Shape-coordinate and tensor analysis of skeletal changes in children with treated Class III malocclusions

Shape-coordinate and tensor analysis of skeletal changes in children with treated Class III malocclusions Tiziano Baccetti, DDS, PhD, a and Lorenzo Franchi, DDS, PhD b

Florence, Italy The current study was undertaken to evaluate both maxillary and mandibular shape/size changes in children with Class III malocclusions, treated with a functional appliance (removable mandibular retractor). Nonconventional cephalometric methods (Bookstein's shape coordinate analysis, centroid size analysis, and tensor analysis) were applied to a maxillary triangle (T-FMN-A) and to a mandibular triangle (Co-Go-Pg). A group of 30 children with treated Class III malocclusions were compared with a matched group of 30 children with untreated Class III malocclusions. Treatment with the functional appliance produced a significantly increased growth of the maxilla, featuring a more downward and forward displacement of the region of point A and a significantly more upward and forward direction of condylar growth, leading to a "shrinkage" of total mandibular length. The biologic significance of some conventional cephalometric measurements used for the assessment of both maxillary and mandibular position and structure was tested. (Am J Orthod Dentofac Orthop 1997; 112:622 -33.)

T h e r e are several methodologic concerns in studying dentoskeletal effects of functional/orthopedic treatment on growing subjects. These typically relate to: (1) the availability of treated and control (untreated) groups, homogeneous as to type of malocclusion, craniofacial pattern before therapy, sample size, age, and sex distribution, and duration of observation periods; (2) a cephalometric reference system appropriate for the longitudinal evaluation of skeletal modifications, i.e., a reference system that should remain stable with time; (3) a method for the representation and quantification of both shape and size changes in the skeletal structures involved; and (4) effective measurements for the description of main treatment effects on those structures. Significant craniofacial changes induced by early treatment of Class III malocclusion with a functional appliance (removable mandibular retractor) were assessed in previous studies, t-3 In keeping with basic method requirements of conventional cephalometrics in such investigations, our research design emphasized the comparison between matched treated and untreated Class III samples. We used a basicranial reference system, which proved to re-

main "stable" along with growth changes. Unfortunately, among the drawbacks of conventional cephalometrics, there is the inability to provide information about separate shape and size contributions to total skeletal changes due to treatment. In many cases, biologic bases for the choice of linear and angular measurements may be fallacious. Therapeutically induced deformations of skeletal structures may be localized far from conventionally traced lines and angles. New descriptive methods of shape and shape changes have been developed and implemented as major improvements, when compared with conventional cephalometrics. 4-9 Among these methods, Bookstein's innovations have been used to investigate modifications in shape, related both to facial growth and to treatment. 9-2° The aim of this study was to evaluate both maxillary and mandibular shape/size changes in children with treated Class III malocclusions by means of two coordinate-free biometric methods (Bookstein's shape-coordinate analysis and tensor analysis), to substantiate the modus operandi of the removable mandibular retractor in the correction of mandibular protrusion.

From the Department of Orthodontics, University of Florence. aPostdoctoral resident. bLecturer in Orthodontics. Reprint requests to: Dr. Tiziano Baccetti, Via E. Pistelli, 11, 50135 Florence, Italy. Copyright © 1997 by the American Association of Orthodontists. 0889-5406/97/$5,00 + 0 8/1/75435

TENSOR ANALYSIS

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The application of tensor analysis to cephalometries was developed by Fred Bookstein in the late 1970s and refined by him in following years? -~5 The effect of deformation or shape change can be de-

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scribed by constructing a series of triangles between homologous landmarks. A circle may be drawn within the initial or reference triangle that is tangent to all three sides. Deformation of this first triangle into a second one (after a period of growth and/or of treatment) transforms the circle into an ellipse. The longest and shortest diameters of the ellipse are the "principal strains" or representations of maximum stretch (dilation or shrinkage) that result from this deformation. The calculation of the ratio of the two principal dilatations provides a measure of change in shape. The intersections between the principal axes (or biorthogonal grids, as they lie at about 90° of each other) are to be considered as tensors, coordinate-free representations of shape change. Change in size of the triangles can be described by the product of the principal axes. Bookstein's tensor analysis is then a coordinate-free method that separates the observed change of a triangle (three landmarks) in one component for size change and a second component for shape change. SHAPE COORDINATES FOR LANDMARK CONFIGURATIONS

