ONLINE ONLY

Integration of parts in the facial skeleton and

cervical vertebrae
Brendan McCanea and Martin R. Keanb
Dunedin, New Zealand

Introduction: The purpose of this study was to undertake an exploratory analysis of the relationship among
parts in the facial skeleton and cervical vertebrae and their integration as 2-dimensional shapes by determining
their individual variations and covariations. The study was motivated by considerations applicable to clinical
orthodontics and maxillofacial surgery, in which such relationships bear directly on pretreatment analysis and
assessment of posttreatment outcome. Methods: Lateral radiographs of 61 adolescents of both sexes without
major malocclusions were digitized and marked up by using continuous outline spline curves for 8 defined parts
in the facial skeleton, including the cervical vertebrae. Individual part variation was analyzed by using principal
components analysis, and paired part covariation was analyzed by using 2-block partial least squares analysis in
2 modes: 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 mandibular
dentition (r 5 0.74); the maxillary dentition and mandibular dentition were highly correlated (r 5 0.70); the
mandible was highly correlated with the bony profile (r 5 0.72), soft-tissue profile (r 5 0.79), and, to a lesser
extent, the cranial base (r 5 0.67); the bony profile was highly correlated with the cranial base (r 5 0.70) and
soft-tissue profile (r 5 0.80); the soft-tissue profile was highly correlated with the nasomaxillary dentition (r 5
0.81). Covariation of shape only was much weaker with significant covariations found between bony profile
and mandible (r 5 0.53), bony profile and mandibular dentition (r 5 0.65), mandibular dentition and
soft-tissue profile (r 5 0.54), mandibular dentition and maxillary dentition (r 5 0.55), and bony profile and
soft-tissue profile (r 5 0.69). Conclusions: We found that integration of the shape of parts in the facial skeleton
and cervical vertebrae is weak; it is the relative size, position, and orientation of parts that form the strongest
correlations. (Am J Orthod Dentofacial Orthop 2011;139:e13-e30)

A
standardized lateral radiographic image of the surgery, in which such relationships bear directly on pre-
craniofacial skeleton shows a series of discrete, treatment analysis and assessment of posttreatment
though related, parts comprising skeletal struc- outcome. In particular, the ultimate goal was for this
tures and spaces, and the maxillary and mandibular type of analysis to become a useful addition and in
dentitions. The purpose of this study was to undertake some cases a replacement for the current uses of
an exploratory analysis of the relationship among these standard cephalometric software.
parts and their integration as 2-dimensional (2D) shapes The pursuit of associations between and among parts
by determining their individual variations and covaria- in the facial skeleton is not a new endeavour. Using stan-
tions. The study was motivated by considerations dard analyses of lateral headfilms, Solow1 sought to dis-
applicable to clinical orthodontics and maxillofacial tinguish between the topographic and nontopographic
associations of 2D variables. The former arose from using
common reference points or landmarks in measuring each
From the University of Otago, Dunedin, New Zealand.
a
Associate Professor, Department of Computer Science. variable. Nontopographic associations, in which common
b
Emeritus professor, School of Dentistry. (Deceased.) reference points or landmarks were not used, were con-
The authors report no commercial, proprietary, or financial interest in the sidered to have biologic significance, although caution
products or companies described in this article.
Reprint requests to: Brendan McCane, Department of Computer Science, University was urged to exclude circumstances in which reference
of Otago, PO Box 56, Dunedin, New Zealand; e-mail, mccane@cs.otago.ac.nz. points might be placed on a common reference structure.
Submitted, August 2009; revised and accepted, June 2010. More recently, Enlow and Hans2 introduced the “counter-
0889-5406/$36.00
Copyright Ó 2011 by the American Association of Orthodontists. part principle” in postnatal facial development studied in
doi:10.1016/j.ajodo.2010.06.016 sequential lateral head radiographs. This principle states
e13
e14 McCane and Kean

