Int. J. Oral Maxillofac. Surg. 2015; 44: 1441–1450
http://dx.doi.org/10.1016/j.ijom.2015.06.007, available online at http://www.sciencedirect.com
Invited Clinical Paper
The William Bell Series
1,2,3 2,3,4
J. J. Xia , J. Gateno ,
5 1 1
J. F. Teichgraeber , P. Yuan , J. Li ,
Algorithm for planning a double- 1,6 7 7
K.-C. Chen , A. Jajoo , M. Nicol ,
2,3,4
D. M. Alfi
1
Surgical Planning Laboratory, Department of
jaw orthognathic surgery using Oral and Maxillofacial Surgery, Houston
Methodist Research Institute, Houston, TX,
2
USA; Oral and Maxillofacial Surgery, Institute
a computer-aided surgical for Academic Medicine, Houston Methodist 3
Hospital, Houston, TX, USA; Oral and
Maxillofacial Surgery, Weill Medical College,
4
Cornell University, NY, USA; Department of
simulation (CASS) protocol. Oral and Maxillofacial Surgery, Houston
Methodist Hospital, Houston, TX, USA;
5
Division of Pediatric Plastic Surgery,
Part 2: three-dimensional Department of Pediatric Surgery, The
University of Texas Health Science Center at
6
§ Houston, TX, USA; Department of Oral and
Maxillofacial Surgery, National Cheng-Kung
cephalometry University Medical College and Hospital,
7
Tainan, Taiwan; Department of Mathematics,
University of Houston, TX, USA
J. J. Xia, J. Gateno, J. F. Teichgraeber, P. Yuan, J. Li, K. -C. Chen, A. Jajoo, M.
Nicol, D. M. Alfi: Algorithm for planning a double-jaw orthognathic surgery using a
computer-aided surgical simulation (CASS) protocol. Part 2: three-dimensional
cephalometry. Int. J. Oral Maxillofac. Surg. 2015; 44: 1441–1450. # 2015
International Association of Oral and Maxillofacial Surgeons. Published by Elsevier
Ltd. All rights reserved.
Abstract. Three-dimensional (3D) cephalometry is not as simple as just adding a Key words: cephalometry; cephalometric ana-
lysis; three-dimensional; two-dimensional; den-
‘third’ dimension to a traditional two-dimensional cephalometric analysis. There
tofacial deformity; craniomaxillofacial
are more complex issues in 3D analysis. These include how reference frames are
deformity; computer-aided surgical simulation;
created, how size, position, orientation and shape are measured, and how symmetry
surgical planning; orthognathic surgery; plan-
is assessed. The main purpose of this article is to present the geometric principles of ning sequence.
3D cephalometry. In addition, the Gateno–Xia cephalometric analysis is presented;
this is the first 3D cephalometric analysis to observe these principles. Accepted for publication 8 June 2015
Introduction These measurements have been made on sagittal or coronal plane. There are two
standardized plain radiographs called fundamental problems associated with tra-
Since 1931, two-dimensional (2D) radio- 2–6
cephalograms. In these cephalograms, ditional 2D cephalometry. The first is
graphic cephalometry has been used to
which can be lateral or frontal, all the that many important parameters cannot be
measure the shape, size, position, and
1 facial structures are projected onto a single measured. The second is that most 2D
orientation of the different facial units.
§
To the authors’ knowledge, no currently available commercial software package is capable of performing such a 3D cephalometric analysis.
The authors are developing a CASS planning software package, AnatomicAligner; this project is sponsored in part by the United States National
Institutes of Health/National Institute of Dental and Craniofacial Research (NIH/NIDCR). It is designed specifically for orthognathic surgery. The
software will be available freely to broader clinical and research communities in the near future (http://www.AnatomicAligner.com).
0901-5027/01201441 + 010 # 2015 International Association of Oral and Maxillofacial Surgeons. Published by Elsevier Ltd. All rights reserved.
1442 Xia et al.
cephalometric measurements are distorted A simple way of envisioning the differ- The anatomical landmark method is
2,3
in the presence of facial asymmetry. ence between the global and local coordi- simple if the face has perfect symmetry.