In 1986, Bookstein conceived a more powerful way to describe shape changes in any number of landmarks by analysis of shape coordinate variables, which represent the shape of a triangle of landmarks in a manner completely independent of size. 14 The landmark locations of a single triangle are reduced to the descriptor space of its shape: instead of shape "variables," we use just two shape coordinates to some convenient baseline. Therefore these may be thought as having "already transformed" the space of interlandmark distances into an algebraic form suitable to a great variety of biologic explanations in terms of shape change. For purposes of shape measurements, the researcher is free to standardize the scale of a landmark configuration in any way. When a set of triangles is scaled so that the separation between one pair of landmarks is constant, then the triangle may be considered to have been set down with both of those landmarks fixed in position. When two landmarks of the original triangle are restricted in this way, the information about the shape of the original triangle is encoded in the only aspect of the data that remains free to vary: the location of the third landmark. The Cartesian coordinates of that landmark (in the "shape space") are called shape coordinates of the original triangle (Fig. 1). The statistics of these coordinates includes all the information of any other system of shape variables that are measured on the same landmarks.

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The analysis, however very useful for statistical purposes, so far is in a language unrelated to biology. To return to the real scientific context, any direction of change in the space of a pair of shape coordinates can be interpreted as a uniform deformation of the original triangle, i.e., as the tensor description of shape change. After statistical computations (average of changes in groups and their comparison) proceed with shape coordinates, findings are given biologic interpretations by means of the tensors corresponding to those changes, when reinterpreted as smooth deformations. This graphical problem is the concern of the method of thin-plate splines, which were not used in the current analysis. Shape-coordinate analysis allows for a calculation of changes in shape, free from information about size changes. It can be supplemented by a separate measure of changes in size that may be correlated or uncorrelated with effects on shape. In the presence of pure digitizing noise, the same at every landmark in every direction, the size variable uncorrelated with all shape coordinates, and hence all ratios of homologously measured lengths, is "centroid size" (the root-summed-squared set of interlandmark distances). SUBJECTS AND METHODS Subjects

A group of 30 children, with treated Class III malocclusions (18 boys, 12 girls), mean age at the first observation (time 1, immediately before the beginning of treatment) 5.64 -- 1.01 years (standard deviations), mean age at the second observation (time 2) 8.43 _+ 1.73 years was compared with a control group of 30 children with untreated Class III malocclusions (13 boys, 17 girls), mean age at first observation (time 1) 6.06 -+ 1.14 years, mean age at second observation (time 2) 8.45 _+ 1.79 years. Mean observation period was 2.86 _+ 1.08 years for the treated group and 2.39 _+ 1.28 years for the control group. The composition of these groups has been described in a previous artMe. 2 At the time of the first observation, Class III malocclusion was diagnosed in cases manifesting: (1) anterior crossbite, (2) Class III deciduous canine relationship, (3) mesial step deciduous molar relationship or Class III permanent molar relationship. Appliance

A removable mandibular retractor (Fig. 2) was applied to the treated children throughout the observation period. The retractor was intended as a true functional appliance. Its labial arch extended to the cervical edge of the mandibular incisors and was activated to be placed 2 mm in front of these teeth when the mandible was forced into maximum retrusion. Therefore the arch was intended

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Fig. 1. Shape-coordinates for a triangle. When two triangles (triangle at time 1, A1BTC ~, and triangle at time 2, A2B2C2) are scaled so that separation between one pair of landmarks (AIB~ and A2BP is constant (AB, baseline), information about shape of original triangle is given by position of third landmark (C). Its Cartesian coordinates are shape coordinates of triangle to chosen baseline. pliance could be reactivated to maintain the labial arch in the correct position. Thereafter, the children wore the same appliance only at night. Compliance was high for the entire group.

