ORIGINAL ARTICLE
A more accurate soft-tissue prediction model for Class III 2-jaw surgeries Yun-Sik Lee,a Hee-Yeon Suh,a Shin-Jae Lee,b and Richard E. Donatellic Seoul, Korea, and Gainesville, Fla Introduction: The use of bimaxillary surgeries to treat Class III malocclusions makes the results of the surgeries more complicated to estimate accurately. Therefore, our objective was to develop an accurate soft-tissue prediction model that can be universally applied to Class III surgical-orthodontic patients regardless of the type of surgical correction: maxillary or mandibular surgery with or without genioplasty. Methods: The subjects of this study consisted of 204 mandibular setback patients who had undergone the combined surgicalorthodontic correction of severe skeletal Class III malocclusions. Among them, 133 patients had maxillary surgeries, and 81 patients received genioplasties. The prediction model included 226 independent and 64 dependent variables. Two prediction methods, the conventional ordinary least squares method and the partial least squares (PLS) method, were compared. When evaluating the prediction methods, the actual surgical outcome was the gold standard. After fitting the equations, test errors were calculated in absolute values and root mean square values through the leave-1-out cross-validation method. Results: The validation result demonstrated that the multivariate PLS prediction model with 30 orthogonal components showed the best prediction quality among others. With the PLS method, the pattern of prediction errors between 1-jaw and 2-jaw surgeries did not show a significantly difference. Conclusions: The multivariate PLS prediction model based on about 30 latent variables might provide an improved algorithm in predicting surgical outcomes after 1-jaw and 2-jaw surgical corrections for Class III patients. (Am J Orthod Dentofacial Orthop 2014;146:724-33)
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ne of the most important considerations in orthodontic treatment planning is an esthetic posttreatment patient profile.1-4 Many orthodontic residency programs teach this as the primary objective from which all other clinical treatment decisions are based. In patients with significant skeletal discrepancies requiring orthognathic surgery, decisions regarding the details of the surgical methods would be much easier if the exact results of the surgery could be accurately predicted and portrayed while still in the treatment
a Postgraduate student, Department of Orthodontics, School of Dentistry and Dental Research Institute, Seoul National University, Seoul, Korea. b Professor and chair, Department of Orthodontics, School of Dentistry and Dental Research Institute, Seoul National University, Seoul, Korea. c Clinical assistant professor, Department of Orthodontics, College of Dentistry, University of Florida, Gainesville, Fla. All authors have completed and submitted the ICMJE Form for Disclosure of Potential Conflicts of Interest, and none were reported. Supported by the Basic Science Research Program funded by the Korean government (NRF 2012-0007574) and partly by grant number 02-2014-0003 from the Seoul National University Dental Hospital Research Fund. Address correspondence to: Shin-Jae Lee, Department of Orthodontics, Seoul National University School of Dentistry and Dental Research Institute, 101 Daehakro, Jongro-Gu, Seoul 110-749, Korea; e-mail, [email protected]. Submitted, May 2014; revised and accepted, August 2014. 0889-5406/$36.00 Copyright Ó 2014 by the American Association of Orthodontists. http://dx.doi.org/10.1016/j.ajodo.2014.08.010
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planning stage. Consequently, the orthodontic literature is replete with studies regarding the associations between surgical skeletal repositioning and soft-tissue responses. Imagine being able to perform a simulation of various surgeries on a commercially available 3-dimensional software-rendered model that produced predictable results, reliable enough to dictate treatment decisions and accurate enough to show a patient what his or her eventual appearance would be. This would increase a patient's understanding and acceptance of the recommended treatment.5,6 Of course, prediction software is currently available, but anyone who has used it realizes that the predictions are often rudimentary, unrealistic, and unreliable. As analysis techniques advance, orthodontic software should continue to integrate these advances to better serve clinicians. Consequently, clinicians should come to expect more from their orthodontic imaging software programs. As any clinician involved with patients with severe Class III malocclusions quickly realizes, the soft-tissue profile does not directly, 1 to 1, follow the surgical changes of the underlying bony structures.7,8 The partial least squares (PLS) method is a comparatively new way of formulating prediction equations, and
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its application to various scientific and biologic disciplines from chemical engineering to brain image analysis is becoming increasingly widespread.9-13 Applying the PLS method is advantageous when there are many variables and they are highly correlated. The merit of the PLS method is its capability of taking correlation structures into account, controlling not only for the correlations between the predictor and response variables but also for the correlations within the predictor variables and the response variables. Recently, a study applying the multivariate PLS method to mandibular setback surgeries demonstrated considerably more accurately predictions than the conventional ordinary least squares (OLS) method.8 The conventional OLS method was determined to be unsatisfactory when there were many correlated variables. Among the variables considered when predicting the soft-tissue response to surgery are the patient's age,2,3,14-17 sex,2,16-19 time after surgery,20,21 and presurgical soft-tissue characteristics, including tissue thickness measured at various landmarks.4,7,17,22-25 These various factors can be considered in the PLS method through orthogonal linear combinations that can extract a small number of significant components that are combinations of the original variables.26 In addition, the improved accuracy of the PLS method is most likely because the soft-tissue response at a specific point highly depends on its adjacent soft-tissue response: ie, the interdependency of soft-tissue points.13 However, the aforementioned investigation had only been performed for mandibular setback surgeries alone. Class III 2-jaw surgery patients had not yet been included.8 Including an additional surgery has a great influence on the soft-tissue profile changes, greatly complicating the prediction.3,27 Consequently, most studies have reported only 1 specific maxillofacial surgery in their soft-tissue analyses. Because bimaxillary surgery produces more stable results than single-jaw mandibular procedures in Class III correction, clinically, there has been an increase in its use.28,29 However, current prediction programs for bimaxillary surgery are less predictable than for 1-jaw surgery.5,27,30,31 It is especially difficult to accurately determine the changes in the soft-tissue profile when mandibular setback surgeries are combined with other surgical procedures, such as a LeFort I osteotomy or a genioplasty. Our aim in this study was to develop an accurate softtissue prediction method that can be applied to various methods of Class III surgical correction: mandibular surgery and maxillary surgery, and genioplasty. In addition, we discuss the steps in choosing an optimal multivariate PLS prediction model.
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MATERIAL AND METHODS
The institutional review board of Seoul National University School of Dentistry for the protection of human subjects reviewed and approved the research protocol (number S-D 20140018). The subjects consisted of 204 patients (103 women and 101 men with an average age of 24 years) who had undergone surgical correction of a severe Class III malocclusion. All subjects were of Korean ethnicity. All patients received mandibular setback surgery, and 133 patients had LeFort I maxillary osteotomy. Genioplasty was performed for 81 patients. Table I provides further details of the patients with regard to ages at surgery and some cephalometric and occlusal characteristics. All patients had ceased growing and were healthy, without cleft lip and palate, injury, or craniofacial syndrome. Preoperative lateral cephalograms were taken close to the time of surgical correction. The data were collected prospectively with the postoperative radiographs taken at least 4 months (average, 9.1 months) after surgery to allow any residual soft-tissue swelling to resolve.32 Thirty-nine skeletal landmarks and 32 soft-tissue landmarks from glabella to the terminal point were identified. With its origin at sella, the vertical reference was established perpendicular to the sella-nasion line plus 7 . Using a custom digitizing program via C# programming language (Microsoft, Redmond, Wash), the coordinates of every landmark on each tracing were sequentially computed in relation to the x and y reference system. The prediction process had 2 stages: building a prediction equation through model fitting, and validation by applying the equation to each subject and calculating test errors. Figure 1 is a flow diagram for establishing the prediction model in this study. Two hundred twenty-six predictor variables and 64 response variables were entered into the prediction equation. The predictor variables included each patient's age and sex, time after surgery, 1-jaw vs 2-jaw surgery, conjunctive genioplasty, amount of asymmetry, 39 presurgical skeletal landmarks, 32 soft-tissue landmarks before surgery, and 78 (39 3 2) variables measuring the surgical skeletal repositioning in both the anteroposterior and vertical directions. The 64 (32 3 2) response variables were set as the soft-tissue position after surgery in the 32 soft-tissue landmarks in both the x-axis and the y-axis. The training data set was numerically treated via centering and normalization. The centering made the following computations numerically well conditioned.33 The normalization gave each variable equal influence in the initial stage of the data analysis.
