## Abstract

Three-dimensional (3D) cephalometry is not as simple as just adding a ‘third’ dimension to a traditional two-dimensional cephalometric analysis. There are more complex issues in 3D analysis. These include how reference frames are created, how size, position, orientation and shape are measured, and how symmetry is assessed. The main purpose of this article is to present the geometric principles of 3D cephalometry. In addition, the Gateno–Xia cephalometric analysis is presented; this is the first 3D cephalometric analysis to observe these principles.

## Introduction

Since 1931, two-dimensional (2D) radiographic cephalometry has been used to measure the shape, size, position, and orientation of the different facial units. These measurements have been made on standardized plain radiographs called cephalograms. In these cephalograms, which can be lateral or frontal, all the facial structures are projected onto a single sagittal or coronal plane. There are two fundamental problems associated with traditional 2D cephalometry. The first is that many important parameters cannot be measured. The second is that most 2D cephalometric measurements are distorted in the presence of facial asymmetry.

The recent introduction of cone beam computed tomography (CBCT) in an office setting has facilitated the acquisition of three-dimensional (3D) images with lower amounts of radiation. 3D cephalometry can correct the problems associated with its 2D counterpart. However, 3D cephalometry is more complex than just adding a third dimension to a 2D analysis. There are complex issues in 3D cephalometry. These include how reference systems are created, how size, position, orientation and shape are measured, and how symmetry is assessed. Understanding these basic principles is essential for the correct use of 3D cephalometry. The main purpose of this article is to present the geometric principles of 3D cephalometry. In addition, the Gateno–Xia cephalometric analysis is presented; this is the first 3D cephalometric analysis to observe these principles.

## Basic principles of 3D cephalometry

Clinicians use cephalometry to determine the configuration of the face. An ideal cephalometric analysis should measure the five geometric attributes of each facial unit. These include size, position, orientation, shape, and symmetry. Three of these measurements – position, orientation, and symmetry – need a reference frame. The reference frames and how the different geometric attributes should be measured are discussed below.

## Reference frames

In 3D cephalometry, measuring position, orientation, and symmetry requires several different reference frames. The main reference frame, ‘global’, encompasses the whole head (face and cranium). It can also be called the ‘global coordinate system’. Subordinate reference frames, ‘local’, belong 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 coordinate systems’ – one for each facial unit. The local coordinate systems can also be called ‘local reference frames’. Although the terms ‘reference frame’ and ‘coordinate 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 system’ referring to each facial unit, are used in the following text.

A simple way of envisioning the difference between the global and local coordinate system is to imagine a room with a toppled chair. In this scenario, the reference frame of the room is the global coordinate system. It is perfectly aligned to our surroundings: roof is up, floor is down, and walls are east, west, north, and south. The toppled chair, lying in the middle of the room, has its local coordinate system aligned with the top, front, and side of the chair. Only the chair’s features define the local coordinate system, which is independent of its spatial position or orientation. The global reference frame and the chair’s local coordinate systems were aligned with each other when the chair was standing upright and turned south. However, in this example, when the chair is laid down on its side, the global reference frame and local coordinate system are not aligned. The details of how global and local coordinate systems are used in 3D cephalometry are described below.

## Global reference frame for the whole head

The reference frame for the whole head is composed of the axial, coronal, and sagittal planes. These planes are mutually perpendicular ( Fig. 1 ). The sagittal plane divides the head into right and left halves. The axial plane divides the head into upper and lower halves. The coronal plane divides the head into front and back halves. Traditionally, two methods have been used to define the reference frame for the whole head: anatomical landmark and neutral head posture (NHP).

The anatomical landmark method is simple if the face has perfect symmetry. The Frankfort horizontal is constructed using both porions and both orbitales. The sagittal plane is constructed using any three midline landmarks, and the coronal plane, which is made perpendicular to the other planes, is aligned to both porions.

