Arehs oral Bid. Vol. 33, No. 12, pp. 901-906, 1988 0003-9969/88 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright 0 1988 Pergamon Press plc

MATHEMATICAL AND DESCRIPTIVE CLASSIFICATION OF VARIATIONS IN DENTAL ARCH SHAPE IN AN

AUSTRALIAN ABORIGINE POPULATION

J. K. MCKEE’* and S. MOLNAR*

‘Department of Anatomy, University of the Witwatersrand, Johannesburg, Republic of South Africa and ‘Department of Anthropology, Washington University, St Louis, MO 63130, U.S.A.

(Accepted 12 July 1988)

Summary-The ability to describe dental arch shape is necessary for biomechanical studies of occlusion as well as for anthropological studies of human and primate dental variation. A mathematical method of describing and classifying human dental arch shape was used to assess the nature of individual variability. The method involved the calculation of a series of third-degree polynomials which were fitted to coordinate points along the dental arcade. The slopes of the polynomials, evaluated at these coordinate points, provided a multivariate description of shape, independent of arch size. Graphic representations of arch shape could be constructed from the polynomial equations. These mathematical techniques were used in association with multivariate and univariate statistics to explore the types of variability in,dental arch shape among a population of Australian aborigines. The results illustrated the ambiguities of conventional subjective classifications.

INTRODUCTION

Variability in size and shape of the human dental arch has been investigated by dentists and anthropologists alike for many years. The clinical importance of understanding arch shape lies primarily in its value for studying occlusion and the biomechanics of mas- tication (Murphy, 1964; Isaacson et al., 1971; Smith and Bailit, 1977; Harris and Smith, 1980; Demes, Preuschoft and Wolff, 1984; Wolff, 1984; McKee, 1985a; Smith, 1986). Anthropologists use arch shape to describe evolutionary changes in the dentition (Simons and Pilbeam, 1965; Jolly, 1970; Walker and Andrews, 1973; Zwell, 1972) and to characterize modern human dental variation (e.g. Campbell, 1925; Heithersay, 1961; Boyde, 1972). Their studies have described this shape variability both by subjective classification and by mathematical quantification.

Heuristic descriptions of variations in dental arches have traditionally depended on subjective precon- ceptions of shape. The primate maxillary arcades have been described by terms such as V-shaped, U-shaped, or parabolic. However, it is difficult to describe any differences between the U-shape in a gorilla or the U-shape in a chimpanzee using such a subjective system. Furthermore, it creates ambiguities in shape definition due to variation within species. Whereas these terms project an image of the existing shape variation, the descriptions are not standard or comparable. It is this variation which we have now considered in the hope of finding arch shape descriptions that can be used on an intrapopulational assessment as well as an interspecific level.

*Address correspondence to: Dr Jeffrey K. McKee, De- partment of Anatomy, University of the Witwatersrand Medical School, 7 York Rd, Parktown, JHB 2193, Republic of South Africa.

Researchers traditionally have depended on linear measurements to describe objectively variations in dental arch shape (e.g. Murphy, 1964; Moorees and Reed, 1965; Barrett and Brown, 1968; Smith and Bailit, 1977; Harris and Smith, 1980). Ratios of arch widths and lengths provide a general idea of the shape of the area encompassed by the dentition, but a variety of shapes could result in the same ratios:In an attempt to overcome the Iimitations imposed by linear measurements, Lavelle et al. (1978) calculated areas enclosed by the dentition. They! used mulfi- variate analysis of area and the length $nd width dimensions to compare the dental arches of several ethnic groups. Though this approach improved com- parisons of arch variability, it was not as descriptive of shape as CUNe fitting routines.

Effective arch-shape description has been accom- plished through the various curve fitting routines used by some researchers (see Pe

pan 1981). Many of these methods rel? , 19$5: Rudge, eav+y on con-

cepts of an ideal shape and were preconceived or mathematically determined. Other attempts have involved fitting third degree polynomials or conic sections to the entire arch, but they also assumed a symmetry of shape which is rare in complex biolog- ical structures such as the masticatory system.

