Following approval by the institutional review board, a retrospective search was performed to identify all patients at Centre for Orthopaedic Surgery OCON who received a bilateral CT scan for a radius malunion between 2018 and 2024. Patients were excluded if they met any of the following criteria: a non-healthy contralateral side, skeletal immaturity or a CT scan which does not span the entire radius. This selection process resulted in 100 patients.

The retrospectively collected CT scans had slice thicknesses of either 0.6 or 0.75 mm, and a B60s or Br59 CT kernel. The voltage was set at 120 kV, and the scan covered the area from proximal of the ulnar olecranon to distal of the radial styloid, ensuring the entire bone was imaged.

Per identified patient, the side without the malunion was semi-automatically segmented to create 3D bone models, using 3D medical imaging processing software Mimics (version 24.0, Materialise NV, Leuven, Belgium). These 3D bone models were then randomly assigned to one of three sets: training, validation and testing, with an 8:1:1 ratio. Demographic data, including age, sex, and the side of the malunion, were collected of all included patients.

The 3D bone models underwent pre-processing. Initially, all healthy left radii were mirrored to generate a dataset of 100 right radius 3D models. Subsequently, all 3D models were aligned with the coordinate system, as shown in Fig. 1. The coordinate system was defined such that the center of the radial head was positioned at the origin (Fig. 1a). The long axis of the radius running from this origin to the middle of the central ridge of the distal radius was aligned with the Y-axis, and the sigmoid notch was pointing in the Z-direction (Fig. 1b). The X-direction is perpendicular to both the Y and Z directions and points dorsally. For evaluation of the results, a second origin, represented by a yellow dot in the distal radius (Fig. 1b, c), was introduced to assess rotational and translational differences in the distal segment. The coordinate system is similar to the coordinate system described by Smees et al. [7], and an adaptation of the ISB recommended coordinate system [16].

Definition of the coordinate system. a The origin of the coordinate system is located at the center of the radial head and is depicted in black. The edge of the radial head is displayed with a purple circle. b The distal end of the Y-axis is located halfway on the central ridge, highlighted with a purple line, of the distal radius. The Z-axis is perpendicular to the Y-axis and aligns with the middle of the sigmoid notch. Purple arrows indicate the distance between the middle of the sigmoid notch and its volar and dorsal ends. In yellow, a second origin of the coordinate system is represented which is used in the evaluation of the results. c An oblique view of the coordinate system. d The description of the colors of the coordinate system

The creation of the shape completion model began with the development of a statistical shape model (SSM) and was done following the work of Lüthi et al. [14]. The SSM was created using the open-source scripting library for statistical shape modeling and model-based image analysis Scalismo (University of Basel, Graphics and Vision Research Group). The final model is available upon reasonable request.

A downsampled radius 3D model, not included in the set of 100 3D models, was used as the reference for the SSM. Correspondence between each of the 80 3D models in the training set and the reference 3D model was established through nonrigid transformation. This was carried out in iterative steps using iterative closest point matching, where each subsequent step allowed for a larger possibility to nonrigidly deform.

After establishing correspondence between all 80 3D models and the reference, a generalized Procrustes analysis (GPA) was conducted to evaluate the shape variations within the dataset. The GPA provided the deformations between the reference 3D model and each of the 80 3D models. By averaging the deformation fields across all 80 3D models, the average deformation and consequently the mean shape of the dataset were determined. Additionally, principal component analysis was performed to identify the directions of the largest deformation among the 80 3D models, with the principal components ranked from largest to smallest deformation.

Using the mean shape as the reference 3D model, all steps were repeated—including correspondence establishment, GPA, and principal component analysis. The combination of the mean 3D model and the principal components together constituted the SSM.

Once the SSM was created, it was extended into a shape completion model using an isotropic Gaussian process model [14]. Specifically, Gaussian process regression was used, which utilizes the statistical correlations captured by the SSM to predict the missing portion of the bone from known partial shape data. This shape completion model was designed to predict the distal 12% of the radius, based on the proximal 88%. The 12% cutoff value was chosen to include most commonly seen metaphyseal extra-articular radius fracture locations [17]. The location of the cutoff height is depicted in Fig. 2.