The recent introduction of cone beam nate system is to imagine a room with a The Frankfort horizontal is constructed
computed tomography (CBCT) in an of- toppled chair. In this scenario, the refer- using both porions and both orbitales.
fice setting has facilitated the acquisition ence frame of the room is the global The sagittal plane is constructed using
of three-dimensional (3D) images with coordinate system. It is perfectly aligned any three midline landmarks, and the cor-
lower amounts of radiation. 3D cephalom- to our surroundings: roof is up, floor is onal plane, which is made perpendicular to
etry can correct the problems associated down, and walls are east, west, north, and the other planes, is aligned to both porions.
with its 2D counterpart. However, 3D south. The toppled chair, lying in the Unfortunately, no human face is per-
cephalometry is more complex than just middle of the room, has its local coordi- fectly symmetrical. In facial asymmetry, it
4
adding a third dimension to a 2D analysis. nate system aligned with the top, front, is complex to use landmarks to construct
There are complex issues in 3D cephalom- and side of the chair. Only the chair’s the reference planes of the face. Let us first
2,3
etry. These include how reference sys- features define the local coordinate sys- consider the Frankfort horizontal. Four
tems are created, how size, position, tem, which is independent of its spatial points define this plane: right orbitale, left
orientation and shape are measured, and position or orientation. The global refer- orbitale, right porion, and left porion.
how symmetry is assessed. Understanding ence frame and the chair’s local coordi- However, in facial asymmetry, these
these basic principles is essential for the nate systems were aligned with each other points are not coplanar, which creates a
correct use of 3D cephalometry. The main when the chair was standing upright and dilemma. A plane can be built using any
purpose of this article is to present the turned south. However, in this example, three points. Which three of the four
geometric principles of 3D cephalometry. when the chair is laid down on its side, the Frankfort points should be used? Each
In addition, the Gateno–Xia cephalomet- global reference frame and local coordi- combination results in a different plane.
ric analysis is presented; this is the first 3D nate system are not aligned. The details of To solve this problem, a best-fitting plane
cephalometric analysis to observe these how global and local coordinate systems among the four points may be constructed.
principles. are used in 3D cephalometry are described This is feasible when the head has minor
below. asymmetries. However, in deformities like
hemifacial microsomia, this best-fitting
Basic principles of 3D cephalometry
method will alter the axial plane by adding
Global reference frame for the whole head
Clinicians use cephalometry to determine a distorted landmark into the equation.
the configuration of the face. An ideal The reference frame for the whole head is Now let us consider the sagittal plane.
cephalometric analysis should measure composed of the axial, coronal, and sagit- This plane can be constructed using three
the five geometric attributes of each facial tal planes. These planes are mutually per- midline landmarks, three pairs of bilateral
unit. These include size, position, orienta- pendicular (Fig. 1). The sagittal plane landmarks, or even just one pair of bilat-
tion, shape, and symmetry. Three of these divides the head into right and left halves. eral landmarks. In the first case, the mid-
measurements – position, orientation, and The axial plane divides the head into upper line landmarks lie on the sagittal plane. In
2–4,7,8
symmetry – need a reference frame. and lower halves. The coronal plane the second case, the pairs of corresponding
The reference frames and how the differ- divides the head into front and back bilateral landmarks are used to calculate
ent geometric attributes should be mea- halves. Traditionally, two methods have three midpoints that lie on the sagittal
sured are discussed below. been used to define the reference frame for plane. In the last case, a line is drawn
the whole head: anatomical landmark and between the single pair of bilateral land-
neutral head posture (NHP). marks and the sagittal plane is constructed
Reference frames
In 3D cephalometry, measuring position,
orientation, and symmetry requires several
different reference frames. The main ref-
erence frame, ‘global’, encompasses the
whole head (face and cranium). It can also
be called the ‘global coordinate system’.
Subordinate reference frames, ‘local’, be-
long to each facial unit (e.g., maxilla,
mandible, and chin). Both reference
frames are coordinate systems made up
of three mutually perpendicular planes. In
cephalometry, there is only one ‘global
reference frame’ and several ‘local coor-
dinate systems’ – one for each facial unit.
The local coordinate systems can also be
called ‘local reference frames’. Although
the terms ‘reference frame’ and ‘coordi-
nate system’ can be used interchangeably,
in order to avoid confusion for the reader,
the term ‘reference frame’ referring to the
whole head, and the term ‘coordinate sys- Fig. 1. Global reference frame (coordinate system) for the whole head. The reference frame for
tem’ referring to each facial unit, are used the whole head includes three orthogonal planes: midsagittal (green), coronal (pink), and axial
in the following text. (blue).
