Introduction
Overall and anterior Bolton ratios have been well covered in the orthodontic literature; however, little has been reported on posterior Bolton ratios. Considering the frequency of premolar extractions in the specialty, it would be relevant to know how the posterior occlusion is affected by premolar extractions. This study aimed to investigate how the posterior Bolton ratio is affected by the extraction of the 4 first premolars.
Methods
Fifty-five patients with Class I occlusion within 1 standard deviation of ideal anterior and overall Bolton ratios models were selected and digitized. Tooth widths were measured. Virtual extractions of 4 first premolars were performed, and a digital setup of anterior and remaining posterior teeth observing ideal occlusion relationships was executed. When space closure compromised the occlusion, preference was given to the latter. Residual interproximal spacing was digitally measured on the setups. Analysis of variance and linear regression tests were used to identify factors contributing to interproximal spacing.
Results
An average of 1.1 mm of net residual spacing between mandibular second premolars and first molars was observed. In 27% of the sample, at least 1.5 mm of residual space was found. In addition, 16% showed at least 2 mm of residual space. The ratio of the maxillary second premolars to the mandibular second premolars and the width of the maxillary second premolars best explain residual space ( r = 0.554; r 2 = 0.307). A regression equation for predicting residual space is offered.
Conclusions
Ideal anterior, posterior, and overall Bolton ratios treated with extraction of 4 first premolars and ideal occlusion will likely finish with some spacing in the mandible.
Highlights
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This is a study of the effect of premolar extractions on the posterior Bolton ratio.
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The nonextraction ideal posterior Bolton ratio is 105.77 ± 1.99%.
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The expected ideal posterior Bolton ratio is 107.29 ± 2.23% for four first premolar extractions cases.
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These cases could yield over 1 mm of spacing in the mandibular arch without compensatory action.
Wayne A. Bolton developed the Bolton tooth size discrepancy analysis as a diagnostic tool to help identify potential limitations in detailing and finishing a patient’s final occlusion. Bolton’s study from 1952, later published in 1958, included a sample of models from 55 patients with excellent occlusion. He defined specific interarch relationships by creating the following ratios:
- (1)
Anterior: sum of mandibular 1s, 2s, 3s/sum of maxillary 1s, 2s, 3s × 100 = anterior ratio.
- (2)
Overall: sum of mandibular 12 teeth/sum of maxillary 12 teeth × 100 = overall ratio.
Using ratios of mandibular teeth widths (mesial to distal) to maxillary teeth widths, excellent occlusion was described as having an anterior ratio of 77.2% and an overall ratio of 91.3%. The overall ratio was comprised the widths of the anterior and the posterior teeth. It is important to note that an anterior tooth size discrepancy may be masked by a compensating posterior tooth size discrepancy, resulting in a normal overall tooth size ratio. As the sum of its parts, the overall ratio may be less important than the anterior and posterior tooth size ratios individually. The posterior ratio can be defined as:
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Sum of mandibular 4s, 5s, 6s/sum of maxillary 4s, 5s, 6s × 100 = posterior ratio
However, the literature includes little information about a posterior Bolton ratio and how it is affected by extracting posterior teeth. In a recent study by Fallis, posterior tooth size discrepancy methods were evaluated. The existing Bolton ratios and related algebraically calculated posterior ratios were not accurate predictors of actual discrepancies. Kumar et al reviewed the effects of premolar extractions on Bolton ratios and concluded that nonextraction generally results in better Bolton ratios than extraction.
Bolton’s predicted ratio in extraction was based on a mathematical calculation that simply omitted the average widths of extracted teeth but did not take into consideration the articulation of the cusps, fossae, and marginal ridges of the remaining teeth. For accuracy, any newly described ratios for extraction should be calculated from Bolton’s method of using nonextraction—that is, from patients that undergo premolar extraction and result in excellent finished occlusion. Comparing Bolton’s ideal ratios and the ratios describing patients’ dentitions could indicate whether teeth are too large, too small, or just right for proper intercuspation and whether treatment aimed at reducing the discrepancy may be warranted to improve the occlusion.
A study by Kayalioglu et al measured 53 posttreatment models from patients treated with 4 first premolar extractions that resulted in good occlusion and reported a new ideal overall ratio, 89.28 ± 1.07%, on the basis of this sample. With no reported knowledge of the data included in Bolton’s thesis, the newly reported ideal overall ratio was strikingly similar to the ideal overall ratio one would expect to find (89.67%) if 4 first premolars were extracted in Bolton’s patients. However, the difference is statistically different from the 88% overall ratio predicted by Bolton for premolar extraction patients. This may indicate that many studies using Bolton’s predicted value for patients treated with premolar extraction may have had erroneous assumptions and/or conclusions about the occlusal consequences of different premolar extraction patterns.
