Mechanics of tooth movement Richard J. Smith, D.M.D., Ph.D.,* and Charles J. Burstone, D.D.S., MS.** Or Smith Baltimore, Md., and Farmington, Corm.

Orthodontic forces can be treated mathematically as vectors. When more than one force is applied to a tooth, the forces can be combined to determine a single overall resultant. Forces can also be divided into components in order to determine effects parallel and perpendicular to the occlusal plane, Frankfort horizontal, or the long axis of the tooth. Forces produce either translation (bodily movement), rotation, or a combination of translation and rotation, depending upon the relationship of the line of action of the force to the center of resistance of the tooth. The tendency to rotate is due to the moment of the force, which is equal to force magnitude multiplied by the perpendicular distance of the line of action to the center of resistance. The only force system that can produce pure rotation (a moment with no net force) is a couple, which is two equal and opposite, noncolinear but parallel forces. The movement of a tooth (or a set of teeth) can be described through the use of a center of rotation. The ratio between the net moment and net force on a tooth (M/F ratio) with reference to the center of resistance determines the center of rotation. Since most forces are applied at the bracket, it is necessary to compute equivalent force systems at the center of resistance in order to predict tooth movement. A graph of the M/F ratio plotted against the center of rotation illustrates the precision required for controlled tooth movement.

Key words: Force, moment, couple, center of resistance, center of rotation, orthodontic tooth movement

T he literature on orthodontic biomechanics usually concerns either specific applications of interest to clinicians or basic questions primarily of interest to researchers. Few articles have attempted to explain biomechanical principles by an approach that would allow the clinician without a background in engineering to understand the concepts and their potential for clini- cal relevance. In this article, we attempt to review for the clinician the basic relationships between forces and tooth movement.

SCALARSANDVECTORS Physical properties (such as distance, weight, tem-

perature, and force) are treated mathematically as either scalars or vectors. Scalars, including temperature and weight, do not have a direction and are completely described by their magnitude. Vectors, on the other hand, have both magnitude and direction. Forces may be represented by vectors. In addition to magnitude, direction and point of application must be taken into account. Direction consists of two properties-a line of action and a sense.

In order to indicate all of these features, force vec- tors are represented as arrows (Fig. 1). The length of

*Associate Professor and Director, Postgraduate Program, Department of Orthodontics, University of Maryland Dental School. **Professor and Head, Department of Orthodontics, University of Connecticut School of Dental Medicine.

294

the arrow is proportional to the magnitude of the force. The actual length is arbitrary and will vary with each particular diagram, as with the scale on a map. In one case each millimeter of length might be equal to 10 grams and in another case, to 100 grams. What is es- sential is that all the forces in a given diagram have the same scale, particularly if they are to be graphically added or subtracted. Dimensionally, forces are the product of mass and acceleration and are measured in units such as newtons or poundals. This complexity is irrelevant to clinical needs, and the magnitude of forces can be measured in common units of weight (such as ounces) or mass (such as grams).

The point of application of a force is indicated by the origin of the arrow, which in Fig. 1 is at the bracket. This is simply the point of contact between the body being moved and the applied force.

Direction is indicated by the body of the arrow itself and the arrowhead. Without the head of the ar- row, the body alone indicates the line of action. The sense is determined by which end we put the ar- rowhead on.

Almost every force applied in clinical conditions will have effects in three planes of space. Throughout this review, we will deal only with the idealized and simplified analysis of forces and movements in two dimensions. Many mechanical principles become much more complex when three-dimensional aspects are considered.

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Mechanics of tooth movement 295

b cl Magnitude

Point of Application

Fig. 1. Forces may be treated as vectors and are convenienUy represented as arrows. A force vector is characterized by four features: magnitude, point of application, line of action, and sense.

RESULTANTSANDCOMPONENTSOF ORTHODONTIC FORCE SYSTEMS

Teeth are often acted upon by more than one force. Since the movement of a tooth (or any object) is de- termined by the net effect of all forces on it, it is neces- sary to combine applied forces to determine a single net force, or resultant.

At other times there may be a force on a tooth that we wish to break up into components. For example, a cervical headgear to maxillary molars will move the molars in both the occlusal and distal directions. It may be useful to resolve the headgear force into the compo- nents that are parallel and perpendicular to the occlusal plane, in order to determine the magnitude of force in each of these directions.

Resultants from forces with a common point of application

In combining forces, the simplest case is for two forces that have a common point of application (Fig. 2, A). In order to determine the resultant (combined effect of the two forces), the two vectors are considered to be sides of a parallelogram. If the parallelogram is com- pleted (Fig. 2, B), the resultant is the diagonal. Its length indicates the magnitude of the resultant force on the same scale as the original forces (Fig. 2, C).

It is important to understand that the resultant will have the identical effect on movement of the tooth as the two separate forces. A tooth or a set of teeth moves in response to the net effect of all forces. If the resultant is the same, the movement will be the same, regardless of how many forces are applied or the direction of individual forces.

