Measuring 3D shape in orthodontics through geometric morphometrics
Huanca Ghislanzoni et al. Progress in Orthodontics
Measuring 3D shape in orthodontics through geometric morphometrics
Luis Huanca Ghislanzoni 0 1
Roberta Lione 3
Paola Cozza 3
Lorenzo Franchi 2
0 Department of Orthodontics, University of Geneve , Geneve , Switzerland
1 Avenue Belles-Roches 7 , 1004 Lausanne, VD , Switzerland
2 Department of Surgery and Translational Medicine, University of Florence , Florence , Italy
3 Department of Orthodontics, University of Rome Tor Vergata , Rome , Italy
Background: Geometric morphometrics (GMM) has been traditionally applied to the field of biology to study developmental differentiations between species. Orthodontics deals with the shape and size of the face and its components. While several tools have been used to measure size, proportions, and relations between anatomical components, shape has been mainly described by esthetic criteria. The purpose of this paper is to present methods to measure shape of 3D orthodontic data, beyond the conventional tools that have been traditionally used in cephalometrics and in facial and dental cast analysis. Findings: The authors showcase an example of applying geometric morphometrics to measure palates from scanned dental casts. GMM can be used as a useful tool to describe the three-dimensional shape of surfaces of orthodontic interest. A general introduction to the theoretical principles of how to apply GMM is provided. Conclusions: Variability can be measured through the Principal Component Analysis (PCA) and can lead to the identification of shape patterns and sources of variability of the shape, independently from changes in size.
Technological advances have made three-dimensional
(3D) orthodontic diagnostic records much more
accessible. We now have 3D images of the craniofacial
skeleton from CBCT imaging, 3D facial photographs from
stereophotogrammetry camera sets, and dental casts
from intraoral and standard scanner, all available in
digital format. We are, therefore, in the condition to
measure and evaluate what interests us most as
orthodontists, namely shape, in ways that were not previously
]. Potentially, this is a turning point in
orthodontic diagnosis. Since 3D data are not a mere
extension of 2D data to an extra dimension but require
new tools to fully exploit their 3D nature, we can now
grasp this opportunity and radically overhaul our
Geometric morphometrics (GMM) [
] has been
traditionally applied to the field of biology to study
developmental differentiations between species [
several researchers applied this method to orthodontics
]. Orthodontics deals with the shape and size of
the face and its components. While several tools have
been used to measure size, proportions, and relations
between anatomical components, shape has been mainly
described by esthetic criteria. GMM can be used as a
specific tool to describe shape variation between
individuals and to identify patterns of variation of orthodontic
interest (es. hypo/hyper-divergency or sagittal
relationship of the basal bones) [
]. When applied to 2D
cephalometrics, it can provide apparently surprising
results, eg., the main source of difference between patients
classified as Class I and Class II consists of the vertical
growth pattern, rather than the expected difference
along the sagittal plane [
]. GMM can be useful to
understand shape variation, especially in 3D where the
shape complexity gets to its maximum [
The purpose of this paper is to present methods to
measure the shape of 3D orthodontic data, beyond the
conventional tools that have been traditionally used in
cephalometrics and in facial and dental cast analysis.
The authors showcase an example of applying geometric
morphometrics to measure palates [
] from scanned
Geometric morphometrics principles
GMM is a special method to measure shape, as it does not
use traditional angles and linear distances (size measures).
By choosing specific angles or linear measurements, we
arbitrarily choose which part of the shape to measure. In
fact, the selection of some measurements and exclusion of
others can lead to biased result as you consider just one
specific part of the shape, instead of the whole of it. With
GMM, the whole of the shape can be analyzed.
The basic principles of GMM are the following: [
5, 18, 19
An object is a collection of landmarks. Objects such
as teeth, bones, and faces need to be reduced to a
set of landmarks before analysis can proceed, as the
basic tools of GMM tools cannot work directly on
curves or surfaces. Objects of the same class
(e.g., faces) must have the same number of
landmarks and the landmarks must be homologous,
i.e., each landmark must represent the same
anatomical (or functional) feature.
Shape of an object can be measured only in relation
to another object, i.e., shape measurement is actually
the comparison between two shape/measurement of
The “shape” is the morphologic entity that remains
after position, and size differences have been removed
from the analyzed objects (as dimensional differences
are not considered when comparing shapes).
Practically, when analyzing a collection of objects, the
main aim is to find the average shape of the objects and
then analyze the variability of shape in the group, with
regard to the average shape. The variability (i.e., how
much shape varies between the objects and in which
ways) is analyzed through the Principal Component
Analysis (PCA) statistical procedure. In a biological
system such as the craniofacial complex, shape variability
can be clinically translated into shape patterns (e.g., the
dolichofacial–brachyfacial pattern, or the Class II–Class
III anteroposterior pattern). The patterns that are most
dominant are usually identified by clinical experience.
GMM can reveal these patterns and measure the relative
contribution of each to the total shape variability.
