Diagnosis of skeletal form on the lateral cephalogram with a finite element-based expert system M. B. Fine, BDS, MSc," and C. L. B. Lavelle, DDS, MRC Path b
Winnipeg, Manitoba, Canada Orthodontic diagnosis depends largely on the current methods of cephalometric analysis and categorization. It is therefore unfortunate that such analyses tend to use inappropriate metrical systems that lack a rigorous scientific basis. As a consequence, the resultant data have questionable validity, as well as being subject to observer bias. In this study, an alternative, more rigorous, approach is described. A random sample of 126 pretreatment lateral cephalograms were analyzed with the finite element method. These data were then classified to form a training set or reference data base for the subsequent objective diagnosis of new, previously unclassified cases. This new case-classification system was based on matching the new cases to similar cases within the existing training set. The degree of concordance between such computer-matched cases was then evaluated by their subsequent conventional cephalometric analyses. (AMJ ORTHOD DENTOFACORTHOP 1992;101:318-29.)
T h i s introduction is subdivided into a series of stages reflecting (A) our approach to conventi0nal cephalometric appraisals, (B) the reasons for adopting a more rigorous cephalometric appraisal technique, and (C) the use of cluster analysis to place cases into different diagnostic categories. Conventional cephalometries
Cephalometric measurement. The use of linear, angular, or proportional measures in conventional cephalometrics is inappropriate for evaluation of complex craniofacial profile forms? The methods also lack rigorous scientific backing. The is illustrated by the tendency to confound size and shape parameters together, there being no method for their discrete separation. 13 For example, when two cephalometric images are compared, a discrepancy in angle ANB may reflect a change in size or shape or, more likely, varying combinations of size and shape changes. Similar criticisms have also been applied to linear cephalometric dimensional changes. 4'5 As the data derived from conventional cephalometrics cannot withstand rigorous scientific scrutiny, their clinical application remains largely subjective. Comparison with cephalometric means. Conventional cephalomctric analyses ~* employ arbitrary measurement sets to delineate a person's craniofacial relationships. These subjective measurements are subsequently compared on a one-on-one basis with From the University of Manitoba. "Graduate orthodontic student, Department of Preventive Dental Science. bProfessor, Deparlment of Oral Biology. 811127147
318
corresponding population means, that is, the univariate approach is more appropriate for studying populations than for studying persons. More significantly, however, the variably correlated dimensions typical of lateral cephalometric forms are also unsuitable for univariate cephalometric analysis. 9 For instance, mandibular and facial height is variably correlated with mandibular and facial length, so that the separate evaluation of these dimensions t/as no scientific meaning, as the degree of correlation will vary between different persons. Indeed, multivariate statistical techniques are more appropriate for multidimensional cephalometric comparisons. These techniques facilitate cephalometric evaluation as a whole, rather than its arbitrary decomposition into a series of discrete parameters. ~~ Craniofacial classification (diagnosis). Orthodontic clinicians measure a series of discrete cephalometric measurements as a prelude to craniofacial diagnosis. Subsequent diagnosis (classification) for a given patient then hinges on the relationship between these dimensions and the corresponding population univariate means and standard deviations?' Different analyses of the same cephalograph may, however, result in different diagnoses, even when appraised by the same clinician. 12 Furthermore, since most conventional cephalometric classifications have not been rigidly defined, marginalcase diagnoses may be only conjecturally delineated. Current (qualitative) classification methods also do not embrace severity. An inherent weakness in current cephalomeetric diagnoses therefore centers on the clinician's inability to recount the logical steps taken. How can certain vail-
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ables be weighted when one is analyzing a particular cephalometric data set? Will this same weighting system be applied to all cases? In the absence of standardized classification methods, how can particular diagnoses be rigorously confirmed by other clinicians? In summary, conventional cephalometric analyses use scientifically inappropriate linear, angular, or proportionate parameters. The use of univariate statistics is less than ideal. Furthermore, the subjective assessment of individual measurement deviations concerning sample means results in informal methods of classification and arbitrary cephalometric diagnoses." As a result of these fundamental criticisms, this 9study was undertaken to investigate (1) a more rigorous form of lateral cephalometric appraisal, (2) a classification system delineated by both qualitative and quantitative information, and (3) a standardized method for lateral cephalometric diagnosis. The prime objective is an attempt to delineate lateral cephalometric evaluations more rigox'ously.
Rigorous lateralcephalometric meansurement Shape and form are terms that are frequentlyused interchangeably. Yet pure shape change implies constancy in size, whereas form change infers concurrent changes in both size and shape. 5 Shape may also be defined as an outline with landmarks in euclidean space after the removal of all information on position, scale, and orientation. ~ As facial profile orientation is correlated with "natural head position, ''~3 extremal datum points, of which the positions vary with orientation (e.g., menton), may be used to augment the paucity of anatomic points, of which the positions are dependent on anatomic features (i.e., independent of cephalographic orientation to delineate craniofacial form). Lateral cephalometric profiles may thus be defined as oriented shapes. In searching for alternative appraisal methods, ~4"~5 increasing interest has focused on the finite element method (FEM). This method is based on the concepts of continuum mechanics and applies mathematical descriptors from finite element analysis. ~6The FEM permits the quantification of deformations of anisotropic, nonhomogeneous, materially nonlinear bodies of any geometric shape. ~7 Quantified and graphic representations of cephalometric deformations (changes in form). may then be depicted as parameters (tens0rs) derived from the FEM. 5"18't9 A requirement of FEM is that lateral cephalometric form be divided into a number of finite elements. The arbitrarily defined elements are usually triangular, although other polygonal forms may be used to represent the area of cephalometric interest. Measurement of de-
Expert system diagnosis of skeletal form
319
formation of a particular finite element centers on the computation of the degree of deformation of a "target" relative to an analogous "reference" finite element delineated by the homologous datum (nodal) points. The computed deformation may then be described by a series of numeric values of the tensor components. 2~ The degree of cephalometric deformation around datum points may thus be obtained, without reference to arbitrary, fixed, registration points, that is, by simultaneous registration of homologous datum points. 5 This technique has been used to evaluate the mechanisms underlying change in craniofacial form, 5'19"27 in addition to demonstrating the advantages of avoiding the shortcomings of conventional cephalometric appraisals. 2"2s But whereas tensor (FEM) analysis has been used as a morphometric tool, 16application to static diagnostic cephalometry has yet to be investigated.
