Integration of parts in the facial skeleton and cervical vertebrae

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Integration of parts in the facial skeleton andcervical vertebrae

Brendan McCanea and Martin R. Keanb

Dunedin, New Zealand

FromaAssobEmeThe aproduReprinof OtaSubm0889-Copyrdoi:10

Introduction: The purpose of this study was to undertake an exploratory analysis of the relationship amongparts in the facial skeleton and cervical vertebrae and their integration as 2-dimensional shapes by determiningtheir individual variations and covariations. The study was motivated by considerations applicable to clinicalorthodontics and maxillofacial surgery, in which such relationships bear directly on pretreatment analysis andassessment of posttreatment outcome. Methods: Lateral radiographs of 61 adolescents of both sexes withoutmajor malocclusions were digitized and marked up by using continuous outline spline curves for 8 defined partsin the facial skeleton, including the cervical vertebrae. Individual part variation was analyzed by using principalcomponents analysis, and paired part covariation was analyzed by using 2-block partial least squares analysis in2modes: relative size, position, and shape; and shape only.Results: For individual part variations, cranial base,soft-tissue profile, and mandible had the largest variations across the sample. For covariation of relative size,position, and shape, the cervical vertebrae were highly correlated with the cranial base (r 5 0.80),nasomaxillary complex (r 5 0.70), mandible (r 5 0.74), maxillary dentition (r 5 0.70), and mandibulardentition (r 5 0.74); the maxillary dentition and mandibular dentition were highly correlated (r 5 0.70); themandible was highly correlated with the bony profile (r 5 0.72), soft-tissue profile (r 5 0.79), and, to a lesserextent, the cranial base (r 5 0.67); the bony profile was highly correlated with the cranial base (r 5 0.70) andsoft-tissue profile (r 5 0.80); the soft-tissue profile was highly correlated with the nasomaxillary dentition (r 50.81). Covariation of shape only was much weaker with significant covariations found between bony profileand mandible (r 5 0.53), bony profile and mandibular dentition (r 5 0.65), mandibular dentition andsoft-tissue profile (r 5 0.54), mandibular dentition and maxillary dentition (r 5 0.55), and bony profile andsoft-tissue profile (r5 0.69). Conclusions:We found that integration of the shape of parts in the facial skeletonand cervical vertebrae is weak; it is the relative size, position, and orientation of parts that form the strongestcorrelations. (Am J Orthod Dentofacial Orthop 2011;139:e13-e30)

A standardized lateral radiographic image of thecraniofacial skeleton shows a series of discrete,though related, parts comprising skeletal struc-

tures and spaces, and the maxillary and mandibulardentitions. The purpose of this study was to undertakean exploratory analysis of the relationship among theseparts and their integration as 2-dimensional (2D) shapesby determining their individual variations and covaria-tions. The study was motivated by considerationsapplicable to clinical orthodontics and maxillofacial

the University of Otago, Dunedin, New Zealand.ciate Professor, Department of Computer Science.ritus professor, School of Dentistry. (Deceased.)uthors report no commercial, proprietary, or financial interest in thects or companies described in this article.t requests to: BrendanMcCane, Department of Computer Science, Universitygo, PO Box 56, Dunedin, New Zealand; e-mail, [email protected], August 2009; revised and accepted, June 2010.5406/$36.00ight � 2011 by the American Association of Orthodontists..1016/j.ajodo.2010.06.016

surgery, in which such relationships bear directly on pre-treatment analysis and assessment of posttreatmentoutcome. In particular, the ultimate goal was for thistype of analysis to become a useful addition and insome cases a replacement for the current uses ofstandard cephalometric software.

The pursuit of associations between and among partsin the facial skeleton is not a new endeavour. Using stan-dard analyses of lateral headfilms, Solow1 sought to dis-tinguish between the topographic and nontopographicassociations of 2D variables. The former arose from usingcommon reference points or landmarks in measuring eachvariable. Nontopographic associations, in which commonreference points or landmarks were not used, were con-sidered to have biologic significance, although cautionwas urged to exclude circumstances in which referencepoints might be placed on a common reference structure.More recently, Enlow and Hans2 introduced the “counter-part principle” in postnatal facial development studied insequential lateral head radiographs. This principle states

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mailto:[email protected]

e14 McCane and Kean

that “the growth of any given facial or cranial part relatesspecifically to other structural and geometric ‘counter-parts’ in the face and cranium.” Enlow and Hans statedthat, if each regional part and its counterpart enlarge(grow) to the same extent, balance between the partsand within the face will be maintained.

The methodology in this study relied on techniquesused in modern morphometrics, albeit with some differ-ences. The principal object of study was a clinically ordiagnostically meaningful part expressed as a shape.We sought to visualize and measure the patterns ofcovariation among parts in a subject and in a sampleof subjects with balanced facial forms and similarmaturity, and to determine the variability of such mea-surements in the sample. Morphometrically, the partscould be regarded as modules being a set of shapes in-dependent, though integrated, through interaction andcovariance with other structures during facial develop-ment. In this study, a part is a discrete structure, andlandmarks (or pseudolandmarks) are defined in relationto the part. Each part is represented by a cubic splinethat provides a much richer definition of the part thana finite number of landmarks, although much of theanalysis is not crucially dependent on the exact formof the continuous representation of the shape.