For combining more than two forces that have a common point of application, a series of successive parallelograms is constructed. Each time, the resultant from any two forces replaces those forces and is used to construct the next parallelogram. The sequence in which forces are combined is of no consequence.

Fig. 2. The parallelogram method of determining the resultant of two forces with a common point of application.

Different points of application

Different forces on a tooth are not usually applied at the same point, as was assumed in the previous example.

Use of the parallelogram method to combine forces applied at different points depends upon one of the basic laws of static mechanics: The law of trans- missibility of force. This law can be stated as follows:

When considering the external effects of a force on a rigid body, the force may be considered to have a point of application anywhere along its line of action.

The law of transmissibility of force indicates that, to combine forces, a point of application may be se- lected anywhere along the line of action of the force. The line of action may be extended forward or back- ward off the tooth to construct a totally artificial point of application in space. The combination of two forces with different points of application is illustrated in Fig. 3.

Resolving a force into components

Rather than combine two or more forces into a sin- gle resultant, it is often useful to divide a single force into components at right angles to each other. Usually, the objective is to determine how much force is being delivered perpendicular and parallel to the occlusal plane, Frankfort horizontal, or the long axis of the tooth.

Fig. 4, A illustrates the force on a maxillary canine from a Class II elastic. How much of the force is acting to retract the canine, and how much to extrude it?

In this case, the parallelogram procedure for com- bining forces is reversed. Let us consider the Class II elastic force vector to be the diagonal of some parallel- ogram. Because the components are parallel and per- pendicular to the occlusal plane (that is, at right angles to each other), the parallelogram will be a rectangle (Fig. 4, B). When the rectangle is drawn surrounding the diagonal, the force components are indicated by the length and width of the rectangle (Fig. 4, C).

296 Smith and Burstone Am. J. Orthod. April 1984

Fig. 3. The resultant of two forces with different points of appli- cation can be determined by extending the lines of action to construct a common point of application. In A, a maxillary canine is being retracted with a Class I elastic to a vertical hook and a Class II elastic to the bracket. In B, the lines of action have been extended until they intersect. For the purposes of drawing the parallelogram, this will be considered the common point of application. In C, the two vectors are moved along their lines of action to the constructed point of application. The vec- tors must maintain their original lengths in order to properly determine the magnitude of the resultant. In D, the parallelo- gram has been constructed and the resultant force obtained. Its direction (both sense and line of action) and magnitude are now known. Since the point of application is artificial (and of no consequence), the resultant can either be left as drawn or moved forward on its line of action so that the point of applica- tion is on the tooth (E).

With more than one force on a tooth, there are two methods for determining overall vertical and horizontal forces. First, the applied forces can be combined into a single resultant, as described previously, and then this resultant can be resolved into horizontal and vertical components. Alternatively, the horizontal and vertical components for each force can be determined sepa- rately, and these components then combined to deter- mine net horizontal and vertical vectors. Direction must be taken into account. If one force has an intrusive vertical component and a second force has an extrusive vertical component, one direction must be selected as negative and the other as positive, so that addition will result in the correct net value.

FORCE AND MOVEMENT Center of resistance

Every object or free body has one point on which it can (at least in theory) be perfectly balanced. This point

Fig. 4. The parallelogram method for resolving a force into ver- tical and horizontal components. (See text for discussion.)

is known as the center of gravity.* For many physical calculations, it is as if the rest of the object does not exist, and all of the weight is concentrated at this single point.

The movement of a free body depends upon the relationship of the line of action of the force to the center of gravity. However, teeth present an additional complication. They are not free to move in response to a force; rather, they are restrained by periodontal struc- tures which are not uniform around the tooth (involving the root but not the crown).

In a restrained body, such as a tooth, a point analo- gous to the center of gravity is used; this is called the center of resistance. By definition, a force with a line of action passing through the center of resistance produces translation. The center of resistance of a single-rooted tooth is on the long axis of the tooth, probably between one third and one half of the root length apical to the alveolar crest.’ For a multirooted tooth, the center of resistance is probably between the roots, 1 or 2 mm apical to the furcation.”

Two important points are evident from the defi- nition of the center of resistance. First, the position of the center of resistance varies with root length.” Maxil- lary canines, having longer roots than maxillary lateral incisors, will have a center of resistance farther from the bracket. Since (as will be discussed in detail) the tooth movement resulting from a force delivered at the bracket depends upon the distance of the line of action of the force from the center of resistance, identical forces applied to teeth with different root lengths can have different effects. A second important point is that the center of resistance varies with alveolar bone height. The movement of teeth in adults with alveolar bone loss will be different than in adolescents.

*Although technically different, the term ceruer of gravity may be used inter- changeably with the term cenfer of mass in all clinical problems.

