Development and validation of a predictive nomogram for bilateral posterior condylar displacement using cone-beam computed tomography and machine-learning algorithms: a retrospective observational study

Abstract

Background

Temporomandibular disorders (TMDs) are frequently associated with posterior condylar displacement; however, early prediction of this displacement remains a significant challenge. Therefore, in this study, we aimed to develop and evaluate a predictive model for bilateral posterior condylar displacement.

Methods

In this retrospective observational study, 166 cone-beam computed tomography images were examined and categorized into two groups based on condyle positions as observed in the sagittal images of the joint space: those with bilateral posterior condylar displacement and those without. Three machine-learning algorithms—Random Forest, Least Absolute Shrinkage and Selection Operator (LASSO) regression, and Extreme Gradient Boosting (XGBoost)—were used to identify risk factors and establish a risk assessment model. Calibration curves, receiver operating characteristic curves, and decision curve analyses were employed to evaluate the accuracy of the predictions, differentiation, and clinical usefulness of the models, respectively.

Results

Articular eminence inclination (AEI) and age were identified as significant risk factors for bilateral posterior condylar displacement. The area under the curve values for the LASSO and Random Forest models were both > 0.7, indicating satisfactory discriminative ability of the nomogram. No significant differences were observed in the differentiation and calibration performance of the three models. Clinical utility analysis revealed that the LASSO regression model, which incorporated age, AEI, A point-nasion-B point (ANB) angle, and facial height ratio (S-Go/N-Me), demonstrated superior net benefit compared to the other models when the probability threshold exceeded 45%.

Conclusion

Patients with a steeper AEI, insufficient posterior vertical distance (S-Go/N-Me), an ANB angle ≥ 4.7°, and older age are more likely to experience bilateral posterior condylar displacement. The prognostic nomogram developed and validated in this study may assist clinicians in assessing the risk of bilateral posterior condylar displacement.

Background

Temporomandibular disorders (TMD) encompass a range of conditions associated with dysfunctional and degenerative changes in the masticatory muscles and temporomandibular joints (TMJ). The overall prevalence of TMD was approximately 31% among adults and the elderly and 11% among children and adolescents [1]. The incidence of TMD was reported to be 3.9% per year [2]. Posterior displacement of the condyle typically results in intra-articular changes, including disc derangement, joint clicking, and anterior disc displacement. These alterations can subsequently lead to clenching, muscle spasm and pain, and pathological changes within the joint [3]. Several studies have demonstrated a correlation between abnormal condylar position and TMD [4, 5]. However, controversy remains regarding the relationship between imaging changes in joint space and the development of TMD. In adults, adaptive remodeling of the TMJ may result from dysfunctional muscles or malocclusion [6, 7]. Early stages of remodeling are typically characterized by imaging findings, often occurring in the absence of clinical symptoms. Prolonged, intense stimulation that exceeds the capacity for physiological remodeling can contribute to the onset of TMD. Early assessment and intervention for patients with abnormal condylar displacement can significantly reduce the risk of developing TMD. These findings indicate the importance of early detection of condylar displacement.

Changes in tooth morphology and the inclination of the lateral guiding facets can contribute to the onset of TMD symptoms [8]. Occlusal factors play a significant role in influencing mandibular movement. During forward mandibular movement, the posterior teeth disengage from occlusal contact and are guided by the anterior teeth, which serve to protect the TMJ [9]. Hence, relevant occlusal risk factors should be identified early during the treatment.

The structures of the TMJ can be visualized using conventional X-ray, magnetic resonance imaging, spiral computed tomography (CT), and cone-beam computed tomography (CBCT). CBCT is the preferred modality for detecting and assessing degenerative lesions of the TMJ [10–12], as it allows for precise evaluation of joint space, condylar position, condylar volume, height of the articular eminence, and articular eminence inclination (AEI) [13]. While CBCT of the TMJ is comparable to spiral CT, it offers the advantage of a lower radiation dose [14, 15]. Thus, we selected CBCT to measure relevant indices associated with the risk factors.

