Morphometrics for cephalometric diagnosis

ORIGINAL ARTICLE

Morphometrics for cephalometric diagnosisDemetrios J. Halazonetis, DDS, Dr Odont, MSAthens, Greece

This article demonstrates morphometric methods by applying them to an orthodontic sample. A total of 150pretreatment cephalograms of consecutive patients (84 female, 66 male) were traced and digitized. Fifteenpoints were used for the analysis. The tracings were superimposed by the Procrustes method, and shapevariability was assessed by principal component analysis. Approximately 70% of the total sample variabilitywas incorporated in the first 5 principal components. The most significant principal component, accountingfor 29% of shape variability, was the divergence of skeletal pattern; the second principal component,accounting for 20% of shape variability, was the anteroposterior maxillary relationship. It is recommendedthat Procrustes superimposition and principal component analysis be incorporated into routine cephalomet-ric analysis for more valid and comprehensive shape assessment. (Am J Orthod Dentofacial Orthop 2004;125:571-81)

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One of the main applications of cephalometis as a shape descriptor. We use various liand angular measurements to arrive at a

cise and comprehensive description of the craniofpattern and to classify each patient, making it easiidentify treatment goals, choose treatment modaland predict treatment success. However, in additiotheir other disadvantages,1 conventional cephalometrmethods have certain inherent problems regardingapplicability as shape measures; they provide onpartial and localized description of shape andconfounded by our biases regarding the referstructures (cranial base, Frankfort horizontal, orers). For example, the conventional view statesangle SNA describes the anteroposterior position omaxilla, because the cranial base is considered “staHowever, 2 points that define this angle belong tocranial base, and it would seem more logical to assthat most of the variability of the SNA measuremendue to the cranial base than to point A.

The local nature of conventional measurementsour bias regarding the “stability” of the referenstructures result in these problems: (1) the meaments, or rather the interpretations that we ascribthem, are often conflicting, (2) many measurementneeded for comprehensive description and diagnoseach patient, (3) it is not trivial to comparecraniofacial pattern between 2 patients, and (4) clfication of patients is based on a limited subset o

Assistant professor, Orthodontic Department, University of Athens DSchool, Athens, Greece.Reprints requests to: Dr Demetrios Halazonetis, 6 Menandrou St, Kifiss61, Greece; e-mail, [email protected], February 2003; revised and accepted, May 2003.0889-5406/$30.00Copyright © 2004 by the American Association of Orthodontists.doi:10.1016/j.ajodo.2003.05.013

possible measurements and might therefore be bby that particular selection.

Other morphometric methods that address tproblems might be more valid for describing biologshape. Methods such as Procrustes superimposprincipal component analysis (PCA), Euclideantance matrix analysis, finite-element scaling analand thin-plate splines2-10 are actively used in othbranches of the biological sciences but havesurprisingly, limited impact in orthodontics. This acle demonstrates some of these methods with an odontic sample, in an effort to increase the orthodocommunity’s awareness and familiarity with them. Tresults obtained on the present sample could be usbaseline data for clinical applications and future sies.

MATERIAL AND METHODS

The sample consisted of initial (pretreatment) cealometric radiographs of 150 consecutively treapatients (84 female, 66 male). Patient records wselected from a private orthodontic practice irrespecof sex, age, and type of malocclusion. Only radiograof good quality that depicted a reference ruler oncephalostat for exact measurement of the magnificfactor were included. Patients with congenital malmations or syndromes were excluded.

The radiographs were scanned at 150 dpidigitized with Viewbox 3 software (dHAL softwarKifissia, Greece; www.dhal.com). A comprehensiveof points was digitized, but the following 15 were uin this investigation: basion (Ba), sella (S), sphenomoidale (Se), nasion (N), porion (Po), orbitale (anterior nasal spine (ANS), A point (A), posterior naspine (PNS), articulare (Ar), gonion (Go), antego

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American Journal of Orthodontics and Dentofacial OrthopedicsMay 2004

572 Halazonetis

notch (Ag), menton (Me), pogonion (Pg), and B point(B). The x and y coordinates of the points were scaledaccording to the magnification of each radiograph tocorrespond to life size. Descriptive statistics of age andselected cephalometric measurements are given in Ta-ble I.