The purpose of modularity and integration studieswas to investigate the covariation of shape and size dur-ing evolution3-6 or development,3,7-13 or both.14 Mostsuch studies used the method of partial least squares(PLS), also known in the morphometrics literature as sin-gular warp analysis.15 The level of integration betweenhypothesized shape modules is usually determined bythe correlations of corresponding singular warp scoresextracted from landmarks,7,8,10,12 although somestudies also use pseudolandmarks,3,5,9 whereas othersused more traditional statistical tests on distances andangles measured from landmarks.13,16,17 Typicalmodules and the corresponding integrations includethe following.

1. The face and neurocranium, which are highly inte-grated across hominid species.5

2. The cranial vault, cranial base, and face, where thecranial vault and face are highly integrated in bothdevelopment and evolution, more so than the vaultand base, or the base and face.3

3. Integration of the neurobasicranial complex, ethno-maxillary complex, mandible, midline cranial base,lateral cranial floor, and neurocranium.9

4. Midline cranial base, lateral cranial base, and face,8

where the lateral, but not the midline, cranial baseand face are highly integrated in contrast to theresults reported by Bookstein et al.3

January 2011 � Vol 139 � Issue 1 American

5. Midline cranial base, middle cranial fossa, and man-dibular ramus,7 where the midline cranial base andmandibular ramus, and the middle cranial fossaand mandibular ramus showed significant integra-tion, but not the midline cranial base and middlecranial fossa.

6. Mandible, nasomaxillary complex, and neurocra-nium.2

7. cranial base, neurocranium, and face.13

This study was intended to extend understanding ofthe form and architecture of the facial skeleton and in-vesting soft tissues viewed laterally in 2 dimensions,with particular emphasis on parts or modules of clinicalinterest to orthodontists and maxillofacial surgeons.This study differs from previous studies in severalrespects: all parts or modules are defined by usingfree-form curves (cubic splines) because we consideredthat landmarks lack representational power of the par-ticular parts in question; we investigated the integrationof parts in the craniofacial skeleton with the intention ofgaining a more complete understanding of craniofacialform, especially in relation to covariation of the parts;we included cervical vertebrae in our analysis to test theirrelationship to facial form. The ultimate goal was to findthe means to study efficiently and informatively thehuge repository of lateral headfilms already availablefrom earlier growth and clinical studies.

MATERIAL AND METHODS

The sample comprised standard lateral radiographsof a mixed-sex group of North American white adoles-cents with normal skeletal patterns (n 5 61) obtainedfrom a university orthodontic clinic. Patients ranged indental maturation from the late mixed dentition to theearly permanent dentition, with second molars emergingor emerged. Ages and sexes were not given. The head-films were high-quality digitally rendered x-ray images.All headfilms were taken in natural head position withthe lips in repose.18

The method relied on the identification and markingup of a series of major parts or modules of the radio-graphic images of bony structures and the dentition in2D lateral views. The parts were verified by studyinglateral x-ray images of a skull on which metallic markershad been placed to show the most reliable identificationof outlines without having recourse to standard land-marks to determine the outlines. The shapes were thefollowing.

1. Cranial base (CRB): CRB shape follows the outline ofthe superior contour of the cranial base extendingfrom the superior posterior of the frontal sinus to

Journal of Orthodontics and Dentofacial Orthopedics

McCane and Kean e15

the superior outline of the pituitary fossa, to thedorsal outline of the superior surface of the basicra-nium to the tip of the clivus.

2. Nasomaxillary complex (NMC): NMC shape follows3 sides of the composite of the outline of the nasalfossa in the lateral view and the maxillary structurethat can be defined reliably where the bony con-tours are visible radiographically in the lateralview. Beginning at its superior and internal termi-nation, the shape follows down from the upperlimit of the pterygomaxillary fossa through thefossa to the posterior limit of the bony palate. It ex-tends from there to an anterior limit approximatingthe anterior nasal spine and from there superiorlyto a point on the inner margin of the image ofthe nasal bone following the outline of the bonyprofile.

3. Mandible (MAN): Mandibular shape is delineated bystarting at the bony shadow immediately dorsal tothe last molar emerged, usually the second molar.The shape is outlined by following the ramus poste-riorly and then superiorly along the shadow of thecoronoid process, into the mandibular notch, upand around the condylar head and down alongthe posterior shadow of the ramus to the inferiorborder of the mandible, and around the externalsurface of the symphysis to the junction of boneand the foremost incisor.

4. Maxillary dentition (XDE): The shape of the XDE isrepresented by an outline beginning at the mostdorsal point on the shadow of the permanentsecond molar (emerged or not) and following anoutline anteriorly and inferiorly that includes themain features of the crowns of the first molar,premolars, canine, and incisors in profile. Fromhere, the shape includes the labial outline of theforemost incisor to the tip of the root and thenfollows a line along the root apices until reachingthe initial point of the shape.

5. Mandibular dentition (MDE): As in the case of theXDE, this shape is drawn to show the outline ofthe MDE, including the second molar, in generalsize and form. Shape delineation is most easilyaccomplished by following a similar method tothat for XDE, starting at a consistent point such asthe most dorsal point on the crown of the perma-nent second molar and proceeding to completethe outline of the MDE.

6. Cervical vertebrae (CVT): Beginning at the superiormargin of the odontoid process of the secondCVT, the shapes of vertebrae C3 to C5 are followedas a continuous outline from this point inferiorly,

American Journal of Orthodontics and Dentofacial Orthoped

around the dorsal shadows of the vertebrae, andacross inferiorly to pick up the ventral outlines ofthe vertebrae. This outline is continued superiorlyto close the shape at the point where it began.