Few studies have reported the incidence of posterior condylar displacement. One study found that 85% of condyles in cases of anterior disc displacement with reduction (ADDWR) were positioned posteriorly [16]. ADDWR is the most common intra-articular TMD, accounting for 41% of TMD diagnoses [17, 18]. While a posteriorly displaced condyle is not sufficient to diagnose or predict TMD [19], changes in joint space can occur in patients with TMD, with posterior condylar displacement being more prevalent [20–22]. These findings indicate the need to explore the risk factors associated with posterior condylar displacement. Previous studies have examined various factors contributing to condylar displacement, such as AEI [23, 24], age, facial height ratio (S-Go/N-Me), cant of the occlusal plane (OP–FH), overjet, overbite, sex, sagittal skeletal pattern, vertical skeletal pattern [25–27], but have rarely employed visual prediction models.

Nomograms are reliable and efficient tools for quantifying risk. Therefore, in this study, we aimed to develop and validate a nomogram to predict the risk of bilateral posterior condylar displacement.

Methods

In this retrospective observational study, CBCT images obtained at Tianjin Medical University Stomatology Hospital between July 2020 and December 2020 were selected for screening based on the inclusion and exclusion criteria. This study was approved by the Ethics Committee of the Stomatology Hospital of Tianjin Medical University (Project No. TMUhMEC20220805). Informed consent was obtained from all patients for the use of their CBCT images in this study.

Inclusion and exclusion criteria

The inclusion criteria were as follows: bilateral posterior condylar displacement or normal condylar positioning; age, 18–35 years [28]; natural dentition without impacted teeth, except for third molars; good periodontal health with no pathological tooth mobility; absence of occlusal contact involving the third molars; no significant asymmetry in the heights of the left and right mandibular rami on pantomography; minimal crowding or spacing (< 3 mm); no osseous changes in the condyle; and in the growth arrest period.

The exclusion criteria were as follows: congenital abnormal facial growth; history of maxillofacial trauma or surgery; current or previous orthodontic treatment; dentition defects; extensive fillings or crown restorations on the occlusal surfaces; severe caries or abrasion resulting in incomplete occlusal surface morphology; crossbite; anterior open bite or edge-to-edge bite; posterior locked bite.

Sample size calculation

In this study, we determined the sample size based on two established considerations. First, we assessed the “events per predictor” (EPP) ratio to ensure that there were at least 10 events for each variable included in the final model [29]. With 82 observed events and four retained predictors, the EPP ratio was 20.5, exceeding the generally recommended threshold of 10. Second, in accordance with best practices outlined in the literature [30], we considered the ratio of the number of observations (N) to the number of parameters to be estimated (K) when determining the required sample size. In the early planning phase, we accounted for up to 14 potentially estimated parameters to ensure an adequate observations-to-parameters ratio. Ultimately, 143 subjects were included, exceeding this threshold and providing a robust foundation for reliable estimation. These measures helped minimize overfitting, enhance model stability, and ensure sufficient statistical power for the development and validation of our predictive nomogram (Fig. 1).

Fig. 1.

Selection of the participants enrolled in the study

Measurement of the joint space and determination of bilateral posterior condylar displacement

CBCT examinations of the TMJs were conducted using a Kavo 3D exam device (KaVo Group, Biberach, Germany). The CBCT images were acquired with the teeth in occlusion and the head in a standardized posture. The exposure parameters were as follows: 120 kV, 5 mA, field of view (FOV) of 16 cm × 13 cm, exposure time of 7 s, and a voxel size of 0.25 × 0.25 × 0.25 mm.