The purpose of this study was to contribute to the existing literature by reporting a posterior Bolton ratio that can be expected for nonextraction with excellent occlusion as described by Bolton’s ideal anterior and overall ratios and by describing the effect of the extraction of 4 first premolars on the expected posterior Bolton ratio. In addition, this study described the effect of the extractions on the observed posterior Bolton ratio with digital setups with excellent occlusion and correlated discrepancies between expected and observed posterior Bolton ratios to various combinations of tooth widths, proportions, and differences. It should be noted that the clinician would certainly seek to close any residual spaces during treatment, but this study was intended to determine the average amount of tooth size discrepancy that must be addressed for patients undergoing extraction.
Material and methods
This retrospective study involved a sample obtained from archived physical models from patients who had completed orthodontic treatment in a university setting. Institutional review board approval was given to this study by the sponsoring institution. In an attempt to match Bolton’s original sample, after screening and inclusion criteria were met, a final sample of 55 patients within 1 standard deviation (SD) of Bolton’s reported ideal anterior and overall ratios were used.
The preliminary inclusion criteria for each set of physical models were (1) a full set of permanent dentition from the first molar to contralateral first molar, (2) a posttreatment result with good Class I occlusion (models used were from the beginning of the retention phase of treatment), and (3) a patient which had no interproximal alteration such as restorations or stripping.
Plaster models were screened for inclusion by measuring individual tooth widths with an electronic digital caliper, performing an anterior and an overall Bolton analysis, and selecting patients with anterior and overall Bolton ratios within 1 SD of Bolton’s reported ideal ratios. Because extracting teeth from plaster models would be a destructive method of assessing change, it was decided to scan the physical models for each patient to create duplicates in digital format. By using digital models, the investigator was able to section teeth to be extracted without damaging adjacent teeth, thereby avoiding the alteration of tooth dimensions that might affect measurements. Thus, this process of digitally performing extractions was found to be more efficient than reproducing the models in plaster. The physical models were scanned using a 3D scanner (R700; 3Shape A/S, Copenhagen, Denmark), and the software product OrthoAnalyzer (3Shape A/S) was used for digital processing. These digital models were then evaluated by performing an extraction function of the software to remove 4 first premolars.
Bolton defined excellent occlusion to follow the pattern described by Wheeler in his dental anatomy text. Accordingly, maxillary molars were rotated so that a ray drawn from the maxillary first molar’s distobuccal cusp tip passed through the mesiolingual cusp tip and contralateral canine. In addition, the buccolingual inclinations of maxillary posterior teeth were set so that the transverse occlusal plane was flattened. Overbite and overjet followed Wheeler’s descriptions as well. In addition, virtual setup treatment objectives were to achieve Andrew’s Six Keys of Occlusion —tip, torque, rotation, flat curve of Spee, no spaces, and Class I molar position. Figure 1 illustrates the occlusion scheme used for the digital setups, and Figure 2 shows an example of a digital extraction setup. This setup involved moving teeth posterior to the extraction sites mesially to be in proximal and occlusal contact in the following sequential order: mandibular second premolars, maxillary second premolars, maxillary first molars, and mandibular first molars. If complete space closure was not possible without compromising Class I ideal cusp-fossa or cusp-marginal ridge occlusal relationships described by Wheeler’s Dental Anatomy from Ash and Nelson, then the final tooth position was determined by ideal occlusal intercuspation.
Maxillary and mandibular residual spaces were determined in the extraction model by measuring the gap between the 2 teeth parallel to the alveolar ridge when observed on the occlusal view. Figure 3 illustrates an example of mandibular residual space measurement. The clinician would certainly seek to close any residual spaces during treatment. This study was intended to determine the average amount of tooth size discrepancy that must be addressed for patients undergoing extraction, whether through interproximal enameloplasty, restorative treatment, compromising occlusal relationships, or even trying to maintain a stable relationship with spacing.
All measurements, including individual maxillary and mandibular anterior and posterior tooth widths, interproximal spaces, and various combinations of tooth widths, proportions, and differences, were recorded in a Microsoft Excel 2010 (Microsoft Redmond, Wash) spreadsheet for analysis.
Statistical analysis
The data were recorded in Microsoft Excel 2010 and exported into IBM SPSS software (version 23.0; SPSS, Chicago, Ill). The alpha level for all statistical tests was set at 0.05. To test for the accuracy of the screening process in matching the experimental sample to Bolton’s original sample, independent t tests were performed comparing the anterior Bolton ratio, overall Bolton ratio, and individual posterior tooth widths between samples.