7. Bony profile (BPL): Starting from a point on theshadow of the most forward or ventral outline ofthe frontal bone over the center of the developingfrontal sinus, the outline is continued inferiorlythrough nasion and along the ventral shadows ofthe nasal bone and the maxilla, through the tip ofthe anterior nasal spine, and around the mandibularsymphysis to its most inferior point.

8. Soft-tissue profile (SPL): From a starting point onthe surface of the soft-tissue shadow over thedeveloping frontal sinus, an outline is followedinferiorly to show the shape of the profile of thenose and lips, and the soft tissue over the chin be-fore terminating at a point immediately below themost inferior shadow of the bony symphysis. Thelips are in a relaxed state. If the lips touch, thenthe outline of the upper lip is followed to thetouching point, and then on to the lower lip. Ifthe lips do not touch, the upper lip is followedto the foremost incisor and then down to thelower lip.

The method used in this study requires the markingup of the parts defined above on an on-screen x-ray im-age by using software developed especially for thepurpose. A screen shot of the prototype software isshown in Figure 1. When each image was loaded, itsmagnification, brightness, and contrast were adjustedto obtain the most detailed rendition of a shape. Ratherthan basing a part on a series of landmarks, continuousoutline curves were used to represent shape in the formof Catmull-Rom splines.19 Catmull-Rom splines are cu-bic interpolating splines that overcome many limitationsof using landmarks. They have the advantages of ease inspecification, validity at all scales, and continuity. Theycan pass through specific landmarks if required and beapplied to any continuous shape. Because they are con-tinuous, they cannot easily be used to represent a discon-tinuous shape such as a square. Consistent with mostinterpolating spline techniques, the representation isnot unique. That is, the same continuous curve can beclosely approximated by many differently parameterizedCatmull-Rom splines.

A Catmull-Rom spline is defined by a set of controlpoints:

pk5ðxk; ykÞ; k5 0; 1; 2;.; n� 1:

A point on the spline is specified by the parameter0 # u\1:

ics January 2011 � Vol 139 � Issue 1

Fig 1. A screen shot of the prototype software. Shapes can be displayed either closed with a straightline or open, and filled or not. Because this image is viewed at 20% magnification, the control pointsappear close together. They are not so at higher magnifications.

crb spl man bpl xde mde nmc cvt

0100

200

300

400

500

Fig 2. The total variation of each part. Larger numbersindicate greater variations in that part.

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PðuÞ 5 pk�1ð � su3 1 2su2 � suÞ 1 pk½ð2 � sÞu3 1

ðs � 3Þu2 1 1�1 pk11½ðs � 2Þu3 1ð3 � 2sÞu2 1 su�1 pk12ðsu3 � su2Þ

where s 5 0.5, 0 # k \ n, and pk are the controlpoints. Any number of control points can be usedto specify the craniofacial parts, but we found bytrial and error that 50 to 60 control points per shapeof each part are sufficient to represent the variabilityin shape.

To specify the shape, the user places control pointsalong visible margins of the part in the x-ray image.Control points are not landmarks (or at least not neces-sarily landmarks) or pseudolandmarks but, rather, spec-ify points through which the curve must pass. Slidinga control point along the curve often causes little changein the curve itself, and, in this way, a source of error inmarking up images is reduced. Landmarks are subjectto 2D errors, whereas interpolating splines are only sub-ject to 1-dimensional errors. Errors can arise when thecurve is moved off the underlying part perpendicularto the part outline but do not arise when a control pointis moved along the curve of an underlying part.

Each subject is represented as:

Si5fCRBi;NMCi;MANi;XDEi;MDEi;CVTi;BPLi;SPLig;

where CRBi5fpikjpik5ðxik; yikÞ; k5 0; 1; 2;.; ni � 1g isthe spline representation of CRB for subject i. The func-tion CRBiðuÞ; 0 # u\1 returns a 2D point along thecurve of CRB for subject i.


The method of shape registration used here differsfrom previous methods for 2 reasons: since we useda spline curve shape representation, we did not have fixedlandmarks to perform registrations, and we assumed thateach part is a tightly integrated module. This led to a 2-step shape registration procedure: individual part registra-tion, followed by a global generalized Procrustes analysis.The 2-step process was needed to obtain correspondencesbetween pseudolandmarks on the same module.


Fig 3. Principal variations for CRB corresponding to thefirst to fourth principal eigenvectors of PCA. Shown arethe mean shape in black and the mean 63 SD from themean (blue and red) along the principal direction. A, Prin-cipal component (PC) 1, percentage of variation (PV):16%. B, PC 2, PV: 13%. C, PC 3, PV: 11%. D, PC 4,PV: 9%. Also included is the percentage of total variationof each component.

Fig 4. Principal variations for SPL corresponding to thefirst to fourth principal eigenvectors of PCA. Shown arethe mean shape in black, and the mean 63 SD from themean (blue and red) along the principal direction. A, PC1, PV: 14%. B, PC 2, PV: 8.6%. C, PC 3, PV 6.8%.D, PC 4, PV: 5.2%.