that “the growth of any given facial or cranial part relates 5. Midline cranial base, middle cranial fossa, and man-
specifically to other structural and geometric ‘counter- dibular ramus,7 where the midline cranial base and
parts’ in the face and cranium.” Enlow and Hans stated mandibular ramus, and the middle cranial fossa
that, if each regional part and its counterpart enlarge and mandibular ramus showed significant integra-
(grow) to the same extent, balance between the parts tion, but not the midline cranial base and middle
and within the face will be maintained. cranial fossa.
The methodology in this study relied on techniques 6. Mandible, nasomaxillary complex, and neurocra-
used in modern morphometrics, albeit with some differ- nium.2
ences. The principal object of study was a clinically or 7. cranial base, neurocranium, and face.13
diagnostically meaningful part expressed as a shape.
This study was intended to extend understanding of
We sought to visualize and measure the patterns of
the form and architecture of the facial skeleton and in-
covariation among parts in a subject and in a sample
vesting soft tissues viewed laterally in 2 dimensions,
of subjects with balanced facial forms and similar
with particular emphasis on parts or modules of clinical
maturity, and to determine the variability of such mea-
interest to orthodontists and maxillofacial surgeons.
surements in the sample. Morphometrically, the parts
This study differs from previous studies in several
could be regarded as modules being a set of shapes in-
respects: all parts or modules are defined by using
dependent, though integrated, through interaction and
free-form curves (cubic splines) because we considered
covariance with other structures during facial develop-
that landmarks lack representational power of the par-
ment. In this study, a part is a discrete structure, and
ticular parts in question; we investigated the integration
landmarks (or pseudolandmarks) are defined in relation
of parts in the craniofacial skeleton with the intention of
to the part. Each part is represented by a cubic spline
gaining a more complete understanding of craniofacial
that provides a much richer definition of the part than
form, especially in relation to covariation of the parts;
a finite number of landmarks, although much of the
we included cervical vertebrae in our analysis to test their
analysis is not crucially dependent on the exact form
relationship to facial form. The ultimate goal was to find
of the continuous representation of the shape.
the means to study efficiently and informatively the
The purpose of modularity and integration studies
huge repository of lateral headfilms already available
was to investigate the covariation of shape and size dur-
from earlier growth and clinical studies.
ing evolution3-6 or development,3,7-13 or both.14 Most
such studies used the method of partial least squares
(PLS), also known in the morphometrics literature as sin- MATERIAL AND METHODS
gular warp analysis.15 The level of integration between The sample comprised standard lateral radiographs
hypothesized shape modules is usually determined by of a mixed-sex group of North American white adoles-
the correlations of corresponding singular warp scores cents with normal skeletal patterns (n 5 61) obtained
extracted from landmarks,7,8,10,12 although some from a university orthodontic clinic. Patients ranged in
studies also use pseudolandmarks,3,5,9 whereas others dental maturation from the late mixed dentition to the
used more traditional statistical tests on distances and early permanent dentition, with second molars emerging
angles measured from landmarks.13,16,17 Typical or emerged. Ages and sexes were not given. The head-
modules and the corresponding integrations include films were high-quality digitally rendered x-ray images.
the following. All headfilms were taken in natural head position with
the lips in repose.18
1. The face and neurocranium, which are highly inte-
The method relied on the identification and marking
grated across hominid species.5
up of a series of major parts or modules of the radio-
2. The cranial vault, cranial base, and face, where the
graphic images of bony structures and the dentition in
cranial vault and face are highly integrated in both
2D lateral views. The parts were verified by studying
development and evolution, more so than the vault
lateral x-ray images of a skull on which metallic markers
and base, or the base and face.3
had been placed to show the most reliable identification
3. Integration of the neurobasicranial complex, ethno-
of outlines without having recourse to standard land-
maxillary complex, mandible, midline cranial base,
marks to determine the outlines. The shapes were the
lateral cranial floor, and neurocranium.9
following.
4. Midline cranial base, lateral cranial base, and face,8
where the lateral, but not the midline, cranial base 1. Cranial base (CRB): CRB shape follows the outline of
and face are highly integrated in contrast to the the superior contour of the cranial base extending
results reported by Bookstein et al.3 from the superior posterior of the frontal sinus to