Classification techniques Th e classification process may be divided into two s!ages29: (1) the construction of classifications from initially tinclassified data and (2) the allocation of further cases to groups within such a classification system. Construction of classifications: Chlster analysis. Cluster analysis is a conventional mathematical technique for classifying previously unclassified data. This is achieved by sorting similar entities into "clusters," whereas members of different clusters tend to be less similar. Such techniques are useful for solving classification problems when little is known about a data set. 3~ As numeric taxonomic classifications of biologic data may contrast with conventional classifications of the same data sets, 29 however, cluster analysis of facial skeletal form may not necessarily reflect conventional classifications. In fact, for any given set of cases, each described by a cephalometric dimensional array, a variety of classifications may be delineated. Consideration should then be given to those dimensions (variables) that are more useful or descriptive. 29 The key to any classification system therefore centers on variable selection. Also, not all variables should necessarily be equally weighted. 29 Variable exclusion is, in itself, a weighting mechanism; that is, excluded variable are effectively given zero weighting. 31 But a priori variable weighting on the basis of current classifications may result in new classifications, which variably resemble existing classifications. ~2 Although cluster analysis has been applied to conventional orthodontic metrical parameters, 3336 the resuits were compromised by being based on scientifically unsound morphometric techniques. As a result, the interpretation of these data remains largely enigmatic. Allocation of fttrther cases to groups within an ex-
320
Fine
atul Lavelle
Fig. 1. Seventeen traditional datum points delineating lateral cephalometric form?
isting numeric classification: Classify. Classify is a statistical technique that compares data of a new case with a previously produced hierarchical classification (or training set). The technique locates the most similar (nearest) case within this classification. ~7"3'~The derived criterion value then provides a measure o f distance between the new case and the nearest case within the existing classification. A zero criterion value thus indicates that the n e w case is identical to its nearest case within the hierarchical classification, whereas higher values indicate increasing degrees of discrepancy. This study was undertaken to compare lateral cephalometric form relative to a previously classified training data set, by m e a n s of cluster techniques on the tensor c o m p o n e n t s of FEM.
MATERIALS AND METHODS Data Collection
Training set cephalograplffc sample. The training set cephalometric sample comprised a random selection of pretreatment lateral ccphalographs of 126 female patients, 13 to 17 years of age. They ',','ere taken under standardized conditions with a cephalostat and derived from a larger collcction at tile University of Manitoba Graduate Orthodontic Clinic. Selection criteria included ihmge clarity and the absence of gross craniofacial deformity (e.g., clefting). Each cephalo-
Am. J. Ortllod. Dcnt~Jc. Ortllol,. April 1992
gram was then traced onto acetate and orientated on the Frankfort horizontal plane. Seventeen traditional datum points were then digitized (Fig. I). Whereas the degree of deformation of the finite elements of each case could have been calculated v,'ith respect to any standard finite element, some of the calculated deformations would have been extremely large. It v,'as therefore preferable to compare the finite elements '`vith homologous elements of similar shape and size. A mean or standard was the most suitable approach for this purpose. To act as a comparative standard, therefore, homologous datum points were digitized on the Burlington 14-year-female standard (horizontal-vertical grower)3" although any other single standard cephalographic outline could have been used v,'ithout biasing the results. Therefore this study centered on evaluation of finite element deformations of each case relative to those of the Burlington 14-year-old standard. Tensor analysis. There is no theoretical basis for the optimal finite element delineation of an2,' cephalometric form. Therefore in this study areas of greatest orthodontic significance were arbitrarily represented by 1 ! triangular finite elements defined by nine datum points ( Fig. 2, Table 1), ahhough other finite element arrays bctv,'ecn the same points could also ha;,'c been selected. The FEM v,'as then used to quantify the deformation (strain) oi" individual finite elements of the 126patient cephalographic sample relative to homologous elements delineated on the Burlington standard. The resultant tensors (degrees of defonnation) '`','ere specified by a set of numbers (componcnts) of their graphic depiction in terms of principal values and angles. ~'--'~ These data comprised the following components (Fig. 3): I. Maximum strain: The tensor component defining the greatest change within a given finite element. 2. Minimum strain: The tensor component defining tile minimum change within a given finite element. 3. Orientation: A measure of maximum strain direction. For convenience, the angle between the X axis and maximum strain quantifies orientation. '~-'~ As minimum and maximum strain are orthogonal, minimum strain orientation is defined by default. Orientation may, in fact, be deceptive, as an orientation of 0 ~ is identical to one of 180" but very diffcrcnt from one of 90 ~. For the purpose of statistical manipulation, this problem has been partially solved by grouping orientation into horizontal and vertical groups."' However, this resulted in some infommtion loss. In this study, therefore, continuous data for statistical analysis '`,,'ere obtained by defining orientation as sin 0, where 0 was the orieentation angle in degrees (Fig. 3). Thus finite element deformation of the target relative to reference finite elements of the Burlington standard were described by the continuous variables of maximum strain, minimum strain, and sin 0. Training set construction. A prerequisite for new case classification (diagnosis) is the organization of a well-studied hierarchical classification (training set) derived from a rep-
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~.:lpert system diagnosis of skeh'ted fi~rnl 321
Fig. 2. Eleven triangular finite elements used in lateral cephalometric evaluations.