McCane and Kean e17

Individual parts were registered by using a variant ofthe iterative closest point (ICP) algorithm, an iterativemethod for matching parametric shapes.20 The variantof ICP used follows the basic structure of the algorithm,except that an explicit 2D registration procedure wasused that accounted for scale, rotation, and translationas described by McCane et al.21 The method produceda similar result to the method of Procrustes superimpo-sition of semilandmarks22 but, along with the spline rep-resentation, has an advantage over the semilandmarkmethod: part specification and matching is independent


of the number of pseudolandmarks used in thesubsequent analysis. For example, using the samemarked-up sample, one can extract different numbersof pseudolandmarks depending on the analysis beingdone, without having to mark up the original samplesagain.

The ICP algorithm was applied as follows:

1. a sample is chosen as the initial mean shape2. While the mean has not converged, do:

(a) register all samples to the mean using ICP(b) calculate a new mean as the average of all

registered samples

End points of open curves were not explicitlymatched. An initial rough scaling was performed on allsamples by normalizing the maximum diameter ofeach shape. This mitigates against a whole curve match-ing to only a small part of another. The new mean wascalculated as follows:

1. newMeanPts)fg2. meanPts)extract control points frommean shape3. for each meanPt in meanPts do:

(a) newMeanPt)xi;where x i)ClosestPointðsi;meanPtÞ, and si is the ith sample underconsideration.

(b) append newMeanPt to newMeanPts

4. newMean)a new spline with newMeanPts as thecontrol points.


Fig 6. Principal variations for BPL corresponding to thefirst to fourth principal eigenvectors of PCA. Shown arethe mean shape in black and the mean 63 SD from themean (blue and red) along the principal direction. A, PC1, PV 12%. B, PC 2, PV 12%. C, PC 3, PV 9.4%. D, PC4, PV 6.9%.

Fig 5. Principal variations for MAN corresponding to thefirst to fourth principal eigenvectors of PCA. Shown arethe mean shape in black and the mean 63 SD from themean (blue and red) along the principal direction. A, PC1, PV: 13%. B, PC 2, PV: 10%. C, PC 3, PV: 8.6%.D, PC 4, PV: 7%.

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After each part was registered, N equally spacedpoints were extracted from the mean and the closestpoints on each sample were extracted as correspondingpseudolandmarks. These pseudolandmarks were thenfixed along the spline for all subsequent analyses. Thatis, they assumed the role of landmarks in the next stageof the analysis. A straightforward generalized Procrustesanalysis across the whole set of shapes was then per-formed using the frozen pseudolandmarks from theindividual shape registration. Each sample consisted ofN extracted landmarks from each part, resulting in a totalof 8N landmarks per sample. The resulting shape vari-ables were projected onto Kendall’s tangent space forstatistical analysis.12 To test that bias was not introducedby the number of extracted pseudolandmarks, weproduced results with N 5 31, 41, 51 and 61.

Statistical analysis

Three analyses were performed.

1. Principal component analysis for individual parts.2. Two-block PLS between parts with common

superimposition (ie, with shape, relative size, andposition). Klingenberg23 called this the “simulta-neous-fit” approach.


3. Two-block PLS between parts with separate super-imposition (ie, with shape only). Klingenberg23

called this the “separate-subsets” approach.

For individual part analysis, principal componentswere used to measure the shape variation in each partacross the sample. For covariation analysis, a 2-blockPLS3,5,7,9,10,12,24 on each pair of parts was performedto establish the extent of covariation or integration ofparts in the facial skeleton. Mutliple PLS3 is also possible,but we have not performed that analysis because wewanted to focus on paired covariation. PLS analysiswas performed between each pair of shape variablesusing the singular value decomposition as outlined byBookstein et al.25 Significance values were estimated us-ing a permutation test with 10,000 permutations, andconfidence intervals were estimated using 200 bootstrapsamples. Two PLS analyses were performed: the firstmaintained the relative size and position of each part;the second factored out relative size and position andconsidered shape only.

RESULTS

For the 3 types of analysis, the number of pseudo-landmarks used had little effect. For individual variations,the direction of variation was in close agreement, taking


Table I. Correlation coefficents and their standard deviations for paired PLS for relative size, shape, and orientation

NMC MAN XDE MDE CVT BPL SPLCRB 0.61 (0.05) 0.67 (0.06) 0.61 (0.06) 0.65 (0.06) 0.80 (0.05) 0.70 (0.05) 0.66 (0.05)NMC 0.51 (0.06) 0.60 (0.05) 0.66 (0.06) 0.70 (0.05) 0.60 (0.06) 0.81 (0.05)MAN 0.59 (0.06) 0.57 (0.07) 0.74 (0.05) 0.72 (0.05) 0.79 (0.05)XDE 0.70 (0.05) 0.70 (0.07) 0.65 (0.05) 0.59 (0.05)MDE 0.74 (0.05) 0.63 (0.05) 0.69 (0.06)CVT 0.63 (0.06) 0.62 (0.07)BPL 0.80 (0.06)

crb

cvt

man

cvt

Fig 7. Visualizations of singular warps for relative size, position, and shape. The level of covariation isshown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). Thepattern of covariation is shown with mean shapes andmean63 SD along the singular warp vectors.A,Level of covariation for relative size, position, and shape of CVT versus CRB. B, Pattern of covariationfor relative size, position, and shape (mean and direction). C, Level of covariation for relative size,position, and shape of MAN versus CVT. D, Pattern of covariation for relative size, position, and shape(mean and direction).

McCane and Kean e19

into account the change in number of pseudolandmarks.For covariation experiments, the correlation, significance,and direction of variation were all in close agreement.Because the results were largely independent of the num-ber of pseudolandmarks, all following results are reportedby using 61 pseudolandmarks because of the higherresolution of visualization.