Even when using objective mathematical_ ap- proaches, individual variation in arch s pe is best described by methods which produce cu r& based on individual arches, not on preconceived shapes. TO-

ward this goal, Barrett and Brown (1968, p. 385) have suggested that “coordinate systems of measurements and analysis provide a more promising method of describing and comparing dental arch shapes.” An example of such a method is given by ffiole (1979, l%O), who utilized a cubic spline technique in de- scribing arch shape from coordinate. data. [Cubic

901

902 J. K. MCKEE and S. MOLNAR

splines were also suggested by Pepe (1975) as a possible alternative to fitting single polynomials to the entire arch.] A set of incisal and cuspal points were represented on a Cartesian coordinate system, and a series of third-degree polynomials, or cubic splines, were fitted to these points. Another approach was taken by Lu (1966), who used a coordinate system to assess asymmetry in the dental arch, in a method that involved fitting a fourth-degree poly- nomial to coordinate data which were equally spaced along the arch based or X-axis. Indices of symmetry and asymmetry were then indicated by the coefficients of the equations.

Each of these objective descriptive methods thus has limitations in accurate representation of arch shape. In considering individual shape variability, the study of dental arches requires a descriptive method with a minimum of assumptions and a flexibility that allows broad applications. Here we present the details of such a technique, apply it to an investigation of variation in maxillary arch shape among a sample of Australian aborigines, and compare the results with commonly used subjective categories of shape.

MATERIALS AND METHODS

A photograph of the occlusal surface was enlarged and printed to an approx. 1.5 scale factor for each cast of the maxillary dental arch. The camera was kept at a constant distance from the occlusal planes to avoid problems of differential degrees of parailax. Fifteen coordinate points were marked by pen on the prints (Fig. 1A). These homologous points were chosen for their morphological relevance to occlusal arch shape and tooth position as well as for ease and consistency of location. Points I and 2 were deter- mined on the basis of a line approximating a tangent to the mesial and distal grooves of the M2, extending mesio-distally to the approximal surface of the tooth. Points 3, 4, and 5 were similarly based on the mesial and distal grooves of M’, and central grooves of p4 and P’ respectively. The incisal points (6 and 7) and mid-incisal point (8) were located at the occlusal edges of the teeth as illustrated. Final records of the coordinate values for each point were collected and scaled to the nearest millimetre by a graphics tablet connected to a microcomputer. Data for each arch were digitized twice, and coordinate points were averaged for the two determinations.

Facial structures of extant Australian aborigines Points representing landmarks on the arches had show sufficient variation in shape to be useful in to be rotated into a standardized orientation for testing numeric descriptions of arch shape variability comparability of shape description by mathematical (Barrett and Brown, 1968; Brown, Carroll and Bur- functions. The orientation method used by Pepe gess, 1983). The arch coordinate data were obtained (1975) and BeGole (1979) relied on location of the from a collection of serial dental casts made from a mid-palatal raphe, and the arch coordinates were group of Australian aborigines in Central Australia rotated so that this line would lie on the Y-axis of who had been participants in a longitudinal growth the Cartesian coordinate system. Because the mid- study from 1951 to 1971 (Brown and Barrett, 1973). palatal raphe was not always distinguishable on our The dental impressions and casts were made at the photographs (and is often indete~inate on fossil time of dental examinations during annual visits by the &search team, A cross-sectional sample of 55

specimens), we developed a mathematical method to standardize arch orientation for comparative studies.

adults from this collection provided the data base for We selected two sets of anteriorly located points

the present study. No samples were accepted if both second molars were unerupted or if any other teeth were missing. This sample included 21 males and 34 females, with an average age of 18.7 yr.

Fig. 1. (A) Location of landmarks used for digitization of coordinate points. (B) Four linear slopes are used in the calculation of slope which is tangent to the curve. (C) The resulting curve as calculated and

produced by the quasi-cubic hermite interpolation technique.

Variations of dental arch shape 903

which would minimize the variation due to the mesial drift that occurs during growth. Points on the anti- merit premolars (P3 and p) were used to define lines of orientation across the arch. The amount of rota- tion necessary to bring each of these lines parallel to the X-axis was calculated and then averaged. This average became the angle of rotation for all of the points, with points being translated so that the mid- incisal point would lie on the Y-axis. Visual in- spection of plots of the coordinates was made to confirm a proper orientation. The resulting axis of orientation does not necessarily correspond to that achieved using the mid-palatal raphe, but provides a standardized basis for arch shape comparisons.