322
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Am. J. Orthod. Dentofac. Orthop. April 1992
Table I. Finite element delineation Finite element
Datum point definition
I 2 3 4 5 6 7 8 9 10 11
N-Me-ANS S-Me-ANS N-Go-Me S-Go-Me Co-Ba-A A-N-B N-S-A N-S-B N-P-A N-B-P B-Me-P
I
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9 9 l0 10 7 3 2 2 8 7 9
4 4 9 9 6 7 6 7 6 8 8
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no.
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I
3
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5
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resentative population sample from which new cases are to be dra~wn.37 To this end, cluster analysis of the finite element
6
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data from the 126 cephalographs was carried out. Initial cluster analyses, on the basis o f the previously described 11 finite elements, showed that the cluster procedure discriminated similar cases from one another when many less important parameters were discrepant and fewer but more relevant parameters were of similar magnitude, that is, equal weighting of all tensor parameters yielded unconventional cephalometri'c grouping (classification) patterns. The validity Of numeric classifications derived from cluster analysis is indisputable. They hinge on precise mathematical descriptors, ttowever, this does not imply that the resultant classifications will make sense. Yet, as a numeric classification must be sensible to be useful as a training set classification (cluster) construction, variable parameter weights may be specified by the user. 38 A priori differential variable weighting (Table II) was therefore used in this study in an attempt to emulate current conventional classification in the training set?-"
New cases. Five new cases (i.e., additional pretreatment cephalograms not included in the original sample) of the same sex and age groups were then traced and digitized. Analysis with the aid of the FEM was carried out, and these data were subjected to the CLASSIFY routine (a component of the cluster software) to investigate their diagnosis relative to the cluster groupings of the training set.
RESULTS
Training
set
The weighted tensor components for the 11 finite elements of the 126 training set cases were subjected to hierarchical cluster analysis by Ward's method, and the results were evaluated from the derived dendrogram (Fig. 4). This dendrogram showed patient case numbers along the horizontal axis, with betx~/een-case or case group distances being depicted on the vertical axis-in standard units. Two adjacent cases 9 by a horizontal line are termed nearest neighbors, when the in-
I
Variable
Iv= 'e weight
Maximum strain Minimum strain sm 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sm 0 Maximum strain Minimum strain sm 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sm 0
norE: Point number = datum point number; definition = cephalometric datum point.
terneighbor distance is the value on the vertical axis at the level of the horizontal line. Thus different nearestneighbor groups have contrasting distances between them. Although within-group homogeneity was not consistent, nearest neighbors were consistently similar when intercase distances were small. This was confirmed from subjective visualization of triangulated plots of the 11 finite elements as well as comparison of their conventional cephalometric parameters. For instance, the dciadrogram showed nearest neighbors 27 and 89 to be more similar than cases 20 and 42. Subjective evaluation of triangulated plots and conventional cephalometrics pointed to cases 27 and 89 as being more similar than cases 20 and 42 (Fig. 5). New case classification
Classify, a component of the cluster software, provided the case number within the training set that would
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have been the nearest neighbor of the new case had it bee included in the original sample. Furthermore, the derived criterion vahle provided a measure of the distance between a new case and the nearest case within the training set, that is, the smaller the criterion value, the more similar are the two cases. The new case was then compared with the nearest original case (i.e., nearest neighbor) both visually from finite clement plots and conventional cephalometric appraisals, the latter being the only numerical comparative basis currently.. available. The criterion values for each new case are listed in Table I11.
The cephalometric values tended to be most similar when the criterion value was the smallest (new case 2, original case I1; Fig. 6). As the criterion values increased, the case pairs appeared less similar on the basis 9of conventional cephalometric appraisals and subjective comparisons of finite element plots (Fig. 6). As new cases could be objectively allocated nearest neighbors that were part of an existing hierarchical classification, this study demonstrated a rigorous method to classify new cases with data obtained from tensor analysis (FEMi. As.diagnoses for preexisting cases can be established, this method represents the basis for ob-
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jective diagnosis on the basis of currently established diagnoses. DISCUSSION
Despite compromising two-dimensional images of complex three-dimensional forms, several dimensions are required to delineate the structurally complex craniofacial skeleton. Yet their continued evaluation as discrete entities, when such dimensions are variably intercorrelated, is surprising. 9'4~Moreover, statistical cephalometric appraisals require multivariate approaches to compensate for varied dimensional correlations/ I The use of multivariate analysis for taxonomic problems was first suggested by Fisher 4-"and Rao. 43 For more than four decades, the need for numeric taxonomic systems has been recognized, "~ but they have yet to be applied to cephalometric appraisals. The FEM was used in this study because of its rigorous theoretical background. The nature of FEM allowed for measurement of form change between individual cephalogram relative to a reference standard (the Burlington 14-year-old female standard), using 1 1 finite elements delineated by nine datum points. The use of nine datum points was necessarily cursory, although the rigorous delineation of additional datum points to describe lateral cephalometric form more precisely has yet to be investigated.
Table III. Criterion value listing of the five new
cases to the nearest cases of the original training set
New case
[
Nearest case
I
Criterionvalue
2
II
0.00160
I
57
0.00185
3 5 4
6 78 53
0.00407 0.01154 0.02656
New case 2 and nearest case I1 are most similar (criterion value smallest), and new case 4 a n d nearest case 53 least similar (highest criterion value).