To estimate variation within individual parts, princi-pal components analysis was used on the tangent spacecoordinates of the 61 pseudolandmarks extracted fromeach part. Figure 2 shows the total variation as measuredby the sum of the eigenvalues of the principal compo-nent analysis for each part. This figure shows CRBto be the most variable part followed by SPL and


xde

cvt

mde

cvt

Fig 8. Visualizations of singular warps for relative size, position, and shape. The level of covariation isshown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). Thepattern of covariation is shown with mean shapes andmean63 SD along the singular warp vectors.A,Level of covariation for relative size, position, and shape of CVT versus XDE. B, Pattern of covariationfor relative size, position, and shape (mean and direction). C, Level of covariation for relative size,position, and shape of CVT versus MDE. D, Pattern of covariation for relative size, position, and shape(mean and direction).

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MAN. A group comprising BPL, XDE, MDE, and NMCcomes next, with CVT showing the least variation.

Figures 3 to 6 show how CRB, SPL, MAN, and BPL varyacross the sample. Each figure shows the 4 principaldirections of variation: the mean shape is shown in black,and shapes63 SD from the mean in the direction of theeigenvector are shown in blue and red, respectively. Ineach case, the shape variations are exaggeratedcompared with shape variation in the sample to displaymore clearly the direction of the variation.

Table I shows the corresponding correlation coeffi-cients for relative size, position and shape covariationamong parts between the resultant first PLS latentvariables. All correlations greater than 0.7 were highlysignificant according to the permutation test to theP\0.0001 level.

Figures 7 to 12 show the relationships between thepaired latent variables with the strongest correlations;


the scatterplot indicates the strength of covariance,and the shape plot indicates the pattern of covariation.Only first singular warps are shown. Some explanation ofthe graphs is warranted. The axes represent the singularwarp scores. That is, a dot product was performedbetween the tangent space representation of the shapeand the first singular vector. Rather than display the rawdot product values that are not informative, the axeswere labeled with the corresponding shape of the partthat would have that dot-product score. The pair of over-layed shapes near the axis titles are the 2 extreme ends ofthe axes superimposed to give the reader a good clue asto the direction of shape change represented by that partic-ular singular warp axis.

Table II shows the correlation coefficients betweenthe resultant first PLS latent variables for shapecovariation only. Only the correlations between BPLand MAN, BPL and MDE, MDE and SPL, MDE and


xde

mde

man

bpl

Fig 9. Visualizations of singular warps for relative size, position, and shape. The level of covariation isshown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). Thepattern of covariation is shown with mean shapes and mean 63 SD along the singular warp vectors.A, Level of covariation for relative size, position, and shape of XDE versus MDE. B, Pattern of covari-ation for relative size, position, and shape (mean and direction).C, Level of covariation for relative size,position, and shape of BPL versus MAN. D, Pattern of covariation for relative size, position, and shape(mean and direction).

McCane and Kean e21

XDE, and SPL and BPL were significant to theP \0.05 level.

Figures 13 and 14 show graphically the relationshipbetween the paired latent variables with the strongestcorrelations. Only first singular warps are shown. Sincesize and relative position were factored out of theanalysis, the size and position of the shapes shown inthe pattern of covariation plots are not to scale andare approximate only.

DISCUSSION

There is a common argument against the use ofpseudolandmarks based on the question of biologicalhomology. How can we be sure that pseudolandmark23, say, on CRB of specimen 1 is homologous to pseudo-landmark 23 on CRB of specimen 2? The short answer isthat we cannot, but there are several factors in favor of


using pseudolandmarks. First, the structures themselvesare homologous. Unfortunately, there were no easilyidentifiable landmarks on such smooth shapes, andtherefore the only option for measuring shape changewith current techniques was to use a pseudolandmarkmethod. Figures 15 and 16 show the correspondencesbetween the pseudolandmarks on the samples and thepseudolandmarks on the mean shape for CRB andMAN. These 2 shapes are shown because they are themost difficult to match. Three example matches areshown ordered by increasing Procrustes distancebetween pseudolandmarks: the best match, a middlingmatch, and the worst match. As can be seen from thefigures, the correspondences matched well. In somespecific cases, it could be argued that the match wasnot optimal—eg, the tip of the posterior clinoid process.However, since the tip of the posterior clinoid processis roughly circular, it cannot be localized to a specific


bpl

crb

bpl

spl

Fig 10. Visualizations of singular warps for relative size, position, and shape. The level of covariation isshown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). Thepattern of covariation is shown with mean shapes and mean 63 SD along the singular warp vectors.A, Level of covariation for relative size, position, and shape of CRB versus BPL. B, Pattern of covari-ation for relative size, position, and shape (mean and direction).C, Level of covariation for relative size,position, and shape of SPL versus BPL. D, (d) Pattern of covariation for relative size, position, andshape (mean and direction).

e22 McCane and Kean

landmark in a mathematical sense (ie, it is ill defined).This is exactly the reason to use pseudolandmarkmethods; overall, the homology was maintained ina well-defined manner.