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Table I. Subjects' sex, age, and other characteristics Variable Age (y) Female (n 5 103) Male (n 5 101) Time after surgery (mo) Maxillary surgery No (n 5 71) Yes (n 5 133) Genioplasty No (n 5 123) Yes (n 5 81) Overjet before surgery (mm) Overbite before surgery (mm) Amount of surgical repositioning at Point A (mm)* (n 5 133) Anteroposterior repositioning Vertical repositioning Amount of surgical repositioning at Point B (mm)* (n 5 204) Anteroposterior repositioning Vertical repositioning
Mean
SD
Minimum
Maximum
23.8 23.6 9.1
5.1 3.5 3.9
16.0 18.8 3.7
50.5 39.1 29.4
5.8 0.2
3.8 1.8
19.6 5.4
2.2 5.9
1.4 1.1
1.9 2.3
4.4 7.1
7.0 4.4
7.3 2.9
3.8 4.2
24.1 14.8
4.1 11.4
*A negative value indicates either a posterior direction or a superior direction during surgical repositioning.
Fig 1. Flow diagram of establishing the prediction model in this study.
Two multivariate methods of constructing prediction equations were developed using the conventional OLS and the PLS methods. First, the conventional OLS method covers from a simple ratio statistics or correlation analysis to a more complex multiple linear regression. In this study, the multivariate multiple linear regression with forward variable selection was used. The model was selected with the Akaike information criterion; the lower this criterion, the better the model will be.34 Second, the PLS method is a multivariate regression with latent variables (also called components). This latent
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variable method was used to focus the information of a large data set into a few underlying components or factors, leaving most of the measurement noise behind as residuals. Although some mathematical details would have been needed, we emphasized the methods and results rather than their theoretical background. With regard to more detailed mathematical properties and interpretations for the PLS method, please refer to the studies of Hastie et al12 and Wehrens.11 In the training data set, the absolute error values after fitting the prediction equation (also called training error) were measured.
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Fig 2. Training errors (top) and test errors (bottom) in the mean absolute values showing the fitting quality of the OLS and the PLS predictions. Top, the training errors at the stage of constructing the equation were negligible and close to zero. The training error values were not perfectly the same but almost coincident. To graphically give a little jitter to eliminate ties, the OLS errors (blue dots) included artificial additions. Bottom, however, after applying the prediction equation to each subject, the test errors in the form of mean absolute values were calculated, showing considerable differences between the OLS and PLS methods.
After constructing the prediction equation in the training data set, the validation was performed in the test data set (also called the validation data set), and test errors (also called validation errors) were calculated for each subject. We applied the leave-1-out crossvalidation technique. The test errors were also used to determine the optimal number of PLS components. From the statistical and clinical viewpoints, the test error is the most important criterion in determining the best prediction model. To restate, the lower the test errors, the better the prediction model becomes. Plus and minus errors of the prediction could cancel each other out when mean errors are being calculated.35,36 Therefore, test errors in the forms of absolute values and root mean square error values of prediction were used as the decision criteria for the predictive performance. Having chosen the final prediction model, we compared the prediction accuracy between the OLS and PLS methods, and between the 1-jaw and 2-jaw surgery patients.
The free statistics software language R was used. It runs on a wide variety of UNIX platforms, Windows (Microsoft), and MacOS (Apple Inc, Cupertino, Calif).37 The entire data and complete results without personal patient information will be available upon request to the authors. RESULTS
At the stage of building prediction equations, the goodness of fit or the quality of model fitting can be expressed as the extent of training errors. The training errors from both the OLS and PLS methods were negligible or trivial, as shown in Figure 2 (top). However, after applying the prediction equation to each subject in all response variables, the test errors showed significantly lower errors in the PLS method than in the conventional OLS method (Fig 2 [bottom]). A root mean square error of prediction curve was used to select the best prediction model. Figure 3
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Fig 3. Root mean square errors of prediction curve in the training data set (top) and in the test data set (bottom) after validation of the prediction equation to each subject. The increasing number of PLS components decreased the training error (top). However, increased complexity of the prediction equation also increased the test error (bottom). The PLS prediction equation with 30 components was selected as the final PLS prediction model.
shows the root mean square error of prediction curves in both the training and the test data set for a selected landmark. In building prediction equations with the training set, the more components that were included, the smaller the predicton errors obtained. The full prediction model with the entire components had no error in the training set. However, when validating the equation to each subject, there was an optimum number of components to minimize prediction error in the test set (Fig 3). Typically, we chose the smallest model that minimized the expected prediction error.12 Therefore, the PLS prediction equation with 30 components was selected as the final prediction model.
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Soft-tissue prediction accuracy was compared between 1-jaw (mandibular setback) and 2-jaw surgery patients; this is given in Table II. Among all 32 softtissue landmarks, 12 were chosen to concisely report the results. There was no statistically significant difference in the test errors between the 1-jaw and 2-jaw surgeries. For illustrative purpose, 4 patients were selected to visualize the prediction results between the OLS and PLS methods (Fig 4). The PLS method appeared to perform better than the OLS method in simulating both the mandibular setback and the 2-jaw surgeries. Superimposition and comparison with the actual outcome showed that the PLS predictions were closer to the actual outcome than the OLS predictions, especially in describing lip curvatures.
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Table II. Comparison of test errors between 1-jaw (mandibular setback) and 2-jaw surgery patients Anteroposterior error (x direction, mm) Soft-tissue landmark Soft-tissue A-point Superior labial sulcus Labrale superius Upper lip Stomion Lower lip Labrale inferius Soft-tissue B-point Protuberance menti Pogonion Gnathion Menton
1-jaw (n 5 71) 0.86 0.85 1.02 1.07 1.43 1.29 1.21 1.14 1.17 1.36 1.92 3.96
2-jaw (n 5 133) 0.92 1.03 1.23 1.28 1.54 1.26 1.18 1.26 1.23 1.33 1.74 4.16
Vertical error (y direction, mm) P value 0.517 0.059 0.072 0.092 0.491 0.828 0.857 0.346 0.649 0.822 0.401 0.692
1-jaw (n 5 71) 1.05 1.50 1.38 1.37 0.96 1.60 1.85 1.93 1.79 2.08 1.69 1.61
2-jaw (n 5 133) 1.19 1.40 1.35 1.32 1.00 1.48 1.58 1.92 2.05 2.45 1.58 1.45
P value 0.265 0.535 0.828 0.698 0.717 0.471 0.196 0.962 0.258 0.208 0.625 0.390
The unit of test errors was calculated as absolute values in millimeters after validation of each subject. The t tests were performed to compare the errors.