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Mechanics of tooth movement 297

Fig. 5. Types of tooth movement. A is translation, B is rotation, and C is a combination of translation and rotation.

Types of movement

A tooth can move in one of three general ways: (1) translation, (2) pure rotation, or (3) combined rotation and translation. Translation, or bodily movement, oc- curs when all points on the tooth move an equal dis- tance in the same direction (Fig. 5, A).

Rotation is used here in a restricted sense. As one of the three basic types of movement, rotation indicates movement of points of the tooth along the arc of a circle, with the center of resistance being the center of the circle (Fig. 5, B).

Any movement that is not pure translation or rota- tion can be described as a combination of these two forms of movement (Fig. 5, C).

Effect of forces

Translation: Ifthe line of action of an appliedforce passes through the center of resistance of a tooth, the tooth will respond with pure bodily movement (transla- tion) in the direction of the line of action of the applied force.

Note that the point of application of the force does not affect its line of action. For example, consider a vertical hook welded to the bracket of a maxillary canine (Fig. 6). If the hook has been positioned so that its height is at the center of resistance, an elastic force attached to the hook will translate the tooth, even though the hook itself is attached to the bracket.

Moment of a force. If the line of action of an

Fig. 6. The line of action of the force passes through the center of resistance. This tooth will translate, even though the point of attachment to the tooth is at the bracket.

applied force does not pass through the center of resis- tance, the force will produce some rotation. The poten- tial for rotation is measured as a moment, and the mag- nitude of the moment is equal to the magnitude of the force multiplied by the perpendicular distance of the line of action of the force to the center of resistance. Thus, if the force is measured in grams and the distance in millimeters, the moment of the force will have units of ‘ ‘gram-millimeters. ’ ’ Fig. 7 illustrates how to de- termine the perpendicular distance from the line of ac- tion to the center of resistance.

Note that either changing the magnitude of the force or changing the perpendicular distance of the line of action of the force to the center of resistance has an equal effect on the magnitude of the moment. Doubling the force or keeping the force constant but doubling its distance from the center of resistance will have iden- tical effects on the tendency of the tooth to rotate.

If forces are indicated by straight arrows, moments can be symbolized by curved arrows. With two- dimensional diagrams, clockwise moments will be ar- bitrarily defined as positive and counterclockwise mo- ments negative. Values can then be added together to determine the net moment on a tooth relative to a par- ticular point, such as the center of resistance. Point of application and line of action are not needed; nor are graphic methods of addition.

The direction of a moment can be determined by continuing the line of action of the force around the center of resistance, as shown in Fig. 8.

In summary, the moment of a force is equal to the magnitude of the force multiplied by the perpendicular distance from the line of action to the point of reference

298 Smith and Burstone Am. .I. Orthod. April 1984

Fig. 7. The moment of a force is equal to the magnitude of the force multiplied by the perpendicular distance from its line of action to the center of resistance.

Fig. 8. The direction of the moment of a force can be deter- mined by continuing the line of action around the center of resistance.

and is measured in units such as gram-millimeters. As with forces, there is only a single net moment on a tooth, which can be calculated by summing individual clockwise and counterclockwise moments.

Combined translation and rotation. We now come to an important and potentially confusing concept. The forces illustrated in Figs. 7 and 8 will not produce the “pure rotation” described earlier. Rather, the center of resistance will rotate but will also translate through space.* In fact, the translation of the center of resis- tance will be similar (and perhaps, in theory, identical) to that which would occur if the force passed through the center of resistance.

Fig. 9 illustrates two 100 gm forces applied to a canine. In both Fig. 9, A and Fig. 9, B, there is a distal force. Both of these forces will have a similar effect in moving the center of resistance distally. What is differ- ent about the forces is that in Fig. 9, B there is also a moment tending to rotate the tooth, while in Fig. 9, A there is not. The result is that the type of tooth move- ment from these forces might be as illustrated. If the line of action of a force passes through the center of resistance, the tooth will translate along the line of action. If the line of action does not pass through the center of resistance, the center of resistance will trans- late as if the force did pass through it, but the tooth will

*Technically, the center of resistance is a point and, by definition, points do not rotate. The concept of rotation of the center of resistance is a clinically useful simplification.

Fig. 9. The force in A, passing through the center of resistance, will result in translation of the tooth. The force in B, at the bracket, will also translate the tooth but, in addition, will cause a rotation because of the moment created at the center of re- sistance.

also rotate, since the applied force produces a moment about the center of resistance. The result is a combina- tion of translation and rotation.

Rotation, couples

Three types of motion were originally defined: pure translation, pure rotation, and a combination of the two. Translation occurs when the line of action of the net force on a tooth passes through the center of resis- tance, and the combined motion occurs when the line of action does not pass through the center of resistance.

A single force, therefore, cannot produce pure ro- tation, meaning a tooth spinning about its center of resistance. Rather, the only system of forces that can produce pure rotation of a tooth is called a couple.

