Facial and dental arch asymmetries in Class II subdivision malocclusion Debra G. Alavi, D.D.S., #I&,* EWn A. Bernard J. St+hnMw, D.D.S., MS,‘**
, Ph.D.,** and
The purpose of this investigation was to determine if any significant C$#WWIWSexiated with regard to dental arch and facial asymmett%s behnsen parsona wmw%lt-d%ftned asymmetric character&tics, spec#kWy [email protected]
&s C&as tl s usions, and persons having nom-tat ocdusions. Setcwtdarily, it was sought to &&ermine the nature of the dWerer~~+sthat contributed to the asymmetric occlusal relationship of buccai segments observed in Class II subdivision malocekafons. The sampte consfated of 28 subjects in each of th possessed a full complement of permanent teeth [email protected] [email protected]
n-&a%. The awe in both groups was 17 years. lMasur8me#ts p&war& of a linear nature were posteroanterior and lateral cephaiometric r in addition to dental mod&s. S measurements were summed to produce of asymmetry. Va with multivariate discrimMnt analysis. A sigM dads b&wn variables describing asymmetry of the derWM&ar region of the mandible appearad to be the primary contributors to the difference obtained. Aayrmwtry of the maxWry appeared to make a secondary cont&uf~n. (AM J ORTWO DENTOFAC Owrwop 1
lass II subdivision malocclusions with their asymmetric occlusal relationships often pose treatment difficulties. The nature of the occlusal asymmetry may be due to dentoalveolar or skeletal asymmetries, or a combination of these factors, and it is thought that these underlying factors complicate the attainment of a symmetric occlusion. The anthropologic investigations of Woo,’ Tildesley,* Pearson and Woo,’ Bjiirk and Bj6rk,4 Gundara and Zivanovic,’ and White6 have all shown craniofacial asymmetry to be common in most persons. Vazquez, Grostic, and Ponder7 compared indices of skull asymmetry and malocclusion, and found asymmetry to be related to severe malocclusion. Several studies showed a tendency for the maxilla to be more asymmetric than the mandible or dentoalveolar regions.*-” Vig and Hewitt,’ and Shah and Joshi” have shown that in normal occlusion the den-
From the Department of Orthodontics, University of Illinois at Chicago, College of Dentistry. *This article is based on research submitted by Dr. Alavi in partial fulfillment of the requirements for the degree of master of science in oxthadoatics, University of Illiiis at Chicago. **Associate Professor of Biostatistics, Department of Orthodontics. ***Professor, Department of Orthodontics. 38
toalveolar region shows less asymmetry than the skeletal areas of the face and these authors suggest that an adaptive mechanism may be present to explain this, Most studies of dentoalveolar asymmetry have used dental models and most often only the maxillary arch. Using the median raphe as an axis of symmetry, a number of studies reported some degree of dental arch asymmetry even in persons with normal occlusion.1*-‘6 He&~ter’~ analyzed asymmetry of the dental arches in normal and malocclusion subjects, and reported greater asymmetry in the mandibuhu arch for both groups. In ad&ion, he found an increase in asymmetry in both arches wben malocclusion was present. Using mathematic functions, investigations of dental arch form have also demonstrated asymmetries. ‘*-*O With the advent of cephalometric radiography, studies were conducted that attempted to relate occlusion to skeh%al asymmetry. Shore,’ by means of posteroanterior cephalometric radiographs, compared normal and malocclusion groups, and found that occiusion was independent of skeletal asymmetry. Letzer and Kronman*’ also compared malocclusion and exceiient occlusion groups using posteroanterior cephalometric racfi~hs and found no statistical evidence to corroborate any relationship between the occlusion and facial asymmetries. However, they found both the mandible
Facial and dental arch asymmetries
Volume 93 Number 1
Fig. 1. Diagram showing the linear and angular measurements made from bilateral landmarks to the constructed midline on the PA cephalometric radiographs.