Unfortunately, no human face is perfectly symmetrical. In facial asymmetry, it is complex to use landmarks to construct the reference planes of the face. Let us first consider the Frankfort horizontal. Four points define this plane: right orbitale, left orbitale, right porion, and left porion. However, in facial asymmetry, these points are not coplanar, which creates a dilemma. A plane can be built using any three points. Which three of the four Frankfort points should be used? Each combination results in a different plane. To solve this problem, a best-fitting plane among the four points may be constructed. This is feasible when the head has minor asymmetries. However, in deformities like hemifacial microsomia, this best-fitting method will alter the axial plane by adding a distorted landmark into the equation.

Now let us consider the sagittal plane. This plane can be constructed using three midline landmarks, three pairs of bilateral landmarks, or even just one pair of bilateral landmarks. In the first case, the midline landmarks lie on the sagittal plane. In the second case, the pairs of corresponding bilateral landmarks are used to calculate three midpoints that lie on the sagittal plane. In the last case, a line is drawn between the single pair of bilateral landmarks and the sagittal plane is constructed perpendicular to this line, passing through the midpoint located between the right and left landmarks. In facial asymmetry, any combination of three midline points, any combination of three pairs of bilateral points, or any pair of bilateral points will produce a different sagittal plane. In facial asymmetry, the landmark method may create hundreds of sagittal planes. To solve this problem, a best-fit plane may be used to average all the planes, but the distorted landmarks will skew the plane, making it useless. Besides, in facial asymmetry, a landmark-based sagittal plane will not necessarily be perpendicular to Frankfort horizontal, a fundamental requirement of a Cartesian coordinate system. Finally, there is no point in discussing the coronal plane because if the axial and sagittal planes cannot be constructed accurately, the entire reference frame will be inaccurate.

The NHP method can solve the problems associated with the anatomical landmark method. This posture is unaltered by most facial deformities and can readily be used to create a reference frame. In NHP, a subject stands upright looking straight forward (visual axis is parallel to the ground). The head is not tilted, flexed, nor rotated. Once in this posture, the reference frame of our surroundings (i.e., our world) is transferred to the patient. A plane that parallels the ground (horizontal) turns into the axial plane. Two orthogonal planes, aligned with gravity, turn into the sagittal and coronal planes. The horizontal plane is moved up and down until it divides the face into the upper and lower halves, becoming the axial plane. The vertical planes are moved right and left, and rotated around their vertical axes until one of them divides the face into right and left halves, becoming the midsagittal plane. The other is moved forward and backward until it divides the face into anterior and posterior halves, becoming the coronal plane.

In 3D cephalometry, there are two ways of orienting a head image to the NHP. The first is to scan the head while in the NHP. The second is to scan the head in an arbitrary orientation, and then reorient the image to the NHP. Spiral computed tomography (CT) scanners require the patient to be in a supine position during image acquisition. Thus, it is impossible to position the patient in the NHP during scanning. CBCT scanners are better, as the patient’s head can be upright. Yet chin rests or forehead holders, which are necessary to ensure immobility during scanning, may distort this posture. Therefore, it is more practical to reorient the CT or CBCT model to the NHP after scanning. Currently, three methods are used for this purpose: standardized photographs, laser levels, and digital orientation sensors.

In the first method, standardized frontal and lateral facial photographs are taken with the patient standing in the NHP. A plumb line hangs in the background as a true vertical line. A camera is calibrated so its focal plane is perpendicular to the ground. Before the photos can serve as visual guides to manually reorient the 3D-CT models, they are rotated to the true vertical using the plumb line in the background. While this method is subjective, it is valuable for verifying the outcome of the more advanced methods described below.

In the second method, a self-levelling laser is used to simultaneously project horizontal and vertical crosshair lines. When the patient is in NHP, the horizontal laser is projected onto the malar eminence level, while the vertical laser is simultaneously projected onto the midsagittal plane. Two radiopaque stickers are taped on the skin serving as reference markers along the horizontal laser line, and the other two are taped on the vertical line. The patient then undergoes CBCT along with the reference markers. After the 3D head model is generated, the reference markers are connected to the lines, which are then used to build an anatomical reference frame. A potential disadvantage of this method is that it relies on skin markers that can easily be displaced.