RESULTSDefinition of shape and Procrustessuperimposition

Shape can be defined as the geometric informationthat remains after we have removed any effects due totranslation, rotation, and scale.5 This definition can beused as an operational definition for shape measure-ment: to compare 2 shapes, we adjust for size andsuperimpose them. Then, any differences that remainare due to shape dissimilarity. The discrepancy betweenthe 2 shapes can be measured by assessing the distancebetween corresponding points of the superimposedshapes.

In most cephalometric studies, the aspect of size iscircumvented by the use of angular measurements.When linear measurements are used, size adjustment isusually done by scaling according to age or the size ofa specific structure (eg, the cranial base). This is not avery satisfactory solution, because the structure usedfor size adjustment might also be the one that differsbetween patients. An alternative method is to scale thecephalometric tracings according to centroid size. Thecentroid of a shape, which is composed of landmarkpoints, is the average of all the points (the “center ofgravity” of the shape), and centroid size is the squareroot of the sum of the squared distances of each pointto the centroid.5

After adjusting for size, we need to align the shapes,so that any effect of translation and rotation are removed.In orthodontics, we have many superimposition methodsfor aligning 2 cephalometric tracings. Figure 1 shows 2tracings superimposed at an arbitrary position (Fig 1, A)

Table I. Descriptive statistics for age and selectedcephalometric measurements

Mean SD Range Median

Age (y) 12.2 3.77 6.4–33.6 11.5SNA (°) 80.1 3.22 71.7–87.9 80.2SNB (°) 75.9 3.51 66.6–86.4 75.5ANB (°) 4.2 2.71 �5.8–10.2 4.6Wits (mm) 1.8 3.65 �13.1–10.7 2.3SN-GoGn (°) 34.9 5.25 21.5–46.6 35.5Overjet (mm) 5.4 2.84 �4.8–12.9 5.3Overbite (mm) 3.9 2.06 �1.6–9.2 4.1

SD, standard deviation.

and on the anterior cranial base (Fig 1, B). The interpre-tation of shape differences will depend on the choice ofthe superimposition, a well-known fact in the orthodonticliterature. Returning to the definition of shape givenabove, shape differences can be measured by the distancebetween corresponding landmarks of the superimposedtracings. It is evident that the superimposition in Figure 1,A, leads to a larger apparent shape difference than that inFigure 1, B, because the sum of the distances betweencorresponding points is larger. Obviously, this is becausea remaining translation and rotation factor has not beenadjusted for. This observation leads to a mathematicsolution of the problem of the “correct” or “optimum”superimposition: translation and rotation are adjusted so asto minimize the distances between points. There arevarious minimization criteria, but the most widely usedcriterion is that which minimizes the sum of the squareddistances between corresponding points. This criterion isused by the Procrustes* superimposition,5,10 a widelyused morphometric method. Procrustes superimpositiontakes 2 shapes, resizes them to the same centroid size, andthen aligns them to minimize this sum.

An important distinction should be made herebetween comparing the tracings of the same patient atdifferent ages and comparing the tracings of differentpatients. When comparing tracings of the same patient,it is valid to use biological structures that are known toremain stable. However, when comparing shape be-tween different patients, such considerations do notapply. We cannot assume that the cranial base, or anyother part, constitutes a preferred common referencestructure between the patients, even though it is bio-logically homologous. For this reason, there is no“correct” superimposition method: each method willlead to a different interpretation of shape differences.7

Procrustes superimposition is just a method that at-tempts to minimize the apparent effect of translation,rotation, and scale and thus bring forward pure shapediscrepancies. The Procrustes method differs fromconventional cephalometric superimpositions because(1) the shapes are resized to achieve a better fit, (2) allpoints are treated equally instead of privileged statusbeing given to some, and (3) there is no requirementthat any corresponding points should coincide.