To ensure quality control, all data were obtained by a single investigator. The same index was measured within 1 week, and 10% of the sample was selected for repeated measurements after 2 weeks. A consistency test of the two measurements was performed using intra-class correlation coefficient (ICC) analysis. The ICC values ranged from 0 to 1, with values > 0.75 indicating good consistency. In this study, the ICC values for all measurements were > 0.85, indicating high consistency and reliability of the results.

The joint space was measured from the sagittal CBCT images according to Pullinger [31]. The relative position of the condyle within the fossa in the sagittal view was calculated using the following formula: In [posterior space (PS) / anterior space (AS)]. Values below − 0.25 indicated posterior condylar displacement. Based on this cutoff, the CBCT images were classified into two groups: those with bilateral posterior condylar displacement and those with a centered condyle.

The CBCT images were uploaded into Dolphin Imaging software (Version 11.8; Dolphin Imaging & Management Solutions, Chatsworth, CA, USA). The “orientation” tool was utilized to adjust the coronal, sagittal, and transverse planes for AEI measurement [32]. For the constructed radiographs, the TMJ view was selected, and the sagittal joint space was measured using the Kamelchuk method [33, 34] (Fig. 2).

Fig. 2.

Measurement of the dimensions of the condylar joint space in the sagittal view. The horizontal plane was used as the reference plane in the sagittal view of the condyle. The horizontal plane intersected with the articular fossa at its most superior point (SF). A vertical line was drawn across SF in the horizontal plane, and it intersected with the condyle at its most superior point (SC). A tangent was drawn across SF, which intersected with the most anterior (AC) and posterior (PC) points of the condyle. Vertical lines were drawn through AC, SC, and PC on the tangent, respectively. The distances from AC, SC, and PC to the fossa were the anterior space (AS), superior space, and posterior space (PS), respectively

Seven cephalometric parameters to be measured

Three-dimensional images were converted into two-dimensional images after planar calibration using Dolphin Imaging 11.8 software. Subsequently, lateral cephalograms were obtained, saved in JPG format, and uploaded into Uceph (Chengdu Dental Communication Technology Co., Ltd., Chengdu, China) for measurement of the following parameters: sagittal intermaxillary angle (ANB angle), angle between the axes of the upper and lower central incisors (U1-L1), OP–FH, Frankfort mandibular plane angle (FMA), angle between the SGn and FH (Y axis), S-Go/N-Me, and angle between the S-Ar and Ar-Go (S-Ar-Go’) (Fig. 3). These measurements were used to identify potential risk factors for bilateral posterior condylar displacement. Seven cephalometric parameters were adjusted according to the Tweed, Downs, Steiner, KNU, and Jarabak’s measurements.

Fig. 3.

Seven measurements of cephalometric parameters. P, Porion; Or, Orbitale; N, Nasion; S, Sella; A, Subspinate; B, Supramental; Gn, Gnathion; Me, Menton; Go, Gonion; Ar, Articulare. ∠1: ANB angle; ∠2: FMA (Frankfort mandibular plane angle); ∠3: Y-axis angle; ∠4: S–Ar–Go angle; ∠5: OP–FH angle (occlusal plane to Frankfort horizontal plane); ∠6: U1–L1 angle (angle between the axes of the upper and lower central incisors); the gray lines represent S–Go and N–Me, used to calculate the posterior-to-anterior facial height ratio

Statistical analysis

Statistical analyses were performed using R software version 3.6.0 (R Foundation for Statistical Computing; http://www.r-project.org/, Vienna, Austria) and R Studio software version 1.1.463 (R Studio, Inc., Boston, MA, USA). The normality of the measurement data variance was examined using the Shapiro-Wilk test. Subsequently, one-way ANOVA and the Kruskal-Wallis test were performed to determine whether significant differences existed among the groups for all parameters. For categorical data, comparisons between the groups were made using the chi-square test. The discriminatory ability of the model was assessed using the area under the curve (AUC). The difference between the predicted and true value distributions was evaluated using the unreliability U test. Decision curve analysis (DCA) was performed to assess the clinical usefulness of the model, and net benefit was used to determine whether the prediction model provided a benefit to patients. For model validation, bootstrapping was employed, and the root mean squared error (RMSE) was used to measure the deviation between the predicted and actual values. Two-sided tests were applied with α = 0.05, and P values < 0.05 were considered statistically significant.