Double-jaw orthognathic surgery planning—part 2 1443
perpendicular to this line, passing through CBCT model to the NHP after scanning. It is important to note that there also are
the midpoint located between the right and Currently, three methods are used for this problems with the NHP method. Some
4
left landmarks. In facial asymmetry, any purpose: standardized photographs, laser patients may have difficulty aligning their
10
combination of three midline points, any levels, and digital orientation sen- heads in the NHP. This is particularly true
4,8,9
combination of three pairs of bilateral sors. in young children and in patients who have
points, or any pair of bilateral points will In the first method, standardized frontal neuromuscular disorders, torticollis, or
produce a different sagittal plane. In facial and lateral facial photographs are taken eye muscle imbalances. In addition, even
asymmetry, the landmark method may with the patient standing in the NHP. A within the same patient, there are temporal
create hundreds of sagittal planes. To plumb line hangs in the background as a variations in the NHP. Based on published
solve this problem, a best-fit plane may true vertical line. A camera is calibrated so studies, the intra-class reproducibility er-
be used to average all the planes, but the its focal plane is perpendicular to the ror of establishing NHP is less than 28, an
distorted landmarks will skew the plane, ground. Before the photos can serve as error that is not considered clinically sig-
16–18
making it useless. Besides, in facial asym- visual guides to manually reorient the 3D- nificant. However, these studies only
metry, a landmark-based sagittal plane CT models, they are rotated to the true examined the reproducibility in pitch. The
will not necessarily be perpendicular to vertical using the plumb line in the back- reproducibilities in roll and yaw have not
Frankfort horizontal, a fundamental re- ground. While this method is subjective, it yet been studied formally. If the same is
quirement of a Cartesian coordinate sys- is valuable for verifying the outcome of true for roll, a 28 variation may cause
tem. Finally, there is no point in discussing the more advanced methods described significant problems. For example, if a
the coronal plane because if the axial and below. symmetrical patient during NHP record-
sagittal planes cannot be constructed ac- In the second method, a self-levelling ing rolls her head around nasion by 28, it
curately, the entire reference frame will be laser is used to simultaneously project will cause the upper incisal midpoint and
11
inaccurate. horizontal and vertical crosshair lines. the pogonion to shift 2.8 mm and 4.2 mm,
4,8,9
The NHP method can solve the When the patient is in NHP, the horizontal respectively (assuming that the distance
problems associated with the anatomical laser is projected onto the malar eminence between nasion and the upper central in-
landmark method. This posture is unal- level, while the vertical laser is simulta- cisor is 80 mm, and the distance between
tered by most facial deformities and can neously projected onto the midsagittal nasion and pogonion is 120 mm). Thus
readily be used to create a reference frame. plane. Two radiopaque stickers are taped one should use the NHP method with
In NHP, a subject stands upright looking on the skin serving as reference markers caution.
straight forward (visual axis is parallel to along the horizontal laser line, and the In the authors’ clinical practice, the
the ground). The head is not tilted, flexed, other two are taped on the vertical line. digital orientation sensor method is used
nor rotated. Once in this posture, the ref- The patient then undergoes CBCT along routinely to place the CT head model in
erence frame of our surroundings (i.e., our with the reference markers. After the 3D the NHP. At the same time, calibrated
world) is transferred to the patient. A plane head model is generated, the reference photographs are also used to cross-vali-
that parallels the ground (horizontal) turns markers are connected to the lines, which date the correctness of the CT model NHP.
into the axial plane. Two orthogonal are then used to build an anatomical ref- Currently, the Surgical Planning Labora-
planes, aligned with gravity, turn into erence frame. A potential disadvantage of tory of the Houston Methodist Research
the sagittal and coronal planes. The hori- this method is that it relies on skin markers Institute is developing a new method to
zontal plane is moved up and down until it that can easily be displaced. calculate the reference frame for the face
divides the face into the upper and lower The third method of orienting a head to mathematically. The goal is to eliminate
halves, becoming the axial plane. The NHP is to reorient a CT/CBCT model to the errors caused by the existing methods.
4,8,9
vertical planes are moved right and left, NHP using a digital orientation sensor.
and rotated around their vertical axes until The orientation sensor is attached to a bite-
Local coordinate system for each facial
one of them divides the face into right and jig. The attachment of the sensor to the
unit
left halves, becoming the midsagittal bite-jig is designed to ensure that the
plane. The other is moved forward and sensor has the same orientation as the Measuring the orientation of the different
backward until it divides the face into bite-jig frame. While the patient is in facial units (e.g., maxilla, mandible)
anterior and posterior halves, becoming NHP with the bite-jig between the teeth makes it necessary to build local coordi-
the coronal plane. and the sensor in front of the face, the nate systems for each unit. A simple meth-
In 3D cephalometry, there are two ways pitch, roll, and yaw of the bite-jig frame od for building local coordinate systems
2,4
of orienting a head image to the NHP. The are recorded. After a CT/CBCT facial for the jaws uses an occlusal triangle.