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McCane and Kean e15

the superior outline of the pituitary fossa, to the around the dorsal shadows of the vertebrae, and
dorsal outline of the superior surface of the basicra- across inferiorly to pick up the ventral outlines of
nium to the tip of the clivus. the vertebrae. This outline is continued superiorly
2. Nasomaxillary complex (NMC): NMC shape follows to close the shape at the point where it began.
3 sides of the composite of the outline of the nasal 7. Bony profile (BPL): Starting from a point on the
fossa in the lateral view and the maxillary structure shadow of the most forward or ventral outline of
that can be defined reliably where the bony con- the frontal bone over the center of the developing
tours are visible radiographically in the lateral frontal sinus, the outline is continued inferiorly
view. Beginning at its superior and internal termi- through nasion and along the ventral shadows of
nation, the shape follows down from the upper the nasal bone and the maxilla, through the tip of
limit of the pterygomaxillary fossa through the the anterior nasal spine, and around the mandibular
fossa to the posterior limit of the bony palate. It ex- symphysis to its most inferior point.
tends from there to an anterior limit approximating 8. Soft-tissue profile (SPL): From a starting point on
the anterior nasal spine and from there superiorly the surface of the soft-tissue shadow over the
to a point on the inner margin of the image of developing frontal sinus, an outline is followed
the nasal bone following the outline of the bony inferiorly to show the shape of the profile of the
profile. nose and lips, and the soft tissue over the chin be-
3. Mandible (MAN): Mandibular shape is delineated by fore terminating at a point immediately below the
starting at the bony shadow immediately dorsal to most inferior shadow of the bony symphysis. The
the last molar emerged, usually the second molar. lips are in a relaxed state. If the lips touch, then
The shape is outlined by following the ramus poste- the outline of the upper lip is followed to the
riorly and then superiorly along the shadow of the touching point, and then on to the lower lip. If
coronoid process, into the mandibular notch, up the lips do not touch, the upper lip is followed
and around the condylar head and down along to the foremost incisor and then down to the
the posterior shadow of the ramus to the inferior lower lip.
border of the mandible, and around the external
The method used in this study requires the marking
surface of the symphysis to the junction of bone
up of the parts defined above on an on-screen x-ray im-
and the foremost incisor.
age by using software developed especially for the
4. Maxillary dentition (XDE): The shape of the XDE is
purpose. A screen shot of the prototype software is
represented by an outline beginning at the most
shown in Figure 1. When each image was loaded, its
dorsal point on the shadow of the permanent
magnification, brightness, and contrast were adjusted
second molar (emerged or not) and following an
to obtain the most detailed rendition of a shape. Rather
outline anteriorly and inferiorly that includes the
than basing a part on a series of landmarks, continuous
main features of the crowns of the first molar,
outline curves were used to represent shape in the form
premolars, canine, and incisors in profile. From
of Catmull-Rom splines.19 Catmull-Rom splines are cu-
here, the shape includes the labial outline of the
bic interpolating splines that overcome many limitations
foremost incisor to the tip of the root and then
of using landmarks. They have the advantages of ease in
follows a line along the root apices until reaching
specification, validity at all scales, and continuity. They
the initial point of the shape.
can pass through specific landmarks if required and be
5. Mandibular dentition (MDE): As in the case of the
applied to any continuous shape. Because they are con-
XDE, this shape is drawn to show the outline of
tinuous, they cannot easily be used to represent a discon-
the MDE, including the second molar, in general
tinuous shape such as a square. Consistent with most
size and form. Shape delineation is most easily
interpolating spline techniques, the representation is
accomplished by following a similar method to
not unique. That is, the same continuous curve can be
that for XDE, starting at a consistent point such as
closely approximated by many differently parameterized
the most dorsal point on the crown of the perma-
Catmull-Rom splines.
nent second molar and proceeding to complete
A Catmull-Rom spline is defined by a set of control
the outline of the MDE.
points:
6. Cervical vertebrae (CVT): Beginning at the superior
margin of the odontoid process of the second pk 5ðxk ; yk Þ; k 5 0; 1; 2; .; n  1:
CVT, the shapes of vertebrae C3 to C5 are followed A point on the spline is specified by the parameter
as a continuous outline from this point inferiorly, 0 # u\1:

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e16 McCane and Kean

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

PðuÞ 5 pk1 ð  su3 1 2su2  suÞ 1 pk ½ð2  sÞu3 1

500

ðs  3Þu2 1 1 1 pk11 ½ðs  2Þu3 1

ð3  2sÞu2 1 su 1 pk12 ðsu3  su2 Þ
400

where s 5 0.5, 0 # k \ n, and pk are the control

points. Any number of control points can be used
to specify the craniofacial parts, but we found by
300

trial and error that 50 to 60 control points per shape

of each part are sufficient to represent the variability
in shape.
200

To specify the shape, the user places control points

along visible margins of the part in the x-ray image.
Control points are not landmarks (or at least not neces-
100

sarily landmarks) or pseudolandmarks but, rather, spec-

ify points through which the curve must pass. Sliding
a control point along the curve often causes little change
0

in the curve itself, and, in this way, a source of error in crb spl man bpl xde mde nmc cvt
marking up images is reduced. Landmarks are subject
to 2D errors, whereas interpolating splines are only sub-
ject to 1-dimensional errors. Errors can arise when the Fig 2. The total variation of each part. Larger numbers
indicate greater variations in that part.
curve is moved off the underlying part perpendicular
to the part outline but do not arise when a control point
is moved along the curve of an underlying part. The method of shape registration used here differs
Each subject is represented as: from previous methods for 2 reasons: since we used
a spline curve shape representation, we did not have fixed
Si 5fCRBi ;NMCi ;MANi ;XDEi ;MDEi ;CVTi ;BPLi ;SPLi g; landmarks to perform registrations, and we assumed that
each part is a tightly integrated module. This led to a 2-
where CRBi 5fpik jpik 5ðxki ; yki Þ; k 5 0; 1; 2; .; ni  1g is step shape registration procedure: individual part registra-
the spline representation of CRB for subject i. The func- tion, followed by a global generalized Procrustes analysis.
tion CRBi ðuÞ; 0 # u\1 returns a 2D point along the The 2-step process was needed to obtain correspondences
curve of CRB for subject i. between pseudolandmarks on the same module.