Bilateral posterior condylar displacement (defined as In (PS/AS) <-0.25) was considered the positive outcome, and the following 12 indices were used as independent variables (X): age, sex, ANB angle, U1-L1, OP–FH, FMA, Y axis, S-Go/N-Me, S-Ar-Go’, overbite, overjet, and AEI. Skeletal classification was determined based on the ANB angle as follows: skeletal class I (ANB angle between 0.7° and 4.7°), skeletal class II (ANB angle ≥ 4.7°), and skeletal class III (ANB angle ≤ 0.7°).

The 12 independent variables influencing bilateral condylar displacement were screened and modeled using the Least Absolute Shrinkage and Selection Operator (LASSO), Random Forest, and Extreme Gradient Boosting (XGBoost) methods. The potential risk factors identified by these three machine-learning (ML) algorithms were further analyzed through nonlinear modeling. A restricted cubic spline was plotted to explore nonlinear relationships using the rms package. The model was evaluated in terms of discrimination, calibration, goodness of fit, and clinical applicability. Finally, the most effective risk evaluation model was presented in the form of a nomogram.

Results

A total of 166 CBCT images were screened based on sagittal joint space measurements. Bilateral posterior condylar displacement (n = 82) was designated as the positive outcome, while the group without posterior condylar displacement (n = 84) served as the control group. The 12 characteristic indices were screened for analysis. The baseline characteristics of the images are presented in Table 1.

Table 1.

Baseline characteristics of the posteriorly displaced and centered condyles

	Bilateral posterior condylar displacement (n = 82)	Centered condyle (n = 84)	P
AEI	47.25 (41.88–52.13)	45.25 (39.50–52.38)	0.457
ANB (%)			0.096
1	37 (45.1%)	31 (36.9%)
2	29 (35.4%)	26 (31.0%)
3	16 (19.5%)	27 (32.1%)
U1-L1	123.10 (113.78–130.30)	122.70 (115.90–129.68)	0.998
FMA (%)			0.045*
1	30 (36.6%)	22 (26.2%)
2	30 (36.6%)	27 (32.1%)
3	22 (26.8%)	35 (41.7%)
Y-axis angle	61.50 (58.75–64.10)	61.25 (57.78–63.58)	0.260
OP-FH	8.40 (4.53–11.32)	7.15 (4.13–11.13)	0.451
S-Go/N-Me	65.40 (62.70–68.73)	67.80 (63.53–70.90)	0.089
S-Ar-Go’	143.00 (138.18–147.63)	153.85 (136.53–148.10)	0.774
Overbite (%)			0.185
1	23 (28.0%)	33 (39.3%)
2	17 (20.7%)	14 (16.7%)
3	42 (51.2%)	37 (44.0%)
Overjet (%)			0.234
1	16 (19.5%)	30 (35.7%)
2	38 (46.3%)	24 (28.6%)
3	28 (34.1%)	30 (35.7%)
Gender (%)			0.206
Male (0)	62 (75.6%)	56 (66.7%)
Female (1)	20 (24.4%)	28 (33.3%)
Age (years)	24.00 (22.00,28.00)	24.00 (21.00,28.00)	0.764

AEI, articular eminence inclination; ANB, A point-nasion-B point angle; U1-L1, angle between the axes of the upper and lower central incisors; FMA, Frankfort mandibular plane angle; Y-axis angle, angle between SGn and FH; SGn, a line through point S and point Gn; OP-FH, cant of the occlusal plane; S-Go/N-Me, posterior and anterior facial height ratio; S-Ar-Go’, angle between S-Ar and Ar-Go