and anterior cranial base to be more symmetric in the excellent occlusion group. It is not surprising that these studies failed to demonstrate differences between groups since the malocclusion groups studied were nonspecific and only posteroanterior cephalometric radiographs were used. Williamson and Simmons** used both posteroanterior and submental-vertical cephalometric radiographs to study mandibular asymmetry as it relates to pain dysfunction. In relating occlusal and skeletal characteristics, they found several subjects in their sample to be more Class II dentally on the short side of the mandible. The present study compared a group of subjects having normal occlusion with a group having welldefined asymmetric relationships of the buccal segments, classically known as Angle’s Class II subdivision malocclusion. The primary objective of the study was to characterize asymmetries of the dental arch and facial regions in both groups and to discriminate between the groups on the basis of a set of measurements. Secondarily, it was sought to find which combination of variables contributed most to the discrimination. To accomplish this, a multivariate statistical approach was used that allowed complex interrelationships among many variables to be explored. METHODS AND MATERIALS
Two groups of 28 individuals were used in the study: a normal occlusion group and a malocclusion group consisting of persons having Class II subdivision mal-
Fig. 2. Diagram showing the linear measurements made from bilateral landmarks to the constructed vertical reference line on the lateral cephalometric radiographs.
occlusions. Mean ages were 17.1 years for both groups with standard deviations of 6.4 and 6.7 years, respectively. Records consisted of posteroanterior and lateral cephalometric radiographs of good quality, and dental models. These were obtained from duplicated records of the Child Research Council, University of Colorado School of Medicine in Denver, for the normal group, and from the records of the University of Illinois Orthodontic Department for the malocclusion group. The criterion for selection was a full complement of permanent teeth through the first permanent molars. Additional criteria for the malocclusion group included (1) a full Class I molar relationship on one side of the dental arch with a full Class II on the contralateral side, (2) the absence of any severely malaligned or blocked out teeth, and (3) no apparent functional mandibular shifts as reported in the clinical history. Dental models
On the maxillary dental model, two pointsnamely, one at the distal aspect of the incisive papilla and the second at the posterior border of the raphe near the fovea centralis-were used to define the median raphe that was used as the dental midline. These two points were transferred to the occluded mandibular model with a device similar to that described by Hechter.17 Buccal cusp tips of the posterior teeth and the midincisal edges of the anterior teeth were then marked
Fig. 4. Diagram illustrating the relationship between the normal and maloccWon groups for the sk&eWen tal analysis.
_ Fig. 3. Diagram showing the finear measurements made from bilateral tooth landmarks to the horkontai and vertical reference lines on the maxillary and mandiiular den&if models.
on the casts with pencil. The models were photographed with a 35 mm telephoto setup at a standard object film distance and life size prints of the models were then produced. The accuracy of the method has been previously confirmed by BeGole.“’ All points on the prints and an additional point between the central incisors at the level of the incisal edge were then digitized using a G&/Pen sonic instrument to record coordinates in an X,Y system. From the coordinate data, a computer program of Cleall and Chebib’” was used to calculate measurements. Linear measurements were made from each of the seven bilateral tooth landmarks in a transverse direction to the median raphe line. An anterior reference line was constructed perpendicular to the raphe line through the midpoint between the central incisors and similar measurements in an anteroposterior direction were made to this line. Fig. 1 shows the measurements that were made on the dental models. Asymmetries were calculated by determining the absolute difference between homologous measurements as related to both reference lines. Measurements of transverse asymmetry for the incisors and canines were combined to produce an index of
transverse anterior segment asymmetry. An index of transverse buccal segment asymmetry was formed as a combination of measurements from premolar and molar points. The same procedure was followed to compute anteroposterior indices of anterior and buccal segment asymmetries. Dental midline deviations were assessed by making linear measurements from the midpoint between the central incisors to the median raphe line or its analog on the mandibular model. Thus, a total of ten dental model variables resulted since the measurements were made on both maxillary and mandibular models. Lateral