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McCane and Kean e17

Fig 4. Principal variations for SPL corresponding to the

first to fourth principal eigenvectors of PCA. Shown are
the mean shape in black, and the mean 63 SD from the
mean (blue and red) along the principal direction. A, PC
1, PV: 14%. B, PC 2, PV: 8.6%. C, PC 3, PV 6.8%.
D, PC 4, PV: 5.2%.

of the number of pseudolandmarks used in the

subsequent analysis. For example, using the same
marked-up sample, one can extract different numbers
of pseudolandmarks depending on the analysis being
done, without having to mark up the original samples
again.
The ICP algorithm was applied as follows:
1. a sample is chosen as the initial mean shape
2. While the mean has not converged, do:
(a) register all samples to the mean using ICP
Fig 3. Principal variations for CRB corresponding to the
(b) calculate a new mean as the average of all
first to fourth principal eigenvectors of PCA. Shown are
registered samples
the mean shape in black and the mean 63 SD from the
mean (blue and red) along the principal direction. A, Prin-
cipal component (PC) 1, percentage of variation (PV): End points of open curves were not explicitly
16%. B, PC 2, PV: 13%. C, PC 3, PV: 11%. D, PC 4, matched. An initial rough scaling was performed on all
PV: 9%. Also included is the percentage of total variation samples by normalizing the maximum diameter of
of each component. each shape. This mitigates against a whole curve match-
ing to only a small part of another. The new mean was
calculated as follows:
Individual parts were registered by using a variant of
the iterative closest point (ICP) algorithm, an iterative 1. newMeanPts)fg
method for matching parametric shapes.20 The variant 2. meanPts)extract control points from mean shape
of ICP used follows the basic structure of the algorithm, 3. for each meanPt in meanPts do:
except that an explicit 2D registration procedure was
(a) newMeanPt)x i ; where x i)ClosestPoint
used that accounted for scale, rotation, and translation
ðsi ; meanPtÞ, and si is the ith sample under
as described by McCane et al.21 The method produced
consideration.
a similar result to the method of Procrustes superimpo-
(b) append newMeanPt to newMeanPts
sition of semilandmarks22 but, along with the spline rep-
resentation, has an advantage over the semilandmark 4. newMean)a new spline with newMeanPts as the
method: part specification and matching is independent control points.

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Fig 6. Principal variations for BPL corresponding to the

Fig 5. Principal variations for MAN corresponding to the first to fourth principal eigenvectors of PCA. Shown are
first to fourth principal eigenvectors of PCA. Shown are the mean shape in black and the mean 63 SD from the
the mean shape in black and the mean 63 SD from the mean (blue and red) along the principal direction. A, PC
mean (blue and red) along the principal direction. A, PC 1, PV 12%. B, PC 2, PV 12%. C, PC 3, PV 9.4%. D, PC
1, PV: 13%. B, PC 2, PV: 10%. C, PC 3, PV: 8.6%. 4, PV 6.9%.
D, PC 4, PV: 7%.

3. Two-block PLS between parts with separate super-

imposition (ie, with shape only). Klingenberg23
After each part was registered, N equally spaced
called this the “separate-subsets” approach.
points were extracted from the mean and the closest
points on each sample were extracted as corresponding For individual part analysis, principal components
pseudolandmarks. These pseudolandmarks were then were used to measure the shape variation in each part
fixed along the spline for all subsequent analyses. That across the sample. For covariation analysis, a 2-block
is, they assumed the role of landmarks in the next stage PLS3,5,7,9,10,12,24 on each pair of parts was performed
of the analysis. A straightforward generalized Procrustes to establish the extent of covariation or integration of
analysis across the whole set of shapes was then per- parts in the facial skeleton. Mutliple PLS3 is also possible,
formed using the frozen pseudolandmarks from the but we have not performed that analysis because we
individual shape registration. Each sample consisted of wanted to focus on paired covariation. PLS analysis
N extracted landmarks from each part, resulting in a total was performed between each pair of shape variables
of 8N landmarks per sample. The resulting shape vari- using the singular value decomposition as outlined by
ables were projected onto Kendall’s tangent space for Bookstein et al.25 Significance values were estimated us-
statistical analysis.12 To test that bias was not introduced ing a permutation test with 10,000 permutations, and
by the number of extracted pseudolandmarks, we confidence intervals were estimated using 200 bootstrap
produced results with N 5 31, 41, 51 and 61. samples. Two PLS analyses were performed: the first
maintained the relative size and position of each part;
Statistical analysis the second factored out relative size and position and
Three analyses were performed. considered shape only.