A couple consists of two forces of equal magnitude, with parallel but noncolinear lines of action and oppo- site senses.

The wire in Fig. 10, A is pictured with a couple. The forces are the same magnitude, each being 50 gm; they are parallel to each other but not coincident; and they face in opposite directions.

In order to demonstrate that this wire will undergo pure rotation, consider the effects of each force inde- pendently and then combine their effects. Each force will tend to move the center of resistance of the wire

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Mechanics of tooth movement 299

Fig. 10. A, Two equal and opposite, parallel, noncolinear forces form a couple. B, The translational effects of the forces cancel each other out, but the moments of each force combine. The result is a moment with no net force.

t

B ,,----.,

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I \ L I 1 B I I

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C I

Fig. 12. In each figure the solid line indicates the original posi- tion of the tooth. With the center of rotation indicated by the solid circle, the type of tooth movement that could result is indicated by the dotted outlines. (See text for discussion.)

i

,/’ I

Fig. 11. The couple illustrated is identical to that in Fig. 10, A except that it is displaced 20 mm to the right. If the moments of each force are added together, the result is identical to that in Fig. 10. (See text for details.)

through space just as if its line of action passed through the center of resistance. The upper force will push the wire down with a force of 50 gm, while the lower force will push the wire up with a force of the same mag- nitude. These translational effects cancel each other out, resulting in no net translational force.

Will the moments of these forces also cancel each other out? The magnitude of each moment about the center will be 50 gm multiplied by the perpendicular distance of each force to the center of resistance, which in the example is 10 mm, for a moment of 500 gm-mm. However, each force produces a 500 gm-mm moment in the same direction (clockwise), for a total moment of 1000 gm-mm (Fig. 10, B). There is no net translational force but a large net moment, which will result in pure rotation. This result also demonstrates that the moment of a couple is equal to the magnitude of one of the

center of rotation

Fig. 13. A and B represent the cusp tip before and after move- ment. A line has been drawn connecting these points. At the midpoint of this line a perpendicular has been constructed. The point at which this perpendicular intersects any other perpen- dicular constructed in a similar manner (in the figure the apex has been selected as the other point) is the center of rotation.

forces multiplied by the perpendicular distance between them, or 50 gm x 20 mm in this example.

It will now be demonstrated that it does not matter where on a rigid object a couple is applied; the external effect is the same.

Consider a force system of two 50-gm forces sepa- rated by a distance of 20 mm, as in Fig. 10, A, except that, instead of being centered around the center of resistance, the forces are displaced off to one side (Fig. 11, A).

300 Smith and Burstone Am. J. Orthod. April 1984

/ 4 / / //’ / / /

center of rotation

Fig. 14. The center of rotation during incisor intrusion (or extru- sion) is perpendicular to the long axis of the tooth. The amount of palatal root torque accompanying the intrusion decreases as the center of rotation moves farther from the tooth along the same perpendicular axis.

As in the symmetrical case, the translational effects of the vertical forces cancel each other out. The mo- ments of these forces about the center of resistance equal the magnitude of each force multiplied by its perpendicular distance to this point. The upward di- rected force will have a moment of 50 gm x 10 mm, or 500 gm-mm (Fig. 11, B). The downward-directed force will produce a moment of 50 gm X 30 mm, or 1,500 gm-mm (Fig. 11, C). However, the 500 gm-mm mo- ment is negative (counterclockwise) and the 1,500 gm-mm moment is positive (clockwise). The sum of these two moments will be 1,000 gm-mm (in a clock- wise direction), exactly the same as when each of the forces was 10 mm away from the center of resistance.

It does not matter where a couple is applied to an object; the net effect is a moment equal to the mag- nitude of one of the forces multiplied by the distance between them. The irrelevance of position is sometimes stated as the fact that a couple is a “free vector. ”

It is important to distinguish between forces, the moment of a force, and the moment of a couple. Only forces are actually applied to an object as a physical entity. Moments are a measure of the “turning ten- dency” produced by a force. The moment of a force is always relative to a point of reference, so that the mo- ment of a force will be low relative to a point close to

M

v- center of rotation

center /-- I ‘\ \ of rotation

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( (

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1

I ‘-, 4’

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Fig. 15. The concept of a center of rotation can be used to define any type of tooth movement in any plane of space. Illus- trated in this figure are the centers of rotation for twc types of first-order corrections.

the line of action and high for a point with a large perpendicular distance to the line of action. A couple is no more than a particular configuration of forces which have an inherent moment. This moment is not relative to any point.

For additional discussions of the concepts reviewed in this section, we refer readers to the works of Mulii- gan,4 Hocevar,” and Thurow.6

THE CENTER OF ROTATION

The movement of a tooth can be described with more precision than is indicated by the three general categories of rotation, translation, and combined mo- tion. For a more complete description, the concept of a center of rotation is used.