The LASSO, Random Forest, and XGBoost algorithms were used to develop and evaluate the models. For the LASSO modeling, the predictor variables included AEI, ANB angle, S-Go/N-Me, and age (Fig. 4). In the Random Forest modeling, the predictor variables were AEI, S-Go/N-Me, and age. For the XGBoost modeling, the predictor variables were S-Ar-Go’, AEI, age, and Y axis. The results from all three ML models indicated that AEI and age were predictive factors for bilateral posterior condylar displacement. The AUCs for the LASSO, Random Forest, and XGBoost prediction models were 0.723 (95% confidence interval [CI], 0.636–0.810), 0.702 (95% CI, 0.612–0.792), and 0.679 (95% CI, 0.586–0.773), respectively (Fig. 5). While the LASSO model exhibited the highest AUC, the differences in AUCs among the three models were not statistically significant (P > 0.05).

Fig. 4.

Risk factor screening for posterior displacement of bilateral condyles based on LASSO regression. (A) LASSO coefficient distributions for the 12 variables. Four potential predictor variables (AEI, ANB, S-Go/N-Me, Age) were selected from the 12 variables based on the penalty coefficient profiles generated by Log (λ). (B) The penalty coefficient λ is cross-validated in LASSO regression using 10-fold cross-validation with the minimum criterion. The root mean square error is plotted against Log (λ). The dashed line represents the optimal λ according to the minimum error (Lambda.min) criterion and the 1 standard deviation of the minimum (Lambda.1se) criterion

Fig. 5.

ROC curves of the three models. The vertical ordinate is the true positive rate and the abscissa is the false positive rate. Model 1 is a model based on variables screened by LASSO, model 2 is a model based on variables screened by random forest, and model 3 is a model based on variables screened by XGBoost. ROC, receiver operating characteristic; LASSO, least absolute shrinkage and selection operator; XGBoost, extreme gradient boosting

Calibration curves were plotted based on the unreliability U test to assess the differences between the distributions of the predicted and true values. The three models exhibited identical U values (U = -0.015), with no significant differences (Fig. 6). DCA was used to evaluate the clinical utility of the models (Fig. 7). If the curve was above both the horizontal black line and the left oblique gray line, the prediction model was considered beneficial. All models presented an NB with threshold probabilities of 20–60%. At threshold probabilities of > 45%, the LASSO model exhibited the best NB.

Fig. 6.

Calibration curves for the three models. The x-axis represents the probability of bilateral posterior condylar displacement predicted by the model, the y-axis represents the rate of bilateral posterior condylar displacement, and the black diagonal indicates the ideal state predicted by the model. The dark blue line is model 1, representing the model based on variables screened by LASSO, the red line is model 2, representing the model based on variables screened by random forest, and the light blue line is model 3, representing the model based on variables screened by XGBoost. The closer the model is to the dotted black line, the more accurately the model predicts bilateral posterior condylar displacement. LASSO, least absolute shrinkage and selection operator; XGBoost, extreme gradient boosting

Fig. 7.

Clinical decision curve analysis for the three models. LASSO, random forest, and XGBoost were used to construct the DCA models and obtain the blue, red, and green curves, respectively. The vertical ordinate is the NB of bilateral posterior condylar displacement, the abscissa is the probability threshold, the horizontal black line (None) assumes that no patient is diagnosed with the net benefit, and the gray line (All) assumes that all patients are diagnosed with the net benefit. If a curve is above the horizontal black and left oblique gray lines, the prediction model is considered beneficial. Model 1 has better net benefit than the other models when the probability threshold is > 45%. LASSO, least absolute shrinkage and selection operator; XGBoost, extreme gradient boosting; DCA, decision curve analysis; NB, net benefit

Model validation included an evaluation of distinguishability, calibration, and other performance indicators. The RMSE was employed to measure the deviation between actual and predicted values, with lower RMSE values indicating better model performance. The RMSEs were 0.516, 0.510, and 0.520 for the LASSO, Random Forest, and XGBoost methods, respectively, with no statistically significant differences (P > 0.05).