From Cartesian coordinates to the shape–space
The procedure that GMM follows to compute average
shape and shape variability is a sequence of the following
Landmarks are placed on the objects at homologous
positions. When all the landmarks are placed, we
can call the group of landmarks that describe an
object as a landmark configuration.
The landmark configurations are aligned and scaled
using a best-fit procedure that minimizes differences
between them. This is called Procrustes
superimposition. After Procrustes superimpositions, we lose all
information about size, and we deal with shape only.
The Cartesian coordinates (x, y, z) of the aligned/
scaled objects are called Procrustes coordinates. The
Procrustes coordinates of an object define its
position in a system, called shape-space.
The shape-space extends along many dimensions,
since each object has many landmarks. For
3-dimensional objects, the number of dimensions of
the shape-space is equal to three times the number
of landmarks minus seven (degrees of freedom).
Each object can be considered as a single point in
shape-space. The distance between two objects in
shape-space is equal to their shape difference.
The average shape of all the objects is the shape at
the center of the shape-space and can be easily
computed by averaging the Procrustes coordinates of
To determine the shape patterns of the population,
the shape-space is rotated in such a way that its
main axes are aligned to the directions of major
variability of the population. This is achieved by
applying the PCA. PCA describes differences between
shapes through determination of the main sources
of variability when comparing different shapes.
Landmark identification and placement
When applying GMM to objects of orthodontic interest,
such as bone and soft tissue surfaces, palates, or teeth,
we cannot study curves and surfaces directly but we
need to place landmarks on them. Thus, the most
challenging problems are how many landmarks to place,
which criteria (anatomical? geometrical?) to use to place
landmarks, how densely should we cover each surface,
and finally how can we ascertain that landmarks are
homologous from one object to the other.
The last question is particularly troublesome because
in orthodontics we come across some very extensive
areas, such as the cheeks on the facial soft-tissue surface
that do not possess any distinguishing/non-ambiguous
markers. A similar problem occurs in 2D data, e.g., the
outline of the mandible on a lateral cephalogram is a
smooth line and there is no anatomical structure to
guide us in placing points such as the gonion.
To manage this problem, Bookstein identified three
types of landmarks: [
▪ Type I landmarks are those that are located at the
juxtaposition of anatomical features, such as the
confluence of three sutures meeting at a single point. Type I
landmarks are defined by features in their immediate
vicinity and can be confidently assumed homologous, at
least in the anatomical sense.
▪ Type II landmarks are defined as the maxima of
curvature of an anatomical structure, e.g., the anterior
nasal spine defined as the point of highest curvature of
the maxillary outline.
▪ Type III landmarks are defined as points along a
curve or surface, in relation to some other more distant
structure. For example, the menton is located on the
mandibular outline but needs other structures (e.g., the
Frankfurt horizontal) or an external vertical direction
(e.g., the true vertical) to define its precise location.
Curves or surfaces that do not provide explicit
information for precise location of landmarks are ubiquitous. The
simple solution is to place landmarks at predetermined
intervals along the curve, e.g., at equidistant intervals. Points
placed with such a criteria that stay on the curve/surface
are called semi-landmarks [
]. The geometry is easier
for curves in 2D or 3D but it is not so easy to define
semilandmarks for non-planar surfaces in 3D. In any case,
semi-landmarks after first placement are not guaranteed
to be 100% homologous.
GMM approach the semi-landmark placement
problem by changing their position until the additional shape
variability is reduced to the minimum possible. This is
achieved by sliding the semi-landmarks in the direction
that reduces shape variance but always constraining
them on the curve or surface. Once slid (the procedure
needs to be repeated at least three times), the
semilandmarks can be considered as homologous points, and
the shapes are ready to be analyzed through Principal
Anatomical variability between subjects in the sample
may lead to a debatable controversy: what to do if an
anatomical trait (e.g., a cusp, a ridge on a tooth, a ruga on the
palate) is absent in one of the individuals? Two solutions
are possible: the first one is to ignore it in all subjects. The
second is to place the landmark at the same anatomical
spot, effectively making the trait zero size. No ideal
solution exists. However, we can experience the same problem
with conventional measurements, e.g., how do you
measure overjet if there are no incisors [
]. In this latter case,
the solution would likely be to exclude the patient from
the sample, while in the case of geometric morphometrics,
it may be acceptable to give to the anatomical trait a
“zero” value and include it in calculations, if the missing
anatomical trait is not the main clinical question to solve.
Advantages and disadvantages of GMM
When using GMM, we renounce to have any information
on size, as all the shapes are “averaged” and size
information is left out of the Procrustes space. This can be seen as
a disadvantage as only change in shape patterns can be
outlined through GMM. Anyway, this limitation can turn into
an advantage. In fact, there is no need to arbitrarily select a
special part of the shape to be measured as all parts can be
compared as far as a landmark fits the area. While looking
at palates, we can get much more information through a
GMM procedure rather than with standardized measures.