The techniques used in this study were based onleast-squares principles (principal components andPLS) and were therefore robust to Gaussian errors inthe data. This led to a weaker requirement for homologyin pseudolandmark methods—pseudolandmarks needonly be homologous in the sense of having Gaussian er-rors. Furthermore, since we only looked at the principaldirections of variation, measurement errors were smallcompared with changes in those directions except forextreme errors. Extreme cases can easily be checked byvisualizing the matches as in Figures 15 and 16. Noextreme matches were evident for the shapes we tested


here. We also tested the (univariate) normality of allpseudolandmark coordinates in the 61 pseudolandmarkexperiment using the Shapiro-Wilk test of normality.26

The null hypothesis of the Shapiro-Wilk test is that thedistribution of the sample is normal; therefore, smallvalues of the statistic indicate nonnormality. We foundthat 1.5% were not normal at the P\0.01 level. Sincewe would expect 1% of the tests to return a nonnormalresult if the data were normal, this is strong evidencefor the normality of the data.

For these reasons, we believe that pseudolandmarkmethods are a useful tool for measuring shape changesin the absence of well-defined landmarks.

In this study, CRB was a much more detailed expres-sion of cranial base shape than the conventional nasion-sella-basion (N-S-Ba) angle derived from 3 standard


nmc

spl

man

spl

Fig 11. Visualizations of singular warps for relative size, position, and shape. The level of covariation isshown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). Thepattern of covariation is shown with mean shapes and mean 63 SD along the singular warp vectors.A, Level of covariation for relative size, position, and shape of NMC versus SPL. B, Pattern of covari-ation for relative size, position, and shape (mean and direction).C, Level of covariation for relative size,position, and shape of MAN versus SPL. D, Pattern of covariation for relative size, position, and shape(mean and direction).

McCane and Kean e23

landmarks. This angle differs significantly among vari-ous racial groups27; within a group, its variability is un-likely to be significant as shown by Solow.1 In this study,however, the shape of CRB was derived from a muchricher outline. Moreover, it extended beyond the basicN-S-Ba angle to include details of the contours of thecranial base and the hypophyseal fossa housing thepituitary gland. The first 2 principal variations in CRBwere not related to the N-S-Ba angle but were focusedon the sella turcica. This is consistent with the range ofvariations found in the sella turcica by Axelsson et al,28

who found 6 different morphologic types. However, theboundaries between the different types are not clear,and, although not pursued in our study, this methodol-ogy might be of benefit in making such distinctionssystematic.


Similar comments can be made about SPL, MAN,and BPL. The SPL has, in the past, been studied exten-sively in relation to development in the context of ortho-dontics by using standard cephalometric analyses.1,29-32

Landmarks were used in those studies, and lengths andangles were calculated to describe the development ofthe profile. Information regarding the outline of theSPL obtained in this study, however, was typicallyqualitative, rather than quantitative. In our sample,most variation occurred first in the shape of the noseand the vertical extent of the SPL; second, in theprominence of the chin and the contour over thefrontal sinus; and third, in the fullness of the lips.Because the subjects in the sample were still in theclosing stages of adolescent development, expressionof the external size and shape of the nose was yet to


crb

man

nmc

man

Fig 12. Visualizations of singular warps for relative size, position, and shape. The level of covariation isshown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). Thepattern of covariation is shown with mean shapes and mean 63 SD along the singular warp vectors.A, Level of covariation for relative size, position, and shape of CRB versus MAN. B, Pattern of covari-ation for relative size, position, and shape (mean and direction).C, Level of covariation for relative size,position, and shape of NMC versus MAN. D, Pattern of covariation for relative size, position and shape(mean and direction).

Table II. Correlation coefficients and their standard deviations for paired PLS shape covariations for shape only

NMC MAN XDE MDE CVT BPL SPLCRB 0.36 (0.07) 0.41 (0.07) 0.42 (0.07) 0.38 (0.07) 0.36 (0.06) 0.45 (0.06) 0.40 (0.06)NMC 0.39 (0.06) 0.49 (0.07) 0.49 (0.05) 0.33 (0.06) 0.43 (0.08) 0.37 (0.08)MAN 0.48 (0.07) 0.41 (0.06) 0.35 (0.07) 0.53 (0.06) 0.36 (0.07)XDE 0.55 (0.08) 0.45 (0.05) 0.46 (0.07) 0.39 (0.07)MDE 0.41 (0.06) 0.65 (0.07) 0.54 (0.06)CVT 0.41 (0.07) 0.33 (0.06)BPL 0.69 (0.05)

e24 McCane and Kean

be completed.33 Some variation was caused by lips ineither a touching or nontouching state; this was notstandard across the sample, although the principalvariations noted here seemed largely independent ofthis lack of standardization.

The first principal components of variation in MANoccurred in the length of the corpus, the height of the


ramus, the ramus-corpus angle, and the relative positionof the coronoid process. These results are consistent withdevelopment processes of MAN and are likely due todifferent development stages of the subjects. Duringadolescence, the corpus and ramus lengthen, the ramusbecomes more upright, and there is a subsequentanterior rotation of the coronoid process.2


man

bpl

mde

bpl

Fig 13. Visualizations of singular warps for shape only. The level of covariation is shown as a scatter-plot of singular warp scores (cosine of the angle between singular warp vectors). The pattern ofcovariation is shown with mean shapes and mean 63 SD along the singular warp vectors. Relativeposition is only approximate in these plots, and relative size is not to scale; it is the shape variationthat is important. A, Level of covariation for shape only of BPL versus MAN. B, Pattern of covariationfor shape only (mean and direction). C, Level of covariation for shape only of BPL versus MDE.D, Pattern of covariation for shape only (mean and direction).