With bilateral posterior condylar displacement being the positive outcome, steep AEI and age were identified as risk factors. The models were developed using LASSO regression, Random Forest, and XGBoost methods. There were no significant differences in the calibration of the three prediction models. All three models provided net benefits within a probability threshold of 20–60%, with the LASSO regression model providing superior net benefits when the threshold exceeded 45%. A beeswarm plot was created to illustrate the SHAP (SHapley Additive exPlanations) values, combining feature importance and feature effects for the LASSO model variables (Fig. 8).

Fig. 8.

Visual interpretation of the LASSO models using SHAP values

This figure presents SHAP values for four key features—age, AEI, ANB angle, S-Go/N-Me—that contributed to the machine learning classifier’s predictions of bilateral posterior condylar displacement. Features are ranked in descending order based on their impact on model output. SHAP dependency analysis was performed to visualize the influence of individual variables on the prediction outcome. Yellow dots indicate higher feature values, whereas purple dots represent lower values. Positive SHAP values suggest an increased risk of posterior condylar displacement, while negative values indicate a decreased risk. For instance, lower S-Go/N-Me ratios were strongly associated with an elevated displacement risk.

The LASSO model was identified as the best risk evaluation model (Figs. 5, 6 and 7), and a nomogram was constructed based on the selected risk factors: AEI, ANB angle, S-Go/N-Me, and age (Fig. 9).

Fig. 9.

Nomogram of the optimal clinical risk assessment model for bilateral posterior condylar displacement. The first row is the point assignment for each variable. Rows 2–5 represent the variables included in the model. For an individual patient, each variable is assigned a point value (uppermost scale, Points). The assigned points for all variables are summed, and the total is found in row 6 (Total Points). The total points projected to the bottom scale indicate the probability of bilateral posterior condylar displacement in %. For example, a 30-year-old man with type II ANB, AEI and S-Go/N-Me of 40 and 60, respectively has a total score of 110, corresponding to a 50% probability of posterior condylar displacement. AEI, articular eminence inclination; ANB, A point-nasion-B point angle; S-Go/N-Me, facial height ratio

Discussion

In this study, we developed a nomogram to predict bilateral posterior condylar displacement. Three ML algorithms were utilized to construct predictive models by screening 12 demographic and radiological variables. To gain a deeper understanding of the model, we employed the SHAP beeswarm plot for visualization. The results indicated that the model based on the LASSO algorithm exhibited optimal discrimination and demonstrated substantial net benefits in clinical practice.

The nomogram indicated that patients with lower S-Go/N-Me values, as well as those classified as skeletal classes I and II, were more likely to have higher risk scores. The S-Go/N-Me primarily reflects growth patterns. In hyperdivergent patients, narrowing of the posterior joint space is most commonly observed [35]. A previous study demonstrated that the masticatory force vector is oriented closer to the TMJ in high-angle patients than in low-angle patients [36]. The reduced posterior facial height in these individuals contributes to the adaptive posterior rotation of the mandible due to inadequate mandibular support, thereby increasing the risk of posterior condylar displacement. The ANB angle was used to describe the sagittal skeletal pattern; skeletal class II (ANB angle of ≥ 4.7°) represents mandibular recession. Some studies have explored the relationship between dentoskeletal malocclusion in the sagittal plane and joint space position, yielding diverse results [25, 27]. Consistent with our findings, the condyle in patients with skeletal class II is typically positioned posteriorly [37, 38]. In contrast, another study found that patients with skeletal class I exhibited the smallest posterior space among patients with the three skeletal patterns [39].

