## 2 Materials and Methods
### Distance Measurement  
For this retrospective study, first, a group of relevant cephalometric points to calculate the differences between the planning and postoperative results were selected, similarly to previous publications [4; 8; 9]. These included: point A, point B, pogonion, both maxilla first molars, and both mandibular first molars (all molar landmarks were selected at the bone margin). Then, the DICOM files of the postoperative CBCT were imported into 3D Slicer (v 4.10.2, [[https://www.slicer.org/](https://www.slicer.org/)]([https://www.slicer.org/](https://www.slicer.org/)), accessed on 19 October 2021) [10] and segmented using a threshold algorithm. A 3D virtual model of the bone was generated in STL format, which was further refined and trimmed using Meshmixer (v 3.5, Autodesk Inc., Mill Valley, CA, USA) for an improved comparison to the virtual planning model. Then, both planning and postoperative STL models were loaded into 3D Slicer for superimposition. Several articles have reported the use of 3D Slicer in different studies, with good results and satisfactory performance for both the segmentation and superimposition of 3D models, and its validity has been established [11; 12; 13]. First, a gross approximation was done using the \"Transforms\" tool (Figure 2), to initialize the algorithm. Then, the \"Surface registration\" tool was used for the superimposition of both models. For this registration, the program uses an iterative closest point (ICP)  
Figure 1: Preoperative (**left**), virtual surgical planning (**center**), and postoperative (**right**) models for an example patient. algorithm, using the planning model as a reference. This superimposition was performed based on a subvolume that comprised the anterior cranial base to compare only those structures affected by the surgery. Once the superimposition was complete, the \"Model to model distance\" tool was used to calculate the distance between both models. In the data input, the postoperative model is defined as the \"source\" and the virtual planning as the \"target\", i.e., the reference. This method measures the Hausdorff distance of the postoperative model to the planning model and generates a color-coded distance map, similar to other publications [14; 15; 16]. The \"Shape Population Viewer\" tool displays a color-coded map. Here, the generated color-coded map shows the corresponding signed distances in different colors attending to the sign and magnitude of this value. In particular, a continuous color scale was defined ranging from blue (\(-4\) mm) to red (\(+4\) mm), where green stays for \(0\) mm. Positive colors (yellow-red) depict regions of the model that are in front of the reference surface (indicating outward movements), while negative colors (blue) indicate areas that are behind the reference surface (meaning backward movements). In Figure 3, an example of a color map is presented, showing the distances from the postoperative model to the virtual planning model. The cranial base appearing green (i.e., a \(0\) mm distance) indicates that the quality of the superimposition is good. Figure 2: Approximation of the postoperative model (beige) to the virtual planning model (white) using the “Transforms” tool, in the frontal (**left**) and lateral (**right**) views. The VTK file of the postoperative model containing the information of the computed distances was exported to the open-source visualization software Paraview (v 5.8.1, Kitware, Inc., New York, NY, USA) [17]. Then, the \"Hover points on\" tool was used to display the information of the distances of the cephalometric points selected in the three axes: \"\(x\)\" (mediolateral), \"\(y\)\" (interoposterior), and \"\(z\)\" (inferosuperior) (Figure 4). The 3D distances for the following cephalometric points were also calculated: point A, point B, and Pog. For both the upper and lower first molars, only the mediolateral distances (\(x\) axis) were computed.