Fig 4. Prediction results for 4 patients (A-D): presurgical (left) and actual outcome photos after surgery (right). The prediction profiles of the OLS (black dashed lines) and the PLS (red dotted lines) methods were superimposed on the actual photos. Comparisons with the actual outcomes showed that the PLS predictions were closer to the actual outcome than the OLS predictions, especially in describing lip curvature. The predicted points were connected by the Bezier line-smoothing function. Refer to the text for the explanation. DISCUSSION
The PLS method in this study demonstrated significantly more accurate predictions than did the conventional OLS method for the mandibular setback and bimaxillary surgery patients (Figs 2 and 4). The training error can be used as a measure of goodness of fit,
whereas the test error implies the validity of a prediction model. The training errors from the 2 methods were almost null. However, the accuracy of test error in the test data set is more important than the training error. The OLS method fitted the training data set perfectly but failed to predict the test data well. When a prediction
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model fits well during building a prediction equation but fails to predict in a new data set, this phenomenon is called overfitting.12 To restate, the OLS method had the disadvantage of overfitting. This phenomenon of the OLS overfitting implies that conventional OLS methods are not satisfactory for complicated softtissue predictions. The PLS prediction showed improved accuracy when compared with current commercial software programs, Quick Ceph (Quick Ceph Systems, San Diego, Calif) and V-Ceph (Osstem, Seoul, Korea). These programs provide a 1:1 soft-tissue ratio setting for the movement of the corresponding hard tissues. Although the exact algorithms for these programs are unknown and would be confidential, based on their less accurate results, the software programs probably use the simple OLS method as their algorithm. After we determined the superiority of the PLS method over the OLS method, the next step was choosing the best prediction model. The best prediction model can be defined as the simplest model that minimizes the test error. For the model selection criteria, a square type of error—root mean square error of prediction—was used (Fig 3). Root mean square error of prediction has been frequently used to assess prediction performance and to choose the optimal number of components in principal components regression.38,39 Most of the reported inaccuracies in the soft-tissue predictions were the upper lip,7,17 lower lip,5,16,30,40-42 and labiomental fold.27,31,43 This was not the case for the PLS results, as shown in Figure 2. Although the results of the PLS predictions in this study were not perfect, the improved accuracy seems obvious. Furthermore, the authors of previous studies could not analyze and interpret more than 1 type of surgery or more than 1 vector of movement in their investigations. Since conventional methods could not properly handle the complex data structures, only identical surgical procedures could be analyzed. Isolated mandibular prognathism occurs in a relatively small portion of Class III patients.2,29 Therefore, the combination of LeFort I osteotomy of the maxilla and mandibular setback surgery seems to be the current trend for skeletal Class III treatment.28,44 With previous prediction methods, even an additional genioplasty was considered a confounding variable when predicting soft-tissue responses.3 For patients who undergo maxillary surgery or genioplasty, the vectors of movement are not uniform. Thus, patients undergoing an additional jaw surgery would have less predictable results than those undergoing a relatively simple mandibular setback surgery.5,27,30 Even though the prediction of 2-jaw surgery has the potential for greater errors, in our use of the
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PLS method, there was no statistically significant difference in the prediction errors between 1-jaw and 2-jaw surgeries, in either the vertical or the anteroposterior direction. From the values obtained and the comparison between the 2 groups, apparently the type of surgery did not influence the accuracy of prediction (Table II). After surgery, a hard-tissue landmark will most likely not match the corresponding presurgical landmark. For example, the most anterior point in the chin area (pogonion) after surgery will probably not indicate pogonion before mandibular surgery. By the same token, because of repositioning and rotation of the mandible, the new most anterior soft-tissue point of the chin (soft-tissue pogonion) may not reflect the presurgical soft-tissue pogonion.13,23 As in most commercial software, a Bezier spline function is used to connect the landmark points, creating a gentle smooth curve.6 By using the Bezier spline function, a prediction error (Figs 2 and 3) that is located along the Bezier curve (Fig 4) becomes more tolerable. We explored the complex relationship between predictor variables and soft-tissue responses by invoking an intricate multivariate statistical analysis. As an example, to identify factors that might influence the soft-tissue response, we depicted a loading plot for the anteroposterior lower lip response. As depicted in Figure 5, the loading value indicates the magnitude of each predictor variable in predicting the response. The loading values are useful not only in determining the influence of each variable, but also in developing computer algorithms. The loading pattern showed that several predictor variables had higher values of influence than others. For example, the sex predictor variable had an important role among factor variables. Consistent with the previous report, this finding signifies that soft-tissue movement in response to skeletal repositioning is somewhat greater in female than in male subjects.17 Three additional conclusions can also be plausibly drawn from the loading plot. (1) Both the presurgical skeletal and soft-tissue characteristics, as well as the amount of surgical repositioning, contributed to predicting the soft-tissue response after surgery. (2) When predicting a soft-tissue response in the x-axis, anteroposterior variables had higher loading values than did vertical predictor variables. However, the vertical predictor variables did not have a minor or trivial role but had considerable influence on the anteroposterior outcome. (3) It was also notable that some neighboring soft-tissue landmarks and all skeletal landmarks as a whole had a greater influence on the predictions of specific softtissue landmarks than did the presurgical landmark of that actual soft-tissue point (ie, presurgical lower lip
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Fig 5. A loading plot of the first 3 major PLS components for the anteroposterior position (x-axis) of the lower lip. The loading value indicates the magnitude of the predictor variables in predicting the response. The number of components indicates the order of power of influence among the PLS components. Components are the latent variables that showed which predictor variables played important roles in predicting the lower lip position. Predictor variables were sorted by anteroposterior variables and vertical axes for convenience. Among the factor variables, the sex variable (blue arrow) had an important role. Both the presurgical skeletal and soft-tissue characteristics, as well as the amount of surgical repositioning, contributed to predicting the soft-tissue response after surgery. It was obvious that when the response variable was an anteroposterior response, the anteroposterior variables exerted more influence than did the vertical variables. To restate, when predicting the lower lip response in the x-axis, in addition to the anteroposterior variables, the vertical variables participate to some extent. It was also notable that some neighboring soft-tissue landmarks and all skeletal landmarks as a whole had a greater influence on the predictions of specific soft-tissue landmarks than did the presurgical landmark of that actual soft-tissue point (ie, the presurgical lower lip predictor variable indicated with the red arrow at the top of the figure).
predictor variable indicated with a red arrow at the top of Fig 5). The complexity of these relationships is the reason that the overly simplistic conventional OLS predictions or the simple 1:1 ratio statistics on which current software programs have depended demonstrate lower accuracy. The multivariate PLS method has been developing rapidly with the advent of high-speed computers. Computer-assisted predictions have become an integral part of surgical-orthodontic treatment planning.40 However, existing software predictions still result in considerable errors. One cause of the errors might be oversimplistic OLS algorithms that were integrated into most commercially available computer programs. These programs have never been clearly published or opened to the public. We hope that the soft-tissue prediction method presented in this study will provide a practical algorithm to improve surgical treatment simulation programs, and that orthodontic clinicians will expect their contemporary orthodontic imaging software to incorporate such advances. Detailed algorithms written in language R will be opened to the public through general public licensure or by request to the authors. We hope that this soft-tissue prediction method will provide a practical algorithm in developing
a more accurate surgical and orthodontic treatment simulation program. A limitation of this study was that all subjects were of Korean ethnicity. The Korean population demonstrates approximately a 19% rate of Class III malocclusion; this provided an ample sample from which to derive Class III subjects.45 However, postsurgery soft-tissue characteristics and behavior may vary among ethnicities. Therefore, a greater sample of the world's ethnic diversity should be examined and is worthy of further investigation. CONCLUSIONS
We observed that the more sophisticated PLS mathematical method predicted better than the conventional OLS method, and by applying the multivariate PLS method, prediction errors can be minimized. Among the methods and variables tested, the multivariate PLS prediction model based on about 30 latent variable components showed the best prediction quality. Based on our findings, we propose that the PLS method might provide an improved algorithm in predicting surgical outcomes after mandibular setback or bimaxillary surgical correction for Class III patients.