1. Principal component analysis for individual parts.

2. Two-block PLS between parts with common RESULTS
superimposition (ie, with shape, relative size, and For the 3 types of analysis, the number of pseudo-
position). Klingenberg23 called this the “simulta- landmarks used had little effect. For individual variations,
neous-fit” approach. the direction of variation was in close agreement, taking

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McCane and Kean e19

Table I. Correlation coefficents and their standard deviations for paired PLS for relative size, shape, and orientation
NMC MAN XDE MDE CVT BPL SPL
CRB 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)

cvt

crb

cvt

man

Fig 7. Visualizations of singular warps for relative size, position, and shape. The level of covariation is
shown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). The
pattern 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 CVT versus CRB. B, Pattern of covariation
for 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).

into account the change in number of pseudolandmarks. To estimate variation within individual parts, princi-
For covariation experiments, the correlation, significance, pal components analysis was used on the tangent space
and direction of variation were all in close agreement. coordinates of the 61 pseudolandmarks extracted from
Because the results were largely independent of the num- each part. Figure 2 shows the total variation as measured
ber of pseudolandmarks, all following results are reported by the sum of the eigenvalues of the principal compo-
by using 61 pseudolandmarks because of the higher nent analysis for each part. This figure shows CRB
resolution of visualization. to be the most variable part followed by SPL and

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e20 McCane and Kean

cvt

xde

cvt

mde

Fig 8. Visualizations of singular warps for relative size, position, and shape. The level of covariation is
shown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). The
pattern 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 CVT versus XDE. B, Pattern of covariation
for 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).

MAN. A group comprising BPL, XDE, MDE, and NMC the scatterplot indicates the strength of covariance,
comes next, with CVT showing the least variation. and the shape plot indicates the pattern of covariation.
Figures 3 to 6 show how CRB, SPL, MAN, and BPL vary Only first singular warps are shown. Some explanation of
across the sample. Each figure shows the 4 principal the graphs is warranted. The axes represent the singular
directions of variation: the mean shape is shown in black, warp scores. That is, a dot product was performed
and shapes 63 SD from the mean in the direction of the between the tangent space representation of the shape
eigenvector are shown in blue and red, respectively. In and the first singular vector. Rather than display the raw
each case, the shape variations are exaggerated dot product values that are not informative, the axes
compared with shape variation in the sample to display were labeled with the corresponding shape of the part
more clearly the direction of the variation. that would have that dot-product score. The pair of over-
Table I shows the corresponding correlation coeffi- layed shapes near the axis titles are the 2 extreme ends of
cients for relative size, position and shape covariation the axes superimposed to give the reader a good clue as
among parts between the resultant first PLS latent to the direction of shape change represented by that partic-
variables. All correlations greater than 0.7 were highly ular singular warp axis.
significant according to the permutation test to the Table II shows the correlation coefficients between
P \0.0001 level. the resultant first PLS latent variables for shape
Figures 7 to 12 show the relationships between the covariation only. Only the correlations between BPL
paired latent variables with the strongest correlations; and MAN, BPL and MDE, MDE and SPL, MDE and

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McCane and Kean e21

mde

xde

bpl

man

Fig 9. Visualizations of singular warps for relative size, position, and shape. The level of covariation is
shown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). The
pattern 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).

XDE, and SPL and BPL were significant to the using pseudolandmarks. First, the structures themselves
P \0.05 level. are homologous. Unfortunately, there were no easily
Figures 13 and 14 show graphically the relationship identifiable landmarks on such smooth shapes, and
between the paired latent variables with the strongest therefore the only option for measuring shape change
correlations. Only first singular warps are shown. Since with current techniques was to use a pseudolandmark
size and relative position were factored out of the method. Figures 15 and 16 show the correspondences
analysis, the size and position of the shapes shown in between the pseudolandmarks on the samples and the
the pattern of covariation plots are not to scale and pseudolandmarks on the mean shape for CRB and
are approximate only. MAN. These 2 shapes are shown because they are the
most difficult to match. Three example matches are
shown ordered by increasing Procrustes distance
DISCUSSION between pseudolandmarks: the best match, a middling
There is a common argument against the use of match, and the worst match. As can be seen from the
pseudolandmarks based on the question of biological figures, the correspondences matched well. In some
homology. How can we be sure that pseudolandmark specific cases, it could be argued that the match was
23, say, on CRB of specimen 1 is homologous to pseudo- not optimal—eg, the tip of the posterior clinoid process.
landmark 23 on CRB of specimen 2? The short answer is However, since the tip of the posterior clinoid process
that we cannot, but there are several factors in favor of is roughly circular, it cannot be localized to a specific

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bpl

crb

spl

bpl

Fig 10. Visualizations of singular warps for relative size, position, and shape. The level of covariation is
shown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). The
pattern 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, and
shape (mean and direction).

