WO2011062473A1

WO2011062473A1 - Regression method for modeling widths of teeth

Info

Publication number: WO2011062473A1
Application number: PCT/MY2010/000269
Authority: WO
Inventors: Bin Mohd Rijal Omar; Mohd Isa Zakiah
Original assignee: Universiti Malaya
Priority date: 2009-11-17
Filing date: 2010-11-10
Publication date: 2011-05-26
Also published as: MY146243A

Abstract

The present invention relates to a regression method for modeling the width of teeth, for use in the absence of pre-extraction records, particularly for rehabilitating edentulous patients or subjects. The method establishes relationships between the widths of the maxillary anterior teeth and certain facial dimensions, with high correlation coefficient. The facial dimensions used are the interpupillary distance (150) and the interalar width (140). The regression method of modeling includes elimination of outliers and correcting non-constant variance by a weighted least square method, from the measured variables used for the regression method.

Description

REGRESSION METHOD FOR MODELING WIDTHS OF TEETH

FIELD OF INVENTION

The present invention relates to the field of modeling the widths of teeth, particularly the maxillary anterior teeth, in relation to certain facial dimensions. Such models are helpful in the absence of pre-extraction records, particularly for rehabilitating edentulous patients or subjects.

BACKGROUND OF THE INVENTION

In the absence of pre-extraction records for edentulous patients, various guides are used to select the size of the teeth. Among the guides proposed, are the relationships between facial measurements and the natural tooth dimensions. As illustrated in Fig. l, facial anatomical landmarks that have been used as guides include the interpupillary width (IPD) (150), interalar width (IA) (140) and the innercanthal distance (ICD) (160). The width between the tips of the left and the right canines has been shown to be similar to the width of the alae of the nose, although this relationship varies with gender and ethnic groups. It is also necessary to add approximately 3-7 mm to the width of the alae of the nose or to multiply the IA (140) by a specific factor to obtain the required widths of the six maxillary anterior teeth to compensate for the curvature of the maxillary residual ridge. The six maxillary anterior teeth specifically include the right and left central incisors (110a,b), right and left lateral incisors (120a,b), right and left canines (130a,b). The use of the ICD (160) in determining the width of the maxillary anterior teeth for edentulous patients have also been considered by various investigators, either by using direct proportions or by adding a multiplying factor to the ICD (160). Computer- generated studies for the correlation between tooth, face, arch forms and palatal contour are also known. The superimposition of outline forms of the face, tooth and arch forms have revealed an insignificant correlation. Subjects of different ethnic groups may also exhibit varying teeth sizes. Therefore, whether the facial dimensions can serve as guides to the selection of appropriate size of teeth for edentulous patients for various ethnic groups or not, needs to be investigated.

SUMMARY OF THE INVENTION

The present invention proposes models based on regression methods, between certain facial dimensions and the widths of the maxillary anterior teeth, to potentially provide a guide for tooth selection. Regression constants depicting the relationships of facial dimensions and anterior teeth widths may be used as aids in the selection of the sizes of artificial anterior teeth, for edentulous patients. Regression methods applied for this invention shows that the data requires verification, such as investigating the effects of outliers, before the actual calculation of the correlation between the widths and the facial dimensions. Certain assumptions for statistical approximations have also been made, that leads to significantly different values for the correlation coefficient. With the invented method, for the population studied, the sizes of maxillary central incisors, lateral incisors and canines are shown to be highly correlated to the IPD and a combination of the IPD and IA. The widths may be predicted by incorporating the relationships established by the regression models discussed herein.

The present invention provides a method of correlating a size of a tooth of a subject with facial dimension(s), by determining statistical data for at least one facial dimension and at least one tooth width from a population, correcting non-constant variance in residual plots therein by a weighted least square method, eliminating outliers and then determining at least one regression model with a corresponding correlation coefficient (r), relating the facial dimension(s) and the tooth width. The tooth width is particularly for a maxillary anterior tooth. The regression method assumes that the residuals are normally distributed. The normality of the residuals may be assessed by statistical tests such as Kolmogorov-Smirnov goodness-of-fit test. The regression method further assumes the facial dimension(s) and the width(s) for the population is an overdetermined system being suitable for regression analysis and for simplicity, being with uncorrelated error terms.

According to the present invention, the regression method generates each of the regression models as a simple regression model (SRM) at a predefined significance level, using a single facial dimension.

In a further development of the present invention, the regression method generates each of the regression models as a multiple regression model (MRM) at a predefined significance level, using multiple facial dimensions, to incorporate the combined effects when appropriate.

The facial dimensions may be digitally derived from one or more digital image(s) for determining dimensions with higher accuracy and with higher reliability. The widths in the models may be the mesiodistal diameters of the teeth, which is a standard parameter in dentistry. In a further development of the present invention, a t-statistic value for the regression model is calculated for determining if a correlation exists between a width and the facial dimension(s).