Estimates of the curves between coordinate points were accomplished using a method of curve fitting described as a “one-dimensional quasi-cubic hermite interpolation”. Details of this interpolation technique can be found in Akima (1970) and it is available as the FORTRAN-based IMSL routine IQHSCU (IMSL, 1982). Though it is very similar to the cubic spline technique used by BeGole (1979, 1980), this method better approximates a curve which would be drawn by hand. The smoothness of a cubic spline will depend on the degree of the spline, as well as on the somewhat arbitrary selection of slope values for the endpoints. The quasi-cubic interpolation is somewhat smoother (Akima, 1970), resulting in a more accurate representation of shape for purposes of description or shape interpolation, and provides an algorithm for estimating the slopes at the endpoints.

Representation of arch shape by the quasi-cubic spline technique involves three major steps. First, the slope at a given point is determined by the coordi- nates of five points, including the point in question as the centre point and the two points on either side (Fig. IB). This slope, representing the first derivative or slope of a curve at the respective point, is thus a measure of shape which is independent of the size of the arch. Step two is a mathematical interpolation between a set of points using the slopes and coordi- nates as conditions for determining a unique third- degree polynomial. This curve-fitting routine does not require a continuous second derivative as does the cubic spline. The third step involves the esti- mation of two more points beyond each end point. This is necessary so that there will be five points with which to determine the slope at the penultimate and endpoints (Akima, 1970). Though this slope depends on an extrapolation from mesially located points, it does provide a reasonable estimate of the respective slopes at the endpoints.

For this study, in which we were interested in looking at overall arch shape in a small population, arch asymmetry can add a considerable amount of variability. To avoid these ambiguities of asymmetry and to limit the number of variables used in the multivariate statistics described below, the absolute values of the antimeric slopes were averaged. Two other slopes and a point were excluded to further the conservative nature of this analysis: these were the MZ endpoint slopes, which are based on extrapolated points, and the mid-incisal point, which is essentially a measure of anterior arch asymmetry. Consequently, the number of variables investigated was limited to six.

Sratistical analysis

Each of’ 24 randomly selected arches were digitized twice for a test of reliability. Duplicate calculations of the Six slope measurements were then compared in a paired I-test of the original and test samples.

Common sources of variability in arch shape were assessed by a principal components type of factor analysis, one of a group of multivariate techniques used in morphometric analyses (Kowalski, 1972; Brown, 1973). This can explain the correlations among the variables in terms of a limited number of hypothetical variables, or factors, which can tell us something about the natural variation in arch shape. The factor analysis begins with a correlation matrix based on linear correlations between pairs of vari- ables (Kim, 1975; Brown, 1973; SAS Institute Inc., 1982). Consequently it must be assumed that the relationships between any two arch-shape variables are linear, as in a simple linear regression, Our shape variables, the slopes of lines tangent to the arch curves, did not progress in a linear fashion, but could be linearly transformed by taking the logarithm of the slope. The logarithmic transformation converts the slopes into an approximate linear relationship while preserving the natural variation of the data.

The individuals were grouped on the basis of their orthogonally rotated factor SCOIW for purposes of illustrating the types of shape variation. These group- ings were based on individual factor si”bres by assign- ing individuals to groups corresponding’to the factor for which they had the highest score. (T%is could be done because the mean value of the scoring coefficients for each factor was standardized to a value of zero.) The highest score woul? reflect a pattern with which the individual is most closely associated, providing us with a basis of bomparison. By using the factor scores in this manqer, we maxi- mized the group differences as expressed alo@ the various sources of variation. These groupings were then tested for their mathematical validity with a discriminant analysis (using the original shape mea- surements as discriminating variables), and individu- ally compared by a Student’s r-test.

RESULTS

Paired t-tests of the six log-transformed slopes of the 24 individuals measured twice showed no significant differences between the duplicate mea- surements (p > 0.05 for all tests). The maximum and mean absolute differences in log-transformed slopes are listed in Table I, together with the mean original measurements of the test sample.