Differential weighting of tensor components in cluster analysis permitted persons with similar cephalometric forms to be paired as nearest neighbors. This contrasts with traditional methods in which persons are compared with means and may be more relevant to the clinician who treats persons rather than means. Orthodontists "soon learn that the 'average' patient is a myth. Even the individual subjects whose measurements have provided our 'norms' did not fit those norms. The prob9lems.,of the patient confronting us at any given moment rarely match the central tendencies . . . . -45 The pairing of a new patient with a like person within the data base
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Volume I01 Number4
Expert system diagnosis of skeletal form
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is a better indicator of facial skeletal form than conventional methods that indicate what the skeletal pattern of the new patient is not like. Rigorous classification of pretreatment orthodontic cases as a formal diagnostic tool could lead to more logical and objective treatment planning. 3'~ The subjective selection of weighting values for the classification o f facial form may appear empirical. However, most objective assessment techniques initially require some degree o f subjectivity. However, while the initial classification was subjectively weighted, the second classification stage required no subjective input. Subjective weighting can be criticized on the basis of empiricy, but the differential weighting figures may be scrutinized by any who wish to do so. However, conventional classification schemes are vagu e in that they are devised without explicit instructions on variable weighting, so that their interpretation largely hinges on the prior experience o f the observer. The use of a training set or data base of cases resuited in categories matched at the nearest-neighbo r level that were (a) precisely defined (through the use Of numeric taxonomic techniques), (b) Useful (in that they related to current classification of the facial skeLeton), and (c) diagnostically precise and operator independent, in that the categorization o f a new case using the same training set would result in the same pairing of nearest neighbors each time the classification procedure was performed. Further study is needed, however, to (a) identify the relationship between objective pretreatment classification and diagnosis, (b) analyze and classify the curving form of hard and soft tissues in two and three dimensions, (c) provide a formal method of weighting variables, (d) expand the diagnostic input and derive a classification based on other metric and parametric variables pertinent to orthodontic diagnosis and classification (e.g., pretreatment and posttreatment arch form, dental relationships, soft tissue, muscle tone, and compliance assessments,* (e) determine the maximum acceptable criterion value for new case classification, and (f) allow for three-dimensional imaging o f both hardand soft-tissue forms through better imaging systems. CONCLUSIONS
1. Cluster analysis o f differentially weighted parameters from 1 1 finite elements spanning 9 datum points resulted in a rigorous classification of skeletal form in two dimensions in a sample of 126 female patients. *New patients may tfien be compared ",s'iththe nearest eases before treatment. For instance, perhaps a comprehensive classification will aid in the selection of suitable functional appliance patients.
Am. J. Orthod. Dentt(ac. Or~hop. April 1992
2. This classification scheme facilitated the categorization of five new patients not previously included in the original sample. 3. More research is required before objective diagnosis can be clinically employed in orthodontics. REFERENCES I. Bookstein FL. The measurement of biol~ical shape and shape change: lecture notes in biomathematics. Berlin: SpringerVerlag, 1978:24. 2. Moyers RE, Bookstein FL. The inappropriateness of conventional cephalometrics. Ast J OR'mOt) 1979;75:599-617. 3. Corruccini RS. Shape in morphomctrics:comparative analyses. Am J Phys Anthropol 1987;73:289-303. 4. Mosimann JE, James FC. New statistical methods for allometry with application to Florida red-winged blackbirds. Evolution 1979;33:444-59. 5. Richtsmeier J, Cheverud J. Finite element scaling analysis of human craniofacial growth. J Craniofac Genet Dev Biol 1986; 6:289-323. 6. Riedel RA, The relation of maxillary structures to cranium in malocclusion and in normal occlusion. Angle Orthod 1952;22: 142-45. 7. Downs WB. Analysis of the dentofacial profile. Angle Orthod 1956;26:191-212. 8. Riolo ML, Moyers RE, McNantara JA, tlunter WS. An atlas of craniofacial growth: cephalometric standards from the university school growth study, The University of Michigan. Monograph No. 2. Craniofacial Growth Series. Ann Arbor. Center for Human Growth and Development, The University of Michigan, 1974. 9. Solow B. The pattern of craniofacial associations. Acta Odontol Scand 1966i24:1-120. 10. tIowells WW. The use of multivariate techniques in the study of skeletal populations. Am J Phys Anthropol 1969;31:311-14. 1I. Ricketts RM. A foundation for cephalometric communication. AM J ORnXOt) 1960;46:330-57. 12. Wylie GA, Fish LC, Epker BN. Cephalometrics: a comparison of 5 analyses currently used in the diagnosis of dentofacial deformities. Int J Adult Orthod Orth~ Surg 1987;2:!5-36. 13. Moorrees CF, Kean MR. Natural head position: a basic consideration in the interpretation of cephalometric radiographs. Am J Phys Anthropol 1958;16:213-34. 14. Thompson DAW. On growth and form. Cambridge, England: The University Press, 1917. 15. BIum H. Biological shape and visual science. J Theor Biol 1973;38:205. 16. Bookstein FL. Size and shape spaces for landmark data in two dimensions. Stat Sci 1986;1:i81-242. 17. Oden JT. Finite elements of nonlinear continua. New York: McGraw Hill, 1972. 18. Bookstein FL. On the cephalomctries of skeletal change. AM J ORTttOD 1982;82:177-98. 19. Moss ML, Skalak R, Patel tt, Sen K, Moss-Salentijn L, Shinozuka M, Vilmann tl. Finite-element modeling of craniofacial growth. Ast J OR'rltoD 1985;87:453-72. 20. Cheverud J, Lewis JL, Bachrach W, Lew W. The measurement of form and variation in form: an applicationof three-dimensional quantitative morpholog,y by finite element methods. Am J Phys Anthropol 1983;62."151-65. 21. McCarthy JG, Grayson B, Bookstein FL, Vickery C. Zide B.