Cephalometric films were traced and landmarks designated before digitization. For the lateral cephdomettic radiographs, right and left side landmarks were distinguished with the aid of the Broadbent-Golden orientator.% It should be noted that while right and left sides may be difficult to distinguish in asymmetric cases, this did not affect the outcome of the study because only absolute values of the difference between the sides as opposed to signed differences were used in the group comparisons. In addition, reference was made to the original models when using the orientator to identify left and right sides. The following landmarks were digitized: sella, nasion, basion, the bilateral orbitale, PTM, antegonia, gonia, articulare, points of maximum concavity on the anterior surface of the ramus, and the most posterior points on the distal surfaces of the maxillary and mandibular first permanent molars. A reference line was determined by constructing a perpendicular to the selia-nasion line through badon. Linear measurements were made in a horizontal direction from the bilateral orbitale and PTM, articulare, anterior ramal point, antegonion , and gonion , to the constructed vertical line. Fig. 2 shows measurements taken from the lateral radiograph. Skeletal asymmetry was evalu-
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Table 1. Means and standard deviations of asymmetry measurement for posteroanterior cephalometric films
(in millimeters) Sample Normal Variable
Transverse maxillary asymmetry Transverse mandibular asymmetry Incision superior-midline Incision inferior-midline Menton-midline ANS-midline Angular maxillary asymmetry* Angular mandibular asymmetry* *Measurements
1.87 2.89 1.09 1.45 1.89 0.84 18.58 18.21
1.22 1.92 1.24 1.09 1.34 0.71 13.46 13.81
1.82 3.50 1.34 1.64 2.16 0.90 17.85 20.59
1.67 2.44 1.10 1.33 1.67 0.62 12.92 15.45
ated by determining the absolute difference between measurements for right and left side landmarks. Individual measurements of asymmetry were then combined to produce indices of anteroposterior maxillary and mandibular asymmetries, producing two skeletal variables. Linear measurements were also made from the bilateral maxillary and mandibular molar points to the vertical reference line. Again, the absolute difference between right and left side landmarks was determined to arrive at measurements of anteroposterior maxillary and mandibular molar asymmetries, resulting in two skeletodental variables.
and left sides for both linear and angular measurements resulting in four skeletal measurements: angular mandibular and maxillary asymmetries, and transverse mandibular and maxillary asymmetries. Midline deviations of the maxilla and mandible were measured by determining the absolute distance from the two anatomically determined midpoints, ANS and menton, to the constructed facial midline, resulting in two additional skeletal variables. Midline deviations of the dental arches were measured by determining the absolute distance from incision superior and inferior to the constructed facial midline, resulting in two skeletodental variables.
The 22 variables-ten dental models, eight skeletal, and four skeletodental-were analyzed. For both groups means and standard deviations were calculated for each variable. To fulfill the major objective of the study, which was to characterize the groups, a multivariate discriminant analysis was performed using all the variables. Discriminant analysis is a multivariate statistical method in which combinations of variables are used to distinguish between groups. It allows group differences to be studied with respect to several variables simultaneously, thus avoiding the type I errors common to separate univariate analyses. The first purpose in using the analysis is to determine the optimal subset of variables that best separate or characterize the groups of interest. The method selects variables on the basis of ability to add to the discrimination until further variables no longer make a significant contribution. Once the variables have been selected, a discriminant function that maximizes group differences may be computed.
On the posteroanterior cephalometric radiographs, as shown in Fig. 3, the landmark points-namely, root of crista galli, ANS, incision superior, incision inferior, menton, and bilateral gonia, maxillare, and superior orbital points-were digitized. A reference midline was determined geometrically so that it passed through the root of crista galli perpendicular to a line connecting the most superior points on the orbital contours. This method of midline construction was chosen because it did not rely upon the use of any maxillary or mandibular landmarks and has been well documented in the literature. Linear measurements were made in a horizontal direction from the bilateral maxillary landmark maxillare and the bilateral mandibular landmark gonion to the constructed midline. Angular measurements were also made from the same bilateral landmarks to the constructed midline with the root of crista galli as the vertex of the angle. Skeletal asymmetry was evaluated by determining the absolute difference between right
Am. .I. Orthod.