landmark in a mathematical sense (ie, it is ill defined). here. We also tested the (univariate) normality of all
This is exactly the reason to use pseudolandmark pseudolandmark coordinates in the 61 pseudolandmark
methods; overall, the homology was maintained in experiment using the Shapiro-Wilk test of normality.26
a well-defined manner. The null hypothesis of the Shapiro-Wilk test is that the
The techniques used in this study were based on distribution of the sample is normal; therefore, small
least-squares principles (principal components and values of the statistic indicate nonnormality. We found
PLS) and were therefore robust to Gaussian errors in that 1.5% were not normal at the P \0.01 level. Since
the data. This led to a weaker requirement for homology we would expect 1% of the tests to return a nonnormal
in pseudolandmark methods—pseudolandmarks need result if the data were normal, this is strong evidence
only be homologous in the sense of having Gaussian er- for the normality of the data.
rors. Furthermore, since we only looked at the principal For these reasons, we believe that pseudolandmark
directions of variation, measurement errors were small methods are a useful tool for measuring shape changes
compared with changes in those directions except for in the absence of well-defined landmarks.
extreme errors. Extreme cases can easily be checked by In this study, CRB was a much more detailed expres-
visualizing the matches as in Figures 15 and 16. No sion of cranial base shape than the conventional nasion-
extreme matches were evident for the shapes we tested sella-basion (N-S-Ba) angle derived from 3 standard

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McCane and Kean e23

spl

nmc

spl

man

Fig 11. Visualizations of singular warps for relative size, position, and shape. The level of covariation is
shown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). The
pattern 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).

landmarks. This angle differs significantly among vari- Similar comments can be made about SPL, MAN,
ous racial groups27; within a group, its variability is un- and BPL. The SPL has, in the past, been studied exten-
likely to be significant as shown by Solow.1 In this study, sively in relation to development in the context of ortho-
however, the shape of CRB was derived from a much dontics by using standard cephalometric analyses.1,29-32
richer outline. Moreover, it extended beyond the basic Landmarks were used in those studies, and lengths and
N-S-Ba angle to include details of the contours of the angles were calculated to describe the development of
cranial base and the hypophyseal fossa housing the the profile. Information regarding the outline of the
pituitary gland. The first 2 principal variations in CRB SPL obtained in this study, however, was typically
were not related to the N-S-Ba angle but were focused qualitative, rather than quantitative. In our sample,
on the sella turcica. This is consistent with the range of most variation occurred first in the shape of the nose
variations found in the sella turcica by Axelsson et al,28 and the vertical extent of the SPL; second, in the
who found 6 different morphologic types. However, the prominence of the chin and the contour over the
boundaries between the different types are not clear, frontal sinus; and third, in the fullness of the lips.
and, although not pursued in our study, this methodol- Because the subjects in the sample were still in the
ogy might be of benefit in making such distinctions closing stages of adolescent development, expression
systematic. of the external size and shape of the nose was yet to

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man

crb

man

nmc

Fig 12. Visualizations of singular warps for relative size, position, and shape. The level of covariation is
shown as a scatterplot of singular warp scores (cosine of the angle between singular warp vectors). The
pattern 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 SPL
CRB 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)

be completed.33 Some variation was caused by lips in ramus, the ramus-corpus angle, and the relative position
either a touching or nontouching state; this was not of the coronoid process. These results are consistent with
standard across the sample, although the principal development processes of MAN and are likely due to
variations noted here seemed largely independent of different development stages of the subjects. During
this lack of standardization. adolescence, the corpus and ramus lengthen, the ramus
The first principal components of variation in MAN becomes more upright, and there is a subsequent
occurred in the length of the corpus, the height of the anterior rotation of the coronoid process.2

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McCane and Kean e25

bpl

man

bpl

mde

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 of
covariation is shown with mean shapes and mean 63 SD along the singular warp vectors. Relative
position is only approximate in these plots, and relative size is not to scale; it is the shape variation
that is important. A, Level of covariation for shape only of BPL versus MAN. B, Pattern of covariation
for 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).