If a model of a tooth is attached to a piece of paper by a pin, the point with the pin in it cannot move, and this point becomes the center of rotation about which the tooth can spin. If the pin is placed at the incisal edge, only movement of the root is possible (Fig. 12, A); if it is placed at the root apex, movement is limited to crown tipping (Fig. 12, B). In each case, the center of rotation is determined by the position of the pin. Thus, in two dimensional figures, the center of rotation may be defined as a point about which a body appears to have rotated, as determined from its initial and final positions.

The center of rotation can be at any position on or off a tooth. In Fig. 12, C, for example, when it is a few millimeters incisal to the apex, some movement of the root apex will occur in the direction opposite the crown movement. Retraction of the maxillary canine with in- sufficient rcot movement has a center of rotation above the apex of the tooth (Fig. 13). The more nearly trans- lational the movement, the farther apically the center of rotation would be located. In the extreme case, with perfect translation, the center of rotation can be defined as being an infinite distance away.?

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Mechanics of tooth movement 301

A B C D

Fig. 16. “Stick figures” can be used to predict tooth movement and centers of rotation. The effects of the force and couple are considered separately. (See text for discussion.)

A simple method for determining a center of rota- tion is to take any two points on the tooth and connect the before and after positions of each point with a line. The intersection of the perpendicular bisectors of these lines is the center of rotation (Fig. 13).“, 8, s

Although a single center of rotation can be con- structed for any starting and ending positions of a tooth, it does not follow that the single point actually acted as the center of rotation for the entire movement. The tooth might have arrived at its final position by follow- ing an irregular path, tipping first one way and then another. As a tooth moves, the forces on it continu- ously undergo slight changes, so that a changing center of rotation is the rule rather than the exception. In de- termining the relationship between a force system and the center of rotation of the resulting movement, all that can really be determined is an instantaneous center of rotation.‘O Nevertheless, this calculation can be clini- cally useful.

Centers of rotation and tooth movement

A center of rotation can be determined for any two-dimensional tooth movement. One of the require- ments for thinking mechanically about tooth movement is to consider the centers of rotation for different types of tooth movement, in addition to common descriptions such as bodily movement or torque. Table I shows some examples of the relationship between typical de- scriptions of tooth movement and centers of rotation.”

Centers of rotation do not need to be along the long axis of a tooth. During intrusion, for example, the cen- ter of rotation falls on a line which is approximately perpendicular to the long axis (Fig. 14). Centers of rotation can also be defined for rotation of a tooth as seen from the occlusal aspect (Fig. 15). In an interest- ing analysis, Hurd and Nikolai” have proposed that a center of rotation for combined vertical and transverse movement can be defined as the point which moves the least on the line of the long axis of the tooth. In agree- ment with Hocevar,” we prefer not to restrict the center of rotation to the long axis but, rather, to find the

Table I

Descriptive terminology for types of movements

Translation (bodily movement) Uncontrolled tipping

Controlled tipping Root movement

Position of the center of rotation

Infinity Slightly apical to the

center of resistance Apex Incisal (occlusal) edge

geometric point about which no moment has occurred. This is more consistent with general use of the term.“, g However, as NikoW3 points out, the actual movement of a tooth does involve a series of changing centers of rotation.

Hocevals suggests that centers or rotation imply curvilinear motion and that the concept of a center of rotation has never been proved. However, GoldsteinI provides proof that such a point exists for any two- dimensional movement. Full documentation of two- dimensional movement with respect to another body or a coordinate system requires at least three independent measurements, since a body moving in a plane has three degrees of freedom.s One such set of mea- surements consists of the two coordinates of the center of rotation plus the angle of rotation about that center.” Another set, as suggested by Hocevar,” is a translation vector plus an angle of rotation. A third set involves translation vectors for two points on the object. All of these approaches to describing movement are valid and useful, although a description of motion using the concept of a center of rotation is probably the most common throughout the literature on clinical biome- chanics.“, g It should be noted that all of these ap- proaches describe only the final change in position of an object, not the actual path of movement.

Force systems and centers of rotation

Controlling the center of rotation gives precise control over the type (but not the extent) of tooth movement.‘” Since all the forces on a tooth can be

302 Smith and Burstone Am. J. Orthod. April 1984

l center of rotation

Fig. 17. The couple is in the opposite direction to the couple illustrated in Fig. 16. As a result, the apex moves less than the crown, and the center of rotation is apical to the tooth.

summed to a single force, and all the moments to a single moment, it follows that a single force plus a single couple can produce any type of tooth move- ment-in other words, any center of rotation. In this section we will review the relationship between the net moment and the net force at the center of resistance and the resulting center of rotation.

The extreme situations have been discussed: If the net effect on the tooth at the center of resistance is a force only, the center of rotation will be “at infinity” (the tooth will translate), and if there is a net moment only, the center of rotation will be at the center of resistance (the tooth will rotate). The only means of producing a moment with no net force is with a couple.