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ACKNOWLEDGMENTS
We thank Dr Ho-Jin Lee for assistance with the database, Dr Soo-Heang Eo for statistical consultation, and the Seoul National University Dental Hospital oral and maxillofacial surgeons for their contributions through the years. REFERENCES 1. Joss CU, Joss-Vassalli IM, Kiliaridis S, Kuijpers-Jagtman AM. Soft tissue profile changes after bilateral sagittal split osteotomy for mandibular advancement: a systematic review. J Oral Maxillofac Surg 2010;68:1260-9. 2. Joss CU, Vassalli IM, Thuer UW. Stability of soft tissue profile after mandibular setback in sagittal split osteotomies: a longitudinal and long-term follow-up study. J Oral Maxillofac Surg 2008;66: 1610-6. 3. Joss CU, Joss-Vassalli IM, Berge SJ, Kuijpers-Jagtman AM. Soft tissue profile changes after bilateral sagittal split osteotomy for mandibular setback: a systematic review. J Oral Maxillofac Surg 2010;68:2792-801. 4. McCollum AG, Dancaster JT, Evans WG, Becker PJ. Sagittal softtissue changes related to the surgical correction of maxillarydeficient Class III malocclusions. Semin Orthod 2009;15:172-84. 5. Eckhardt CE, Cunningham SJ. How predictable is orthognathic surgery? Eur J Orthod 2004;26:303-9. 6. Smith JD, Thomas PM, Proffit WR. A comparison of current prediction imaging programs. Am J Orthod Dentofacial Orthop 2004; 125:527-36. 7. Kasai K. Soft tissue adaptability to hard tissues in facial profiles. Am J Orthod Dentofacial Orthop 1998;113:674-84. 8. Suh HY, Lee SJ, Lee YS, Donatelli RE, Wheeler TT, Kim SH, et al. A more accurate method of predicting soft tissue changes after mandibular setback surgery. J Oral Maxillofac Surg 2012;70: e553-62. 9. Krishnan A, Williams LJ, McIntosh AR, Abdi H. Partial least squares (PLS) methods for neuroimaging: a tutorial and review. Neuroimage 2011;56:455-75. 10. Lee SJ. Modified partial least squares method incorporating mixed effect model [Dissertation]. Seoul, Korea: Korea University; 2012. p. 1-50. 11. Wehrens R. Chemometrics with R: multivariate data analysis in the natural sciences and life sciences. Heidelberg, Germany: Springer; 2011. 12. Hastie T, Tibshirani R, Friedman J. The elements of statistical learning. Data mining, inference, and prediction. New York: Springer Verlag; 2009. 13. Lee HJ, Suh HY, Lee YS, Lee SJ, Donatelli RE, Dolce C, et al. A better statistical method of predicting postsurgery soft tissue response in Class II patients. Angle Orthod 2014;84:322-8. 14. Alves PV, Mazucheli J, Vogel CJ, Bolognese AM. How the lower face soft tissue changes after mandibular advancement or setback. J Craniofac Surg 2008;19:593-8. 15. Burden D, Johnston C, Kennedy D, Harradine N, Stevenson M. A cephalometric study of Class II malocclusions treated with mandibular surgery. Am J Orthod Dentofacial Orthop 2007;131:7.e1-8. 16. Naoumova J, Soderfeldt B, Lindman R. Soft tissue profile changes after vertical ramus osteotomy. Eur J Orthod 2008;30:359-65. 17. Mobarak KA, Krogstad O, Espeland L, Lyberg T. Factors influencing the predictability of soft tissue profile changes following mandibular setback surgery. Angle Orthod 2001;71:216-27.
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18. Chou JI, Fong HJ, Kuang SH, Gi LY, Hwang FY, Lai YC, et al. A retrospective analysis of the stability and relapse of soft and hard tissue change after bilateral sagittal split osteotomy for mandibular setback of 64 Taiwanese patients. J Oral Maxillofac Surg 2005;63:355-61. 19. Kolokitha OE. Validity of a manual soft tissue profile prediction method following mandibular setback osteotomy. Eur J Dent 2007;1:202-11. 20. Dolce C, Hatch JP, Van Sickels JE, Rugh JD. Five-year outcome and predictability of soft tissue profiles when wire or rigid fixation is used in mandibular advancement surgery. Am J Orthod Dentofacial Orthop 2003;124:249-56. 21. Kau CH, Cronin A, Durning P, Zhurov AI, Sandham A, Richmond S. A new method for the 3D measurement of postoperative swelling following orthognathic surgery. Orthod Craniofac Res 2006;9: 31-7. 22. Ksiezycki-Ostoya BK, McCollum AG, Becker PJ. Sagittal soft-tissue changes of the lower lip and chin associated with surgical maxillary impaction and consequent mandibular autorotation. Semin Orthod 2009;15:185-95. 23. McCollum AG, Gardener GJ, Evans WG, Becker PJ. Soft-tissue changes related to mandibular advancement surgery. Semin Orthod 2009;15:161-71. 24. Gjorup H, Athanasiou AE. Soft-tissue and dentoskeletal profile changes associated with mandibular setback osteotomy. Am J Orthod Dentofacial Orthop 1991;100:312-23. 25. Stella JP, Streater MR, Epker BN, Sinn DP. Predictability of upper lip soft tissue changes with maxillary advancement. J Oral Maxillofac Surg 1989;47:697-703. 26. Chun H, Keles S. Sparse partial least squares regression for simultaneous dimension reduction and variable selection. J R Stat Soc Series B Stat Methodol 2010;72:3-25. 27. Jones RM, Khambay BS, McHugh S, Ayoub AF. The validity of a computer-assisted simulation system for orthognathic surgery (CASSOS) for planning the surgical correction of class III skeletal deformities: single-jaw versus bimaxillary surgery. Int J Oral Maxillofac Surg 2007;36:900-8. 28. Johnston C, Burden D, Kennedy D, Harradine N, Stevenson M. Class III surgical-orthodontic treatment: a cephalometric study. Am J Orthod Dentofacial Orthop 2006;130:300-9. 29. Bailey LJ, Cevidanes LH, Proffit WR. Stability and predictability of orthognathic surgery. Am J Orthod Dentofacial Orthop 2004;126: 273-7. 30. Kaipatur NR, Flores-Mir C. Accuracy of computer programs in predicting orthognathic surgery soft tissue response. J Oral Maxillofac Surg 2009;67:751-9. 31. Enacar A, Taner T, Toroglu S. Analysis of soft tissue profile changes associated with mandibular setback and double-jaw surgeries. Int J Adult Orthodon Orthognath Surg 1999;14:27-35. 32. Dolce C, Van Sickels JE, Bays RA, Rugh JD. Skeletal stability after mandibular advancement with rigid versus wire fixation. J Oral Maxillofac Surg 2000;58:1219-27. 33. Lindberg W, Persson JA, Wold S. Partial least-squares method for spectrofluorimetric analysis of mixtures of humic-acid and ligninsulfonate. Anal Chem 1983;55:643-8. 34. Akaike H. A new look at the statistical model identification. IEEE Trans Automatic Control 1974;19:716-23. 35. Donatelli RE, Lee SJ. How to report reliability in orthodontic research: part 2. Am J Orthod Dentofacial Orthop 2013;144: 315-8. 36. Donatelli RE, Lee SJ. How to report reliability in orthodontic research: part 1. Am J Orthod Dentofacial Orthop 2013;144: 156-61.