According to an exemplary embodiment of the invention, a regression model is proposed for predicting a maxillary central incisor width y2 for a subject, the formula being of a form: y2 = 4.22 + 0.07 (IPD), where the facial dimension is the IPD for the same subject. The values of the fitting constants in the equation may vary by a small percentage.

According to an exemplary embodiment of the invention, a regression model is proposed for predicting a maxillary lateral incisor width 3 for a subject, the formula being of a form: y^"3= 2.24 + 0.07 (IPD) + 0.02 (LA), where the facial dimensions are the IPD and the LA for the same subject. The values of the fitting constants in the equation may vary by a small percentage.

According to an exemplary embodiment of the invention, a regression model is proposed for predicting a maxillary canine width y6 for a subject, the formula being of a form: y6= 4.16 + 0.05 (IPD) + 0.02 (IA), where the facial dimensions are the IPD and the IA for the same subject. The values of the fitting constants in the equation may vary by a small percentage.

The present invention consists of certain novel features and a combination of parts hereinafter fully described and illustrated in the accompanying drawings and particularly pointed out in the appended claims; it being understood that various changes in the details may be possible without departing from the scope of the invention or sacrificing any of the advantages of the present invention.

BRIEF DESCRIPTION OF THE DRAWINGS

In the following drawings, same reference numbers generally refer to the same parts throughout. The drawings are not necessarily to scale, instead emphasis is placed upon illustrating the principles of the invention. The various embodiments and advantages of the present invention will be more fully understood when considered with respect to the following detailed description, appended claims and accompanying drawings wherein:

Fig. 1 illustrates the various facial dimensions commonly used to relate to the widths of the various maxillary anterior teeth as shown therein.

Fig. 2 provides an exemplary table with a measured set of statistical values for a popuiation, for the facial dimensions and the mesiodistal widths of the maxillary anterior teeth as illustrated in Fig. l , according to an embodiment of the present invention. Fig. 3 provides a table with fitted simple regression models for each of the widths in Fig. 2 in relation with each of the facial dimensions, each model presenting a corresponding correlation coefficient (r).

Fig. 4 provides a table with t-statistics values for testing null hypothesis for the models in Fig.3.

Fig.5 provides a table with the results of Kolmogorov-Smirnov goodness-of-fit test for testing normality of residuals for each of the models in Fig.3.

Fig.6 provides a table with fitted regression models and correlation coefficients (r) after using weighted least square estimation on the residual plots of the variables in Fig.3.

Fig.7 provides a table presenting the effect of removing outliers after using weighted least squares, as in Fig.6.

Fig.8 provides a table with the best fitted models and the corresponding correlation coefficients ( r ), from the results in Fig.7.

DETAILED DESCRIPTION OF THE INVENTION

The following description presents several preferred methods of the present invention in sufficient detail such that those skilled in the art can make and use the invention.

Before describing in detail embodiments that are in accordance with the present invention, it should be noted that all of the figures are drawn for ease of explanation of the basic teachings of the present invention only. The extension of the figures with respect to the number, position, relationship and dimension of the parts of the preferred embodiment will be within the skill of the art after the following teachings of the present invention have been read and understood. Further, the exact dimensions and dimensional proportions to conform to specific force, weight, strength and similar requirements will likewise be within the skill of the art after the following teachings of the present invention have been read and understood.

There have been many attempts to use facial measurements as guides to estimating the size of artificial teeth for edentulous patients.^' Establishing precise or exact relationship between facial measurements and the widths of the anterior teeth is the primary focus of this invention. The method of arriving at relations between the maxillary anterior teeth widths and the facial dimensions, according to an embodiment of the invention, is illustrated as follows. The relations are arrived at through regression analysis, where the focus is on the relationship between a dependent variable (width) and one or more independent variable (facial dimension).