For any tooth movement other than these simple conditions, a force and a couple will be present at the center of resistance.

In order to understand these combined effects, “stick diagrams ’ ’ are helpful. Fig. 16, A illustrates a tooth with a force and couple at the center of resistance. The general type of movement that will occur can be estimated by replacing the tooth with a simple line (Fig. 16, B) and applying the force and couple separately. In Fig. 16, C the effect of the force alone would be to translate the stick, and in Fig. 16, D rotating the al- ready translated stick in the direction indicated by the couple will result in more root movement than crown movement compared with the original position.

Let us consider the effects of changing the ratio between the force and the couple. Suppose we start with a pure force producing translation (Fig. 16, C) and

JHHI

e e C

zb

_,_. . . . .._.,,, _.. . . .- _, . . . . . . -- ._

0

B C Fig. 18. Teeth move according to the forces and movements acting at the center of resistance. Most orthodontic forces are applied to the tooth at the bracket. Understanding the relation- ship between force systems at the bracket and the center of resistance requires using the rules for equivalent force systems. The force in A is equivalent to the force system in C, not to the one in 8.

then add a very small counterclockwise couple. With the force only, the center of rotation is an “infinite” distance away. As the couple increases, it brings the center of rotation in toward the tooth from infinity. If the couple is large enough relative tc?the force, the amount of translation could become almost negligible in comparison to the rotation, and the center of rotation would be near the center of resistance. With the mag- nitude of the counterclockwise couple increasing, the center of rotation moves in from beyond the occlusal aspect of the tooth.

If the couple added in Fig. 16, C produced a clockwise moment rather than a counterclockwise one, the center of rotation would be apical to the tooth rather than incisal. Again, as the couple increases, the center of rotation moves closer, but this time, it is approach- ing from beyond the apex (Fig. 17).

EQUIVALENT SYSTEMS

The preceding discussion has primarily concerned forces applied at the center of resistance and couples. Occasionally, attachments are used that allow forces to act directly in this area. For the most part, however, forces are applied on the crown of the tooth. It has already been mentioned that the moment of a force with respect to the center of resistance depends upon the perpendicular distance of its line of action to the center of resistance, and therefore otherwise identical forces placed at different positions on the tooth have different effects on tooth movement. In order to determine how a tooth will move, it is useful to evaluate the force system

Volume 85 Number 4

f-3 Q 1400g-mm 1 ZOOg-mm

14mm .a 4 12mm

OOQ 1oog &

A B

Mechanics of tooth movement

rc 0 f-a 1200g-mm + ZOOg-mm = 1400g-mm

303

Fig. 19. Force systems at the apex (or any other arbitrary point) change when bracket height is altered (See text for discussion.)

at the bracket to determine the equivalent force system at the center of resistance.

Consider a distal force at the bracket of a maxillary canine (Fig. 18, A). This tooth will not exhibit pure translation, because the force also produces a moment about the center of resistance. Thus, Fig. 18, B, where the force has been moved to the center of resistance, is not equivalent to Fig. 18, A because if a force is applied as in Fig. 18, B, this tooth will translate. What else needs to be applied at the center of resistance in Fig. 18, B to have the tooth move the same way that it would from the force in Fig. 18, A? The answer is a clockwise couple (Fig. 18, C). The size of this couple, measured in gram-millimeters, depends upon the mag- nitude of the force in Fig. 18, A and its perpendicular distance to the center of resistance. Thus, the force systems pictured in Fig. 18, A and Fig. 18, C are equivalent, meaning that they would produce identical movement. There are three simple rules that allow the calculation of equivalent force systems. Two force sys- tems are equivalent if (1) the sums of forces in the x direction are identical, (2) the sums of forces in the y direction are identical, (3) the sums of moments about any point are identical.

Before proceeding with some examples, the third requirement for equivalence requires a brief explana- tion. So far, the moment of a force has been measured as the perpendicular distance to a specific point, the center of resistance. But the moment of a force can be calculated about any point by simply measuring the perpendicular distance from the line of action of the force to the point in question and calculating the mo- ment as the measured distance multiplied by the mag- nitude of the force. The third requirement listed for equivalent force systems requires that at any point, the moments be equal in magnitude and direction. Some examples should serve to clarify the concept of equivalence.

8mm

1200g-mm

B Fig. 20. For bodily movement (translation), a pure force is re- quired at the center of resistance. The equivalent force system at the bracket requires a large moment from a couple to coun- teract the moment of the force.

Fig. 19 illustrates two identical maxillary canines. The canine in Fig. 19, A is being retracted with a lOO-gm Class I elastic force. The manner of retraction has been satisfactory, and we would like to continue, but in rebonding the tooth after the bracket breaks, the bracket has been placed 2 mm more gingival (Fig. 19, B). If the fact that there will be a slight change in angulation of the Class I force is discounted, does the force system have to be changed in order for the tooth to continue moving as before? In other words, what is the force system that will result in equivalent tooth movement at the new bracket position?