Cutoff location at 88% of the length of the radius, depicted in red. The radius is displayed from an ulnar (top) and a volar view (bottom)

A key step in creating the shape completion model was determining optimal hyperparameter values for the Gaussian process. These hyperparameters influence how well the model predicts missing shapes and must be manually tuned. They include noise, which represents the level of noise in the data; sigma, which controls the amount of correlation between neighboring points; and scale, which defines the size of the Gaussian kernel [14]. This hyperparameter tuning was performed systematically by evaluating noise levels of 5, 10, 15, and 20; sigma values of 50, 100, 150, 200, 250, and 300; and scale values of 5, 10, and 15. The optimal hyperparameter combination was identified as the one minimizing the mean prediction error.

To determine the values for the hyperparameters, the validation set was used. First, the distal 12% of each 3D bone model from the validation set was discarded. Next, each remaining proximal 88% of these 3D bone models was registered to the SSM’s mean shape to establish correspondence. Then, the shape completion model predicted the missing distal portion. The predicted complete 3D bone model including the 12% distal part was compared to its actual reference 3D model. The error between predicted and reference was quantified by calculating the mean difference as a Euclidean distance between both 3D models. This procedure was repeated for all ten 3D models of the validation set.

Once the optimal hyperparameter values were identified, the test set 3D models were used to evaluate the shape completion model. The test set was processed in the same manner as the validation set, but only with the optimal hyperparameter settings. The resulting predicted 3D models were then compared to the actual reference 3D models to determine the overall accuracy of the shape completion model.

Evaluation of the shape completion model was conducted using measures of accuracy, compactness, generalization and specificity, as previously described by Audenaert et al. [18]. The accuracy of the shape completion model was determined as the mean distance between the shape completion model-generated 3D model and the reference 3D model for each point on the model’s surface. An accuracy smaller than the slice thickness of the CT scan, which was 0.6 mm, was deemed acceptable. The compactness of the shape completion model was evaluated by determining how many principal components were required to describe at least 95% of variation, as well as evaluation whether 100% of variation could be described. Generalization was assessed by incrementally determining the accuracy of the shape completion model as the number of 3D models in the training set increased. If the accuracy no longer improved, the shape completion model was considered population covering with that number of training 3D models. Specificity was evaluated by using the test set to generate 10 new 3D models and determining their accuracy and maximum Hausdorff distance. Additionally, heatmaps displaying the local differences between the shape completion model-generated 3D model and the test set actual reference 3D model were generated to evaluate local surface differences.

The results of the shape completion model were also expressed in clinically relevant measures: Three rotational (radial inclination; φx, volar tilt; φz, axial rotation; φy) and three translational (dorsal shift Δx, radial shift; Δz and lengthening; Δy) measurements were assessed. For each degree of freedom, the difference between the shape completion model-generated 3D model and the reference 3D model was determined. The origin of the coordinate system was aligned with the center of gravity of the distal segment of the reference 3D model (Fig. 1, yellow dot), and the orientation was aligned following the previously described coordinate system (Fig. 1). The distal segment was aligned with the shape completion model-generated 3D model through global registration and subsequent manual adjustment, similar to alignment in 3D surgical planning. The change in rotation and translation of the distal segment was used to determine the degrees of rotational and translational difference. Additionally, the 3D angle and 3D distance were calculated. These describe the directionally independent shortest rotation and translation between the shape completion model-generated 3D model and the test set 3D model.

To enable comparison with previous studies [13, 15], a second evaluation was conducted using a method similar to that employed in those studies. In this evaluation, the 3D model generated by the shape completion model was cut halfway along the Y-axis, which also served as the origin of the coordinate system for this analysis. The distal 50% of the predicted model was then aligned with the corresponding region of the reference 3D model using global registration. Rotational and translational differences were subsequently quantified based on the resulting transformation.