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35. Petrovic AG, Lavergne JM, Stutzmann JJ. Tissue-level growth and responsiveness potential: growth rotation and treatment decision. In: Vig P, Ribbens KA, eds. Science and clinical judgement in orthodontics. Mon~raph No 19. Craniofacial Growth Series. Ann Arbor: Center for Human Growth and Development, University of Michigan, 1986. 36. Moyers RE. Handbook of orthodontics. 4th ed. Chicago: Year Book Medical Publishers, 1988:577. 37. Wishart D. Cluster analysis in information retrieval and diagnosis. In: Bock till, ed. Classification and related methods of data analysis. North Holland: Elsevier Science Publishers, 1988:717-24. 38. Wishart D. Estimation of missing values and diagnosis using heirarchical classifications. Comput Stat 1985;2:124-34. 39. Popovich F, Thompson GW. Craniofacial templates for orthodontic case analysis. AM J ORTIIOD 1977;71:406-20. 40. Baunuind S, Frantz RC. The reliability of head film measurements, l, landmark identification. AM J ORTtIOD 1971;60:11127. 41. Goodall C. Comment on Bookstein FL: Size and shape spaces for landmark data in two dimensions. Stat Sci 1986;1:234. 42. Fisher RA. Tile use of multiple measurements in taxonomic problems. Ann Eugenics 1936;7:179-88. 43. RaoCR. The utilization of multiple measurements in problems of biol~ical classification. J Royal Stat Soc, Series B, 1948; 10:159-203. 44. Simpson GG. The pdnciple_s of classification and a classification of mammals. Bull Am Museum Natural Ilistory 1945;85:!-350. "45. Thurow RC. Chances are that the odds are wrong: the sample of one. Angle Orthod 1986;56:178-80.
Reprint requests to: Dr. C. L. B. Lavelle Department of Oral Biol~y. University of Manitoba 780 Bannatyne Ave. Winnipeg. Manitoba Canada R3E 0W2
Winnipeg, Manitoba, Canada Orthodontic diagnosis depends largely on the current methods of cephalometric analysis and categorization. It is therefore unfortunate that such analyses tend to use inappropriate metrical systems that lack a rigorous scientific basis. As a consequence, the resultant data have questionable validity, as well as being subject to observer bias. In this study, an alternative, more rigorous, approach is described. A random sample of 126 pretreatment lateral cephalograms were analyzed with the finite element method. These data were then classified to form a training set or reference data base for the subsequent objective diagnosis of new, previously unclassified cases. This new case-classification system was based on matching the new cases to similar cases within the existing training set. The degree of concordance between such computer-matched cases was then evaluated by their subsequent conventional cephalometric analyses. (AMJ ORTHOD DENTOFACORTHOP 1992;101:318-29.)
T h i s introduction is subdivided into a series of stages reflecting (A) our approach to conventi0nal cephalometric appraisals, (B) the reasons for adopting a more rigorous cephalometric appraisal technique, and (C) the use of cluster analysis to place cases into different diagnostic categories. Conventional cephalometries
Cephalometric measurement. The use of linear, angular, or proportional measures in conventional cephalometrics is inappropriate for evaluation of complex craniofacial profile forms? The methods also lack rigorous scientific backing. The is illustrated by the tendency to confound size and shape parameters together, there being no method for their discrete separation. 13 For example, when two cephalometric images are compared, a discrepancy in angle ANB may reflect a change in size or shape or, more likely, varying combinations of size and shape changes. Similar criticisms have also been applied to linear cephalometric dimensional changes. 4'5 As the data derived from conventional cephalometrics cannot withstand rigorous scientific scrutiny, their clinical application remains largely subjective. Comparison with cephalometric means. Conventional cephalomctric analyses ~* employ arbitrary measurement sets to delineate a person's craniofacial relationships. These subjective measurements are subsequently compared on a one-on-one basis with From the University of Manitoba. "Graduate orthodontic student, Department of Preventive Dental Science. bProfessor, Deparlment of Oral Biology. 811127147
318
corresponding population means, that is, the univariate approach is more appropriate for studying populations than for studying persons. More significantly, however, the variably correlated dimensions typical of lateral cephalometric forms are also unsuitable for univariate cephalometric analysis. 9 For instance, mandibular and facial height is variably correlated with mandibular and facial length, so that the separate evaluation of these dimensions t/as no scientific meaning, as the degree of correlation will vary between different persons. Indeed, multivariate statistical techniques are more appropriate for multidimensional cephalometric comparisons. These techniques facilitate cephalometric evaluation as a whole, rather than its arbitrary decomposition into a series of discrete parameters. ~~ Craniofacial classification (diagnosis). Orthodontic clinicians measure a series of discrete cephalometric measurements as a prelude to craniofacial diagnosis. Subsequent diagnosis (classification) for a given patient then hinges on the relationship between these dimensions and the corresponding population univariate means and standard deviations?' Different analyses of the same cephalograph may, however, result in different diagnoses, even when appraised by the same clinician. 12 Furthermore, since most conventional cephalometric classifications have not been rigidly defined, marginalcase diagnoses may be only conjecturally delineated. Current (qualitative) classification methods also do not embrace severity. An inherent weakness in current cephalomeetric diagnoses therefore centers on the clinician's inability to recount the logical steps taken. How can certain vail-
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ables be weighted when one is analyzing a particular cephalometric data set? Will this same weighting system be applied to all cases? In the absence of standardized classification methods, how can particular diagnoses be rigorously confirmed by other clinicians? In summary, conventional cephalometric analyses use scientifically inappropriate linear, angular, or proportionate parameters. The use of univariate statistics is less than ideal. Furthermore, the subjective assessment of individual measurement deviations concerning sample means results in informal methods of classification and arbitrary cephalometric diagnoses." As a result of these fundamental criticisms, this 9study was undertaken to investigate (1) a more rigorous form of lateral cephalometric appraisal, (2) a classification system delineated by both qualitative and quantitative information, and (3) a standardized method for lateral cephalometric diagnosis. The prime objective is an attempt to delineate lateral cephalometric evaluations more rigox'ously.