The first rule of equivalence is that the sum of forces in the x direction is equal in both systems. In Fig. 19, A there is a lOO-gm distal force in the x direc- tion, so in Fig. 19, B there must be one also. Next, the forces in the y direction must be equal; there are none in Fig. 19, A so there must be none in Fig. 19, B. Finally, the sums of moments about any point must be equal.

304 Smith and Burstone Am. J. Orrhod. April 1984

Fig. 21. If the moment of the couple in Fig. 20, B were not large enough to counteract the moment of the force, there would be a net moment at the center of resistance (Fig. 21, A). The resulting tooth movement is analyzed in B and C, in which the force produces translation and the net moment produces rotation. The larger the underestimation of the required couple, the greater the resulting rotation (D).

1400g-mm

A B C Fig. 22. The previous illustration (Fig. 21) demonstrated the consequences of underestimating the counterclockwise couple needed for bodily movement. If the couple is overestimated, as in this figure, the rotation of the tooth is in the opposite direction

Arbitrarily, we can sum moments about the apex. If in Fig. 19, A the bracket is 14 mm from the root apex, the moment of the force about the apex will be 100 gm x 14 mm or 1,400 gm-mm. But when a lOO-gm force is applied to the bracket in Fig. 19, B, the mo- ment of the force about the apex will be 100 gm x 12 mm or 1,200 gm-mm. To make Fig. 19, A and B equivalent, an additional 200 gm-mm clockwise mo- ment about the apex must be added to Fig. 19, B. A single force cannot be used to produce this moment because it would violate one of the first two require- ments for equilibrium. With a couple, however, the x and y resultant forces are unchanged, and the couple has the same effect wherever it is applied. Thus, a system equivalent to Fig. 19, A would involve applica- tion at the bracket of the same lOO-gm force plus the 200 gm-mm couple (Fig. 19, C).

If the bracket in Fig. 19, B had been placed 2 mm to the occlusal instead of 2 mm to the gingival, the lOO-gm force would have a moment of 1,600 gm-mm about the apex. To make the system equivalent to Fig. 19, A where there is a 1,400 gm-mm clockwise mo- ment, a 200 gm-mm counterclockwise couple would be required.

We now return from the academic exercise of com- puting moments about the apex to the clinically impor- tant issue of the relationship between force systems at the center of resistance and force systems at the bracket. For bodily movement, the force system at the bracket must be equivalent to a force with no couple at the center of resistance. Fig. 20, A illustrates the de- sired force system. What force system applied at the bracket is equivalent to a pure force at the center of resistance? First, forces in the x and y directions must be equal to those in Fig. 20, A, so a 150-gm force at the bracket is needed in the x direction. The final require- ment is that the moments about any point be equal. To make calculations simple, let us consider moments about the center of resistance. The force in Fig. 20, A has no moment about the center of resistance, so that the system applied at the bracket must do the same. But the 150-gm force at the bracket, multiplied by its 8 mm distance to the center of resistance, results in a 1,200 gm-mm moment. This moment of the force is in a clockwise direction. To make the net moment about the center of resistance equal to zero, a 1,200 gm-mm couple in a counterclockwise direction must be added, as in Fig. 20, B. Thus, the 150-gm force at the bracket

Mechanics of tooth movement 305 Volume 85 Number 4

A

3 c ..-. . ..‘....._ B

Fig. 23. The force system in A is rarely indicated in edgewise technique. Almost all centers of rotation for maxillary canine retraction result from a moment of a couple in the opposite direction to the moment of the force, as in B.

plus a 1,200 gm-mm counterclockwise couple will pro- duce pure bodily movement in this example.

Suppose that a couple had been added as described, but its moment was not quite enough, perhaps 1,000 gm-mm instead of 1,200 gm-mm. At the center of re- sistance the net effect would be as in Fig. 21, A. The net 200 gm-mm moment at the center of resistance will be clockwise, because the 1,000 gm-mm counter- clockwise couple does not completely balance the clockwise 1,200 gm-mm moment from the applied 150-gm force.

Stick figures can be used to explain how this tooth will move; Fig. 21, B shows the translation due to the force, and Fig. 21, C shows the rotation due to the couple. Note that in comparing the before-movement and after-movement positions, the apices are closer to- gether than the crowns. The center of rotation is apical to the center of resistance of the tooth. If the moment had been underestimated by 400 gm-mm rather than only 200 gm-mm, there would be more of a rotational effect (Fig. 21, D) and the center of rotation would move occlusally. Next let us consider the effects of eliminating the counterclockwise couple altogether, so that the full 1,200 gm-mm clockwise moment due to the 150-gm force operates at the center of resistance. Since the center of resistance “feels” a 150-gm trans- latory force and a 1,200 gm-mm clockwise moment, the tooth rotates more than it translates and the center of rotation approaches the center of resistance.