Construction of Maxillary and Mandibular Triangles

Fig. 2. Removable mandibular retractor. to merely work as a stop for sagittal movement of the mandible. The retractor was attached to the second deciduous molars in the upper jaw by Adams' clasps. It might incorporate auxiliary devices such as an expansion screw or springs for proclination of the upper incisors, when required. Treated children wore the appliance at least 14 hours a day (nighttime included) until there was evidence of a corrected anterior crossbite. During this period, the ap-

Lateral standardized cephalograms of the 30 children with treated Class III malocclusions and of the 30 children with untreated Class III malocclusions were taken at both first and second observations. The same x-ray device was used for the 120 radiograms, which were performed by a single technician. The focus-median plane distance was 152 cm and the film-median plane distance was 10 cm for the lateral cephalometric films with an enlargement of 7%. No correction was made for this radiographic enlargement as it affected all the cephalograms of both groups in the same way. Landmarks were located directly on the radiographs by digitizer (Numonics 2210, Numonics). A maxillary triangle and a mandibular triangle for each subject were constructed with the aid of digitizing software (Viewbox 1.8, as described by Halazonetis21). Each landmark was digitized twice to reduce method error, and the average location of each cephalometric point was computed and used. Maxillary triangle: point T (the most superior point of the anterior wall of the sella turcica at the junction

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with tuberculum sellae as described by Viazis22); point FMN (fronto-maxillary-nasal suture, according to Riolo et al.23); Downs' point A (Fig. 3). Mandibular triangle: point Co (condylion); point Go (gonion); point Pg (pogonion) (Fig. 3). The shape of a set of six landmarks is completely represented by four pairs of shape coordinates. Here we used only two pairs corresponding to two specific areas of clinical concern. These represent the shapes of maxilla and mandible separately, but not their relative positions, which are not, of course, rigidly specified.

Co

Shape Coordinate Analysis The segment from point T to point FMN and the segment from point Go to point Pg were chosen as baselines for maxillary and mandibular triangles, respectively. The two baselines were positioned in a Cartesian coordinate system, so that one extremity (point T or point Go) corresponded to the origin (0,0) and the other extremity (point FMN or point Pg) was located at point (1,0), one unit to the right along the x-axis. Shape coordinates for the third point (point A or point Co) in both treated and control groups were computed after this transformation of the original triangles into "shape-coordinate space" in a conventional statistical package. 24 The procedure was repeated for both triangles in both groups at time 2. Because of the homogeneity of the two groups as to age at time 1 and at time 2, type of malocclusion, craniofacial pattern at time 1, gender distribution, and as to observation period, the comparison between groups was performed on the annualized mean differences between time 2 and time 1 for the values of the third point. All shape changes in "shape space" can be visualized as vectors connecting individual starting points (time 1) and ending points (time 2). A graphic reduction to a common starting point with a statistical distribution of ending points is possible. A vector representing mean shape change in each group of observations (treated and untreated groups) will originate from the common starting point for that group. The vector difference between these two mean vectors (still originating from a mean common starting point) expresses the mean differences in shape change between the two groups. Adequate statistical method for comparison of vectors is Hotelling's T 2 test. For the matched part of a study such as ours, an alternative statistic is available that can speed computation, i.e., the comparison of the length of the mean vector difference to the mean of the lengths of the individual vectors difference that make up the sample. Values of this ratio greater than about 2/~,fN are significant for mean shape change difference at about the conventional 0.05 level. 12

Fig. 3. Maxillary and mandibular triangles.

groups. Centroid size was calculated as the root sum of squares of the sides of triangles. Size changes between time 1 and time 2 in treated and untreated groups were compared by t test (p < 0.05). In some studies, 13 it is appropriate to test for growth allometry, the existence of correlations between size change and shape change. In the current data, where both maxillary and mandibular triangles have a biologically stable baseline, test for growth allometry reduces to a test for simple vertical growth, and hence need not to be carried out separately.