Statistical data from an exemplary population of sixty fully dentate Malaysian adults (18- 36 years) from two ethnic groups with well aligned maxillary anterior teeth and minimal attrition are used for the modeling, as shown in Fig.2. Standardized digital images of the faces viewed from the front of these subjects are recorded. Using image analyzer software, the images are used to determine the IPD (150), ICD (160) and IA width (140). Widths of the six maxillary anterior teeth are measured directly from casts of these subjects using digital calipers. Other methods for these measurements may also be used. Regression analyses are conducted to measure the strength of the associations between the variables. The null hypothesis is that there is no relationship between the facial measurements and the widths of the maxillary anterior teeth. The sample consists of Chinese (3 men and 1 1 women) and Malay (19 men and 27 women) adults. Those with a history of orthodontic treatment, asyrnmetry of the face, anterior restorations or fixed dental prosthesis in the maxilla or mandible, are excluded for consideration of only natural and . standard data. The photographs for the digital images are made by the same examiner for better consistency in measurements, with the subjects seated upright and looking straight ahead, with their heads being supported by the chin rest, such as of a panoramic radiograph machine (Planmeca Proline PM2002 CC; Planmeca Oy. Helsinki, Finland), set for minimized distortions. Using an image analyzer software such as Leica Qwin Lite Vers. 2 of Leica Microsystems Imaging Solutions, Cambridge, UK, the computer images are used to determine: the IPD, the ICD and the IA. Impressions of the maxillary arches of the subjects are made with an irreversible hydrocolloid, such as Duplast Fast Set, Dentsply Ltd., Addlestone, England. Casts are made using Type III dental stone such as Heraeus Kuzler GmbH and Co., Germany. The maximum mesiodistal widths of the individual anterior teeth, specifically the maxillary right and left central incisors: yl and y2 (110a,b), right and left lateral incisors : y3 and y4 (120a,b) and right and left canines : y5 and y6 (130a,b) are measured using digital calipers such as from Mitutoyo, Tokyo, directly from the maxillary casts. Each parameter is measured and recorded 3 times, and the average value is calculated. All of the measurements are performed by the same operator for better consistency of measurement. The descriptive statistics including mean, standard deviation, minimum and maximum values of the measured data are listed in Fig.2.

The measured data is analyzed with a statistical software, such as S-Plus; Vers.8.0 from Insightful Corporation. The simple regression model (SRM) investigates the relationship between the widths of the anterior teeth with a particular facial measurement. The SRM is normally given in the following form

y. = α + P x_j + Sj i=(l,2 ,n) (1) where y. are the widths which are the dependent parameters, x. are the facial dimensions which are the independent parameters, a and β are the fitting constants, ε. is an error constant and the subscript 'i' is indexing a particular observation.

The fitted simple regression models performed on the measured data and their corresponding correlation coefficients (r) are given in Fig.3. The (r ) value measures the strength and direction of a relationship between two random variables and Fig.3 suggests weak linear relationships between all pairs of the variables studied. In certain situations, for example, the variables ICD (160) and y_$ are possibly independent (r =

0.18). The highest r is 0.48 between ICD (160) and Correlation coefficient values close to zero may imply either independence of the variables or possibly the existence of a more complicated non-linear relationship of the variables. Small correlation values may imply that the population correlation, p, may be equal to zero. It is of interest to test the validity of the hypothesis H_Q : p = 0 with a test statistic. The values are compared to a critical value of t(58) = 1.672 at the predefined significance level. If the value of t is larger than 1.672, then the null hypothesis is rejected. Fig.4 shows the results of this hypothesis testing. The general result of this test shows that H_Q is rejected except for one situation of y₅ against ICD (160). Therefore, what the test has shown is that although the correlation values were small, they were significantly different from zero.

One of the important assumptions of the SRM made is that the residuals are normally distributed. It is also assumed that the measured data is an overdetermined system being suitable for regression analysis and the error terms are uncorrelated.

Even though the correlation is significantly different from zero, the standard assumptions need to be verified. A statistical test such as Kolmogorov-Smirnov goodness-of-fit test is applied to confirm the normality of the residuals, and the results are presented in Fig.5 These results are important since the confirmation of normality ensures the use of optimal estimates derived from regression models. The critical value of the Kolmogorov-Smirnov statistic used is D = 0.1575 for the predefined significance level. The main result of Fig. is that the normality of the residuals is satisfied for all the regression models considered. Residual plots are generated to indicate the general problems, namely the existence of outliers and non-constant variance. The weighted least square (WLS) method is applied which effectively eliminates the problem of non-constant variance. The result from the WLS method is shown in Fig, 6. The correlation coefficient (r )values obtained after applying the WLS method were considerably larger ranging from 0.67 to 0.98, with the exception of the model y₅ in relation to ICD (r =0.31). Since the WLS estimation involves linear transformation of the data, it is not necessary to perform any further test of normality. The large range of correlation coefficient values (r) is partially attributed to the existence of outliers. Outliers indentified for five SRMs from the residual plots are removed and the fitted regression models and their corresponding correlation coefficients are recalculated and are shown in Fig.7. Removal of outliers resulted in high values for the correlation coefficients (r).

Eighteen SRMs are considered, for studying pair-wise relationship between (yl ,y2...y6) and (ICD, IA and IPD). Residual plot for each of the eighteen SRMs indicates the appropriateness of using subsequent sample correlation. Multiple regression model (MRM) is subsequently used to investigate the possible combined linear effects of ICD, IA and IPD on the prediction of the widths of the anterior teeth, at the predefined significance level. Regression methods yielded six regression models for predicting the six widths from a combination of IPD, LA and ICD. For example, instead of using only one facial dimension to predict, four combinations are considered namely IPD-IA-ICD, IPD-IA, IPD-ICD and IA-ICD. Twenty-four regression models are considered which cover all possible combinations of (IPD-IA-ICD, IPD-IA, IPD- ICD, IA-ICD) and (y_(> y₂, y₃, y₄ ,y₅,y₆ ).

