure precise alignment of condylar head and dental landmarks [Figure 8]. The source point cloud, comprising condylar head landmarks and the central fossa of mandibular molars, was aligned with the target point cloud, consisting of original condylar landmarks and the mesiopalatal cusp of maxillary molars. A 4 × 4 transformation matrix was computed and applied to the repositioned fragments to ensure accurate TMJ articulation and dental occlusion.

The quantification of the overlap between two bone fragments can be achieved by computing the volume of their intersecting region. This volume of overlap could serve as a key performance indicator (KPI). A higher overlap volume may signal a false alarm, indicating suboptimal repositioning in which the meshes interpenetrate excessively and deviate from anatomically or mechanically correct alignment.

The fit between repositioned bone fragments can be quantitatively assessed by measuring the direct contact area, which represents the extent of surface adjacency between the fragments following alignment. This metric serves as a KPI for evaluating the effectiveness of the repositioning algorithm, where a higher contact area indicates a greater degree of fit, suggesting that the meshes have been repositioned with enhanced accuracy and anatomical fidelity. To establish typical threshold values for this KPI, a statistical analysis of cross-sectional areas from a diverse cohort of healthy mandibles and varied fracture line morphologies is recommended, enabling the identification of benchmarks that correlate with successful repositioning outcomes.

The overlap volume and contact area were computed in Python using the trimesh library (version 4.5.2), with Boolean mesh intersection performed via the Manifold3d backend. The fracture fragments were loaded as individual STL meshes following the GA-based optimization and ICP registration steps. Pairwise Boolean intersections were computed between the adjacent fragments. All input meshes were confirmed to be watertight (manifold, closed surfaces) prior to the Boolean operation, satisfying the precondition required for valid volume computation via the Divergence Theorem.

The overlap volume in the repositioned simulated fracture was 167.50 mm³ and the corresponding direct contact area was 754.44 mm².

The synthetic validation model provided a controlled environment for quantitative analysis, allowing a direct comparison between the algorithmically reduced and the original unfractured mandible geometry. An illustrative clinical application was subsequently performed on three mandible fracture cases to demonstrate the algorithm's technical feasibility across varying fracture patterns and degrees of displacement. Computation time for each reduction averaged 25-28 min on a standard workstation (Intel Core i7, 32 GB RAM). The algorithm showed robust convergence behavior and stable accuracy across all tested fracture configurations.

To evaluate the accuracy of the proposed repositioning algorithm, the repositioned 3D mandibular model was superimposed onto the original pre-fracture model. A surface distance map was generated with the Python library PyVista (version 0.44.1)^[40] to quantify spatial discrepancies between the repositioned and original configurations, providing both visual and numerical assessments of anatomical alignment. The maximum deviation between the repositioned model and the pre-fracture reference was 1.58 mm, with minimum deviations approaching 0 mm. Notably, deviations along the fracture cut surfaces were predominantly within 1 mm, aligning with the acceptable interfragmentary gap for ORIF as reported in the literature^[26]. These results indicate that the algorithm achieves high anatomical accuracy, particularly in regions critical for fracture alignment and bone healing [Figure 9].

The quality of dental occlusion was assessed by measuring the Euclidean distance between the mesiopalatal cusp of the maxillary molars and the central groove of the mandibular molars, comparing the pre-fracture and repositioned models. Table 2 summarizes the molar relationship metrics for both sides of the mandible. The deviations in molar relations were minimal, with 0.07 mm on the right side and 0.15 mm on the left side. These results demonstrate that the repositioning algorithm successfully restored occlusal relationships comparable to the pre-fracture state, ensuring functional bite mechanics and supporting the structural integrity of the mandibular fracture reduction.

The algorithm was applied to three clinical cases to evaluate its feasibility. Specifications of the fractures are as follows:

Objective evaluation of fracture repositioning in clinical cases presents inherent challenges, as no pre-fracture reference model exists for comparison. To address this limitation, several corresponding anatomical landmarks were placed on the repositioned bone segments. The mean Euclidean distance between these landmarks was calculated for each fracture. The resultant values ranged from 0.22 to 0.70 mm across the five fracture segments in the three cases, with an overall mean of 0.50 ± 0.19 mm (95%CI: 0.27-0.74 mm), indicating consistent alignment. This value lies well within the clinically acceptable threshold of 1 mm, thereby indicating accurate repositioning. The results are summarized in Table 3.

To further analyze repositioning accuracy, the fracture segments were examined for potential overlaps. A color-coded intersection colormap was generated using Geomagic Freeform 2024 software (v2024.0.87, Oqton Inc.) to visualize the degree of congruence along the fracture contours [Figures 10C, 11C and 12C]. Overlaps within a 2 mm threshold were plotted, revealing that most outer contours showed high congruence, with deviations typically ranging between 0.60 and 1.00 mm. The KPIs for each case were calculated as described in section 2.10 and summarized in Table 4. The algorithm runtime for each case is given in Table 5.