Assessing the impact of occlusal plane rotation on facial aesthetics in orthodontic treatment: a machine learning approach

Cephalometric radiographs of patients visited West China Stomatology Hospital from January, 1st, 2020 to July, 1st, 2021 for orthodontic treatments were selected for dataset construction. We mainly included data of 18–35 years old young adults without congenital deformity, infection, trauma, tumor, surgery history. Meanwhile, patients with obvious skeletal malformation with indication of surgery were excluded to ensure the applicability of our network. Lateral cephalometry were obtained from a Cephalometer (Veraviewepocs, Morita, Kyoto, Japan) when subject was positioned with the Frankfort plane paralleled to the horizontal line with their teeth in centric occlusion and lips lightly closed.

The automatic cephalometric landmarks localization function of Uceph (China) was adopted to label the lateral cephalometry and two trained orthodontists manually checked and revised the marks. The landmarks and cephalometric measurements were illustrated by Fig. 1 and Table 1. Patients were classified by sagittal and vertical skeletal type based on criteria listed in Table 2. The Intra-class Correlation Coefficient (ICC) was used to assess inter-operator and intra-operator reliabilities. Briefly, 50 cephalograms were randomly selected and measured by three orthodontists. Each orthodontists repeated the measurements after 2 weeks. Both the inter- and intra-operator agreements of measurement were pretty good, indicated by the ICC scores of 15 parameters ranged from 0.90 to 0.98 and 0.88 to 0.94.

Inputs were chosen from skeletal measurement parameters as N-ANS, S-N, SNA, NA-FH, Ptm-A, PP-FH for cranio-maxilla complex, Go-Po, Go-Co, S-Go for mandible, and SN-OP (Table 1). Noticeably, the input except SN-OP stand as the indicators of basic anatomic structure of cranium, maxilla and mandible, which could largely remain constant during orthodontic therapies regardless of mandibular rotation (Fig. 1.A, B). By excluding the parameters which have overlapping function with SN-OP in representing the mandible rotation, we separated the invariables into two main parts as: a) constant ones representing basic craniofacial structure, b) SN-OP standing for maxillo-mandible relationship and occlusion rotation. By this means, we ensured that alternation of SN-OP in the mathematical function was independent with other factors and represented the relative position of mandible. The corresponding values of P-A Face Height ratio (%) (FHR) and FH-NPo (Facial Angle, FA) were used as target, representing the esthetic improvement vertically and sagittally, individually. Variables were then all normalized to the range from − 1 to 1 by the mapminmax function of MATLAB. A three-layer feedforward BP-ANN model was built by MATLAB R2018a neural network toolbox. Samples were divided randomly into training, validation and testing dataset with a proportion of 75:15:15. The number of hidden layer nodes was set to 10 and the topological structure of the model was shown in Fig. 2A.

To predict the correlation between P-A Face Height (%) and FH-NPo (Facial Angle) with SN-OP, we conducted simulations using our dataset. Specifically, we varied the SN-OP angle within a range of 9 to 25 degrees while keeping all other fundamental craniofacial parameters constant for each patient, resulting in an individualized input matrix for each case (Fig. 2). Using this simulated input matrix, which combined the patient’s baseline anatomical information with the altered SN-OP value, we employed model FHR to predict the corresponding FHR values across various SN-OP angles for each patient. Subsequently, we conducted a linear regression analysis to quantify the correlation between input SN-OP and out FHR. This analysis allowed us to calculate both the vertical intercepts (b) and slopes (k). In interpretation, the vertical intercepts (b) signify the extent of variation range attributed to the patient’s inherent craniofacial condition, while the slopes (k) represent the degree of efficiency in which changes in SN-OP influence alterations in FHR values. Similarly, the simulated strategy was illustrated by Fig. 2B and was also adopted to model FA to elucidate the correlation between FA and SN-OP changes.