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37. R Development Core Team. R: a language and environment for statistical computing. Vienna, Austria: R Foundation for Statistical Computing; 2014. 38. Martens H, Næs T. Multivariate calibration. Chichester, United Kingdom: Wiley; 1989. 39. Mevik BH, Cederkvist HR. Mean squared error of prediction (MSEP) estimates for principal component regression (PCR) and partial least squares regression (PLSR). J Chemometrics 2004; 18:422-9. 40. Pektas ZO, Kircelli BH, Cilasun U, Uckan S. The accuracy of computer-assisted surgical planning in soft tissue prediction following orthognathic surgery. Int J Med Robot 2007;3:64-71. 41. Sameshima GT, Kawakami RK, Kaminishi RM, Sinclair PM. Predicting soft tissue changes in maxillary impaction surgery: a comparison of two video imaging systems. Angle Orthod 1997;67:347-54.
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42. Donatsky O, Bjorn-Jorgensen J, Hermund NU, Nielsen H, Holmqvist-Larsen M, Nerder PH. Immediate postoperative outcome of orthognathic surgical planning, and prediction of positional changes in hard and soft tissue, independently of the extent and direction of the surgical corrections required. Br J Oral Maxillofac Surg 2011;49:386-91. 43. Power G, Breckon J, Sherriff M, McDonald F. Dolphin imaging software: an analysis of the accuracy of cephalometric digitization and orthognathic prediction. Int J Oral Maxillofac Surg 2005;34: 619-26. 44. Jung MH. Age, extraction rate and jaw surgery rate in Korean orthodontic clinics and small dental hospitals. Korean J Orthod 2012;42:80-6. 45. Lee SJ, Kim TW, Suhr CH. Recognition of malocclusion and orthodontic treatment. Korean J Orthod 1994;24:193-8.
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A more accurate soft-tissue prediction model for Class III 2-jaw surgeries Yun-Sik Lee,a Hee-Yeon Suh,a Shin-Jae Lee,b and Richard E. Donatellic Seoul, Korea, and Gainesville, Fla Introduction: The use of bimaxillary surgeries to treat Class III malocclusions makes the results of the surgeries more complicated to estimate accurately. Therefore, our objective was to develop an accurate soft-tissue prediction model that can be universally applied to Class III surgical-orthodontic patients regardless of the type of surgical correction: maxillary or mandibular surgery with or without genioplasty. Methods: The subjects of this study consisted of 204 mandibular setback patients who had undergone the combined surgicalorthodontic correction of severe skeletal Class III malocclusions. Among them, 133 patients had maxillary surgeries, and 81 patients received genioplasties. The prediction model included 226 independent and 64 dependent variables. Two prediction methods, the conventional ordinary least squares method and the partial least squares (PLS) method, were compared. When evaluating the prediction methods, the actual surgical outcome was the gold standard. After fitting the equations, test errors were calculated in absolute values and root mean square values through the leave-1-out cross-validation method. Results: The validation result demonstrated that the multivariate PLS prediction model with 30 orthogonal components showed the best prediction quality among others. With the PLS method, the pattern of prediction errors between 1-jaw and 2-jaw surgeries did not show a significantly difference. Conclusions: The multivariate PLS prediction model based on about 30 latent variables might provide an improved algorithm in predicting surgical outcomes after 1-jaw and 2-jaw surgical corrections for Class III patients. (Am J Orthod Dentofacial Orthop 2014;146:724-33)
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ne of the most important considerations in orthodontic treatment planning is an esthetic posttreatment patient profile.1-4 Many orthodontic residency programs teach this as the primary objective from which all other clinical treatment decisions are based. In patients with significant skeletal discrepancies requiring orthognathic surgery, decisions regarding the details of the surgical methods would be much easier if the exact results of the surgery could be accurately predicted and portrayed while still in the treatment
a Postgraduate student, Department of Orthodontics, School of Dentistry and Dental Research Institute, Seoul National University, Seoul, Korea. b Professor and chair, Department of Orthodontics, School of Dentistry and Dental Research Institute, Seoul National University, Seoul, Korea. c Clinical assistant professor, Department of Orthodontics, College of Dentistry, University of Florida, Gainesville, Fla. All authors have completed and submitted the ICMJE Form for Disclosure of Potential Conflicts of Interest, and none were reported. Supported by the Basic Science Research Program funded by the Korean government (NRF 2012-0007574) and partly by grant number 02-2014-0003 from the Seoul National University Dental Hospital Research Fund. Address correspondence to: Shin-Jae Lee, Department of Orthodontics, Seoul National University School of Dentistry and Dental Research Institute, 101 Daehakro, Jongro-Gu, Seoul 110-749, Korea; e-mail, [email protected]. Submitted, May 2014; revised and accepted, August 2014. 0889-5406/$36.00 Copyright Ó 2014 by the American Association of Orthodontists. http://dx.doi.org/10.1016/j.ajodo.2014.08.010
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planning stage. Consequently, the orthodontic literature is replete with studies regarding the associations between surgical skeletal repositioning and soft-tissue responses. Imagine being able to perform a simulation of various surgeries on a commercially available 3-dimensional software-rendered model that produced predictable results, reliable enough to dictate treatment decisions and accurate enough to show a patient what his or her eventual appearance would be. This would increase a patient's understanding and acceptance of the recommended treatment.5,6 Of course, prediction software is currently available, but anyone who has used it realizes that the predictions are often rudimentary, unrealistic, and unreliable. As analysis techniques advance, orthodontic software should continue to integrate these advances to better serve clinicians. Consequently, clinicians should come to expect more from their orthodontic imaging software programs. As any clinician involved with patients with severe Class III malocclusions quickly realizes, the soft-tissue profile does not directly, 1 to 1, follow the surgical changes of the underlying bony structures.7,8 The partial least squares (PLS) method is a comparatively new way of formulating prediction equations, and
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its application to various scientific and biologic disciplines from chemical engineering to brain image analysis is becoming increasingly widespread.9-13 Applying the PLS method is advantageous when there are many variables and they are highly correlated. The merit of the PLS method is its capability of taking correlation structures into account, controlling not only for the correlations between the predictor and response variables but also for the correlations within the predictor variables and the response variables. Recently, a study applying the multivariate PLS method to mandibular setback surgeries demonstrated considerably more accurately predictions than the conventional ordinary least squares (OLS) method.8 The conventional OLS method was determined to be unsatisfactory when there were many correlated variables. Among the variables considered when predicting the soft-tissue response to surgery are the patient's age,2,3,14-17 sex,2,16-19 time after surgery,20,21 and presurgical soft-tissue characteristics, including tissue thickness measured at various landmarks.4,7,17,22-25 These various factors can be considered in the PLS method through orthogonal linear combinations that can extract a small number of significant components that are combinations of the original variables.26 In addition, the improved accuracy of the PLS method is most likely because the soft-tissue response at a specific point highly depends on its adjacent soft-tissue response: ie, the interdependency of soft-tissue points.13 However, the aforementioned investigation had only been performed for mandibular setback surgeries alone. Class III 2-jaw surgery patients had not yet been included.8 Including an additional surgery has a great influence on the soft-tissue profile changes, greatly complicating the prediction.3,27 Consequently, most studies have reported only 1 specific maxillofacial surgery in their soft-tissue analyses. Because bimaxillary surgery produces more stable results than single-jaw mandibular procedures in Class III correction, clinically, there has been an increase in its use.28,29 However, current prediction programs for bimaxillary surgery are less predictable than for 1-jaw surgery.5,27,30,31 It is especially difficult to accurately determine the changes in the soft-tissue profile when mandibular setback surgeries are combined with other surgical procedures, such as a LeFort I osteotomy or a genioplasty. Our aim in this study was to develop an accurate softtissue prediction method that can be applied to various methods of Class III surgical correction: mandibular surgery and maxillary surgery, and genioplasty. In addition, we discuss the steps in choosing an optimal multivariate PLS prediction model.