first is to scan the head while in the NHP. model is generated, the recorded sensor This method is best illustrated using an
The second is to scan the head in an orientation is applied to the bite-jig frame, example (Fig. 2). In this example, a local
arbitrary orientation, and then reorient reorienting the head to the recorded NHP. coordinate system is built for the maxilla.
the image to the NHP. Spiral computed Afterwards, the reference frame can easily In the first step (Fig. 2A), a triangle is
tomography (CT) scanners require the be defined. The axial plane is a horizontal constructed on the maxillary dentition.
patient to be in a supine position during plane that passes through the right and left The vertices of the triangle are the right
image acquisition. Thus, it is impossible to porions, the sagittal plane is the vertical and the left mesiobuccal cusps of the first
position the patient in the NHP during plane that best divides the head into right molars and the incisal midpoint (the point
scanning. CBCT scanners are better, as and left halves, and the coronal plane is a located on the dental midline at the level
the patient’s head can be upright. Yet chin vertical plane that is perpendicular to the of the incisal edges). The incisal midpoint
rests or forehead holders, which are nec- other two planes, passing through the right is selected as the origin of the local coor-
essary to ensure immobility during scan- and left porions. The advantage of this dinate system. The computer then calcu-
ning, may distort this posture. Therefore, it method is that it has been validated in lates a vector that arises from the origin
9,12,13 8,14,15
is more practical to reorient the CT or vitro and clinically. and is perpendicular to the surface of the
1444 Xia et al.
Fig. 2. Defining the local coordinate system for a facial unit using the triangular method. This example illustrates how the triangular method is
completed for the maxilla. (A) In the first step, a maxillary occlusal plane is defined by three points: right mesiobuccal cusp (U6R), left
mesiobuccal cusp (U6L), and central dental midline (U0). U0 is used as the origin of the maxillary local coordinate system. (B) In the second step,
the computer calculates a normal vector for this plane. This vector represents the vertical (z) axis of the local coordinate system. (C) In the third
step, the computer calculates the local anteroposterior (y) axis. This axis passes through U0 and bisects a line that goes from U6R to U6L. (D) In the
last step, the computer calculates the transverse (x) axis of the local coordinate system. This axis is perpendicular to z and y axes. (E) 3D view of the
maxillary local coordinate system.
triangle (occlusal plane, Fig. 2B). This volumes. In 3D cephalometry, linear mea- distance and two angles (spherical coordi-
vector becomes the vertical (z) axis of surements of size are calculated as the nate system). Mathematically, these three
the local coordinate system. The computer distance between two points (landmarks) coordinate systems are all convertible.
next calculates the local anteroposterior located in 3D space. For example the Since the spherical coordinate system is
(y) axis. This axis originates at the incisal maxillary width can be measured as the more complex than the other two and is
midpoint and bisects the base of the trian- distance between the two first molars. seldom used in 3D cephalometry, it is not
gle (the line that goes from one first molar discussed in this article.
to the other) (Fig. 2C). In the final step A 3D Cartesian coordinate system con-
Position measurements
(Fig. 2D and E), the computer calculates sists of three mutually perpendicular axes
the transverse (x) axis of the local coordi- Position refers to the location of a point in that cross each other at an origin. The
nate system. This axis rises from the inci- space. Its measurement is always relative transverse axis is x, the anteroposterior
sal midpoint pointing perpendicular to z and requires a reference frame. In 3D axis is y, and the superoinferior axis is
and y axes. cephalometry, we need to determine the z. Each pair of axes forms a reference
Currently, the Surgical Planning Labo- position of the jaws, which are complex plane. The absolute position of any land-
ratory is investigating an approach that 3D objects made up of thousands of mark is expressed using three coordinates
uses a principal component analysis points. Each point that makes up a jaw (x, y, z), which represent the transverse,
(PCA) to automatically construct local has a different position. In order to mea- anteroposterior, and vertical positions, re-
coordinate systems. Unlike the triangle sure the position, we must select a point to spectively. The relative position between
method, which uses only three points, represent the whole jaw. However, since two landmarks can be calculated easily.