Figure 2 shows the 2 tracings in Figure 1 superim-posed by the Procrustes method. Note that the tracingsseem much more similar in Figure 2 than in Figure 1, B,and it becomes apparent that the discrepancies in Figure

*Procrustes was a robber in ancient Greek mythology. He laid his victims ona bed. If they were too short, he would stretch them; otherwise, he would chopoff their legs. In either case, they usually died. Procrustes was killed byTheseus, who was on his way to Athens to become king.

Procrustes method.

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Halazonetis 573

1, B, are due more to different cranial base orientationsthan to overall shape differences.

Average shape and shape variability

As with any biological variable, measurement ofshape entails (1) calculating the average shape of thepopulation and (2) estimating the variability of shape.Calculation of the average is trivial and can be accom-plished by calculating for each point the average x andy coordinates from the corresponding coordinates of alltracings. Figure 3 shows the 150 tracings of the samplesuperimposed by the Procrustes method together withthe average shape.

Although an estimate of average shape is important,the most important, and most difficult to estimate, is thevariability of shape. The average is of little use byitself, because it does not indicate how much a specificpatient differs from it. The variability of the samplearound the mean is represented by the scatter of thepoints in Figure 3. The position of each point mightvary along both the x and y directions; thus there are 2kvariables to describe variability, where k is the numberof points. Principal component analysis is a method todecrease the number of variables. This statistical pro-cedure takes advantage of the fact that the points do notbehave independently, because they all belong to the

Fig 1. Superimposition of 2 tracings: A, at arbbase (S-N).

Fig 2. Tracings of Fig 1 superimposed according to

itrary position and orientation; B, on anterior cranial


574 Halazonetis

same biological entity. Thus, for example, a patientwith vertical growth pattern is expected to have Nlocated higher than the average and Me lower than theaverage, whereas a patient with low vertical height willprobably have the opposite arrangement. Such intercor-relations between the positions of the points are used byPCA to arrive at a different set of variables (namedprincipal components [PCs]), each of which describes aspecific shape pattern. One component, for example,might represent vertical facial height, whereas anothermight represent the anteroposterior relationship of themaxillae. The components of PCA are arrived atstatistically, not from biological considerations. Theyhave the following properties: (1) they are statisticallyunrelated to each other; (2) each component represents,in decreasing order, part of the variability of the sample(ie, the first component represents the largest part of thevariability, the second component the second-largestpart, and so on); and (3) each component is a linearcombination of the original variables (ie, each compo-nent incorporates to some extent the variability of everypoint along the x and y direction). However, each pointcontributes a different amount. For example, if a PC

Fig 3. Average shape of sample and scattertracings is according to Procrustes method.

represents a vertical shape pattern, the variability of Meand N along the y direction would have a higherweight, or loading, than the variability along the xdirection.

The usefulness of PCA is that the PCs are sorted inorder of decreasing importance. Therefore, one canretain only the PCs that describe the most significantpart of the overall variability and discard the rest,thereby significantly reducing the total number ofvariables. In this study, 15 points were used, whichwould produce 30 PCs. However, because some de-grees of freedom are lost because of the alignment ofthe shapes10 (the explanation for this is beyond thescope of this article), the total number of PCs is 26.Calculating the PCs was performed with APS 2.41software (Xavier Penin, Caen, France; www.cpod.com/monoweb/aps). The loadings of the first 3 PCs for eachpoint are shown in Table II. Table III shows thepercentage of total shape variability accounted for byeach of the first 8 components. Approximately 70% ofthe variability was incorporated into the first 5 compo-nents.

The average shape of the sample has, by definition,

ints of 150 tracings. Superimposition of 150
of po


Halazonetis 575

all PCs equal to 0. To visualize the pattern of shapevariability represented by each PC, the average shapecan be warped by moving the points according to theirloadings on the PC. Figure 4 shows the average shapeof the sample and the warped shapes obtained bysetting the first 3 PCs to a value equal to 3 standarddeviations in the negative and positive directions. Fromthis figure, we can interpret the first PC as representingthe divergence of the skeletal pattern. Low values showa high-angle skeletal pattern, and high values show alow-angle pattern. The second component is associatedwith the anteroposterior relationship of the maxillae. Alow value corresponds to a Class II pattern, and a highvalue to a Class III pattern. The third PC relates to thegonial angle. A low value is associated with a smallgonial angle and an increased posterior facial height,whereas the reverse is true for a high value of the PC.