The foregoing shapes were more complex relative to is correlated with a smaller CRB angle; this did not agree
the remaining group of BPL, XDE, MDE, NMC, and CVT with the findings of Solow and Tallgren,36 who reported
and could be expected to show greater variability than a significant positive correlation between the angle
the latter. XDE and MDE, shapes representing the max- made by the nasion-sella line and the cervical vertebrae
illary and mandibular dentitions, and NMC, representing tangent (NSL/CVT, defined as the posterior tangent to
the nasomaxillary complex, are not quite complete at the odontoid process through the most posterior-
this stage, whereas CVT has reached its essentially adult inferior point on the corpus of the fourth cervical verte-
size and form. bra) and the N-S-Ba. However, in the case of Solow and
One of the most interesting results of this study Tallgren, the correlation coefficient was relatively weak
involved the CVT. We found that the anteriorly rotated (0.21) at the P \0.05 level, and, given the number of
cervical column (cervical vertebrae in extension) corre- correlations tested in that study, some significant
lated most strongly with shorter mandibular body and correlations were likely produced by chance. Moreover,
ramus, smaller XDE, and smaller MDE (Figs 7 and 8) in this disagreement in results strengthens the case for
agreement with Solow and Sandham34 and Solow the sort of analysis we report in this article. Rather
et al.35 Although we did not measure angles per se, than testing for many correlations from derived variables
Figure 7 shows that an anteriorly rotated cervical column (angles and distances), we sought to find the principal

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e26 McCane and Kean

spl

mde

spl

bpl

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 of
covariation is shown with mean shapes and mean 63 SD along the singular warp vectors. Relative
position is only approximate in these plots, and relative size is not to scale; it is the shape variation
that is important. A, Level of covariation for shape only of SPL versus MDE. B, Pattern of covariation
for 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).

correlations in the fundamental shape variables. Never- the facial skeleton: middle cranial fossa, ramus orienta-
theless, this study reinforces the results of Solow and tion, posterior maxillary vertical height, maxillary and
Sanham,34 who showed the importance of the relative mandibular skeletal arch lengths, and ramus width.
position and orientation of CVT to craniofacial develop- Bhat and Enlow reported that an anteriorly rotated mid-
ment. Similarly, Karlsen37 showed that the position of dle cranial fossa produces a mandible rotated downward
gonion in the face is determined at an early age and and subsequently a mandibular retrusive effect. In
concluded that there is a mutual relationship between contrast, we found that the strongest covariation had
incremental growth of the upper cervical spine, espe- an anteriorly rotated CRB covarying with a longer corpus
cially the lower face. We found that not only is CVT cor- and a larger ramus-corpus angle. However, the second
related with the relative size and position of craniofacial strongest covariation was also significant (P 5 0.02),
shapes, but also it has stronger correlations than be- but not as strong (r 5 0.49), in which the posterior of
tween any other pairs of shapes, indicating that cervical the CRB rotated anteriorly was correlated with a smaller
posture might be crucially important in facial morphol- mandible. No other covariations were significant. Not
ogy (in this study, only correlation and not causation surprisingly, we also found that a more horizontal
was established). ramus was correlated with a more protrusive mandible
The counterpart principle of Enlow et al38 and Bhat (Figs 9 and 11); however, as noted above, the
and Enlow39 established 5 prominent counterparts in mandible itself had significant variations not related

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McCane and Kean e27

of Enlow and McNamara.40 Interestingly, more recent

results showed only weak evidence that basicranial flex-
ure is correlated with head form type.8,13,41 Figures 11, C
and 11, D also show a correlation between a posteriorly
rotated mandible and brachycephalic head forms,
again in close agreement with the results of Enlow and
McNamara.40
A note is required on the interpretation of these
results and the sample. In particular, the sexes and the
ages 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 morphology
are few and small, and that the female skull is signifi-
cantly smaller than the male.” Since we factored out
absolute size in all of our analyses, it is unlikely that
any variations were due to sex differences. Development,
however, might play a role in these results. During
adolescence, the ramus lengthens2 and becomes more
upright, and this might explain the second principal
variation 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. BPL
shows a strong correlation with MAN, suggesting that
mandibular shape, size, and position are most highly
Fig 15. The correspondences between the mean shape
(blue) and samples (red) for CRB. Corresponding pseu-
integrated with the shape of the bony profile. In sharp
dolandmarks are indicated by the grey lines joining the contrast, SPL appears to be highly integrated with
2 curves. Matches are ordered according to Procrustes NMC and MAN. High correlation with MAN was shared
distance. A, Best match; B, middling match; C, worst with BPL and could be expected to be, because the
match. form, size, and position of the bony chin would be
most likely to influence the shape of the overlying soft
tissue and the bony profile overall. The high correlation
to this effect. For the NMC, we found that a short between SPL and NMC is less easily explained. Possibly,
maxillary arch length was highly correlated with the anterior or ventral limits of NMC influenced the
a generally smaller mandible (both corpus length and shape of the superior terminus of SPL through variations
ramus height), but we did not find that the posterior in the form of the underlying bony contours. It might be
maxillary vertical height was the most significant that with such internal variations no compensation is
covariation with the mandible. Similarly, we noted made in the overlying soft tissues, which retain their
that a narrow ramus and protrusive mandible were not usual thickness over bone, thus reflecting the high asso-
the most significant covariations between MAN and ciation between BPL and SPL, but there is insufficient
BPL or MAN and SPL. However, as seen in Figure 5, evidence to make an assured statement about this
a narrow ramus tended to be correlated with a larger relationship.
ramus-corpus angle and a retrusive mandible, although Although they used landmarks, we found similar
this was not measured in relation to the pharyngeal results in terms of covariation of the CRB and MAN
space as in Bhat and Enlow.39 (r 5 0.66 6 0.07), compared with the results of Bastir
Figures 10, A and 10, B show a correlation between and Rosas,7 who found reasonably strong correlations
basicranial flexure and head form type, with more open between landmarks of the mandibular ramus and the
CRB correlated with a longer BPL and a shorter antero- middle cranial fossa (r 5 0.61 6 0.06). Correlation
posterior facial depth with a dolichocephalic form. scores were not directly comparable because of different
Conversely, a more closed CRB is correlated with numbers of coordinates in each study; rather, it is
a more brachycephalic form; this agrees with the results the fact of high correlations in both studies that is