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MATERIAL AND METHODS
The institutional review board of Seoul National University School of Dentistry for the protection of human subjects reviewed and approved the research protocol (number S-D 20140018). The subjects consisted of 204 patients (103 women and 101 men with an average age of 24 years) who had undergone surgical correction of a severe Class III malocclusion. All subjects were of Korean ethnicity. All patients received mandibular setback surgery, and 133 patients had LeFort I maxillary osteotomy. Genioplasty was performed for 81 patients. Table I provides further details of the patients with regard to ages at surgery and some cephalometric and occlusal characteristics. All patients had ceased growing and were healthy, without cleft lip and palate, injury, or craniofacial syndrome. Preoperative lateral cephalograms were taken close to the time of surgical correction. The data were collected prospectively with the postoperative radiographs taken at least 4 months (average, 9.1 months) after surgery to allow any residual soft-tissue swelling to resolve.32 Thirty-nine skeletal landmarks and 32 soft-tissue landmarks from glabella to the terminal point were identified. With its origin at sella, the vertical reference was established perpendicular to the sella-nasion line plus 7 . Using a custom digitizing program via C# programming language (Microsoft, Redmond, Wash), the coordinates of every landmark on each tracing were sequentially computed in relation to the x and y reference system. The prediction process had 2 stages: building a prediction equation through model fitting, and validation by applying the equation to each subject and calculating test errors. Figure 1 is a flow diagram for establishing the prediction model in this study. Two hundred twenty-six predictor variables and 64 response variables were entered into the prediction equation. The predictor variables included each patient's age and sex, time after surgery, 1-jaw vs 2-jaw surgery, conjunctive genioplasty, amount of asymmetry, 39 presurgical skeletal landmarks, 32 soft-tissue landmarks before surgery, and 78 (39 3 2) variables measuring the surgical skeletal repositioning in both the anteroposterior and vertical directions. The 64 (32 3 2) response variables were set as the soft-tissue position after surgery in the 32 soft-tissue landmarks in both the x-axis and the y-axis. The training data set was numerically treated via centering and normalization. The centering made the following computations numerically well conditioned.33 The normalization gave each variable equal influence in the initial stage of the data analysis.
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Table I. Subjects' sex, age, and other characteristics Variable Age (y) Female (n 5 103) Male (n 5 101) Time after surgery (mo) Maxillary surgery No (n 5 71) Yes (n 5 133) Genioplasty No (n 5 123) Yes (n 5 81) Overjet before surgery (mm) Overbite before surgery (mm) Amount of surgical repositioning at Point A (mm)* (n 5 133) Anteroposterior repositioning Vertical repositioning Amount of surgical repositioning at Point B (mm)* (n 5 204) Anteroposterior repositioning Vertical repositioning
Mean
SD
Minimum
Maximum
23.8 23.6 9.1
5.1 3.5 3.9
16.0 18.8 3.7
50.5 39.1 29.4
5.8 0.2
3.8 1.8
19.6 5.4
2.2 5.9
1.4 1.1
1.9 2.3
4.4 7.1
7.0 4.4
7.3 2.9
3.8 4.2
24.1 14.8
4.1 11.4
*A negative value indicates either a posterior direction or a superior direction during surgical repositioning.
Fig 1. Flow diagram of establishing the prediction model in this study.
Two multivariate methods of constructing prediction equations were developed using the conventional OLS and the PLS methods. First, the conventional OLS method covers from a simple ratio statistics or correlation analysis to a more complex multiple linear regression. In this study, the multivariate multiple linear regression with forward variable selection was used. The model was selected with the Akaike information criterion; the lower this criterion, the better the model will be.34 Second, the PLS method is a multivariate regression with latent variables (also called components). This latent
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variable method was used to focus the information of a large data set into a few underlying components or factors, leaving most of the measurement noise behind as residuals. Although some mathematical details would have been needed, we emphasized the methods and results rather than their theoretical background. With regard to more detailed mathematical properties and interpretations for the PLS method, please refer to the studies of Hastie et al12 and Wehrens.11 In the training data set, the absolute error values after fitting the prediction equation (also called training error) were measured.
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Fig 2. Training errors (top) and test errors (bottom) in the mean absolute values showing the fitting quality of the OLS and the PLS predictions. Top, the training errors at the stage of constructing the equation were negligible and close to zero. The training error values were not perfectly the same but almost coincident. To graphically give a little jitter to eliminate ties, the OLS errors (blue dots) included artificial additions. Bottom, however, after applying the prediction equation to each subject, the test errors in the form of mean absolute values were calculated, showing considerable differences between the OLS and PLS methods.
After constructing the prediction equation in the training data set, the validation was performed in the test data set (also called the validation data set), and test errors (also called validation errors) were calculated for each subject. We applied the leave-1-out crossvalidation technique. The test errors were also used to determine the optimal number of PLS components. From the statistical and clinical viewpoints, the test error is the most important criterion in determining the best prediction model. To restate, the lower the test errors, the better the prediction model becomes. Plus and minus errors of the prediction could cancel each other out when mean errors are being calculated.35,36 Therefore, test errors in the forms of absolute values and root mean square error values of prediction were used as the decision criteria for the predictive performance. Having chosen the final prediction model, we compared the prediction accuracy between the OLS and PLS methods, and between the 1-jaw and 2-jaw surgery patients.