Table II. Loadings of first 3 PCs for each point in x an

Point

PC1

x y

Ba �0.2458 0.1944 0Se 0.0549 �0.1581 �0N 0.1104 �0.3148 �0S �0.0072 0.0234 0Po �0.1176 0.2146 0Or �0.0060 �0.2556 �0PNS �0.1226 �0.0502 �0ANS �0.0573 �0.2473 �0A �0.0529 �0.1747 �0B 0.1646 0.2085 0Pg 0.3122 0.3281 0Me 0.3417 0.2197 0Ag �0.0212 �0.0713 0Go �0.1392 �0.0155 0Ar �0.2140 0.0985 0

Large absolute values signify that PC has relatively large effect on varat warped shapes of Figure 4.

Table III. Standard deviation, variance, percentvariance, and cumulative variance explained by eachof 8 first PCs

Component SD Variance%

varianceCumulative variance

(%)

PC1 0.03406 0.00116 28.7 28.7PC2 0.02809 0.00079 19.6 48.3PC3 0.01922 0.00037 9.2 57.4PC4 0.01519 0.00023 5.7 63.2PC5 0.01424 0.00020 5.0 68.2PC6 0.01314 0.00017 4.3 72.5PC7 0.01227 0.00015 3.7 76.2PC8 0.01143 0.00013 3.2 79.4

Percent variance was calculated on total variance of 26 components,which was equal to 0.00404.

Figure 5 shows scatterplots of the first 5 PCs againsteach other. These plots show that the sample cannot bedivided into discrete shape categories and should beregarded as a homogeneous group.

Shape space

Shape is not a discrete variable but is a continuumof smoothly varying patterns. The collection of allpossible shape patterns is called a “shape-space.” Eachpatient of the sample is located at a particular point inthis shape-space, much as each person has a uniquehome address on the map. The PCs can be thought of asrepresenting the coordinates that pinpoint the patient tothe particular location. Because successive PCs repre-sent ever-decreasing parts of shape variability, they canbe used to describe shape in increasing detail, much asan address describes location from the global to thespecific (country, town, street, number).

The difference in shape between 2 patients can bemeasured by the distance between them in shape space.A popular measure of distance is the Procrustes dis-tance, calculated as the square root of the sum ofsquared distances between corresponding points whenthe shapes are aligned.

Application in cephalometrics

The craniofacial shape of a patient can be assessedby calculating the value of the PCs and placing thepatient at the appropriate position in the population’sshape-space. The first PCs describe the shape pattern ingeneral terms, and successive PCs concentrate on finerdetail. This scheme allows us to adjust the number of

rections

PC2 PC3

y x y

0.2087 �0.0435 �0.05700.1236 �0.1707 �0.0111

�0.0245 0.0925 �0.17180.1648 �0.2074 �0.03550.1598 �0.3170 �0.06150.0683 �0.0624 �0.15170.0616 0.0151 �0.0054

�0.0670 0.0058 �0.0329�0.0657 0.0224 �0.0481�0.1718 0.1716 �0.1568�0.2303 0.0617 �0.1620�0.2641 0.0287 �0.1398�0.1172 0.2004 0.4615�0.0664 0.2199 0.5844

0.2201 �0.0172 �0.0124

of point in corresponding direction. These values were used to arrive

d y di

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.1210

.2611

.0264

.2381

.1509

.1584

.3554

.3608

.0895

.1827

.2398

.1872

.2166

.1290

iability


576 Halazonetis

PCs used to describe a patient, according to the detailwe are seeking. For broad classification, 2 or 3 PCsmight be sufficient. In such a case, the shape space is 2-or 3-dimensional, and the scatterplots of Figure 5 canbe used to obtain a visual representation of the locationof the patient in the shape space.