A steep AEI was identified as a risk factor for bilateral posterior condylar displacement. The condyle makes firm contact with the articular eminence, and as it moves along the posterior plane of the articular node, patients with a greater AEI experience more anterior extension or opening, leading to a vertical downward displacement of the condyle [40]. Patients with steeper articular eminences have been reported to be more prone to internal dysfunctions, such as anterior disc displacement, compared to those with flatter articular eminence [41]. Several studies have indicated that patients with skeletal class II exhibit a steeper AEI and posteriorly positioned condyles [23, 24], which is consistent with our findings. We hypothesize that the relationship between AEI and posterior condylar displacement may not be purely linear. In the early stages of articular disc displacement, a steeper AEI is associated with an increased risk of both articular disc and posterior condylar displacement. However, in the later stages of articular disc displacement, the posterior plane of the articular eminence tends to flatten due to bone remodeling, while the condyle continues to maintain its posterior position [42, 43].

In our study, age was found to influence the spatial relationship between the condyle and the fossa, with the prevalence of posterior condylar displacement increasing in patients aged 35–44 years [44]. In patients aged 8–30 years with open bites, the condyles were positioned significantly more posteriorly as age increased [45]. The position of the condyle may change with age [46], possibly due to bone remodeling in the TMJ and an increase in the mean joint space in older patients [47]. The flattening of the superior part of the condyle and remodeling of the articular eminence [48] leads to an increase in the superior and anterior joint spaces, resulting in posterior condylar displacement. In edentulous patients, the reduction in vertical distance and the loss of occlusal relationships cause posterior and superior displacement of the condyles. However, when occlusion is rehabilitated, the condyles shift to a relatively anterior and inferior position [49].

In this study, bilateral posterior condylar displacement was selected as the positive outcome, while facial asymmetry was excluded due to its potential influence on the results. Abnormal development of the maxilla is a known cause of functional posterior cross-bite (FPXB), and FPXB contributes to the asymmetrical positioning of the condyle within the articular fossa [50, 51]. Tun et al. [52] found that condylar motion and the condyle-fossa relationship were influenced by the asymmetry of the bony structures of the articular fossa in individuals with jaw asymmetry. As the condyle is central to the development of the mandible and serves as the structure linking the maxilla and mandible, it plays a significant role in facial morphology. These findings support the validity of our study design, leading to the exclusion of patients with facial asymmetry.

In this study, the SHAP (Fig. 8) analysis and the nomogram (Fig. 9) reveal opposite predictive trends regarding the relationship between age and the disease. According to the SHAP graph, the probability of bilateral condylar posterior displacement decreases with age, whereas the nomogram suggests an increase in probability with age. This discrepancy may be attributed to the fact that LASSO and SHAP rely on distinct algorithms. LASSO tends to select fewer variables that significantly contribute to the model, often prioritizing a representative feature when multiple relevant variables are present, thereby simplifying the model [53, 54]. In contrast, SHAP decomposes the marginal contribution of each feature, offering a more detailed and intuitive understanding of each feature’s role in predicting individual samples [55, 56]. Considering the clinical context of our study, we chose the Lasso-based model, as it provides greater interpretability and aligns more closely with our clinical objectives.

CBCT is capable of three-dimensional imaging and offers advantages such as high image resolution, minimal artifacts, short scanning time, and high image accuracy, making it highly suitable for craniofacial imaging [57–59]. The accuracy and reliability of CBCT images have been well-established [60, 61]. Numerous studies have employed CBCT to assess the morphology and position of the condyle within the glenoid fossa [57–59]. CBCT serves as a crucial bridge between artificial intelligence (AI) and dentistry, as it allows computer systems to directly process digitally encoded image data [62]. By analyzing CBCT image data, ML—a core branch of AI—can learn from and make predictions based on this information. One study analyzed bone texture features and bone morphometric variables through radiomics analysis of high-resolution CBCT scans, providing a research foundation for the identification of early-stage conditions through image-specific biomarkers [63]. The integration of CBCT and ML demonstrates significant potential for applications in medical imaging. However, several challenges remain. The mandible may appear distorted, or there may be a false impression of articular fossa displacement if there are patient positioning errors during image acquisition. Additionally, the annotation or validation of the ML dataset may be compromised by inadequate CBCT resolution [64].