The third method of orienting a head to NHP is to reorient a CT/CBCT model to NHP using a digital orientation sensor. The orientation sensor is attached to a bite-jig. The attachment of the sensor to the bite-jig is designed to ensure that the sensor has the same orientation as the bite-jig frame. While the patient is in NHP with the bite-jig between the teeth and the sensor in front of the face, the pitch, roll, and yaw of the bite-jig frame are recorded. After a CT/CBCT facial model is generated, the recorded sensor orientation is applied to the bite-jig frame, reorienting the head to the recorded NHP. Afterwards, the reference frame can easily be defined. The axial plane is a horizontal plane that passes through the right and left porions, the sagittal plane is the vertical plane that best divides the head into right and left halves, and the coronal plane is a vertical plane that is perpendicular to the other two planes, passing through the right and left porions. The advantage of this method is that it has been validated in vitro and clinically.

It is important to note that there also are problems with the NHP method. Some patients may have difficulty aligning their heads in the NHP. This is particularly true in young children and in patients who have neuromuscular disorders, torticollis, or eye muscle imbalances. In addition, even within the same patient, there are temporal variations in the NHP. Based on published studies, the intra-class reproducibility error of establishing NHP is less than 2°, an error that is not considered clinically significant. However, these studies only examined the reproducibility in pitch. The reproducibilities in roll and yaw have not yet been studied formally. If the same is true for roll, a 2° variation may cause significant problems. For example, if a symmetrical patient during NHP recording rolls her head around nasion by 2°, it will cause the upper incisal midpoint and the pogonion to shift 2.8 mm and 4.2 mm, respectively (assuming that the distance between nasion and the upper central incisor is 80 mm, and the distance between nasion and pogonion is 120 mm). Thus one should use the NHP method with caution.

In the authors’ clinical practice, the digital orientation sensor method is used routinely to place the CT head model in the NHP. At the same time, calibrated photographs are also used to cross-validate the correctness of the CT model NHP. Currently, the Surgical Planning Laboratory of the Houston Methodist Research Institute is developing a new method to calculate the reference frame for the face mathematically. The goal is to eliminate the errors caused by the existing methods.

## Local coordinate system for each facial unit

Measuring the orientation of the different facial units (e.g., maxilla, mandible) makes it necessary to build local coordinate systems for each unit. A simple method for building local coordinate systems for the jaws uses an occlusal triangle. This method is best illustrated using an example ( Fig. 2 ). In this example, a local coordinate system is built for the maxilla. In the first step ( Fig. 2 A), a triangle is constructed on the maxillary dentition. The vertices of the triangle are the right and the left mesiobuccal cusps of the first molars and the incisal midpoint (the point located on the dental midline at the level of the incisal edges). The incisal midpoint is selected as the origin of the local coordinate system. The computer then calculates a vector that arises from the origin and is perpendicular to the surface of the triangle (occlusal plane, Fig. 2 B). This vector becomes the vertical ( *z *) axis of the local coordinate system. The computer next calculates the local anteroposterior ( *y *) axis. This axis originates at the incisal midpoint and bisects the base of the triangle (the line that goes from one first molar to the other) ( Fig. 2 C). In the final step ( Fig. 2 D and E), the computer calculates the transverse ( *x *) axis of the local coordinate system. This axis rises from the incisal midpoint pointing perpendicular to *z *and *y *axes.

Currently, the Surgical Planning Laboratory is investigating an approach that uses a principal component analysis (PCA) to automatically construct local coordinate systems. Unlike the triangle method, which uses only three points, PCA can estimate the *x *, *y *, and *z *coordinate axes using a larger group of points, e.g., all the cusps of the maxillary teeth. Thus, the PCA approach may estimate more accurately the local coordinate systems of the jaws.