PCA can estimate the x, y, and z coordinate there is no perfect point, clinicians have For example, if the anteroposterior coor-
axes using a larger group of points, e.g., all had to use several points to represent the dinate of pogonion is 62 mm and the
the cusps of the maxillary teeth. Thus, the jaws. For example, the anteroposterior anteroposterior coordinate of nasion is
PCA approach may estimate more accu- position of the maxilla has been measured 60 mm, pogonion is then 2 mm in front
rately the local coordinate systems of the at the anterior nasal spine (ANS), point A, of nasion (62 60 = 2 mm). The same is
À
jaws. and the upper incisal midpoint. true for relative vertical and transverse
Measuring position in one, two, and positions.
three dimensions requires one, two, and In 2D cephalometry, we often use the
Size measurements
three numbers. Thus, any system that combination of one angle and one distance
Measuring size in 3D cephalometry is measures jaw position in 3D space must to determine the position of a point. This
straightforward. Size is an intrinsic prop- use three numbers. The position of an type of measurement typifies a ‘polar’
erty of an object that is independent of the object can be expressed either by three coordinate system (Fig. 3). A polar system
space it occupies. We can measure size in distances (3D Cartesian coordinate sys- resides on a plane and consists of a fixed
3D space using linear measurements (e.g., tem), or by two distances and one angle point (the pole) and a ray (the polar axis).
length, width, and height), areas, or (cylindrical coordinate system), or by one From the pole, the polar axis points in a
Double-jaw orthognathic surgery planning—part 2 1445
plane, the first two coordinates (r, u) are
expressed in the same fashion as in the
polar system. The third coordinate is the
distance (z) between the point and its
projected point on the plane of the polar
system. Thus, the position of a point in 3D
cephalometry is expressed using three
coordinates (r, u, z), including two dis-
tances and one angle.
Orientation measurements
Orientation is relative. Although there are
many different ways to measure the orien-
tation in 3D cephalometry, we usually
measure the orientation as three rotations
following a specific order. The measure-
ment is done by rotating a facial unit from
Fig. 3. Measuring the 2D position of a point using the polar coordinate system. In this example,
its reference alignment to its current place-
the 2D position of pogonion is being measured in relation to sella–nasion. The origin of the
coordinate system, the pole, is nasion. The polar axis is the ray that originates on nasion, pointing ment. The reference alignment is the
at sella. The angle u of sella–nasion–pogonion represents the anteroposterior position of the chin (global) coordinate system of the whole
relative to the anterior cranial base (sella–nasion). The radius r is the distance between nasion head, while the current placement is the
and pogonion, representing the vertical position of the chin. (local) coordinate system of a facial unit.
The sequential rotations are yaw, roll, and
pitch. In our approach, these rotations
fixed direction. To measure the position of In 3D cephalometry, the same measure-
occur around the axes of the local coordi-
a point in a polar system, a line segment ments are completed in a cylindrical sys-
nate system, which is the one that belongs
(called the radius, r) is first drawn from the tem by adding a cylindrical axis to the 2D
to the facial unit being measured. Yaw is
point that we are locating to the pole. The polar system (Fig. 4). The cylindrical axis
the rotation around the vertical (z) axis.
position of the point is then determined by is perpendicular to the plane of the polar
Roll is the rotation around the anteropos-
measuring the length of the radius (r) and system and passes through the pole. In
terior (y) axis. Pitch is rotation around the
the angle theta (u) between the radius and order to measure the position of a point
transverse (x) axis.
the polar axis. The position of a point in in a cylindrical system, the point is first
The order in which one measures pitch,
2D cephalometry is expressed using two projected onto the plane of the polar sys-
roll, and yaw is important. These rotations
coordinates (r, u). tem, along the cylindrical axis. On this
are not commutative and the result of
Fig. 4. Measuring the 3D position of a point using the cylindrical coordinate system. The cylindrical system is the 2D polar system with the
addition of a cylindrical axis. In this example, the 3D position of pogonion is being measured in relation to sella–nasion. As in the 2D polar system,
the origin (pole) of the cylindrical system is nasion. The polar axis is the ray that originates on nasion and points at sella, and the cylindrical axis is
perpendicular to the sagittal plane and crosses nasion. The position of pogonion is determined by first projecting it onto the sagittal plane, then by
measuring the radius (r) of nasion–pogonion distance and theta (u) sella–nasion–pogonion on the plane. The third coordinate (z) is the positive or
negative transverse distance between pogonion and its projection on the plane of the polar system. (A) The 3D position of pogonion being
measured in the cylindrical system. (B) Details of the same cylindrical system by hiding the midface.
1446 Xia et al.
Fig. 5. Measuring the yaw, roll, and pitch of the facial unit. In this example, yaw, roll, and pitch of a maxilla are measured by comparing the
orientation of the local coordinate system (coloured arrows) with the global coordinate system (black lines). The origin of the local coordinate
system is U0. The yellow arrow indicates the x-axis, the red arrow indicates the y-axis, and the blue arrow indicates the z-axis. (A) The first step is
to measure the yaw (angle between x-axis of the local coordinate system and x-axis of the global coordinate system as they project on the axial
plane, submental vertex view). (B) After yaw has been calculated, the algorithm rotates the local coordinate system around the origin aligning local
x-axis to global x-axis (removing yaw, submental vertex view). (C) Frontal view of maxilla at the same orientation as in (B). Once yaw is removed,
the next step is to measure the roll (angle between local z-axis and global z-axis as they project on the coronal plane). (D) After roll has been
calculated, the algorithm rotates the local coordinate system around the origin aligning local z-axis to global z-axis (removing roll, frontal view).