Dentoja~ Whop Januurv 1988
Table II. Means and standard deviations of asymmetry measurements for lateral cephalometric films
(in. millimeters) Sample -Normal Variable
AP AP AP AP
maxillary asymmetry mandibular asymmetry maxillary molar asymmetry mandibular molar asymmetry
1.77 3.93 0.59 0.87
0.98 3.21 0.62 0.72
2.41 8.05 1.39 2.10
2.23 6.87 1.28 1.34
Table HI. Means and standard deviations of asymmetry measurements for maxillary dental models (in millimeters) Sample Normal Variable
Transverse buccal segment asymmetry Transverse anterior segment asymmetry AP buccal segment asymmetry AP anterior segment asymmetry Mandibular midpoint-raphe
3.95 4.48 7.85 2.72 0.94
5.66 4.99 6.95 3.23 1.03
The second phase of the discriminant analysis involves calculation of a discriminant score and probability of group membership for each individual in the sample using the mathematic function. The subject is classified into the group for which the probability is highest and a table is printed showing the actual and theoretical group membership based on the use of the function. These results indicate the percentage of cases correctly classified based on the use of the function. The use of Wilks’ lambda, which is the multivariate test statistic for group differences, indicates the discriminating power of the variables. In addition, the tau statistic, which is a part of the analysis, provides a measure of improvement in classification over that which might be expected through random assignment of subjects to groups. The discriminant analysis also produces a set of standardized coefficients, which are used to compute discriminant scores in standard deviation units. These are useful in determining the contribution of each variable in the discriminating set. The larger the absolute magnitude of the coefficient, the greater the contribution of the variable to the discrimination. RESULTS
Means and standard deviations for all variables in both groups are listed in Tables I through IV. Tables I
3.17 5.25 1.98 0.56
4.80 5.76 2.95 1.09
and II provide the statistics for measurements taken on the posteroanterior and lateral cephafometric radiographs, respectively. Overall, the mean values for asymmetry tend to be greater for the malocclusion group as compared with those of the normal group. This was an anticipated finding since the malocciusion group was selected on the basis of asymmetric occlusal relationships. The variabIes measured o&e lateral film showed the greatest asymmetry. This was especially true for anteroposterior mandibular asymmetry, which showed a mean difference of 4.1 mm between the groups. ANS-midline and transverse maxillary asymmetries, both of which were measured on posteroanterior cephalometric radiographs, had group means that were nearly equal to the malocclusion group or, in the case of transverse maxillary asymmetry, slightly smaller by an insignificant amount. Although this could be interpreted as a function of unequal distances of the right and left sides of the structures from the cassette on the lateral radiograph, such a condition should influence computation of asymmetry equally for both groups under study. Thus, any reported difference between the groups shouki be meaningful. Tables III and IV list statistics for dental model measurements. Again, the mean values for asymmetry tended to be consistently larger for the malocclusion group as compared with the normal group with the
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Facial and dental arch asymmetries
Table IV. Means and standard deviations of asymmetry measurements for mandibular dental models (in millimeters) Sample Normal Variable
Transverse buccal asymmetry Transverse anterior segment asymmetry AP buccal segment asymmetry AP anterior segment asymmetry Mandibular midpoint-raphe
27 (96.4%)* 3 (10.7%)**
4.79 4.66 5.81 1.95 1.11
2.84 4.09 5.09 1.33 0.80
Table V. Classification results of discriminant analysis Predicted
1 (3.6%)** 25 (89.3%)*
*Percent of total cases correctly classified: 92.868, tau = 0.857. **Percent of total cases incorrectly classified: 7.14%.
exception of anteroposterior maxillary buccal segment asymmetry. In the malocclusion group, mandibular mean asymmetry values were consistently larger than the maxillary arch asymmetry values. Within the normal group, the values of anteroposterior asymmetry for both buccal and anterior segments tended to be smaller for the mandibular arch; mean values for transverse asymmetry differed only minimally between arches, with the values being slightly larger for the mandibular arch. For both groups, the anterior dental segment displayed greater asymmetry in the transverse dimension. Most likely, this is related to the magnitude of tbe dimensions involved since transverse measurements were larger than anteroposterior measurements. All measurements on both the cephalometric radiographs and dental models displayed consistently high standard deviations. This indicates a large amount of variability within the groups with regard to asymmetry. This was particularly true for the malocclusion group, whose standard deviations exceeded those of the normal group. Results of the discriminant analysis are listed in Tables V and VI. The Wilks’ latnbda of 0.7273 for the analysis was statistically significant. This shows that the combination of variables selected in the analysis was highly effective in distinguishing between the groups. Table V lists the classification results of the discriminant analyses. The percentage of known cases
6.59 8.31 9.60 3.42 1.56
5.61 6.64 6.60 2.35 1.36
Table VI. Standardized coefficients of discriminating variables for skeletodental discriminant analysis Variable
AP mandibular molar asymmetry (lateral cephalometric radiograph AP maxillary molar asymmetry (lateral cephalometric radiograph) Maxillary midpoint-raphe (maxillary model) Incision superior-midline (PA cephalometric radiograph) Transverse anterior segment asymmetry (maxillary model) AP buccal segment asymmetry (mandibular model) ANS-midline (PA cephalometric radiograph) AP mandibular asymmetry (lateral cephalometric radiograph) Transverse buccal segment asymmetry (mandibular model)