Tensor Analysis Tensors for the mean shape change of triangles in each group were constructed, according to the geometric method proposed by Bookstein 9 in 1982. Software (Viewbox, 1.8) performed the geometric construction and calculated the strain along each principal axis. Changes expressed increments/decrements as annualized percentages of initial length (time 1) separately in treated and untreated groups. This procedure allowed for a biologic interpretation of the shape changes that have been determined by shape-coordinate analysis. Further, the "mean treatment effects," computed as the contrast of changes between treated and untreated groups, have been displayed in the same way.

Centroid Size Analysis

RESULTS

Centroid size was measured both on maxillary and mandibular triangles in both treated and untreated

T h e triangles we are studying did not differ in size or shape before the onset of treatment. W e do

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A -.90.

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Fig. 4. Graphic display of shape change in treated group (maxillary triangle): scatterplot representing annualized individual displacements of point A at time 2 relative to common starting point at time 1. Arrow identifies vector of mean shape change for point A relative to baseline T-FMN,

not reproduce this analysis in this article. The results of the comparisons between treated and untreated groups for the annualized mean shape-coordinate differences between time 2 and time 1 are graphically displayed in Figs. 4 through 9. In the maxillary triangle, shape change in both groups can be described as forward displacement of point A along the x-axis (i.e., parallel to the T-FMN baseline) and downward displacement of point A along the y-axis (i.e., away from the baseline) (Figs. 4 and 5). The groups differed significantly (p < 0.05) in this change of shape coordinates with Hotelling's T 2 test. The forward displacement of point A along x-axis was greater in the treated group (Fig. 6). Note that, in Fig. 5, the presence of two cases showing much greater growth rate than the others. In principle, this could be a function of age, dentition, or growth pattern. Also, the shape coordinates produce the identification of these outliers automatically, as could a conventional analysis, but not the tensor analysis, because it does not show changes case by case.

As for the mandibular triangle, shape change in both groups was represented by a backward displacement of point Co along the x-axis (i.e., parallel to the Go-Pg baseline) and by an upward displacement of point Co along the y-axis (i.e., away from the baseline) (Figs. 7 and 8). The displacements exhibited significant differences between the two groups (Fig. 9); the difference is obvious because every single case moved backward in the untreated group and upward-forward in the treated group. As expected, Hotelling's T 2 test was quite significant (p < 0.0001). The treated group showed a smaller backward displacement of point Co along the x-axis, and a greater upward displacement of point Co along the y-axis. The analysis of shape changes displayed as tensors substantiated these results (Figs. 10 and 11). The picture of growth of the maxillary triangle in the treated group shows a direction of greatest rate of change (4.8% per 24 months) oriented about 20° counterclockwise of the line FMN-A, and a direction of least rate of change (1.6% per 24

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months) oriented about 20° clockwise of the line T-FMN (Fig. 10). The maxillary triangle of the untreated group shows a direction of greatest rate of growth (3.2% per 24 months) oriented approximately along the line FMN-A, and a direction of least rate of growth (1.3% per 24 months) oriented approximately along the line T-FMN (Fig. 10). Therefore the "treatment effect" on the shape of the maxilla is the difference between these tensors: an increase in growth rate of 2.2% per 24 months along a direction oriented approximately halfway between the directions of greatest growth rate in treated and untreated groups, but nearly no change (an increase of 0.5% per 24 months) along the perpendicular direction (Fig. 10). For the mandibular triangle, in the treated group, the direction of greatest rate of change (5.1% per 24 months) is oriented about 15° counterclockwise of the line Go-Pg, whereas the direction of least rate of change (3.7% per 24 months) is oriented 20° clockwise of the line Co-Go (Fig. 11). In the untreated group, the direction of greatest rate of growth (2.8% per 24 months) was oriented approx-

imately along the line Go-Pg, whereas the direction of least rate of growth (2.1% per 24 months) was oriented approximately perpendicular to the line Go-Pg (Fig. 11). The "treatment effect" for this mandibular triangle is therefore an increase in the growth rate of a full 8.1% per 24 months along the bisector of the gonial angle Co-Go-Pg, and a decrease in the growth rate of 3.6% per 24 months approximately along the line Co-Pg (Fig. 11). Change in centroid size from time 1 to time 2 differed significantly between treated and control groups for both the maxillary triangle (2.99 +_ 1.74 for treated group and 2.002 -+ 1.3 for control group; t = 2.5, p = 0.015) and the mandibular triangle (4.17 + 3.1 for treated group and 6.27 + 4.5 for control group; t = 2.11,p = 0.038). DISCUSSION