As an example, Figure 6 shows the tracings of 3patients from the present sample. Table IV shows someselected cephalometric measurements and the values ofthe first 3 PCs. Looking at the first PC, we can tell that

Fig 4. Warped shapes depicting effect of varwarped by applying each PC by amount equal t(right) direction.

patients A and B should be roughly similar, belongingto the high-angle group, because PC1 is negative (Fig4), whereas patient C is on the opposite side of thespectrum—definitely low-angle. The second PC showsthat patients A and B are also similar in anteroposteriormaxillary relationship; both are relatively Class II,whereas patient C is again on the opposite side, a ClassIII. The similarity of patients A and B in contrast topatient C is graphically evident by placing them in theapproximate shape-space, as shown in Figure 7, A.

ach of first 3 PCs. Average shape (middle) isndard deviations in negative (left) and positive

ying eo 3 sta


Halazonetis 577

However, further refinement of shape assessment withthe third PC (Table IV) differentiates between patientsA and B at the area of the mandibular angle. Patients Aand B are relatively far apart in the scatterplot of PC2compared with PC3 (Fig 7, B), but this does notconstitute a major difference in overall shape becausePC3 is less influential than PC1 and PC2. All 3 patientsare at approximately the same distance from the originof the graphs, which shows that the degree of skeletaldiscrepancy is similar. This can also be quantitativelyassessed by calculating the Procrustes distance.

DISCUSSION

Conventional cephalometric methods have existedfor more than 60 years, and their limitations have beeninvestigated and appreciated. Other methods for assess-ing shape are under development in biological morpho-metrics, and their applicability in cephalometricsshould be researched. The orthodontic literature con-

Fig 5. Scatterplots of first 5 PCs against eachfew outliers.

tains few relevant articles (see examples in the refer-ence list11-18), so an introductory article presentingbaseline data on a general orthodontic sample wasconsidered important. Because readers are probablyunfamiliar with the methods used, extra details andexplanatory material were included. Considerable de-bate exists concerning the theoretic validity and appro-priateness of the various morphometric methods, andno consensus has been reached (for a critical, butadmittedly biased, comparison, see Richtsmeier et al19).The techniques presented here were selected on thebasis of their wide usage and relative conceptual ease.For more information, the reader is directed to theMorphometrics web site at http://life.bio.sunysb.edu/morph/.

The sample used in this study was not a sample ofideal or normal subjects, nor does it represent thegeneral population, and the results cannot, therefore, beregarded as normal values for use in diagnosis. How-

Sample constitutes homogeneous group with
other.


578 Halazonetis

Fig 6. Tracings of 3 patients and their Procrustes superimpositions.


Halazonetis 579

ever, the size of the sample and the method of patientselection (consecutive patients) lend reasonable supportto the belief that the sample uniformly covers most ofthe shape space, excluding syndromic patients. Thus,the calculated average shape should not be far from thepopulation average. Conventional cephalometric mea-sures of the calculated average form were withinnormal limits. Furthermore, the average form was verysimilar to the templates of the Michigan GrowthStudy.20 Data from the general population are thereforeexpected to be similar but, probably, have smallerstandard deviations.

PCA is a statistical technique for reducing thenumber of variables when a significant correlationbetween the variables is present. The application ofPCA in this sample resulted in 5 PCs explaining

Table IV. Selected measurements and values of first 3 P

Patient A

SNA (°) 86.3SNB (°) 76.2ANB (°) 10.2Wits (mm) 6.4SN-GoGn (°) 43.3Overjet (mm) 10.6Overbite (mm) 3.5PC1 �0.03827 (�1.12)PC2 �0.03262 (�1.16)PC3 0.04541 (2.36)

Numbers in parentheses are z scores of PCs.