McCane and Kean e25

The foregoing shapes were more complex relative tothe remaining group of BPL, XDE, MDE, NMC, and CVTand could be expected to show greater variability thanthe latter. XDE and MDE, shapes representing the max-illary and mandibular dentitions, and NMC, representingthe nasomaxillary complex, are not quite complete atthis stage, whereas CVT has reached its essentially adultsize and form.

One of the most interesting results of this studyinvolved the CVT. We found that the anteriorly rotatedcervical column (cervical vertebrae in extension) corre-lated most strongly with shorter mandibular body andramus, smaller XDE, and smaller MDE (Figs 7 and 8) inagreement with Solow and Sandham34 and Solowet al.35 Although we did not measure angles per se,Figure 7 shows that an anteriorly rotated cervical column


is correlated with a smaller CRB angle; this did not agreewith the findings of Solow and Tallgren,36 who reporteda significant positive correlation between the anglemade by the nasion-sella line and the cervical vertebraetangent (NSL/CVT, defined as the posterior tangent tothe odontoid process through the most posterior-inferior point on the corpus of the fourth cervical verte-bra) and the N-S-Ba. However, in the case of Solow andTallgren, the correlation coefficient was relatively weak(0.21) at the P \0.05 level, and, given the number ofcorrelations tested in that study, some significantcorrelations were likely produced by chance. Moreover,this disagreement in results strengthens the case forthe sort of analysis we report in this article. Ratherthan testing for many correlations from derived variables(angles and distances), we sought to find the principal


mde

spl

bpl

spl

Fig 14. Visualizations of singular warps for shape only. The level of covariation is shown as a scatter-plot of singular warp scores (cosine of the angle between singular warp vectors). The pattern ofcovariation is shown with mean shapes and mean 63 SD along the singular warp vectors. Relativeposition is only approximate in these plots, and relative size is not to scale; it is the shape variationthat is important. A, Level of covariation for shape only of SPL versus MDE. B, Pattern of covariationfor shape only (mean and direction). C, Level of covariation for shape only of BPL versus SPL.D, Pattern of covariation for shape only (mean and direction).

e26 McCane and Kean

correlations in the fundamental shape variables. Never-theless, this study reinforces the results of Solow andSanham,34 who showed the importance of the relativeposition and orientation of CVT to craniofacial develop-ment. Similarly, Karlsen37 showed that the position ofgonion in the face is determined at an early age andconcluded that there is a mutual relationship betweenincremental growth of the upper cervical spine, espe-cially the lower face. We found that not only is CVT cor-related with the relative size and position of craniofacialshapes, but also it has stronger correlations than be-tween any other pairs of shapes, indicating that cervicalposture might be crucially important in facial morphol-ogy (in this study, only correlation and not causationwas established).

The counterpart principle of Enlow et al38 and Bhatand Enlow39 established 5 prominent counterparts in


the facial skeleton: middle cranial fossa, ramus orienta-tion, posterior maxillary vertical height, maxillary andmandibular skeletal arch lengths, and ramus width.Bhat and Enlow reported that an anteriorly rotated mid-dle cranial fossa produces a mandible rotated downwardand subsequently a mandibular retrusive effect. Incontrast, we found that the strongest covariation hadan anteriorly rotated CRB covarying with a longer corpusand a larger ramus-corpus angle. However, the secondstrongest covariation was also significant (P 5 0.02),but not as strong (r 5 0.49), in which the posterior ofthe CRB rotated anteriorly was correlated with a smallermandible. No other covariations were significant. Notsurprisingly, we also found that a more horizontalramus was correlated with a more protrusive mandible(Figs 9 and 11); however, as noted above, themandible itself had significant variations not related


Fig 15. The correspondences between the mean shape(blue) and samples (red) for CRB. Corresponding pseu-dolandmarks are indicated by the grey lines joining the2 curves. Matches are ordered according to Procrustesdistance. A, Best match; B, middling match; C, worstmatch.

McCane and Kean e27

to this effect. For the NMC, we found that a shortmaxillary arch length was highly correlated witha generally smaller mandible (both corpus length andramus height), but we did not find that the posteriormaxillary vertical height was the most significantcovariation with the mandible. Similarly, we notedthat a narrow ramus and protrusive mandible were notthe most significant covariations between MAN andBPL or MAN and SPL. However, as seen in Figure 5,a narrow ramus tended to be correlated with a largerramus-corpus angle and a retrusive mandible, althoughthis was not measured in relation to the pharyngealspace as in Bhat and Enlow.39

Figures 10, A and 10, B show a correlation betweenbasicranial flexure and head form type, with more openCRB correlated with a longer BPL and a shorter antero-posterior facial depth with a dolichocephalic form.Conversely, a more closed CRB is correlated witha more brachycephalic form; this agrees with the results


of Enlow and McNamara.40 Interestingly, more recentresults showed only weak evidence that basicranial flex-ure is correlated with head form type.8,13,41 Figures 11, Cand 11, D also show a correlation between a posteriorlyrotated mandible and brachycephalic head forms,again in close agreement with the results of Enlow andMcNamara.40

A note is required on the interpretation of theseresults and the sample. In particular, the sexes and theages of the subjects were unknown. Hence, some varia-tions found in the sample could be due to sexual dimor-phism or development. In terms of sexual dimorphism,Ingerslev and Solow42 showed that “the average sex de-termined shape differences in cranio-facial morphologyare few and small, and that the female skull is signifi-cantly smaller than the male.” Since we factored outabsolute size in all of our analyses, it is unlikely thatany variations were due to sex differences. Development,however, might play a role in these results. Duringadolescence, the ramus lengthens2 and becomes moreupright, and this might explain the second principalvariation of MAN.2