The free statistics software language R was used. It runs on a wide variety of UNIX platforms, Windows (Microsoft), and MacOS (Apple Inc, Cupertino, Calif).37 The entire data and complete results without personal patient information will be available upon request to the authors. RESULTS
At the stage of building prediction equations, the goodness of fit or the quality of model fitting can be expressed as the extent of training errors. The training errors from both the OLS and PLS methods were negligible or trivial, as shown in Figure 2 (top). However, after applying the prediction equation to each subject in all response variables, the test errors showed significantly lower errors in the PLS method than in the conventional OLS method (Fig 2 [bottom]). A root mean square error of prediction curve was used to select the best prediction model. Figure 3
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Fig 3. Root mean square errors of prediction curve in the training data set (top) and in the test data set (bottom) after validation of the prediction equation to each subject. The increasing number of PLS components decreased the training error (top). However, increased complexity of the prediction equation also increased the test error (bottom). The PLS prediction equation with 30 components was selected as the final PLS prediction model.
shows the root mean square error of prediction curves in both the training and the test data set for a selected landmark. In building prediction equations with the training set, the more components that were included, the smaller the predicton errors obtained. The full prediction model with the entire components had no error in the training set. However, when validating the equation to each subject, there was an optimum number of components to minimize prediction error in the test set (Fig 3). Typically, we chose the smallest model that minimized the expected prediction error.12 Therefore, the PLS prediction equation with 30 components was selected as the final prediction model.
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Soft-tissue prediction accuracy was compared between 1-jaw (mandibular setback) and 2-jaw surgery patients; this is given in Table II. Among all 32 softtissue landmarks, 12 were chosen to concisely report the results. There was no statistically significant difference in the test errors between the 1-jaw and 2-jaw surgeries. For illustrative purpose, 4 patients were selected to visualize the prediction results between the OLS and PLS methods (Fig 4). The PLS method appeared to perform better than the OLS method in simulating both the mandibular setback and the 2-jaw surgeries. Superimposition and comparison with the actual outcome showed that the PLS predictions were closer to the actual outcome than the OLS predictions, especially in describing lip curvatures.
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Table II. Comparison of test errors between 1-jaw (mandibular setback) and 2-jaw surgery patients Anteroposterior error (x direction, mm) Soft-tissue landmark Soft-tissue A-point Superior labial sulcus Labrale superius Upper lip Stomion Lower lip Labrale inferius Soft-tissue B-point Protuberance menti Pogonion Gnathion Menton
1-jaw (n 5 71) 0.86 0.85 1.02 1.07 1.43 1.29 1.21 1.14 1.17 1.36 1.92 3.96
2-jaw (n 5 133) 0.92 1.03 1.23 1.28 1.54 1.26 1.18 1.26 1.23 1.33 1.74 4.16
Vertical error (y direction, mm) P value 0.517 0.059 0.072 0.092 0.491 0.828 0.857 0.346 0.649 0.822 0.401 0.692
1-jaw (n 5 71) 1.05 1.50 1.38 1.37 0.96 1.60 1.85 1.93 1.79 2.08 1.69 1.61
2-jaw (n 5 133) 1.19 1.40 1.35 1.32 1.00 1.48 1.58 1.92 2.05 2.45 1.58 1.45
P value 0.265 0.535 0.828 0.698 0.717 0.471 0.196 0.962 0.258 0.208 0.625 0.390
The unit of test errors was calculated as absolute values in millimeters after validation of each subject. The t tests were performed to compare the errors.
Fig 4. Prediction results for 4 patients (A-D): presurgical (left) and actual outcome photos after surgery (right). The prediction profiles of the OLS (black dashed lines) and the PLS (red dotted lines) methods were superimposed on the actual photos. Comparisons with the actual outcomes showed that the PLS predictions were closer to the actual outcome than the OLS predictions, especially in describing lip curvature. The predicted points were connected by the Bezier line-smoothing function. Refer to the text for the explanation. DISCUSSION
The PLS method in this study demonstrated significantly more accurate predictions than did the conventional OLS method for the mandibular setback and bimaxillary surgery patients (Figs 2 and 4). The training error can be used as a measure of goodness of fit,
whereas the test error implies the validity of a prediction model. The training errors from the 2 methods were almost null. However, the accuracy of test error in the test data set is more important than the training error. The OLS method fitted the training data set perfectly but failed to predict the test data well. When a prediction
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model fits well during building a prediction equation but fails to predict in a new data set, this phenomenon is called overfitting.12 To restate, the OLS method had the disadvantage of overfitting. This phenomenon of the OLS overfitting implies that conventional OLS methods are not satisfactory for complicated softtissue predictions. The PLS prediction showed improved accuracy when compared with current commercial software programs, Quick Ceph (Quick Ceph Systems, San Diego, Calif) and V-Ceph (Osstem, Seoul, Korea). These programs provide a 1:1 soft-tissue ratio setting for the movement of the corresponding hard tissues. Although the exact algorithms for these programs are unknown and would be confidential, based on their less accurate results, the software programs probably use the simple OLS method as their algorithm. After we determined the superiority of the PLS method over the OLS method, the next step was choosing the best prediction model. The best prediction model can be defined as the simplest model that minimizes the test error. For the model selection criteria, a square type of error—root mean square error of prediction—was used (Fig 3). Root mean square error of prediction has been frequently used to assess prediction performance and to choose the optimal number of components in principal components regression.38,39 Most of the reported inaccuracies in the soft-tissue predictions were the upper lip,7,17 lower lip,5,16,30,40-42 and labiomental fold.27,31,43 This was not the case for the PLS results, as shown in Figure 2. Although the results of the PLS predictions in this study were not perfect, the improved accuracy seems obvious. Furthermore, the authors of previous studies could not analyze and interpret more than 1 type of surgery or more than 1 vector of movement in their investigations. Since conventional methods could not properly handle the complex data structures, only identical surgical procedures could be analyzed. Isolated mandibular prognathism occurs in a relatively small portion of Class III patients.2,29 Therefore, the combination of LeFort I osteotomy of the maxilla and mandibular setback surgery seems to be the current trend for skeletal Class III treatment.28,44 With previous prediction methods, even an additional genioplasty was considered a confounding variable when predicting soft-tissue responses.3 For patients who undergo maxillary surgery or genioplasty, the vectors of movement are not uniform. Thus, patients undergoing an additional jaw surgery would have less predictable results than those undergoing a relatively simple mandibular setback surgery.5,27,30 Even though the prediction of 2-jaw surgery has the potential for greater errors, in our use of the
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PLS method, there was no statistically significant difference in the prediction errors between 1-jaw and 2-jaw surgeries, in either the vertical or the anteroposterior direction. From the values obtained and the comparison between the 2 groups, apparently the type of surgery did not influence the accuracy of prediction (Table II). After surgery, a hard-tissue landmark will most likely not match the corresponding presurgical landmark. For example, the most anterior point in the chin area (pogonion) after surgery will probably not indicate pogonion before mandibular surgery. By the same token, because of repositioning and rotation of the mandible, the new most anterior soft-tissue point of the chin (soft-tissue pogonion) may not reflect the presurgical soft-tissue pogonion.13,23 As in most commercial software, a Bezier spline function is used to connect the landmark points, creating a gentle smooth curve.6 By using the Bezier spline function, a prediction error (Figs 2 and 3) that is located along the Bezier curve (Fig 4) becomes more tolerable. We explored the complex relationship between predictor variables and soft-tissue responses by invoking an intricate multivariate statistical analysis. As an example, to identify factors that might influence the soft-tissue response, we depicted a loading plot for the anteroposterior lower lip response. As depicted in Figure 5, the loading value indicates the magnitude of each predictor variable in predicting the response. The loading values are useful not only in determining the influence of each variable, but also in developing computer algorithms. The loading pattern showed that several predictor variables had higher values of influence than others. For example, the sex predictor variable had an important role among factor variables. Consistent with the previous report, this finding signifies that soft-tissue movement in response to skeletal repositioning is somewhat greater in female than in male subjects.17 Three additional conclusions can also be plausibly drawn from the loading plot. (1) Both the presurgical skeletal and soft-tissue characteristics, as well as the amount of surgical repositioning, contributed to predicting the soft-tissue response after surgery. (2) When predicting a soft-tissue response in the x-axis, anteroposterior variables had higher loading values than did vertical predictor variables. However, the vertical predictor variables did not have a minor or trivial role but had considerable influence on the anteroposterior outcome. (3) It was also notable that some neighboring soft-tissue landmarks and all skeletal landmarks as a whole had a greater influence on the predictions of specific softtissue landmarks than did the presurgical landmark of that actual soft-tissue point (ie, presurgical lower lip
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Fig 5. A loading plot of the first 3 major PLS components for the anteroposterior position (x-axis) of the lower lip. The loading value indicates the magnitude of the predictor variables in predicting the response. The number of components indicates the order of power of influence among the PLS components. Components are the latent variables that showed which predictor variables played important roles in predicting the lower lip position. Predictor variables were sorted by anteroposterior variables and vertical axes for convenience. Among the factor variables, the sex variable (blue arrow) had an important role. Both the presurgical skeletal and soft-tissue characteristics, as well as the amount of surgical repositioning, contributed to predicting the soft-tissue response after surgery. It was obvious that when the response variable was an anteroposterior response, the anteroposterior variables exerted more influence than did the vertical variables. To restate, when predicting the lower lip response in the x-axis, in addition to the anteroposterior variables, the vertical variables participate to some extent. It was also notable that some neighboring soft-tissue landmarks and all skeletal landmarks as a whole had a greater influence on the predictions of specific soft-tissue landmarks than did the presurgical landmark of that actual soft-tissue point (ie, the presurgical lower lip predictor variable indicated with the red arrow at the top of the figure).