1.0570 0.8529 0.5594 0.5306 -0.5081 0.4951 0.4057 0.2960 0.2028
correctly classified based on the mathematic function is important in evaluating group differences because it gives me probability of correct classification using the theoretical model. It can be used together with the Wilks’ lambda as a further means of indicating how well the variables discriminated between groups. Using the mathematic function, 92.86% of all cases were correctly classified; thus, the probability of misclassification using the model was 0.9286. The computer value of tau was 0.857, which indicates that classification based on the function produced 85.7% fewer errors man would be anticipated using random assignment. An illustration of this relationship is provided in Fig. 4. Table VI lists the standardized function coefficients for the nine variables of the original 22, which was the set chosen as the best discriminators in the discriminant analysis. These coefficients indicate tbe relative im-
Alavi, BeGole, and Schneider
portance of each of the variables and they are listed in decreasing order of importance on the basis of absolute magnitude. Three skeletodental variables, four dental model variables, and two skeletal variables comprised the set of discriminating measures, indicating that a combination of both skeletal and dental factors was responsible for the separation between the two groups. The highest ranked and first selected variable was anteroposterior mandibular molar asymmetry. This is the single variable that produced the greatest separation between groups. The Wilks’ lambda based on this variable alone was 0.7473, with a significance level of 0.003. Anteroposterior mandibular asymmetry as measured in this study reflects asymmetry in spatial position of the mandibular molar within the craniofacial complex, which could be due to both dentoalveolar and skeletal asymmetries. However, the former seems to make a greater contribution since buccal segment asymmetry of the mandibular arch ranked above asymmetry of the mandible among the discriminating variables. Anteroposterior maxillary molar asymmetry also made a significant contribution, second in order to mandibular molar asymmetry. Beyond this it is difficult to singularly evaluate the individual contribution of any variable because all variables following the first selected in the discriminant analysis were chosen for the improvement they added to the discrimination. It is the unique combination of all the selected variables that is responsible for the total discrimination. DISCUSSION
As demonstrated by the means and standard deviations, asymmetry of both dental arches, maxilla, and mandible was a common finding in both normal and malocclusion groups. This result is not unique and has been reported in the literature on numerous occasions. The discriminant analysis results provided the basis for this study. As might have been expected, the discriminant analysis demonstrated a highly significant difference between normal and Class II subdivision malocclusion groups. A combination of skeletal and dentoalveolar variables was shown to be responsible for the separation of the groups. The relative importance of specific variables in the discrimination demonstrated a somewhat unexpected result. Since the two groups had been originally selected on the basis of dental model characteristics, that is, the relationships of the first permanent molars, it would seem that variables relating to the dental arch should rank highest among the discriminating variables. This was not the case, however, as AP buccal segment asymmetry ranked sixth among the discriminating variables,
while AP mandibular and maxillary molar asymmetries, as determined from the lateral cephalometric films, best characterized the difference between groups. These variables accounted for most of the discrimination even if no other variables had been considered. These variables would not have been discernible on the PA radiographs that were used to evaluate asymmetries in many previous studies. It is of further interest that a malocclusion whose classification is made from study models on the basis of molar relationships is best represented by variables discernible on lateral cephalometric films involving the molar teeth. Apparently, the use of the posterior vertical reference line passing through basion on the lateral cephalometric films offers a reference away from the area of the dental region and results in a more accurate evaluation of the spatial position of the maxillary and mandibular molar teeth. While the spatial position of the molar teeth. especially the mandibular molars, proved to be the most discriminating variable, what cannot be ascertained is whether this was due to their position in their respective jaws, or to the position of the jaws in the craniofacial complex, or a combination of both. There is yet one other consideration affecting mandibular molar position and position of the mandible that should be addressed-namely, that of mandibular repositioning incident to dental relationships. Cephalometric films are typically exposed in the position of maximum intercuspation, be it in centric occlusion or centric relation. If occlusal interferences are present, which is a possibility in the case of malocclusions, these positions may not coincide. Thus, in certain casesthe position of the mandible, and consequently that of the mandibular molar, could be affected by dental reIationships that create mandibular repositioning. Although the preceding would be impossible to determine because the cephalometric films used in this study had been taken previously, this possibility does deserve mention. If it is assumed that no malpositioning of the mandible occurred, it may be concluded that the dentition is a good indicator of asymmetry and that such asymmetry is perhaps best evaluated with references outside the dental region. It should also be noted that while spatial position of both permanent molars was important in distinguishing the groups, it was the position of the mandibular molar that provided the maximum amount of discrimination, with the AP position of the maxillary molar serving to enhance this result. In fact, AP mandibular molar asymmetry, as measured on the lateral cephalometric film, alone discriminated between groups at a highly significant level. While it could be argued that