As might be expected, BPL and SPL were highly cor-related, as was also shown by Halazonetis.43 However,they diverge in their correlations with other parts. BPLshows a strong correlation with MAN, suggesting thatmandibular shape, size, and position are most highlyintegrated with the shape of the bony profile. In sharpcontrast, SPL appears to be highly integrated withNMC and MAN. High correlation with MAN was sharedwith BPL and could be expected to be, because theform, size, and position of the bony chin would bemost likely to influence the shape of the overlying softtissue and the bony profile overall. The high correlationbetween SPL and NMC is less easily explained. Possibly,the anterior or ventral limits of NMC influenced theshape of the superior terminus of SPL through variationsin the form of the underlying bony contours. It might bethat with such internal variations no compensation ismade in the overlying soft tissues, which retain theirusual thickness over bone, thus reflecting the high asso-ciation between BPL and SPL, but there is insufficientevidence to make an assured statement about thisrelationship.

Although they used landmarks, we found similarresults in terms of covariation of the CRB and MAN(r 5 0.66 6 0.07), compared with the results of Bastirand Rosas,7 who found reasonably strong correlationsbetween landmarks of the mandibular ramus and themiddle cranial fossa (r 5 0.61 6 0.06). Correlationscores were not directly comparable because of differentnumbers of coordinates in each study; rather, it isthe fact of high correlations in both studies that is


Fig 16. The correspondences between the mean shape(blue) and samples (red) for MAN. Corresponding pseu-dolandmarks are indicated by the grey lines joining the2 curves. Matches are ordered according to Procrustesdistance. A, Best match; B, middling match; C, worstmatch.

e28 McCane and Kean


interesting. Although not explicitly stated, Bastir andRosas implied that a single Procrustes fit was performedon the landmarks, making that study comparable withour relative size, position, and orientation results. Asthey noted, this provides further evidence for the coun-terpart principle of Enlow and Hans2 that posits that theramus is the counterpart of the middle cranial fossa, andthat both are counterparts of the pharyngeal space.However, we noted that there are many stronger covari-ations in the facial skeleton than this one, according toour results, and more work is required to explain thecovariations in functional terms.

Bastir and Rosas8 found little support for shape, andonly integration of the midline basicranium and theface, and reported correlation scores of r 5 0.37 andP 5 0.09. Again, we found similar results as shown inTable II, with correlations of moderate strength but statis-tically insignificant (P 5 0.57, 0.48, 0.49, and 0.74 forNMC, MAN, XDE, and MDE, respectively). Again, themethodology of Bastir and Rosas8 was similar to theshape-only study used here, with separate Procrustesfits for each module. Unlike their study, which foundmoderate correlations between the lateral basicraniumand the face, we did not include the lateral basicraniumin this study. This would be a useful area for furtherinvestigation.

Aside from the CVT, the most highly covarying partwas the SPL,which covaried stronglywith 2major internalstructures in the upper facial skeleton and mandible, notto mention the underlying BPL. This was an unexpectedfinding, drawing attention to a feature of the face usuallyoverlooked in conventional cephalometric analyses. It iscommonly regarded as an external feature somewhatindependent of internal structures and, not beinga bony or dental structure, beyond the direct influenceof treatment. Changes that occur in the SPL are attributedto changes in deeper structures, albeit rather indirectly.

Because the sample studied here comprised a groupof adolescents without overt malocclusion, the covari-ances of dentition and face were low, but this wouldlikely not apply in a group of subjects with majormalocclusions.

When relative size and position were factored out andthe covariation between shape only was considered, theresults become much less interesting. The only signifi-cant covariations found (Table II) were between partsfor which subparts are physically adjacent (BPL andMAN, BPL and MDE, MDE and XDE, MDE and SPL,SPL and BPL). Furthermore, both the correlations andstatistical significances were weak compared with the re-sults for relative size and position. It appeared that therewas weak (if any) covariation between the shape of theparts in the facial skeleton. Rather, it was the relative


McCane and Kean e29

size and position (including relative orientation) thatappeared to form the basis of the counterpart principle.However, as noted by Klingenberg23 one expects weakercorrelations between shape only comparisons (separatesubsets) than in comparisons involving shape, relativesize, and position (simultaneous fit). We did not followthe methodology of Klingenberg, but believe that theweak statistical significance for covariation of shape-only provides good evidence that each part specified inthis study is an integratedmodule. (Actually, this is a nec-essary but not sufficient condition. To be fully confident,we would also need to test that each part does not con-sist of multiple modules.)

CONCLUSIONS

Studies in which measurements were made on wholeskulls or 3-dimensional images extend the exploration ofcorrelations among parts not possible in the lateral 2Dimages available for measurement in this investigation.Yet, despite the latter limitations, it is still possible to elicitinformation about covariation among parts of use clini-cally using images usually available. This study opens upthe possibility of investigating more productively than inthe past the form and architecture of the face and thedentition in the large archive of lateral head imagesobtained over many decades for the geometric study offacial growth and form, and response of growing childrenand adolescents to orthodontic treatment andmaxillofa-cial surgery. The intention is to extend this methodologyto test its utility diagnostically.

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Integration of parts in the facial skeleton and cervical vertebrae

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