Graphpad Prism 9 was used for the statistical evaluations. Normality and lognormality of the data was tested and evaluated by Kolmogorov-Smirnov test. For normally distributed data, test for homogeneity of variance was carried out. One-way ANOVA and Bonferroni test were applied to evaluate the differences between groups. For data which did not show normal distribution, Kruskal-Wallis test and Dunn’s multiple comparisons test were employed for difference analysis. Means and standard deviations were used for comparison. Statistical significance was set at 5%.

After the exclusion of patients with unclear cephalometry images, 1220 of 1301 samples were collected. In order to get a relatively balance sample size of each skeletal type to avoid fitting deviation of the model, we randomly excluded half of the Class I samples considering that it originally had nearly twice the number of Class II and III. After that, 903 samples were included and distribution of samples in different skeletal classes were displayed in Table 3. The overview of cephalometric measurements was listed in Table 4 in the form of means and standard deviations of each skeletal group. Intriguing results could be revealed by the cephalometric data through intra- and inter-group comparison. First, average values of indicators as SN-MP, FMA and S-Go/N-Me for vertical and SNA, SNB and ANB for sagittal classification showed high consistency with the principle that we followed for grouping, validating the reliability of our classification strategy. Second, specialized growth patterns of each skeletal types could be noticed. For crania and maxilla morphology, our results indicated relatively smaller value of Ptm-A, smaller SNA and NA-FH in hypodivergent group, echoing previous studies which suggested that maxilla of this group was generally shorter and posteriorly positioned than the hypodivergent facial type [19, 20]. Sagittally, maxilla of Class II tends to be longer and more protrusive than Class III [19, 20], as larger value of Ptm-A and NA-FH in Class II group could also be revealed by our study in Table 3. From the aspect of mandible morphology, shorter and retrognathic positioning of mandible was also found in hyperdivergent group represented by shorter Go-Po and Go-Co, smaller FH-NPo and SNB. Intriguingly, the discrepancy of mandible size among different sagittal groups mainly existed in the length of Go-Po, while the length of Go-Co showed closer relationship with vertical classification and remained consistent among sagittal types, indicating that mandibular corpus rather than ramus could show higher heterogeneity in different sagittal groups.

The relationship curve between the performance loss function of the model and iteration training times is shown in Fig. 3A, B, suggesting the training efficacy of the BP-ANN network. The blue, green and red line represents the cross-entropy over the course of training, validation and test, individually. The mean squared error (MSE) of fitting decreased along with the training process which gradually reached convergence during the process of 11 and 10 iterations for FHR model and FA model, respectively. The best performance evaluated by MSE of validation samples was 0.0133, at the 5th iteration for FHR model and 0.0186 at epoch 4 for FA model. Regression coefficients which calculated the relativity between predicted and true values of training, validation and test sets, respectively (Fig. 3C, D), were all above 0.88, representing the accuracy of prediction. For consistency verification, paired-samples T test and Bland-Altman analysis were carried out. The paired-samples T test’s results indicated that the predicted and actual values were well paired as the correlation coefficient being 0.9007 (p < 0.001), while there was no significant difference between predicted and true values as the t value was 0.1994 (p > 0.05). The results of Bland-Altman analysis indicated that almost all paired values were included in the 95% Confidence interval (95%CI) (Fig. 3E, F), testifying that the BP-ANN models built in this study could predict the value of FHR and FA well once given the basic information of the craniofacial structure.

After ensuring that our model could successfully predict the according values of FHR and FA once given the inputs, we went on to explore the impact of SN-OP changes on FHR and FA. In the model, we separated the variables into SN-OP which representing the rotation, and anatomic parameters which remain constant against the alternation. By this means, we ensured the rationality of the simulation strategy as to solely change SN-OP angle without altering basic geometrical values. In other words, once given the anatomical parameters of one specific patient, we could simulate the FHR and FA changes to OP alternation (Fig. 2B), and the according ratio was then quantified by linear regression analysis. In this way, the predicted correlation between SN-OP and FHR, FA changes could be concluded in the form of vertical intercepts (b) and slopes (k). After running the stimulation for each sample we collected, a list containing patient’s skeletal inputs and corresponding k and b could be obtained. Thus, the general trends of k and b, which reflects the efficacy of responding alternation of FHR or FA to SN-OP changes, could be summarized for each skeletal type (Tables 5 and 6), respectively. The inter- and intra-group comparison results among diversified skeletal groups would be detailed in following sections.