Descriptive and frequency statistics were calculated for data in the sample. Data described, which had not been previously reported, included: (1) the nonextraction posterior Bolton ratio, the overall Bolton ratio, and posterior Bolton ratios that could be expected on the basis of a mathematical calculation that omits the widths of the extracted first premolars; and (2) the overall Bolton ratio and posterior Bolton ratios observed after 4 first premolar extractions and setting the teeth in ideal occlusion.
Paired t tests were employed to test the following: (1) that the widths of the individual premolars in the maxillary arch were not different than the widths of the corresponding premolars in the mandibular arch; (2) that the overall and posterior Bolton ratios were not altered by extracting teeth; and (3) that the Bolton ratios determined by the mathematical omission of first premolar widths were not different from Bolton ratios observed as determined by ideal occlusion after extracting 4 first premolars.
Correlation and regression were used to determine the relationship between the amount of residual space and the discrepancy between the posterior Bolton ratio determined by the mathematical omission of the widths of the first premolars and the posterior Bolton ratio observed when ideal occlusion was achieved in preference over complete space closure.
To reduce the number of variables describing tooth widths, proportions, and differences in each patient and isolate only those variables that may most affect the amount of observed residual spaces after extractions and setups in an ideal occlusion, 1-way analysis of variance (ANOVA) was performed. Patients were grouped according to the number of residual spaces identified after the extractions and setup in ideal occlusion.
From these tests, a regression equation was derived that could be used to predict the amount of the residual space remaining in a 4 first premolar extraction case treated to ideal occlusion. A resulting positive number indicates spacing in the mandible, and a negative number indicates spacing in the maxilla. This equation is shown here:
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Net residual space = 12.43 (U5s width/L5s width) + 0.29 (U5s width) − 14.56.
All manual and digital measurements, as well as virtual setups, were performed by the same investigator (A.D.M.). Six of the 55 patients were randomly selected using a number generator ( random.org ) to assess the intraexaminer reliability. Using OrthoAnalyzer software, the 6 patients were digitally remeasured with new setups from canine to canine and the first molar to the first molar with the extraction of the 4 first premolars, performed as described previously. Cronbach alpha set at a level of 0.8 was used to determine the measurement reliability.
Results
Cronbach alpha test for intraexaminer reliability between original and repeated measurements was above 0.8 for all variables, showing that the process for obtaining original measurements and repeated measurements was reliable for accuracy.
The mean anterior and overall Bolton ratios for this sample were 77.23 ± 0.93% and 91.75 ± 0.97%, respectively. Individual tooth widths for the maxillary first premolar, second premolar, and first molar averaged 7.07 ± 0.48 mm, 6.88 ± 0.46 mm, and 10.42 ± 0.54 mm, respectively. Individual tooth widths for the mandibular first premolar, second premolar, and first molar averaged 7.22 ± 0.46 mm, 7.35 ± 0.51 mm, and 11.20 ± 0.63 mm, respectively. Table I shows the descriptive statistics for the anterior Bolton ratio, overall Bolton ratio, and individual posterior tooth widths from this study’s sample and Bolton’s original sample. Independent t tests comparing the mean ratios and mean individual tooth sizes from these samples did not differ significantly.
| Category | Bolton’s original sample | Experimental sample | P | ||||
|---|---|---|---|---|---|---|---|
| Mean | SD | N | Mean | SD | N | ||
| Anterior ratio | 77.20% | 1.65 | 55 | 77.23% | 0.93 | 55 | 0.904 |
| Overall ratio | 91.30% | 1.91 | 55 | 91.75% | 0.97 | 55 | 0.125 |
| Posterior ratio | 105.27% ∗ | 55 | 105.77% | 1.99 | 55 | ||
| U4 | 7.04 mm | 0.46 | 110 | 7.07 mm | 0.48 | 110 | 0.592 |
| U5 | 6.84 mm | 0.39 | 110 | 6.88 mm | 0.46 | 110 | 0.499 |
| U6 | 10.40 mm | 0.58 | 110 | 10.42 mm | 0.54 | 110 | 0.792 |
| L4 | 7.15 mm | 0.38 | 110 | 7.22 mm | 0.46 | 110 | 0.243 |
| L5 | 7.27 mm | 0.39 | 110 | 7.35 mm | 0.51 | 110 | 0.174 |
| L6 | 11.14 mm | 0.62 | 110 | 11.20 mm | 0.63 | 110 | 0.463 |
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