Rigorous lateralcephalometric meansurement Shape and form are terms that are frequentlyused interchangeably. Yet pure shape change implies constancy in size, whereas form change infers concurrent changes in both size and shape. 5 Shape may also be defined as an outline with landmarks in euclidean space after the removal of all information on position, scale, and orientation. ~ As facial profile orientation is correlated with "natural head position, ''~3 extremal datum points, of which the positions vary with orientation (e.g., menton), may be used to augment the paucity of anatomic points, of which the positions are dependent on anatomic features (i.e., independent of cephalographic orientation to delineate craniofacial form). Lateral cephalometric profiles may thus be defined as oriented shapes. In searching for alternative appraisal methods, ~4"~5 increasing interest has focused on the finite element method (FEM). This method is based on the concepts of continuum mechanics and applies mathematical descriptors from finite element analysis. ~6The FEM permits the quantification of deformations of anisotropic, nonhomogeneous, materially nonlinear bodies of any geometric shape. ~7 Quantified and graphic representations of cephalometric deformations (changes in form). may then be depicted as parameters (tens0rs) derived from the FEM. 5"18't9 A requirement of FEM is that lateral cephalometric form be divided into a number of finite elements. The arbitrarily defined elements are usually triangular, although other polygonal forms may be used to represent the area of cephalometric interest. Measurement of de-
Expert system diagnosis of skeletal form
319
formation of a particular finite element centers on the computation of the degree of deformation of a "target" relative to an analogous "reference" finite element delineated by the homologous datum (nodal) points. The computed deformation may then be described by a series of numeric values of the tensor components. 2~ The degree of cephalometric deformation around datum points may thus be obtained, without reference to arbitrary, fixed, registration points, that is, by simultaneous registration of homologous datum points. 5 This technique has been used to evaluate the mechanisms underlying change in craniofacial form, 5'19"27 in addition to demonstrating the advantages of avoiding the shortcomings of conventional cephalometric appraisals. 2"2s But whereas tensor (FEM) analysis has been used as a morphometric tool, 16application to static diagnostic cephalometry has yet to be investigated.
Classification techniques Th e classification process may be divided into two s!ages29: (1) the construction of classifications from initially tinclassified data and (2) the allocation of further cases to groups within such a classification system. Construction of classifications: Chlster analysis. Cluster analysis is a conventional mathematical technique for classifying previously unclassified data. This is achieved by sorting similar entities into "clusters," whereas members of different clusters tend to be less similar. Such techniques are useful for solving classification problems when little is known about a data set. 3~ As numeric taxonomic classifications of biologic data may contrast with conventional classifications of the same data sets, 29 however, cluster analysis of facial skeletal form may not necessarily reflect conventional classifications. In fact, for any given set of cases, each described by a cephalometric dimensional array, a variety of classifications may be delineated. Consideration should then be given to those dimensions (variables) that are more useful or descriptive. 29 The key to any classification system therefore centers on variable selection. Also, not all variables should necessarily be equally weighted. 29 Variable exclusion is, in itself, a weighting mechanism; that is, excluded variable are effectively given zero weighting. 31 But a priori variable weighting on the basis of current classifications may result in new classifications, which variably resemble existing classifications. ~2 Although cluster analysis has been applied to conventional orthodontic metrical parameters, 3336 the resuits were compromised by being based on scientifically unsound morphometric techniques. As a result, the interpretation of these data remains largely enigmatic. Allocation of fttrther cases to groups within an ex-
320
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atul Lavelle
Fig. 1. Seventeen traditional datum points delineating lateral cephalometric form?
isting numeric classification: Classify. Classify is a statistical technique that compares data of a new case with a previously produced hierarchical classification (or training set). The technique locates the most similar (nearest) case within this classification. ~7"3'~The derived criterion value then provides a measure o f distance between the new case and the nearest case within the existing classification. A zero criterion value thus indicates that the n e w case is identical to its nearest case within the hierarchical classification, whereas higher values indicate increasing degrees of discrepancy. This study was undertaken to compare lateral cephalometric form relative to a previously classified training data set, by m e a n s of cluster techniques on the tensor c o m p o n e n t s of FEM.
MATERIALS AND METHODS Data Collection
Training set cephalograplffc sample. The training set cephalometric sample comprised a random selection of pretreatment lateral ccphalographs of 126 female patients, 13 to 17 years of age. They ',','ere taken under standardized conditions with a cephalostat and derived from a larger collcction at tile University of Manitoba Graduate Orthodontic Clinic. Selection criteria included ihmge clarity and the absence of gross craniofacial deformity (e.g., clefting). Each cephalo-
Am. J. Ortllod. Dcnt~Jc. Ortllol,. April 1992
gram was then traced onto acetate and orientated on the Frankfort horizontal plane. Seventeen traditional datum points were then digitized (Fig. I). Whereas the degree of deformation of the finite elements of each case could have been calculated v,'ith respect to any standard finite element, some of the calculated deformations would have been extremely large. It v,'as therefore preferable to compare the finite elements '`vith homologous elements of similar shape and size. A mean or standard was the most suitable approach for this purpose. To act as a comparative standard, therefore, homologous datum points were digitized on the Burlington 14-year-female standard (horizontal-vertical grower)3" although any other single standard cephalographic outline could have been used v,'ithout biasing the results. Therefore this study centered on evaluation of finite element deformations of each case relative to those of the Burlington 14-year-old standard. Tensor analysis. There is no theoretical basis for the optimal finite element delineation of an2,' cephalometric form. Therefore in this study areas of greatest orthodontic significance were arbitrarily represented by 1 ! triangular finite elements defined by nine datum points ( Fig. 2, Table 1), ahhough other finite element arrays bctv,'ecn the same points could also ha;,'c been selected. The FEM v,'as then used to quantify the deformation (strain) oi" individual finite elements of the 126patient cephalographic sample relative to homologous elements delineated on the Burlington standard. The resultant tensors (degrees of defonnation) '`','ere specified by a set of numbers (componcnts) of their graphic depiction in terms of principal values and angles. ~'--'~ These data comprised the following components (Fig. 3): I. Maximum strain: The tensor component defining the greatest change within a given finite element. 2. Minimum strain: The tensor component defining tile minimum change within a given finite element. 3. Orientation: A measure of maximum strain direction. For convenience, the angle between the X axis and maximum strain quantifies orientation. '~-'~ As minimum and maximum strain are orthogonal, minimum strain orientation is defined by default. Orientation may, in fact, be deceptive, as an orientation of 0 ~ is identical to one of 180" but very diffcrcnt from one of 90 ~. For the purpose of statistical manipulation, this problem has been partially solved by grouping orientation into horizontal and vertical groups."' However, this resulted in some infommtion loss. In this study, therefore, continuous data for statistical analysis '`,,'ere obtained by defining orientation as sin 0, where 0 was the orieentation angle in degrees (Fig. 3). Thus finite element deformation of the target relative to reference finite elements of the Burlington standard were described by the continuous variables of maximum strain, minimum strain, and sin 0. Training set construction. A prerequisite for new case classification (diagnosis) is the organization of a well-studied hierarchical classification (training set) derived from a rep-
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~.:lpert system diagnosis of skeh'ted fi~rnl 321
Fig. 2. Eleven triangular finite elements used in lateral cephalometric evaluations.