A factor analysis of the six arch-shape variables resulted in three factors, based on a minimum eigen- value of 1.0, as reported in Table 2. Examination of the greatest absolute value of the factdIr.patterns revealed that the three factors basically represented variations in different parts of the arch. Factor l was loaded the highest on the M’ points (points 2 and 3); factor 2 was most highly loaded on the middle part of the arch where the premolar points (4 and. 5) are located; factor 3 loaded the greatest on the incisai points (6 and 7), but actually contrasted with the slopes of these points.

904 J. K. MCKEE and S. MOLNAR

Table 1. Mean errors in dual trials, and mean values of the loga~t~ically transformed slopes (at points I-8)

Mean Minimum Maximum Min. Max.

Variable error error error Mean SD slope slope

Slope 1 0.072 - 1.286 0.831 3.586 1.032 1.389 5.999 Slope 2 -0.041 -0.261 0.357 2.784 0.494 2.160 4.197 Slope 3 -0.056 -0.541 0.380 2.449 0.453 1.859 3.367 Slope 4 -0.022 -0.421 0.168 2.229 0.393 1.544 3.199 Slope 5 0.037 -0.390 0.347 1.861 0.361 1.211 2.463 Slope 6 0.067 -0.332 0.618 0.095 0.615 - I.435 1.131 SloDe 7 Slope 8

-0.101 -0.610 0.491 -2.418 0.667 -4.451 -1.475 0.263 - I .436 2.219 -3.040 0.924 -5.521 - 1.355

Table 2. Orthogonally rotated factor pattern

Factor 1 Factor 2 Factor 3

Slope 2 0.88533 -0.07361 0.17699 Slope 3 0.88506 0.18837 -0.19033 Slope 4 0.28878 0.83666 -0.28627 Slope 5 -0.08777 0.83028 0.26623 Slope 6 0.06502 -0.13995 0.88199 Slope 7 0.06475 -0.30883 -0.47442

Variance explained by each factor (Eigenvalues) - ‘1.666665 iS4523i - 1.223355

In order to iflustrate the types of shape variation, the three factor patterns were used to define three groups (Table 3). A discriminant analysis of these groupings ,was performed to establish the validity of the groups as based on the original shape variables. The six shape variables (the log-transformed slopes evaluated, at points 2-7) were able to discriminate between these groups and predicted group mem- bership with 95% accuracy. As might be expected, the three individuals who were misclassified were those who &id not have high scores on any factor relative to others in the population. Visual in-

spections of the graphic plots of the individual arches further confirmed the validity of the groupings.

Relationships among the three shape groups were assessed by an analysis of the mean values for the shape variables from each of the groups, as illustrated in Fig. 2. These means were compared to the means for the rest of the population by a Student’s f-test (Table 3). In general, the variables that had the highest loadings on each of the three factors were significantly greater (p < 0.05) for those groups Car-

responding to each respectively factor.

DISCUSSION

Heuristic descriptions of arch shape variation, as characterized by the three groups, can be summarized as follows. Group 1 had relatively steeper (more vertical) slopes in the distal region, resulting in some- what more parallel sides. With the exception of the most anterior slope, the slopes were less steep in the anterior region when compared with the other arches. This gives the appearance of a smoother, more rounded anterior region. Group 2 was characterized by steeper slopes in the mid-arch region of the premolars, while being more flared at the distal end

Table 3. Shape variable comparisons of groups

Max. Min. Sig. Variable N Mean SD value value dif.

Shape group I Slope 2 16 1.685 0.257 2.098 1.275 > Slope 3 16 1.461 0.194 1.816 1.060 1 Slope 4 16 1.093 0.230 1.599 0.713 Slope 5 16 0.829 0.165 1.188 0.550 < Slope 6 16 0.075 0.273 0.565 - 0.443 < Slope 7 16 - 1.052 0.208 -0.137 - l.458 >

Shape group 2 Slope 2 22 1.369 0.181 1.716 1.105 < Slope 3 22 1.194 0.190 1.519 0.852 Slope 4 22 I.175 0.154 1.499 0.975 > Slope 5 22 1.060 0.174 1.591 0.802 > Slope 6 22 -0.028 0.290 0.694 -0.718 < Slope 7 22 - 1.367 0.375 -0.863 -2.407

Shape group 3 Slope 2 17 1.458 0.260 1.976 1.080 Slope 3 17 1.082 0.144 1.368 0.899 < Slope 4 17 0.940 0.121 1.146 0.772 < Slope 5 17 0.888 0.172 1.176 0.638 Slope 6 17 0.429 0.210 0.866 0.119 > Slope 7 17 -1.416 0.471 -0.929 -2.509

A > or c denotes significantly greater slope or significantly lesser slope respectively for the difference between the indicated group and the remainder of the population.