Fig 7. Position of 3 patients in shape spaceestimate of shape of tracing and of distance frovs PC3.

approximately 70% of the total shape variability. Thissmall number of variables is sufficient to reconstructthe approximate shape of a patient’s craniofacial pat-tern (cephalometric tracing), in contrast to conventionalcephalometric measurements, with which this inversefunction, from measurements to tracing, is, in general,not possible. Therefore, the use of PCs as adjunctvariables of a mixed-mode (conventional/morphomet-ric) cephalometric analysis should be seriously consid-ered. This analysis could provide the following advan-tages:

1. Comprehensive description of the overall craniofa-cial shape with a small number of measurements,which are not conflicting because they are unrelatedstatistically. More detailed description is possible by

r example patients

Patient B Patient C

82.6 84.875.1 84.67.5 0.35 �4.9

38.1 25.86.2 2.55.3 1.3

�0.03170 (�0.93) 0.04475 (1.31)�0.03657 (�1.30) 0.05187 (1.85)�0.00117 (�0.06) �0.01047 (�0.54)

ed by first 3 PCs. This position gives roughrage (center of graphs). A, PC1 vs PC2. B, PC2

Cs fo

definm ave


580 Halazonetis

increasing the number of reported variables (PCs).The approximate tracing of a patient can be re-created from the values of the PCs and the averagetracing of the population.

2. Relative insensitivity to errors in landmark identifi-cation. This is a major problem in conventionalanalyses,21-25 because a small error in the location ofthe points that are used for the reference plane (eg,Po in defining the Frankfort horizontal) can have asignificant impact on all measurements that use thatparticular reference plane (eg, Frankfort-mandibularplane angle, Frankfort-mandibular incisor angle,facial angle).23 In contrast, Procrustes superimposi-tion treats all points equally, and an error in thelocation of 1 point will not have as significant aneffect.

3. Easy assessment of the degree of variation from theaverage. This is computed as the Procrustes distanceand consists of a single variable that shows how farfrom the average is the overall skeletal pattern.

4. Easy comparison of shape between patients (againby the Procrustes distance). This might be particu-larly helpful when planning the treatment of a newpatient, because it allows retrieval of data for pre-viously treated patients of a similar pattern toconsider their responses to treatment.

5. More valid selection of patients for research projectsregarding sample homogeneity.

A significant disadvantage of PCA is that theresulting components, because they are derived statis-tically, do not necessarily have a clear biologicalinterpretation. Usually, only the first, or the first few,can be described satisfactorily, and this was true for theresults of this study. Furthermore, the componentsmight be influenced by the number and spatial distri-bution of cephalometric points used to describe thecraniofacial shape. An uneven density of points mightintroduce bias in both the Procrustes superimpositionand the PCA results,9 and, for this reason, an effort wasmade to select points evenly distributed over the wholeshape. Analyses with fewer points (7 total points) werealso conducted, and the results were very similar tothose presented here (results available on request).

The PCA showed that the largest PC was related tovariability of the craniofacial complex in the verticaldirection. This was unexpected, considering the ortho-dontic community’s traditional preoccupation with theanteroposterior direction, although notable exceptionsexist (eg, the work by Shudy26). It seems that, althoughmalocclusions in the anteroposterior direction might bemore apparent, vertical discrepancies of the skeletalcomponent are more pronounced.

The first PC explained approximately 29% of theshape variability, whereas the second PC, related toanteroposterior maxillary discrepancy, accounted for20% of the total sample variability. The second PCdescribes the variability of the anteroposterior positionof the mandible relative to the combined craniomaxil-lary structure. The maxilla and anterior cranial baseseem to remain relatively stable to each other when thesecond PC takes values from �3 to �3 standarddeviations (Fig 4).

A significant hindrance to applying morphometricmethods to cephalometrics is that the points have to bedigitized and special software used to arrive at theresults. The ubiquitous use of computers in orthodonticoffices and the development of new software shouldalleviate these problems in the near future.

CONCLUSIONS

Procrustes superimposition and PCA of a wide-spectrum orthodontic sample showed the following:

1. Seventy percent of shape variability can be de-scribed by the first 5 PCs.

2. The first PC, explaining the largest percentage ofshape variability (29%), described vertical compo-nents, and the second PC (20%) described antero-posterior discrepancies.

3. The sample was a homogeneous group that couldnot be differentiated into discrete subgroups.

Morphometric methods might provide useful ad-junctive information for assessing skeletal pattern.

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Morphometrics for cephalometric diagnosis