predictor variable indicated with a red arrow at the top of Fig 5). The complexity of these relationships is the reason that the overly simplistic conventional OLS predictions or the simple 1:1 ratio statistics on which current software programs have depended demonstrate lower accuracy. The multivariate PLS method has been developing rapidly with the advent of high-speed computers. Computer-assisted predictions have become an integral part of surgical-orthodontic treatment planning.40 However, existing software predictions still result in considerable errors. One cause of the errors might be oversimplistic OLS algorithms that were integrated into most commercially available computer programs. These programs have never been clearly published or opened to the public. We hope that the soft-tissue prediction method presented in this study will provide a practical algorithm to improve surgical treatment simulation programs, and that orthodontic clinicians will expect their contemporary orthodontic imaging software to incorporate such advances. Detailed algorithms written in language R will be opened to the public through general public licensure or by request to the authors. We hope that this soft-tissue prediction method will provide a practical algorithm in developing
a more accurate surgical and orthodontic treatment simulation program. A limitation of this study was that all subjects were of Korean ethnicity. The Korean population demonstrates approximately a 19% rate of Class III malocclusion; this provided an ample sample from which to derive Class III subjects.45 However, postsurgery soft-tissue characteristics and behavior may vary among ethnicities. Therefore, a greater sample of the world's ethnic diversity should be examined and is worthy of further investigation. CONCLUSIONS
We observed that the more sophisticated PLS mathematical method predicted better than the conventional OLS method, and by applying the multivariate PLS method, prediction errors can be minimized. Among the methods and variables tested, the multivariate PLS prediction model based on about 30 latent variable components showed the best prediction quality. Based on our findings, we propose that the PLS method might provide an improved algorithm in predicting surgical outcomes after mandibular setback or bimaxillary surgical correction for Class III patients.
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ACKNOWLEDGMENTS
We thank Dr Ho-Jin Lee for assistance with the database, Dr Soo-Heang Eo for statistical consultation, and the Seoul National University Dental Hospital oral and maxillofacial surgeons for their contributions through the years. REFERENCES 1. Joss CU, Joss-Vassalli IM, Kiliaridis S, Kuijpers-Jagtman AM. Soft tissue profile changes after bilateral sagittal split osteotomy for mandibular advancement: a systematic review. J Oral Maxillofac Surg 2010;68:1260-9. 2. Joss CU, Vassalli IM, Thuer UW. Stability of soft tissue profile after mandibular setback in sagittal split osteotomies: a longitudinal and long-term follow-up study. J Oral Maxillofac Surg 2008;66: 1610-6. 3. Joss CU, Joss-Vassalli IM, Berge SJ, Kuijpers-Jagtman AM. Soft tissue profile changes after bilateral sagittal split osteotomy for mandibular setback: a systematic review. J Oral Maxillofac Surg 2010;68:2792-801. 4. McCollum AG, Dancaster JT, Evans WG, Becker PJ. Sagittal softtissue changes related to the surgical correction of maxillarydeficient Class III malocclusions. Semin Orthod 2009;15:172-84. 5. Eckhardt CE, Cunningham SJ. How predictable is orthognathic surgery? Eur J Orthod 2004;26:303-9. 6. Smith JD, Thomas PM, Proffit WR. A comparison of current prediction imaging programs. Am J Orthod Dentofacial Orthop 2004; 125:527-36. 7. Kasai K. Soft tissue adaptability to hard tissues in facial profiles. Am J Orthod Dentofacial Orthop 1998;113:674-84. 8. Suh HY, Lee SJ, Lee YS, Donatelli RE, Wheeler TT, Kim SH, et al. A more accurate method of predicting soft tissue changes after mandibular setback surgery. J Oral Maxillofac Surg 2012;70: e553-62. 9. Krishnan A, Williams LJ, McIntosh AR, Abdi H. Partial least squares (PLS) methods for neuroimaging: a tutorial and review. Neuroimage 2011;56:455-75. 10. Lee SJ. Modified partial least squares method incorporating mixed effect model [Dissertation]. Seoul, Korea: Korea University; 2012. p. 1-50. 11. Wehrens R. Chemometrics with R: multivariate data analysis in the natural sciences and life sciences. Heidelberg, Germany: Springer; 2011. 12. Hastie T, Tibshirani R, Friedman J. The elements of statistical learning. Data mining, inference, and prediction. New York: Springer Verlag; 2009. 13. Lee HJ, Suh HY, Lee YS, Lee SJ, Donatelli RE, Dolce C, et al. A better statistical method of predicting postsurgery soft tissue response in Class II patients. Angle Orthod 2014;84:322-8. 14. Alves PV, Mazucheli J, Vogel CJ, Bolognese AM. How the lower face soft tissue changes after mandibular advancement or setback. J Craniofac Surg 2008;19:593-8. 15. Burden D, Johnston C, Kennedy D, Harradine N, Stevenson M. A cephalometric study of Class II malocclusions treated with mandibular surgery. Am J Orthod Dentofacial Orthop 2007;131:7.e1-8. 16. Naoumova J, Soderfeldt B, Lindman R. Soft tissue profile changes after vertical ramus osteotomy. Eur J Orthod 2008;30:359-65. 17. Mobarak KA, Krogstad O, Espeland L, Lyberg T. Factors influencing the predictability of soft tissue profile changes following mandibular setback surgery. Angle Orthod 2001;71:216-27.
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