Variations of dental arch shape 905

n 3

Fig. 2. Mean shapes of the three arch types as determined by factor analysis, and verified by Student’s r-test and

discriminant analysis.

and also of a lesser slope at the anterior end. This resulted in the appearance of a somewhat constricted premolar region. Group 3 arches weie steepest in the lateral incisor region, and less steep, or more flared, in the molar-premolar region.

In preliminary study (McKee, 1985b), we subjec- tively arranged the occlusal photographs of the maxillary arches into shape categories following Boyde’s (1972) classification. All individuals fitted into one of three categories: hyperbolic, parabolic, or hypsiloid. Using eight slope measurements (including the endpoints and mid-incisal point) as discrimin- ating variables, a discriminant analysis was then performed in order to classify individuals into the three shape categories. The two resulting discrimi- nant functions were able to classify correctly up to 85.5% of the individuals. Using only the six slope variables employed in our present analysis, and with more conservative statistical criteria (i.e. using a pooled instead of a non-pooled sample), only 65.5% were correctly classified. One should not expect complete concordance between subjective and mathematical classifications because of individual variability within shape categories. This variability is thus better described by the three mathematically determined groups than by the subjective categories.

The mathematical description provides a distinct advantage in that not only can we classify arch shapes, but also discern the location of important shape differences within the arches. For example, a term such as hypsiloid (or U-shaped) can describe the flattened anterior region of one group, or the parallel distal tooth row of another distinct group. Thus we can move from studies of subjective shape categories to rigid mathematical comparisons of the continuum of arch shapes.

Human arch-shape variations which might not have been discovered in subjective appraisals of shape have been exposed by our statistical exercises. In part, the patterns of factor loadings could be expected, due to the way in which the arch shape variables were determined. Because the slope of a point depends upon the location of the adjacent points, the highest loadings for a factor are most likely to concentrate on points which are next to each other. However, under this assumption there should have been more vari- ables loading highly on the first factor (that which

explains the most variation), because slopes are deter- mined by the locations of five adjacent points. Thus arch shape variation does appear to vary somewhat independently in various regions of the arch.

Extrapolation of the types of arch shape variability found in our population to other populations or species has limitations. Our analysis was performed on a limited number of Australian aborigines, and it cannot be expected that the factor patterns or the resulting groupings could adequately describe the types of variation found in other populations. Yet the analysis demonstrates the usefulness of mathematical description and provides a means by which popu- lations can be compared. With a much larger data set, all of the shape variables could be used in multi- variate analyses to give a better assessment of the degrees of arch shape asymmetry.

Reliable assessment of the causes of variation in arch shape awaits further investigation. It cannot be said at this stage what genetic and ontogenetic sources may be responsible for the shape variation. Similarly, we do not know what biomechanical causes and effects may be associated with the dental arch shapes. In such investigation, additional sources of shape variation may be discovered; the groups determined here are certainly not the only possible statistically verifiable shapes. The key point is that with mathematical shape descriptions, multivariate analyses are possible and informative.

An adequate mathematical means of describing arch shape is an essential starting point for clinical, morphological, and anthropological research. Our methods provide an effective means of investigating the individual shape variation of an intrapopula- tional sample. The slopes of the dental arch curves, as calculated by the quasi-cubic interpolation tech- nique, provide measures of arch shape that are independent of size and do not assume arch sym- metry. Because the geometric equations describe ‘the curve of a line, graphic representations of arch shape are easily constructed for subjective analyses and individual comparisons. These mathematical descrip- tions of shape have the further flexibility of providing various measures of arch asymmetry and traditional arch dimensions, in constrast to the limited applica- tion possible with dimensions and ratios alone. Such flexibility allows a wide range of questions to be answered concerning occlusion, growth and devel- opment, as well as phylogenetic and ethnic variation.

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