322
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Am. J. Orthod. Dentofac. Orthop. April 1992
Table I. Finite element delineation Finite element
Datum point definition
I 2 3 4 5 6 7 8 9 10 11
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I
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5
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resentative population sample from which new cases are to be dra~wn.37 To this end, cluster analysis of the finite element
6
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data from the 126 cephalographs was carried out. Initial cluster analyses, on the basis o f the previously described 11 finite elements, showed that the cluster procedure discriminated similar cases from one another when many less important parameters were discrepant and fewer but more relevant parameters were of similar magnitude, that is, equal weighting of all tensor parameters yielded unconventional cephalometri'c grouping (classification) patterns. The validity Of numeric classifications derived from cluster analysis is indisputable. They hinge on precise mathematical descriptors, ttowever, this does not imply that the resultant classifications will make sense. Yet, as a numeric classification must be sensible to be useful as a training set classification (cluster) construction, variable parameter weights may be specified by the user. 38 A priori differential variable weighting (Table II) was therefore used in this study in an attempt to emulate current conventional classification in the training set?-"
New cases. Five new cases (i.e., additional pretreatment cephalograms not included in the original sample) of the same sex and age groups were then traced and digitized. Analysis with the aid of the FEM was carried out, and these data were subjected to the CLASSIFY routine (a component of the cluster software) to investigate their diagnosis relative to the cluster groupings of the training set.
RESULTS
Training
set
The weighted tensor components for the 11 finite elements of the 126 training set cases were subjected to hierarchical cluster analysis by Ward's method, and the results were evaluated from the derived dendrogram (Fig. 4). This dendrogram showed patient case numbers along the horizontal axis, with betx~/een-case or case group distances being depicted on the vertical axis-in standard units. Two adjacent cases 9 by a horizontal line are termed nearest neighbors, when the in-
I
Variable
Iv= 'e weight
Maximum strain Minimum strain sm 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sm 0 Maximum strain Minimum strain sm 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sin 0 Maximum strain Minimum strain sm 0
norE: Point number = datum point number; definition = cephalometric datum point.
terneighbor distance is the value on the vertical axis at the level of the horizontal line. Thus different nearestneighbor groups have contrasting distances between them. Although within-group homogeneity was not consistent, nearest neighbors were consistently similar when intercase distances were small. This was confirmed from subjective visualization of triangulated plots of the 11 finite elements as well as comparison of their conventional cephalometric parameters. For instance, the dciadrogram showed nearest neighbors 27 and 89 to be more similar than cases 20 and 42. Subjective evaluation of triangulated plots and conventional cephalometrics pointed to cases 27 and 89 as being more similar than cases 20 and 42 (Fig. 5). New case classification
Classify, a component of the cluster software, provided the case number within the training set that would
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have been the nearest neighbor of the new case had it bee included in the original sample. Furthermore, the derived criterion vahle provided a measure of the distance between a new case and the nearest case within the training set, that is, the smaller the criterion value, the more similar are the two cases. The new case was then compared with the nearest original case (i.e., nearest neighbor) both visually from finite clement plots and conventional cephalometric appraisals, the latter being the only numerical comparative basis currently.. available. The criterion values for each new case are listed in Table I11.
The cephalometric values tended to be most similar when the criterion value was the smallest (new case 2, original case I1; Fig. 6). As the criterion values increased, the case pairs appeared less similar on the basis 9of conventional cephalometric appraisals and subjective comparisons of finite element plots (Fig. 6). As new cases could be objectively allocated nearest neighbors that were part of an existing hierarchical classification, this study demonstrated a rigorous method to classify new cases with data obtained from tensor analysis (FEMi. As.diagnoses for preexisting cases can be established, this method represents the basis for ob-
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jective diagnosis on the basis of currently established diagnoses. DISCUSSION
Despite compromising two-dimensional images of complex three-dimensional forms, several dimensions are required to delineate the structurally complex craniofacial skeleton. Yet their continued evaluation as discrete entities, when such dimensions are variably intercorrelated, is surprising. 9'4~Moreover, statistical cephalometric appraisals require multivariate approaches to compensate for varied dimensional correlations/ I The use of multivariate analysis for taxonomic problems was first suggested by Fisher 4-"and Rao. 43 For more than four decades, the need for numeric taxonomic systems has been recognized, "~ but they have yet to be applied to cephalometric appraisals. The FEM was used in this study because of its rigorous theoretical background. The nature of FEM allowed for measurement of form change between individual cephalogram relative to a reference standard (the Burlington 14-year-old female standard), using 1 1 finite elements delineated by nine datum points. The use of nine datum points was necessarily cursory, although the rigorous delineation of additional datum points to describe lateral cephalometric form more precisely has yet to be investigated.
Table III. Criterion value listing of the five new
cases to the nearest cases of the original training set
New case
[
Nearest case
I
Criterionvalue
2
II
0.00160
I
57
0.00185
3 5 4
6 78 53
0.00407 0.01154 0.02656
New case 2 and nearest case I1 are most similar (criterion value smallest), and new case 4 a n d nearest case 53 least similar (highest criterion value).