(E) The last step is to measure the pitch (angle between local y-axis and global y-axis as they project on the sagittal plane, right view).
20
following the pitch–roll–yaw order is dif- follows this rule is a Procrustes analysis. aligned with the normal mandible. After
ferent from the result of following the In this analysis, shape is evaluated by the differences in size, position, and ori-
2
yaw–roll–pitch order (Fig. 5). For 3D comparing two objects after both have entation are removed, one can better
cephalometry, we recommend a yaw– been scaled to the same size, placed in appreciate the distortion of the deformed
roll–pitch sequence. Most facial units the same location, and rotated to the best mandible. In this example, the deformed
20
(i.e., maxilla and mandible) normally have possible alignment. mandible has an obtuse gonial angle and a
some degree of pitch, while their ideal roll Figure 6 shows the example of a de- short ramus height.
and yaw is zero. This strategy is helpful in formed mandible (pink) that is being com-
that it minimizes the yaw and roll angles. pared with an average mandible (blue).
Symmetry measurements
Both mandibles differ in size, position,
orientation, and shape. To assess shape, The two elements that relate to symmetry
Shape measurements
a Procrustes superimposition first scales are ‘object symmetry’ and ‘symmetrical
By definition, shape is the geometric attri- the mandibles to the same size. It then alignment’. Object symmetry refers to the
bute of an object that is not size, position, places both mandibles in the same position intrinsic – mirror – symmetry that each
19
or orientation. Thus, the measurement of (both centroids at the origin of the Carte- facial unit should have. Symmetric align-
shape should not be influenced by size, sian coordinate system). Finally, it rotates ment refers to the alignment of each facial
position, or orientation. A method that the deformed mandible until it is best unit with the midsagittal plane of the face.
Double-jaw orthognathic surgery planning—part 2 1447
Fig. 6. Shape analysis using Procrustes superimposition. Human mandibles always have a different size, position, orientation, and shape from an
average normal mandible – the normative data. In this example, the top mandible in blue is an average normal mandible, while the lower one in
pink is deformed. Both mandibles differ in size, position, orientation, and shape. In the first step, the subject’s mandible is scaled to the same size as
the normal mandible. In the second step, the subject’s mandible is translated to the normal mandible by perfectly registering the two centroids
together. In the third step, the subject’s mandible is rotated around its centroid to best fit the normal mandible. After the differences in size,
position, and orientation are removed, the remaining component is the shape difference. In this example, the mandible has an obtuse gonial angle
and relatively narrow body and ramus.
Object symmetry as all the midline landmarks; the left set transformation centres both the half-forms
contains all the left landmarks and again on a Cartesian coordinate system. In order
The intrinsic symmetry of the dental
the midline landmarks. Next, we superim- to centre the half-forms, we first have to
arches can be assessed using a simple
2,4,21,22 pose the two half-forms using a series of determine their centroids. Then, we trans-
occlusal triangle. The triangle is
transformations. However, prior to this late both forms, so that their centroids are
constructed by connecting the incisal mid-
step, we have to pick one half-form to on the origin of the Cartesian system. The
point and the mesiobuccal cusps of the
be the object of the transformations and third transformation rotates the object
first molars. The triangle is isosceles in
the other to be the target. Which half-form half-form around its centroid until the
patients with symmetric dental arches,
(right or left) becomes the object is incon- form is aligned to the target. The final
with two equal lateral sides and two equal
sequential. The first transformation alignment is reached when the sum of
base angles. The triangle is scalene in
reflects (flips) the object half-form around all the distances between the correspond-
patients with asymmetric dental arches,
its anteroposterior axis, creating a ing landmarks of the right and the left half-
with three different sides and three differ-
mirror image. This operation makes the forms is minimal. To quantify the degree
ent angles. The degree of intrinsic asym-
half-forms comparable. The second of asymmetry for each landmark, we
metry of a dental arch is proportional to
the difference between the basal angles:
the larger the difference, the greater the
asymmetry. Alternatively, one can mea- Table 1. Gateno–Xia 3D cephalometric analysis.
sure the intrinsic asymmetry by comparing Mandible
the length of the lateral sides of the trian- Maxilla
Whole Chin
gle. The greater the absolute difference
between the sides of the triangle, the Object symmetry greater the asymmetry. Shape
Size Length
One can also measure more elaborately
Width
the intrinsic symmetry of any facial unit
Height
using a Procrustes analysis of symmetry. It
Position Anteroposterior
begins by dividing the unit into two half- Vertical
forms: right and left. In order to divide the
Transverse Symmetrical alignment
form, the landmarks of the facial unit are Orientation Yaw
divided into two groups: the right set Roll
contains all the right landmarks as well Pitch
1448 Xia et al.
Table 2. An example of 3D cephalometric analysis for orthognathic surgery (a shadowed space indicates input from the clinical examination).