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Facial and dental arch asymmetries
this might be a function of measurement since, with the radiographic technique, the mandibular molar is further away from the central ray through porion, this factor should not have affected the results because the discriminant analysis is carried out between groups, which would make the effect of such measurements insignificant. It is generally accepted that shape, size, and position of the jaws and dentoalveolar region are to a certain extent genetically programmed. Johnson,26 in analyzing Stockard’s classic work on heritability in dogs, concluded that genetic constitution was a vital factor in the development of skull form and dental occlusion. Moore*’ thought that heredity was operative in facial symmetries and found such asymmetry to be three to four times as great in children whose parents were similarly asymmetric. The concept of a genetically predetermined asymmetric pattern of growth and development in Class II subdivision persons might offer some explanation as to why such persons may be difficult to treat orthodontitally. Few reliable methods of delivering asymmetric orthopedic forces are available, especially forces to the mandible, which according to this study was most responsible for the existing asymmetric occlusal relationship. In the case of the dentoalveolar region, where it is often possible to use unilateral mechanics, the asymmetric growth pattern in these cases may tend to undermine or hinder treatment. Brodie2’ suggested that even into the postretention period, the supporting alveolar bone continues to grow in its asymmetric form. Coincident midlines are also often difficult to achieve because of the presence of such asymmetries. Thus, it can be seen that subdivision malocclusion cases caused by the presence of increased skeletal and dentoalveolar asymmetries may often require compromises in treatment or an asymmetric extraction pattern as suggested by Wertz3” and Cheney.“’ As mentioned previously, it was concluded in this study that skeletal and dentoalveolar asymmetries were found to contribute to the asymmetric buccal segment relationship present in Class II subdivision malocclusions. Significant differences in skeletal asymmetry have not always been reported in previous studies of asymmetry comparing normal and malocclusion groups, most notably those of Shore’ and Letzer and Kronman2’ The somewhat contrary finding of the present study can be accounted for by several factors. First, the present study used a malocclusion group with well-defined asymmetric characteristics. The implications of using such a group as opposed to a nonspecific malocclusion sample should be apparent. While
most malocclusions would be expected to display increased asymmetry of the individual dental units, most often they are of a bilateral nature with regard to buccal segment relationships except perhaps in the case of mutilation or blocked out teeth. It does not seem that previous authors should have anticipated finding increased skeletal asymmetry in such malocclusion groups as compared with normals. Second, the use of only PA cephalometric films in previous studies was limiting in that it allowed only transverse and vertical dimensions to be studied. The use of lateral cephalometric radiographs in the present study allowed asymmetry to be studied in an anteroposterior direction, The use of these radiographs did involve the problem of accurate identification of right vs. left side landmarks. However, using the BroadbentGolden orientator and the dental models as a secondary reference, identification was reasonably well made. It should be noted that the use of submental vertical films would also have been quite appropriate for this type of study. In their cephalometric study using submental vertical films, Williamson and Simmons2* found that subjects displaying 3 mm or more of mandibular asymmetry had a tendency toward a Class II buccal segment occlusion on the short side. This correlates well with the results of the present study. Third, the difference in statistical techniques between previous studies and the present one must also be appreciated. Simple univariate analysis allows only for the comparison of variables on a one-to-one basis and is lacking in the characterization of complex interrelationships among the variables. The multivariate statistical approach of the discriminant analysis allows for perturbations of the data that are difficult to evaluate visually and impossible to reveal by measurements of simple univariate analysis alone. Such an approach allowed the information contained in numerous variables to be summarized, producing an expanded picture of dentoalveolar and skeletal relationships. REFERENCES I. Woo TJ. On the asymmetry of the human skull. Biometrika 1931;22:324-52. 2. Tildesley ML. A critical analysis of investigations into facial growth changes. Int J Orthod Oral Surg Radgr 1932;18: 1131-69. of the morphometric 3. Pearson K, Woo TL. Further investigation characters of the individual bones of the human skull. Biometrika 1935;2:424-65. 4. Bjijrk A, Bjdrk A. Artificial deformation and craniofacial asymmetry in ancient Peruvians. J Dent Res 1964;43:353-62. 5. Gundara N, Zivanovic S. Asymmetry in east African skulls. Am J Phys Anthropol 1968;28:331-8. 6. White JC. A study of craniofacial asymmetry [Master’s thesis]. Cleveland: Case Western Reserve University, 1982.