Earlier articles 1-3 applied conventional cephalometrics (angular and linear measurements in association with a reference system) and found significant skeletal changes in both the maxilla and the mandible, induced by early functional treatment of

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Class III malocclusion in these same groups. These comparisons showed greater increments in maxillary protrusion, measured as sagittal displacement of point A, and smaller increments in mandibular total length (Co-Pg). These were explained as more upward-forward direction of condylar growth in treated children. The analyses in this article apply to exactly the same data but did not require the choice of variables in advance, and hence investigate more deeply the mode of action of the removable mandibular retractor. As a rule, shape-coordinate transformation of interlandmark distances often benefits from tensor descriptors of shape change to provide a biologic interpretation of the findings. These interpretations are most helpful when baselines are biologically reasonable, and in fact, in the current study, the shape-coordinate baselines for both triangles corresponded to true biologic baselines, reference lines through stable structures in either the cranial base

or the mandible. 1,2,16,25 We could therefore draw conclusions about the meaning of maxillary and mandibular shape changes directly from the analysis of shape-coordinate variations for point A and for point Co, a benefit that is more commonly found associated only with the classic superimposition methods. Early treatment of Class III malocclusion with a functional appliance in the deciduous dentition appears to determine an increased growth of the maxilla, featuring a more downward and forward displacement of the region of point A, when compared with untreated subjects, One of the main objectives of shape-coordinate analysis (and one of its most remarkable advantages with respect to conventional cephalometrics) is to substantiate the biologic effectiveness of biometric variables (angular and linear cephalometric measurements) in the assessment of treatment effects on skeletal structures. 1l In previous investigations, a,3

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Fig. 7. Graphic display of shape change in treated group (mandibular triangle): scatterplot representing annualized individual displacements of point Co at time 2 relative to common starting point at time 1. Arrow identifies vector of mean shape change for point Co relative to baseline Go-Pg.

the maxillary position in the sagittal plane had been analyzed by means of the linear measurement AVertT. Vertical T (VertT) is a line perpendicular to Stable Basicranial Line (SBL), which is a line passing through point T and tangent to the lamina cribrosa of the ethmoid, and originating from point T (Fig. 12). In the current study, SBL has been substituted by the line T-FMN, as "basicranial edge" of the maxillary triangle. In this context, line T-FMN maintains the same biologic meaning as the SBL. The A-VertT is a measurement performed along a direction parallel to the cranial base, that is, along one of the two directions that proved to coincide with significant treatment effect (Fig. 10). Therefore A-VertT appears a posteriori to have been an appropriate measurement for the evaluation of the sagittal component of maxillary changes in cephalometrics, even in the presence of all the well-known drawbacks of the conventional metric approach that produced it. It is difficult to conceive a direct effect of the

retractor on the maxillary growth pattern. On the contrary, we believe, early changes in occlusal relationships may be the origin of correcting signals for the growth of the maxilla. 26 In fact, it has been suggested that the rapid correction of anterior crossbite could be responsible for a more favorable maxillary growth pattern in the treated Class III malocclusion, a pattern that has become comparable to the Class I pattern. 2'3 As for the mandibular triangle, early treatment of Class III malocclusion produces a significantly more upward and forward displacement of point Co, relative to the "baseline" Go-Pg (Fig. 9). This evidence again confirms previous cephalometric and superimposition studies, 1,2 which revealed that the most important mandibular skeletal modification induced by R M R is a change in the orientation of condylar growth. More upward and forward direction of condylar growth is a mechanism that accounts for significant closure of the gonial angle and for significantly smaller increments in mandibular