## Size measurements

Measuring size in 3D cephalometry is straightforward. Size is an intrinsic property of an object that is independent of the space it occupies. We can measure size in 3D space using linear measurements (e.g., length, width, and height), areas, or volumes. In 3D cephalometry, linear measurements of size are calculated as the distance between two points (landmarks) located in 3D space. For example the maxillary width can be measured as the distance between the two first molars.

## Position measurements

Position refers to the location of a point in space. Its measurement is always relative and requires a reference frame. In 3D cephalometry, we need to determine the position of the jaws, which are complex 3D objects made up of thousands of points. Each point that makes up a jaw has a different position. In order to measure the position, we must select a point to represent the whole jaw. However, since there is no perfect point, clinicians have had to use several points to represent the jaws. For example, the anteroposterior position of the maxilla has been measured at the anterior nasal spine (ANS), point A, and the upper incisal midpoint.

Measuring position in one, two, and three dimensions requires one, two, and three numbers. Thus, any system that measures jaw position in 3D space must use three numbers. The position of an object can be expressed either by three distances (3D Cartesian coordinate system), or by two distances and one angle (cylindrical coordinate system), or by one distance and two angles (spherical coordinate system). Mathematically, these three coordinate systems are all convertible. Since the spherical coordinate system is more complex than the other two and is seldom used in 3D cephalometry, it is not discussed in this article.

A 3D Cartesian coordinate system consists of three mutually perpendicular axes that cross each other at an origin. The transverse axis is *x *, the anteroposterior axis is *y *, and the superoinferior axis is *z *. Each pair of axes forms a reference plane. The absolute position of any landmark is expressed using three coordinates ( *x *, *y *, *z *), which represent the transverse, anteroposterior, and vertical positions, respectively. The relative position between two landmarks can be calculated easily. For example, if the anteroposterior coordinate of pogonion is 62 mm and the anteroposterior coordinate of nasion is 60 mm, pogonion is then 2 mm in front of nasion (62 − 60 = 2 mm). The same is true for relative vertical and transverse positions.

In 2D cephalometry, we often use the combination of one angle and one distance to determine the position of a point. This type of measurement typifies a ‘polar’ coordinate system ( Fig. 3 ). A polar system resides on a plane and consists of a fixed point (the pole) and a ray (the polar axis). From the pole, the polar axis points in a fixed direction. To measure the position of a point in a polar system, a line segment (called the radius, *r *) is first drawn from the point that we are locating to the pole. The position of the point is then determined by measuring the length of the radius ( *r *) and the angle theta ( *θ *) between the radius and the polar axis. The position of a point in 2D cephalometry is expressed using two coordinates ( *r *, *θ *).

In 3D cephalometry, the same measurements are completed in a cylindrical system by adding a cylindrical axis to the 2D polar system ( Fig. 4 ). The cylindrical axis is perpendicular to the plane of the polar system and passes through the pole. In order to measure the position of a point in a cylindrical system, the point is first projected onto the plane of the polar system, along the cylindrical axis. On this plane, the first two coordinates ( *r *, *θ *) 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 *, *θ *, *z *), including two distances and one angle.

## Orientation measurements

Orientation is relative. Although there are many different ways to measure the orientation in 3D cephalometry, we usually measure the orientation as three rotations following a specific order. The measurement is done by rotating a facial unit from its reference alignment to its current placement. The reference alignment is the (global) coordinate system of the whole head, while the current placement is the (local) coordinate system of a facial unit. The sequential rotations are yaw, roll, and pitch. In our approach, these rotations occur around the axes of the local coordinate system, which is the one that belongs to the facial unit being measured. Yaw is the rotation around the vertical ( *z *) axis. Roll is the rotation around the anteroposterior ( *y *) axis. Pitch is rotation around the transverse ( *x *) axis.

The order in which one measures pitch, roll, and yaw is important. These rotations are not commutative and the result of following the pitch–roll–yaw order is different from the result of following the yaw–roll–pitch order ( Fig. 5 ). For 3D cephalometry, we recommend a yaw–roll–pitch sequence. Most facial units (i.e., maxilla and mandible) normally have some degree of pitch, while their ideal roll and yaw is zero. This strategy is helpful in that it minimizes the yaw and roll angles.