All the norms were established for Caucasian subjects with normal facial appearance. All the norms for symmetry analyses were based on the authors’ opinion. The hard tissue norms were
23 24,25
established by Bhatia and Leighton for 20-year-old subjects, with the exceptions of the following: the Holdaway ratio was based on Holdaway’s opinion; intercondylar and intergonial
26 27,28
distances were established by McNamara for 18-year-old subjects; maxillary occlusal plane inclination was established by Downs; the determination of optimal maxillary AP position
29,30 31–33
was inspired by Andrews and based on the authors’ clinical observation; and Ricketts norms were established by Ricketts for 9-year-old subjects. The soft tissue norms were
34
established by Farkas for 20–26-year-old subjects, with the exception of the following: incisal show was based on the authors’ opinion; lip measurements were established by Peck et al. for
35 36
15.5-year-old subjects who were either in orthodontic treatment or post-treatment; and posterior airway space was established by Riley et al. for male subjects. The dental arch norms were
37
established by Moyers et al. for 18-year-old subjects.
a
The gonial angle is measured by projecting the landmarks articulare, gonion, and menton onto the mandibular local sagittal plane.
b
The norm is calculated based on the Frankfort horizontal plane. In our 3D cephalometry, this is calculated based on the true horizontal. Please use with caution.
c
Posterior airway space should be measured as the shortest distance between the tongue base and the posterior pharyngeal wall. The norm was established by Riley for measuring the distance between the two
landmarks: tongue base (TB, the intersection point of a line from point B through gonion and the base of the tongue), and posterior pharyngeal wall (PPWB, the intersection point of a line from point B through
gonion and the base of the posterior pharyngeal wall). Please use with caution.
d
This is the original Ricketts norm established for 9-year-old subjects. No age adjustment is required.
e 27
This is the original Downs norm established for ‘occlusal plane to FH’ (‘cant of occlusal plane’ in Downs’s original manuscript ). The occlusal plane was defined as a line bisecting the occluded
mesiobuccal cusps of the upper and lowers first molars and the incisal overbite. Please use with caution.
Abbreviations: A, point A (the deepest point on the concave outline of the upper labial alveolar process extending from the anterior nasal spine to prosthion); ANS, anterior nasal spine (the tip of the
anterior nasal spine); AP, anteroposterior; AxP, the axial plane; B, point B (the deepest point on the bony curvature between the crest of the alveolus (infradentale) and pogonion); Co, condylion (the
most posterosuperior point of the mandibular condyle); F, female; G0, soft tissue glabella (the most prominent or anterior point of the forehead on the midsagittal plane); Go, gonion (the midpoint at the
mandibular angle); Gn, gnathion (the midpoint between the soft tissue pogonion and menton); L1, long axis of the mandibular central incisors (a line connecting the lower central incisal embrasure and
the midpoint of the right and left lower central incisal apices); L1E, lower central incisal embrasure; L6L, mesiobuccal cusp of the mandibular left first molar; L6R, mesiobuccal cusp of the mandibular
right first molar; LMSP, midsagittal plane of the local coordinate system for a given facial unit; M, male; Me, menton (the lowest point on the lower border of the mandibular symphysis); MP,
mandibular plane (a plane constructed by thelandmarks Me right Go andleft Go); MSP, midsagittalplane of thereference frame for the wholehead; N, nasion (the most anterior point on the frontonasal
suture); nasal floor plane, a plane constructed by the landmarks ANS and right and left greater palatine foramen; Pg, pogonion (the most anterior point on the mandibular symphysis); PM,
suprapogonion (a point at which the shape of the symphysis mentalis changes from convex to concave – also known as protuberance menti); PNS, posterior nasal spine; S, sella (the centre of sella
turcica); U1, long axis of maxillary central incisors (a line connecting the upper central incisal embrasure and the midpoint of right and left upper central incisal apices); U1E, upper central incisal
embrasure; U6L, mesiobuccal cusp of the maxillary left first molar; U6R, mesiobuccal cusp of the maxillary right first molar; Xi, Ricketts Xi point (the midpoint of the right and left inferior alveolar
nerve foramens).