In the linear regression analysis of FHR, the slopes (k) represented the changing rate of FHR to SN-OP. Clinically, larger k meant more sensitive change of FHR to OP rotation. However, k in all skeletal types appeared to be constant and only showed significant difference as k^II-hypo < k^II-hyper (p < 0.01), k^III-hypo < k^III-nor (p < 0.01) (Fig. 4A). The average slope of all samples was about − 0.28 ± 0.05, indicating that by increasing 10 degree of OP angle, the predicted decrease of FHR was about 2.8%.

On the other hand, the vertical intercepts (b) representing the range of FHR decided by skeletal types, showed great divergence among different vertical skeletal types (Fig. 4B). Significantly, the decline tendency as b^hypo > b^nor > b^hyper was similar in every sagittal type (p < 0.01), consistent with the fact that hypodivergent type usually has shorter face and higher FHR. The average value of b in hypo-, nor-, hyper-groups are about 73, 71 and 68, while the average slope (k) of SN-OP of each group are 14, 17, 20 (Table 3). Combined with the function of regression as FHR = b + k × SN-OP, the average outcome of FHR could be around 69, 65 and 61, which highly echoed the normative range and classification basis of FHR for each vertical group, conforming the reliability of our models.

As for FA prediction, the change rate appeared to be discrepant among skeletal types (p < 0.01), but with some regularity to conform. In summary, hypodivergent group and Class III group, tended to have higher |k| in vertical and sagittal comparison, respectively. Vertically, the slopes (k) showed consistent decline trend as k^hypo > k^nor > k^hyper inside every sagittal type (Fig. 4C). Significant differences between each pair in Class I and Class II were also testified. As for Class III, the discrepancy was significant as k^III-hypo > k^III-hyper (p < 0.0001) and k^III-nor > k^III-hyper (p < 0.0001), but not between k^III-hypo and k^III-nor (p > 0.05). Furthermore, the sagittal classification also matters. Though classified as the same vertical types, the k of Class III appeared to be larger than that of Class I or II in all vertical groups (Fig. 4C, a-f, p < 0.05), while there is no significant difference of k between Class I or II in each vertical groups (p > 0.05).

Similarly, the vertical intercepts (b) reflected the tendency of convexity (Fig. 4D), which was largely decided by the skeletal structure. Generally, hyperdivergent and Class II group showed higher tendency to have more convex profile and lower scores of FA. Intriguingly, our results also validated significant differences of FA-b between hypo- and hyper- groups and between Class II and III. In details, a decline tendency as b^hypo > b^nor > b^hyper was shown inside every sagittal group. Significant differences between b^hypo and b^hyper existed in Class I (p < 0.0001), II (p < 0.0001) and III (p < 0.01), while only Class I (p < 0.001) and II (p < 0.05) showed significant differences by comparing b^nor and b^hyper. No significant divergences were found between b^hypo and b^nor in neither Class I, II or III (p > 0.05). For sagittal comparison, discrepancy as b^III > b^II was validated in each vertical groups (p < 0.0001). Significantly b^III > b^I was found in normodivergent group (p < 0.01) and hyperdivergent group (p < 0.0001), but not in hypodivergent group (p > 0.05). No significant difference between b^I and b^II was found in each vertical groups (p > 0.05). As for the clinical meaning of the b value, similar calculation as FA = b + k × SN-OP could be done as for FHR. By the mathematical function, the average values could be calculated which corroborated with the facts that hypodivergent Class III patients tend to have concave face (FA > 90°), while hyperdivergent Class II patients are more apt to convex profiles (FA < 85°). Accordingly, the consistency of our function with the reality knowledge also verified the reliability of our models.