The nomogram presented in this study has significant clinical implications, offering personalized risk predictions based on specific variables (S-Go/N-Me, ANB angle, AEI, and age). Our model can be integrated into medical record systems to provide risk scores, assisting clinicians in more accurately stratifying patients at risk for bilateral posterior condylar displacement and offering immediate decision support. If the nomogram indicates that a patient is at high risk for posterior condylar displacement, clinicians should prioritize clinical occlusal examinations, which may include, but are not limited to, assessments of tooth morphology, cusp inclination [65], presence of unilateral posterior missing teeth [66], and muscle pain [67], to facilitate early intervention for high-risk patients. Such predictions could enhance patient outcomes and may serve as a foundation for prospective interventional trials. However, it is important to note that although ML models are powerful tools for addressing complex problems, they are based on statistical theories and thus inherently probabilistic [68].

This study had some limitations. First, the sample size was relatively small and derived from a single organization, which may limit the generalizability of the results. Second, the retrospective design of the study limits the ability to validate the algorithm’s predictive accuracy through clinical testing. Third, accurately localizing the condyle in CBCT cross-sectional slices is challenging, which complicates the correlation of the results with actual clinical scenarios. Finally, this study relied solely on imaging data from CBCT, and potential influencing factors, such as systemic diseases or bruxism, were not considered. Future studies should include clinical documentation of whether a patient has TMD and the severity of symptoms, supported by more real-world clinical evidence to validate the practical application of predictive models. Clinical occlusal examination results should also be incorporated, particularly by comparing patients with TMD symptoms, to enhance the clinical relevance of the findings. A more comprehensive study design is needed to further investigate the impact of occlusal factors on condylar position and to identify populations at risk.

Conclusion

This study demonstrated that patients with a steeper AEI, insufficient posterior vertical distance (S-Go/N-Me), an ANB angle of ≥ 4.7°, and older age are more likely to exhibit posterior condylar displacement. The nomogram provides a valuable tool for identifying at-risk populations and assessing the risk of condylar displacement.

Abbreviations

LASSO

Least absolute shrinkage and selection operator

AEI

Articular eminence inclination

ANB

A point-nasion-B point angle

TMD

Temporomandibular disorders

TMJ

Temporomandibular joints

CBCT

Cone-beam computed tomography

ICC

Intra-class correlation coefficient

PS

Posterior space

AS

Anterior space

SF

Superior point of fossa

SC

Superior point of condyle

AC

Anterior point of the condyle

PC

Posterior point of the condyle

U1-L1

Angle between the axes of the upper and lower central incisors

OP–FH

Cant of the occlusal plane

FMA

Frankfort mandibular plane angle

Y axis

Angle between the SGn and FH

S-Go/N-Me

Facial height ratio

S-Ar-Go'

Angle between the S-Ar and Ar-Go

AUC

The area under the curve

DCA

Decision curve analysis

NB

Net benefit

RMSE

The root mean squared error

XGBoost

Extreme gradient boosting

Author contributions

H.S., M.X. and L.W. conceived of the idea for the manuscript. X.J. and L.W. acquired the data and drafted the article. J.L. and F.Q. analyzed the data. B.Y. contributed to the data curation and visualization. Y.W. made the rough drafts. L.W. contributed to the development of the drafts, successive revision and refinement of the manuscript.

Funding

This research received the grant from the science and technology popularization project of Tianjin science and technology commission (Grant Number:24KPHDRC00190).

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declarations

Ethics approval and consent to participate

This study was performed in line with the principles of the Declaration of Helsinki. Ethics approval was granted by the Ethics Committee of the Stomatology Hospital of Tianjin Medical University (Project No. TMUhMEC20220805). Oral informed consent was obtained from all the participants.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.