## Shape measurements

By definition, shape is the geometric attribute of an object that is not size, position, or orientation. Thus, the measurement of shape should not be influenced by size, position, or orientation. A method that follows this rule is a Procrustes analysis. In this analysis, shape is evaluated by comparing two objects after both have been scaled to the same size, placed in the same location, and rotated to the best possible alignment.

Figure 6 shows the example of a deformed mandible (pink) that is being compared with an average mandible (blue). Both mandibles differ in size, position, orientation, and shape. To assess shape, a Procrustes superimposition first scales the mandibles to the same size. It then places both mandibles in the same position (both centroids at the origin of the Cartesian coordinate system). Finally, it rotates the deformed mandible until it is best aligned with the normal mandible. After the differences in size, position, and orientation are removed, one can better appreciate the distortion of the deformed mandible. In this example, the deformed mandible has an obtuse gonial angle and a short ramus height.

## Symmetry measurements

The two elements that relate to symmetry are ‘object symmetry’ and ‘symmetrical alignment’. Object symmetry refers to the intrinsic – mirror – symmetry that each facial unit should have. Symmetric alignment refers to the alignment of each facial unit with the midsagittal plane of the face.

## Object symmetry

The intrinsic symmetry of the dental arches can be assessed using a simple occlusal triangle. The triangle is constructed by connecting the incisal midpoint and the mesiobuccal cusps of the first molars. The triangle is isosceles in patients with symmetric dental arches, with two equal lateral sides and two equal base angles. The triangle is scalene in patients with asymmetric dental arches, with three different sides and three different angles. The degree of intrinsic asymmetry of a dental arch is proportional to the difference between the basal angles: the larger the difference, the greater the asymmetry. Alternatively, one can measure the intrinsic asymmetry by comparing the length of the lateral sides of the triangle. The greater the absolute difference between the sides of the triangle, the greater the asymmetry.

One can also measure more elaborately the intrinsic symmetry of any facial unit using a Procrustes analysis of symmetry. It begins by dividing the unit into two half-forms: right and left. In order to divide the form, the landmarks of the facial unit are divided into two groups: the right set contains all the right landmarks as well as all the midline landmarks; the left set contains all the left landmarks and again the midline landmarks. Next, we superimpose the two half-forms using a series of transformations. However, prior to this step, we have to pick one half-form to be the object of the transformations and the other to be the target. Which half-form (right or left) becomes the object is inconsequential. The first transformation reflects (flips) the object half-form around its anteroposterior axis, creating a mirror image. This operation makes the half-forms comparable. The second transformation centres both the half-forms on a Cartesian coordinate system. In order to centre the half-forms, we first have to determine their centroids. Then, we translate both forms, so that their centroids are on the origin of the Cartesian system. The third transformation rotates the object half-form around its centroid until the form is aligned to the target. The final alignment is reached when the sum of all the distances between the corresponding landmarks of the right and the left half-forms is minimal. To quantify the degree of asymmetry for each landmark, we measure the distances between analogous right and left landmarks. These distances are (directly) proportional to the degree of asymmetry.

## Symmetrical alignment

The other element of the symmetry measurement is the symmetrical alignment. Facial asymmetry can be the result of misalignment of a facial unit about the midsagittal plane. When a unit is not symmetrically aligned, the degree of misalignment (i.e., asymmetry) can be quantified by measuring the transformations (manoeuvres) that are necessary to realign it. The three parameters of symmetric alignment are transverse translation, roll, and yaw. Transverse translation places the midpoint on the midsagittal plane. Roll rotates the unit around the midpoint until right and left landmarks are vertically aligned. Yaw rotates the unit around the midpoint, minimizing the distance differences between the corresponding bilateral landmarks and the coronal and midsagittal reference planes. Zero is the ideal value for transverse position, yaw, and roll.