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total length (Co-Pg) in treated children. Tensor analysis clearly showed that these therapeutically induced skeletal changes best expressed the actual statistics of coordinate variations. As a consequence of the results related to shape changes in treated and untreated subjects wtih Class III malocclusion, cephalometric measurements for mandibular total length (Co-Pg) and for the inclination of the mandibular condyle relative to the mandibular body (CondAx-ML) (Fig. 12) appear to be adequate to describe skeletal modifications induced by early functional treatment. With regard to shape-coordinate analysis, two observations should be made. The first one is that the analysis by shape coordinates is more powerful than the analysis by tensors of the same data. The shape coordinates are pure shape, the tensors have size left in. In Fig. 9, treatment effect is about 4 standard deviations; in Fig. 11, it is about 2 standard deviations. This is in practice of enormous consequence for statistical power and sample size. We had an effect large enough, and a sample large enough, that both analyses were significant, but in

samples of 10 versus 10, the tensor analysis might have ceased to be significant, whereas the shapecoordinate analysis remains so. Another crucial point is that the "conventional measures" (distances or angles) that best report the treatment effect are not necessarily those that best report the changes in the groups separately. We see this very clearly in the current data set. In the maxilla (Fig. 6), the group mean changes are rather similar, but the treatment effect is almost perpendicular to one and highly oblique to the other. In the mandible (Fig. 9), the groups have quite different geometries of change, almost perpendicular in the appropriate (shape-coordinate) geometry. The treatment effect is oblique to both. Finally, the current study also provides information about growth changes in untreated subjects with Class III malocclusion in the early developmental phases. Both shape-coordinate and tensor analyses show how, in untreated Class IlI malocclusion, maxillary growth is predominant in the vertical component, with a minimal sagittal component, most probably as a consequence of the persistent anterior crossbite. In the

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Fig. 10. Tensor analysis of changes in maxillary triangle. Numbers indicate annualized percentage variations (with relative standard errors) along principal axes.

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Fig. 12. Conventional cephalometrics: Measurements for sagittal position of maxilla (A-VertT), for mandibular total length (Co-Pg), and for inclination of condyle relative to mandibular line (CondAx-ML). S B L = Stable basicranial line; V e r t T = Vertical T; C o n d A x = Condylar axis (line passing through point Co and point Cs, ie., center of condylar head27).

same children, mandibular growth can be seen as an upward and backward direction of condylar growth, leading to growth increments along the direction of the mandibular total length, increments that make the clinical problem worse. A comparison between untreated Class III and Class I samples is obviously needed to draw definite conclusions about physiologic growth changes in Class lII malocclusion. CONCLUSIONS The findings of the current shape-coordinate and tensor study of skeletal changes induced by early functional treatment of Class III malocclusion may be summarized as follows:

1. As to the analysis of shape changes, the treated group showed a significantly more forward direction of maxillary growth, and a significantly more upward and forward direction of condylar growth, leading to a "shrinkage" of the mandible along the direction of total mandibular length. 2. The biologic effectiveness of some conventional measurements for the sagittal position of the maxilla and the mandibular dimensions and morphology was substantiated. We express our gratitude to Dr. Fred Bookstein for his invaluable assistance in revising the article and in performing the statistical machinery related to shapecoordinate analysis.

American Journal of Orthodontics and Dentofacial Orthopedics Volume 112, No. 6 REFERENCES