Another important aspect is that in orthodontics we
normally compare anatomical features between patients
and controls, assuming that controls are more regular or
“normal.” However, what can be considered normal or
not normal, is controversial and of difficult
interpretation. With GMM, variation of shapes just comes out
from the population, considering all the aspects of the
shape, without the need of pre-selecting some parts of
the population. Variability analysis through PCA allows
to determine shape patterns and can thereafter dictate
which measures to take and not vice versa.
When pre-selecting patients with different anatomical
features (like in the example of palates collected from
oral breathers and standard breathers), GMM has the
role to underline the source of differences between the
two samples. If the samples are really different as for
their space entities, they should appear clusterized, as at
least the group with pathologic problem (oral breathers)
represents an extreme of the population.
GMM and palatal surfaces
Two samples of mouth-breathing (19 subjects, 12 females
and 7 males, with a mean age of 8.5 ± 1.6 years) and
nosebreathing subjects (16 subjects, 8 females and 8 males,
mean age 8.5 ± 1.7 years) in the mixed dentition have
been already selected and studied for anatomical
differences in terms of palatal vault linear dimensions, surface,
and volume [
]. Having 3D data such as surface and
volumes does not give a much better insight on palatal shape
changes, as the value of the surface or of the volume is a
mere number that does not tell where is the source of
variability between patients. GMM was thus applied to
describe the shape of the palates. The use of GMM allows to
understand the changes of shape not only in preselected
areas (i.e., molars and canine transverse distance, palatal
height, palatal depth) but virtually in any point of the
surface where homologous landmarks and semi-landmarks
were positioned, thus allowing for a more comprehensive
understanding of changes in shape.
Palatal vaults were digitized through a template
representing a dataset of homologous landmarks. The template
was created with Viewbox (Viewbox 4, dHAL software,
Kifissia, Greece) and applied for all the patients [
] (Fig. 1).
Three boundary lines were defined for each palate: a
midline passing on the median raphe, a U-shaped line
passing apical to the gingival sulci of each tooth, and a
posterior boundary, perpendicular to the plane defined
by the previous line at the height of the mesial contact
points of upper first molars. The extremes of the lines
were digitized and provisional semi-landmarks were
used to approximate the curvature of each line along the
surfaces. Semi-landmarks were then automatically placed
and distributed at equal distances along the lines and on
the palatal surface delimited by the described lines. A
total of 240 landmarks and semi-landmarks were
registered for each palate. The average of all the dataset was
then calculated and used as a reference to allow the
semi-landmarks of each palate to become more and
more homologous through a repeated (×3) sliding
procedure that minimizes the differences between each
palate and the average template.
Principal Component Analysis (PCA) was calculated
for all the palates of both samples. PCA allows to
determine the source of variability of the shape in between a
sample. A Procrustes distance between groups means a
test was applied and a p = 0.10 was calculated. Even
though not statistically significant, PCA results showed
that the two samples were clusterized (Fig. 2), thus being
different in the shape/form space. Fifty-five percent (PC1)
of the morphological differences may be explained
through a difference in transverse dimension in the lateral
side of the palate (narrower for oral breather and larger
for normal breathers Fig. 3) and in the height of the
palatal vault, while 15% (PC2) of the differences may be
explained with a difference only in the vertical dimension.
Finally, the average morphology of the palate was
calculated for each group (the average shape does not give
information on shape size after performing Procrustes
superimposition). A mesh was adapted through warping
to the average palates both for mouth and nose breather.
Warping consists of “sticking” a 3D surface to a point
configuration (in this case the average configuration for
both groups). A colorimetric map was used to visually
present differences between the two shapes and make
them easily identifiable from a clinical point of view
(Fig. 4). The morphological analysis is coherent with the
linear, surface, and volumetric results reported in the
previous paper [
]. Moreover, it is consistent with the
clinical observation of narrow and higher palatal vault in
oral-breathers when compared to the palatal vault of
nasal-breathers. GMM can be a useful tool to visually
illustrate and describe with a scientific method the
differences in palatal shape between subjects/samples.
Geometric morphometrics (GMM) can be used as a useful
tool to describe the three-dimensional shape of surfaces of
orthodontic interest. A general introduction to the
theoretical principles of how to apply GMM is provided.
Variability can be measured through the Principal
Component Analysis (PCA) and can lead to the
identification of shape patterns and sources of variability of the
shape, independently from changes in size.
The authors would like to thank Prof. Demetrios Halazonetis for his valuable
suggestions and comments on this paper.
LH and LF made the conception or design of the work. RL and PC carried
out the data collection. LH and LF carried out the data analysis and
interpretation. LH, LF, and LR took part in drafting the article. PC did the
critical revision of the article. LH, RL, LF, and PC took part in the final
approval of the version to be published. All authors read and approved the
Ethics approval and consent to participate
Ethical approval was obtained from the Ethical Committee of the University
of Rome “Tor Vergata” and informed consent was obtained from the
subjects’ parents before inclusion in the study.
The authors declare that they have no competing interests.
Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.
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