Fig.8 shows the final selection of SRM and MRM models which may be used. The widths of the central incisors is highly correlated to the IPD (r=0.99) while the widths of the lateral incisors and canines are highly correlated to a combination of IPD and IA (r=0.99 and 0.94, respectively). Since is similar to y₂, y₃ is similar to y₄ and finally y₅ is similar to y₆, knowledge of y₂, y₃ and y₆, is sufficient to determine the widths of all the six anterior teeth.

The results of the dental measurements as in Fig.2 for Malays and Chinese are comparable to other known studies where Chinese, Malay or Caucasian subjects are used. The aim of restoring edentulous patients is to have the maxillary anterior teeth restore optimal dentolabial relations in harmony with the overall facial appearance. There may be no universally reliable method for using facial measurements to determine artificial denture teeth size and the continuing process of alveolar bone resorption may cause difficulties in restoring the lost tissues in approximately the same amounts and in the same positions from which they were lost. However, the use of regression formulas like those obtained in the current proposal as in Fig.8 may aid in the selection of artificial teeth for complete dentures, especially for relatively inexperienced clinicians. The fitted regression models did not specifically investigate differences due to gender or ethnic group. Nevertheless, the results of this study can be considered accurate because careful interpretation of the correlation is performed in the sense that standard assumptions for the regression models were investigated. The main beneficiary of this study is the clinicians whose patients are of Malay and Chinese origin who now have a formal method of determining the teeth size from the regression models for rehabilitating edentulous patients.

The methods of this study can be repeated for a new data set, for example for patients from other ethnic groups and henceforth the appropriate models required may be constructed for the same purpose.

While the foregoing description presents preferred embodiments of the present invention along with many details set forth for purpose of illustration, it will be understoad by those skilled in the art that many variations or modifications in details of design, construction and operation may be made without departing from the present invention as defined in the claims.

Claims

1. A method of correlating a size of a tooth of a patient or subject with facial dimension(s), the method comprising the steps of: determining at least one facial dimension for a population;

determining at least one tooth width for the population;

identifying outliers and non-constant variance in residual plots for the at least one facial dimension and the at least one tooth width for the population, upon assuming that residuals in the residual plots are normally distributed;

correcting the non-constant variance by a least square method;

removing the outliers; and

determining at least one regression model with a corresponding correlation coefficient^), relating the at least one facial dimension and the at least one tooth width, assuming the at least one facial dimension and the at least one tooth width for the population is an overdetermined system being suitable for regression analysis and being with uncorrelated error terms.

2. The method according to claiml wherein the at least one tooth width is an anterior tooth width.

3. The method according to claim2 wherein the at least one tooth width is a maxillary anterior tooth width.

4. The method according to claim! wherein the regression model is a simple regression model (SRM) at a irst predefined significance level, using a single facial dimension from the at least one facial dimension.

5. The method according to claiml wherein the regression model is a multiple regression model (MRM) at a second predefined significance level, when the at least one facial dimension has a plurality of facial dimensions.

6. The method according to claiml wherein the at least one facial dimension is digitally derived from digital image(s).

7. The method according to claiml wherein the width is a mesiodistal diameter.

8. The method according to claiml further comprising a step of:

assessing the normality of the residuals by Kolmogorov-Smirnov goodness-of-fit test.

9. The method according to claiml wherein the least square method is a weighted least square method.

10. A method according to claiml further comprising a step of: calculating a t-statistic value for the at least one regression model, for determining if a correlation exists between the at least one tooth width and the at least one facial dimension.

11. A method of predicting a maxillary central incisor (110a or 110b) width y2 for a subject, using the at least one regression model according to claiml, the formula being of a form: y2= 4.22 + 0.07 (IPD), where the at least one facial dimension is an inter-pupillary distance (IPD) (150) for the same subject. 2.A method of predicting a maxillary lateral incisor (120a or 120b) width y3 for a subject, using a formula derived by claim5, the formula being of a form: y3= 2.24 + 0.07 (IPD) + 0.02 (IA), where the plurality of facial dimensions is an inter-pupillary distance (IPD) (150) and an interalar width (IA)(140) for the same subject. 3.A method of predicting a maxillary canine (130a or 130b) width y6 for a subject, using a formula derived by claim5, the formula being of a form: y6= 4.16 + 0.05 (IPD) + 0.02 (IA), where the plurality of facial dimensions is an inter-pupillary distance (IPD) (150) and an interalar width (IA) (140) for the same subject.