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e28 McCane and Kean

interesting. Although not explicitly stated, Bastir and

Rosas implied that a single Procrustes fit was performed
on the landmarks, making that study comparable with
our relative size, position, and orientation results. As
they noted, this provides further evidence for the coun-
terpart principle of Enlow and Hans2 that posits that the
ramus is the counterpart of the middle cranial fossa, and
that 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 to
our results, and more work is required to explain the
covariations in functional terms.
Bastir and Rosas8 found little support for shape, and
only integration of the midline basicranium and the
face, and reported correlation scores of r 5 0.37 and
P 5 0.09. Again, we found similar results as shown in
Table II, with correlations of moderate strength but statis-
tically insignificant (P 5 0.57, 0.48, 0.49, and 0.74 for
NMC, MAN, XDE, and MDE, respectively). Again, the
methodology of Bastir and Rosas8 was similar to the
shape-only study used here, with separate Procrustes
fits for each module. Unlike their study, which found
moderate correlations between the lateral basicranium
and the face, we did not include the lateral basicranium
in this study. This would be a useful area for further
investigation.
Aside from the CVT, the most highly covarying part
was the SPL, which covaried strongly with 2 major internal
structures in the upper facial skeleton and mandible, not
to mention the underlying BPL. This was an unexpected
finding, drawing attention to a feature of the face usually
overlooked in conventional cephalometric analyses. It is
commonly regarded as an external feature somewhat
independent of internal structures and, not being
a bony or dental structure, beyond the direct influence
of treatment. Changes that occur in the SPL are attributed
to changes in deeper structures, albeit rather indirectly.
Because the sample studied here comprised a group
of adolescents without overt malocclusion, the covari-
ances of dentition and face were low, but this would
likely not apply in a group of subjects with major
malocclusions.
When relative size and position were factored out and
the covariation between shape only was considered, the
results become much less interesting. The only signifi-
Fig 16. The correspondences between the mean shape
cant covariations found (Table II) were between parts
(blue) and samples (red) for MAN. Corresponding pseu- for which subparts are physically adjacent (BPL and
dolandmarks are indicated by the grey lines joining the MAN, BPL and MDE, MDE and XDE, MDE and SPL,
2 curves. Matches are ordered according to Procrustes SPL and BPL). Furthermore, both the correlations and
distance. A, Best match; B, middling match; C, worst statistical significances were weak compared with the re-
match. sults for relative size and position. It appeared that there
was weak (if any) covariation between the shape of the
parts in the facial skeleton. Rather, it was the relative

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McCane and Kean e29

size and position (including relative orientation) that 9. Bastir M, Rosas A, O’Higgins P. Craniofacial levels and the morpho-
appeared to form the basis of the counterpart principle. logical maturation of the human skull. J Anat 2006;209:637-54.
10. Bastir M, Sobral PG, Kuroe K, Rosas A. Human craniofacial sphe-
However, as noted by Klingenberg23 one expects weaker
ricity: a simultaneous analysis of frontal and lateral cephalograms
correlations between shape only comparisons (separate of a Japanese population using geometric morphometrics and
subsets) than in comparisons involving shape, relative partial least squares analysis. Arch Oral Biol 2008;53:295-303.
size, and position (simultaneous fit). We did not follow 11. Klingenberg CP, Zaklan SD. Morphological integration between
the methodology of Klingenberg, but believe that the developmental compartments in the drosophila wing. Evolution
2000;54:1273-85.
weak statistical significance for covariation of shape-
12. Rohlf FJ, Corti M. Use of two-block partial least-squares to study
only provides good evidence that each part specified in covariation in shape. Syst Biol 2000;49:740-53.
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essary but not sufficient condition. To be fully confident, on overall cranial shape. J Hum Evol 2000;38:291-315.
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