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Alavi, BcGole, and Schneider Vazquez F, Grostic JD, Fonder AC. Eccentricity of the skull correlation with malocclusion. Angle Orthod 1982;52: 144-8. Shore IL. A cephalometric study of facial asymmetry IMaster’s thesis]. University of Pittsburgh, 1959. Vig PS, Hewitt AB. Asymmetry of the human facial skeleton. Angle Mod 1975;45:125-9. Svanholt P, Solow B. Assessment of midline discrepancies on the posteroantetior cephalometric radiograph. Tram Eur Orthod Sot 1977:261-70. Shah SM, Joshi MR. An assessmentof asymmetry in the normal craniofacial complex. Angle Ortbod 1978;48:141-7. Haberle FE. A quantitative study of the malocclusion and correction in the posterior segment of the arches of 16 patients [Master’s thesis]. Chicago: Northwestern University, 1937. Hunter WS. Lateral asymmetries of 93 maxillary arches. Acta Odontol Scand 1953;11:95-9. Barr M, Gron P. Configuration of the adult palate. J Can Dent Assoc 1960;26:62-9. Lundstrom A. Some asymmetries of the dental arches, jaws, and skull, and their etiologic significance. AM J ORTHOD1%1;47:81106. Jensen GM. A study of the dentoalveolar morphology and developmental changes in Downs’ syndrome (Trisomy 21) [Master’s thesis]. Winnipeg: University of Manitoba, 1972. Hechter FJ. Symmetry, form and dimension of the dental arches of orthodontically treated patients [Master’s thesis]. Winnipeg: University of Manitoba, 1975. Biggerstaff RH. Three variations in dental arch form estimated by a quadratic equation. J Dent Res 1972;51:1509. Pepe SH. Polynomial and catenary curves fit to human dental arches. J Dent Res 1975;54:1124-32. BeGole EA. Application of the cubic spline function in the description of dental arch form. J Dent Res 1980;59:1549-56. Letzer GM, Kromnan JH. A posteroanterior cephalometric evaluation of craniofacial asymmetry. Angle Orthod 1967;37: 205-l 1.
22. Williamson EH, Simmons MD. Mandibular asymmetry and its relation to pain dysfunction. AM J ORTHOII1979;76:612-7, 23. Cleall JF, Chebib FS. Co-ordinate analysis applied to orthodontic studies. Angle Otthod 1971;41:214-8. 24. Broadbent BH Sr, Broadbent BH Jr, Golden WH. Bolton standards of dentofacial developmental growth. St. Louis: The CV Mosby Company, 1975. 25. Hellman M. Some facial features and their orthodonttc implication. AM J ORTHODORAL SURG1939;25:927-51. 26. Johnson AL. The constitutional factor in skull form and dental occlusion. AM J Oarnon ORAL SURG1940;26:627-63. 27. Stockard CH. The genetic and endocrine basis for differences in form and behavior. Philadelphia: Wistar Institute of Anatomy and Biology. 1941. 28. Moore GR. Heredity as a guide in dentofacial orthopedics. AM J ORTHODORAL SURG1944;30:548-54. 29. Brodie AC. Anatomy and physiology of the head and neck musculature AM J ORTHOB1950;26:831-44. 30.. Wertz RA. Diagnosis and treatment planning of unilateral Class II malocclusions. Angle Orthod 1975;45:85-94. 31. Cheney EA. The influence of dentofacial asymmetries upon treatment procedures. AM J ORTHOD1952;38:934-45. 32. For&erg CT, Burstone CJ, Hanley KJ. Diagnosis and treatment planning of skeletal asymmetry with the submental-vettica1 radiograph. AM J OKTHOD1984;85:224-37. Reprint requests to‘ Dr. Ellen A. BeGole University of Illinois at Chicago College of Dentistry Department of Orthodontics 801 South Paulina St. Box 6998 Chicago, IL 60680