Double-jaw orthognathic surgery planning—part 2 1449
measure the distances between analogous necessary information needed to plan a in the treatment of patients with craniomax-
right and left landmarks. These distances surgery in a single report. In addition, this illofacial deformities. J Oral Maxillofac
are (directly) proportional to the degree of analysis is meant to be modular. The Surg 2011;69:2014–24.
6. Swennen GR, Schutyser F, Hausamen JE.
asymmetry. important aspects of the analysis are its
Three-dimensional cephalometry: a color
concepts and structure. The measurements
atlas and manual. 1st ed. Heidelberg:
presented here are the ones that we have
Symmetrical alignment Springer; 2006.
found useful. We encourage others to
7. Swennen GR, Mollemans W, Schutyser F.
The other element of the symmetry mea- adapt this analysis to their needs by add-
Three-dimensional treatment planning of
surement is the symmetrical alignment. ing, deleting, or replacing measurements
orthognathic surgery in the era of virtual
Facial asymmetry can be the result of (rows) or by adding new elements or units
imaging. J Oral Maxillofac Surg
misalignment of a facial unit about the (columns).
2009;67:2080–92.
midsagittal plane. When a unit is not
8. Xia JJ, McGrory JK, Gateno J, Teichgraeber
symmetrically aligned, the degree of mis-
Funding JF, Dawson BC, Kennedy KA, et al. A new
alignment (i.e., asymmetry) can be quan-
method to orient 3-dimensional computed
tified by measuring the transformations
Dr Chen was sponsored by the Taiwan tomography models to the natural head po-
(manoeuvres) that are necessary to realign Ministry of Education while he was work- sition: a clinical feasibility study. J Oral
it. The three parameters of symmetric ing at the Surgical Planning Laboratory, Maxillofac Surg 2011;69:584–91.
alignment are transverse translation, roll, Department of Oral and Maxillofacial Sur- 9. Schatz EC, Xia JJ, Gateno J, English JD,
and yaw. Transverse translation places the gery, Houston Methodist Research Insti- Teichgraeber JF, Garrett FA. Development of
a technique for recording and transferring
midpoint on the midsagittal plane. Roll tute, Houston, TX, USA. This work was
natural head position in 3 dimensions. J
rotates the unit around the midpoint until supported in part by NIH/NIDCR research
Craniofac Surg 2010;21:1452–5.
right and left landmarks are vertically grants 5R42DE016171, 5R01DE022676,
10. Damstra J, Fourie Z, Ren Y. Simple tech-
aligned. Yaw rotates the unit around the and 1R01DE021863.
nique to achieve a natural position of the
midpoint, minimizing the distance differ-
head for cone beam computed tomography.
ences between the corresponding bilateral
Br J Oral Maxillofac Surg 2010;48:236–8.
landmarks and the coronal and midsagittal Competing interests
11. Bobek S, Farrell B, Choi C, Farrell B, Wei-
reference planes. Zero is the ideal value
Dr James Xia, Dr Jaime Gateno, and Dr
mer K, Tucker M. Virtual surgical planning
for transverse position, yaw, and roll.
John Teichgraeber receive a patent royalty for orthognathic surgery using digital data
from Medical Modeling Inc. through Uni- transfer and an intraoral fiducial marker: the
Gateno–Xia cephalometric analysis versity of Texas Health Science Center at Charlotte method. J Oral Maxillofac Surg
Houston. Dr Xia and Dr Gateno receive a 2014. http://dx.doi.org/10.1016/
In the authors’ 3D cephalometric analysis,
second patent royalty from Medical j.joms.2014.12.008.
the measurements are displayed in a grid
2 Modeling Inc. through Houston Methodist 12. Schatz EC. A new technique for recording
(Table 1). Each row displays a different
Hospital. natural head position in three dimensions.
geometric property: object symmetry,
[MS thesis] Texas: The University of Texas
shape, size, position, and orientation. Health Science Center at Houston; 2006.
The columns represent the individual fa- Ethical approval 13. Weiskircher MN. Accuracy of a new tech-
cial units: e.g., maxilla, whole mandible, nique for recording natural head position in
No human subjects.
and chin. three dimensions. [MS thesis] Texas: The
In the first part of the analysis, object University of Texas Health Science Center
at Houston; 2007.
symmetry, the intrinsic symmetry of each Patient consent
jaw, is first determined by a Procrustes 14. McCormick SU, Drew SJ. Virtual model
No human subjects.
analysis of symmetry or a triangle. In the surgery for efficient planning and surgical
performance. J Oral Maxillofac Surg
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