1. Tollaro I, Baccetti T, Franchi L. Mandibular skeletal changes induced by early functional treatment of Class III malocclusion: a superimposition study. Am J Orthod Dentofac Orthop 1995;108:525-32. 2. Tollaro I, Baccetti T, Franchi L. Craniofacial changes induced by early functional treatment of Class III malocclusion. Am J Orthod Dentofac Orthop 1996;109: 310-8. 3. Baceetti T, Franchi L. Enhanced maxillary growth following early treatment of Class III malocclusion. Fur J Orthod 1995;17:330 (abstract). 4. Blum H. Biological shape and visual science; J Theor Biol 1973;38:205-87. 5. LestreI PE. A Fourier analytic procedure to describe complex morphological shapes. In: Dixon AD, Sarnat BG, editors. Factors and mechanisms influencing bone growth. New York: Alan R. Liss; 1982. p. 393-409. 6. Lavelle CLB. A preliminary study of mandibular shape. J Craniofac Genet Dev Biol 1985;5:159-65. 7. Lestrel PE, Roche AF. Cranial base shape variation with are: a longitudinal study of shape using Fourier analysis. Hum Biol 1986;58:527-40. 8. Cheverud JM, Lewis JL, Bachrach W, Lew WD. The measurement of form and variation in form: an application of three-dimensional quantitative morphology by finite-element methods. Am J Phys AnthropoI 1983;62:151-65. 9. Bookstein FL. On the cephalometrics of skeletal change. Am J Orthod 1982;82: 177-82. 10. Bookstein FL. Geometry of crauiofacial growth invariants. Am J Orthod 1983;83: 221-34. 11. Bookstein FL. Measuring treatment effects on craniofacial growth. In: McNamara JA Jr, Ribbens KA, Howe RP, editors. Clinical alteration of the growing face. Monograph 14. Craniofacial Growth Series. Ann Arbor: Center for Human Growth and Development, University of Michigan; 1983. 12. Bookstein FL. A statistical method for biological shape comparisons. J Theor Biol 1984;107:475-520. I3. Bookstein FL. Tensor biometrics for changes in cranial shape. Ann Hum BioI 1984;11:413-37.

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14. Bookstein FL. Size and shape spaces for landmark data in two dimensions. Stat Sci 1986;1:181-242. 15. Bookstein FL. Morphometrics tools for landmark data. New York: Cambridge University Press; 1991. 16. McNamara JA Jr, Bookstein FL, Shaughnessy TG. Skeletal and dental changes following functional regulator therapy on Class lI patients. Am J Orthod 1985;88: 91-110. 17. Kerr WGS, Tenhave TR. A comparison of three appliance systems in the treatment of Class III malocclusion. Eur J Orthod 198;10:203-14. 18. Ngan P, Scheick J, Florman M. A tensor analysis to evaluate the effect of high-pull headgear on Class II malocclusions. Am J Orthod Dentofac Orthop 1993;103:26% 79. 19. Battagel JM. The aetiolngy of Class III malocclusion examined by tensor analysis. Br J Orthod 1993;20:283-96. 21. HaIazonetis DJ. Computer-assisted cephalometric analysis. Am J Orthod Dentofac Orthop 1994;105:517-21. 22. Viazis AD. The cranial base triangle. J Clin Orthod I991;XXV:565-70. 23. Riolo ML, Moyers RE, McNamara JA Jr, Hunter WS. An atlas of craniofacial growth: cephalometric standards from the University School Growth Study, The University of Michigan. Monograph 2. Craniofacial Growth Series. Ann Arbor: Center for Human Growth and Development; 1974. 24. Statistical Graphics Corporation. Statgraphics. Ver. 2.6. Rockville (MD): STSC, Inc.; 1987. 25. Melsen B. The cranial base. Acta Odontol Scand 1974;32:Suppl. 62. 26. Petrovic A, Stutzmann J, Lavergne J. Mechanism of craniofacial growth and modus operandi of functional appliances: a cell-level and cybernetic approach to orthodontic decision making. In: Carlson DS, editor. Craniofacial growth theory and orthodontic treatment. Monograph 23. Craniofacial Growth Series. Ann Arbor: Center for Human Growth and Development, University of Michigan; 1990: p. 13-73. 27. Odegaard I. Growth of the mandible studied with the aid of metal implants. Am J Orthod 1970;57:145-57.

AAO MEETING CALENDAR 1998 - - Dallas, Texas, May 16 to 20, Dallas Convention Center 1999 - - San Diego, Calif., May 15 to 19, San Diego Convention Center 2000 - - Chicago, II1., April 29 to May 3, McCormick Place Convention Center (5th IOC and 2nd Meeting of WFO) 2001 - - Toronto, Ontario, Canada, May 5 to 9, Toronto Convention Center 2002 - - Baltimore, Md., April 20 to 24, Baltimore Convention Center 2003 - - Hawaiian Islands, May 2 to 9, Hawaii Convention Center 2004 - - Orlando, Fla., May 1 to 5, Orlando Convention Center