Differential weighting of tensor components in cluster analysis permitted persons with similar cephalometric forms to be paired as nearest neighbors. This contrasts with traditional methods in which persons are compared with means and may be more relevant to the clinician who treats persons rather than means. Orthodontists "soon learn that the 'average' patient is a myth. Even the individual subjects whose measurements have provided our 'norms' did not fit those norms. The prob9lems.,of the patient confronting us at any given moment rarely match the central tendencies . . . . -45 The pairing of a new patient with a like person within the data base
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Volume I01 Number4
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is a better indicator of facial skeletal form than conventional methods that indicate what the skeletal pattern of the new patient is not like. Rigorous classification of pretreatment orthodontic cases as a formal diagnostic tool could lead to more logical and objective treatment planning. 3'~ The subjective selection of weighting values for the classification o f facial form may appear empirical. However, most objective assessment techniques initially require some degree o f subjectivity. However, while the initial classification was subjectively weighted, the second classification stage required no subjective input. Subjective weighting can be criticized on the basis of empiricy, but the differential weighting figures may be scrutinized by any who wish to do so. However, conventional classification schemes are vagu e in that they are devised without explicit instructions on variable weighting, so that their interpretation largely hinges on the prior experience o f the observer. The use of a training set or data base of cases resuited in categories matched at the nearest-neighbo r level that were (a) precisely defined (through the use Of numeric taxonomic techniques), (b) Useful (in that they related to current classification of the facial skeLeton), and (c) diagnostically precise and operator independent, in that the categorization o f a new case using the same training set would result in the same pairing of nearest neighbors each time the classification procedure was performed. Further study is needed, however, to (a) identify the relationship between objective pretreatment classification and diagnosis, (b) analyze and classify the curving form of hard and soft tissues in two and three dimensions, (c) provide a formal method of weighting variables, (d) expand the diagnostic input and derive a classification based on other metric and parametric variables pertinent to orthodontic diagnosis and classification (e.g., pretreatment and posttreatment arch form, dental relationships, soft tissue, muscle tone, and compliance assessments,* (e) determine the maximum acceptable criterion value for new case classification, and (f) allow for three-dimensional imaging o f both hardand soft-tissue forms through better imaging systems. CONCLUSIONS
1. Cluster analysis o f differentially weighted parameters from 1 1 finite elements spanning 9 datum points resulted in a rigorous classification of skeletal form in two dimensions in a sample of 126 female patients. *New patients may tfien be compared ",s'iththe nearest eases before treatment. For instance, perhaps a comprehensive classification will aid in the selection of suitable functional appliance patients.
Am. J. Orthod. Dentt(ac. Or~hop. April 1992
2. This classification scheme facilitated the categorization of five new patients not previously included in the original sample. 3. More research is required before objective diagnosis can be clinically employed in orthodontics. REFERENCES I. Bookstein FL. The measurement of biol~ical shape and shape change: lecture notes in biomathematics. Berlin: SpringerVerlag, 1978:24. 2. Moyers RE, Bookstein FL. The inappropriateness of conventional cephalometrics. Ast J OR'mOt) 1979;75:599-617. 3. Corruccini RS. Shape in morphomctrics:comparative analyses. Am J Phys Anthropol 1987;73:289-303. 4. Mosimann JE, James FC. New statistical methods for allometry with application to Florida red-winged blackbirds. Evolution 1979;33:444-59. 5. Richtsmeier J, Cheverud J. Finite element scaling analysis of human craniofacial growth. J Craniofac Genet Dev Biol 1986; 6:289-323. 6. Riedel RA, The relation of maxillary structures to cranium in malocclusion and in normal occlusion. Angle Orthod 1952;22: 142-45. 7. Downs WB. Analysis of the dentofacial profile. Angle Orthod 1956;26:191-212. 8. Riolo ML, Moyers RE, McNantara JA, tlunter WS. An atlas of craniofacial growth: cephalometric standards from the university school growth study, The University of Michigan. Monograph No. 2. Craniofacial Growth Series. Ann Arbor. Center for Human Growth and Development, The University of Michigan, 1974. 9. Solow B. The pattern of craniofacial associations. Acta Odontol Scand 1966i24:1-120. 10. tIowells WW. The use of multivariate techniques in the study of skeletal populations. Am J Phys Anthropol 1969;31:311-14. 1I. Ricketts RM. A foundation for cephalometric communication. AM J ORnXOt) 1960;46:330-57. 12. Wylie GA, Fish LC, Epker BN. Cephalometrics: a comparison of 5 analyses currently used in the diagnosis of dentofacial deformities. Int J Adult Orthod Orth~ Surg 1987;2:!5-36. 13. Moorrees CF, Kean MR. Natural head position: a basic consideration in the interpretation of cephalometric radiographs. Am J Phys Anthropol 1958;16:213-34. 14. Thompson DAW. On growth and form. Cambridge, England: The University Press, 1917. 15. BIum H. Biological shape and visual science. J Theor Biol 1973;38:205. 16. Bookstein FL. Size and shape spaces for landmark data in two dimensions. Stat Sci 1986;1:i81-242. 17. Oden JT. Finite elements of nonlinear continua. New York: McGraw Hill, 1972. 18. Bookstein FL. On the cephalomctries of skeletal change. AM J ORTttOD 1982;82:177-98. 19. Moss ML, Skalak R, Patel tt, Sen K, Moss-Salentijn L, Shinozuka M, Vilmann tl. Finite-element modeling of craniofacial growth. Ast J OR'rltoD 1985;87:453-72. 20. Cheverud J, Lewis JL, Bachrach W, Lew W. The measurement of form and variation in form: an applicationof three-dimensional quantitative morpholog,y by finite element methods. Am J Phys Anthropol 1983;62."151-65. 21. McCarthy JG, Grayson B, Bookstein FL, Vickery C. Zide B.
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