Now consider what would happen if the coun- terclockwise couple had been slightly overestimated, so that 1,400 gm-mm was applied instead of the 1,200

0 5 10 15 20 25 30

M/F ratio at the bracket

Fig. 24. This graph summarizes the relationship between the couple-to-force ratio at the bracket and the center of rotation for a hypothetic tooth with a center of resistance to bracket dis- tance of 10 mm. The directions of the couple and the force are as in Fig. 23, B. The M/F ratio has units of mm. (See text for discussion.)

gm-mm needed to counteract the moment of the force (Fig. 22, A). In this case there is a net counterclockwise moment at the center of resistance (Fig. 22, B). As Fig. 22, C illustrates, there will be both translation and ro- tation, but the rotation will produce less crown move- ment than root movement, so that the center of rotation will be somewhere to the occlusal of the tooth. As this couple gets larger, the center of rotation will move in from the occlusal direction, eventually becoming just occlusal to the center of resistance.

Because it is the force system at the center of resis- tance that determines how a tooth moves, and because the line of action of forces applied at the bracket is usually a large distance from the center of resistance (thereby producing a large moment), simple forces applied at the bracket produce large rotation effects.’ Different centers of rotation along the long axis of a tooth are created by changing the magnitude of a couple in the direction opposite the moment of the force about the center of resistance. For example, in retracting a maxillary canine, it would be unusual to add a clockwise couple (Fig. 23, A). As far as the tooth is concerned, at its center of resistance there is already a large clockwise moment when a single force is applied at the bracket. In edgewise technique, all variations of

306 Smith and Burstone

canine retraction, including pure bodily movement, more crown than root movement, more root than crown movement, etc., are achieved by a force and a counter-tipping couple (Fig. 23, B).

It should be emphasized that the magnitude of forces and couples cannot be directly compared. The translation resulting from a 150-gm force has no rela- tion to the rotation resulting from a 150 gm-mm couple. In fact, roughly 300 to 400 gm is an effective force level for mesial translation of a mandibular molar, but a couple with a moment of about 2,000 gm-mm is neces- sary for efficient molar uprighting.

MOMENT-TO-FORCE RATIOS AND TOOTH MOVEMENT

From the preceding discussion, it follows that the type of movement exhibited by a tooth is determined by the ratio between the magnitude of the couple (M) and the force (F) applied at the bracket.‘. ‘IL ” In terms of direction, the moment of the couple is almost always going to be in the direction opposite the moment of the force about the center of resistance.

Note that moments are measured in gram-millime- ters and forces in grams, so that a ratio of the two has units of millimeters (this represents the distance away from the bracket that a single force will produce the same effect). Unfortunately, it has become conven- tional in orthodontics to ignore these units and just speak of the moment-to-force ratio as a pure number.

It is possible to graph the relationship between the center of rotation and the M/F ratio applied at the bracket. This is for the situation in which a horizontal force (perpendicular to the long axis) is applied at the bracket along with a couple producing a moment in the direction opposite the moment of the force. Adapted from Burstone and Pryputniewicz,’ Fig. 24 illustrates these relationships for a tooth with a lo-mm distance from the bracket to the center or resistance. The abscissa (x axis) gives the moment-to-force ratio at the bracket, and ordinate (y axis) gives the distance from the center of resistance to the center of rotation (in this example, the units of the M/F ratio are millimeters).

When only a force is applied at the bracket (M/F ratio of zero), the center of rotation is just apical to the center of resistance (labeled A on the graph). The re- sulting tooth movement is uncontrolled tipping. As the moment of the couple increases, the center of rotation moves apically (toward the position labeled B). With a distance from the bracket to the center of resistance of 10 mm, the center of rotation approaches infinity as the M/F ratio approaches 10/l (position C). As soon as the M/F ratio passes 10/l, the net moment at the center of resistance changes direction, since the moment of the

Am. J. Orthod. April 1984

couple is now greater than the moment from the applied force. The center of rotation is incisal to the center of resistance. At first it is an “infinite” distance to the incisal (position D). When the M/F ratio is 12 or 13 to 1, the center of rotation will be at the incisal edge (position E, root movement), and as the M/F ratio in- creases up to about 2011, the center of rotation becomes just incisal to the center of resistance (position F), ap- proaching a purely rotational movement.

This graph has important implications.” It is the ratio between the applied couple and force that deter- mines the type of tooth movement, not the absolute magnitudes.“’ (This mechunical principle does not take into account the fact that magnitudes of forces and couples are important in determining the biologic re- sponse to an orthodontic force system.) Except for M/F ratios which result in centers of rotation near the center of resistance, small changes in the M/F ratio have im- portant consequences on the type of tooth movement. An M/F ratio of about 8/ 1 will put the center of rotation at the apex, tooth movement being controlled tipping, while an M/F ratio of 1211 will put the center of rota- tion at the incisal edge, which is usually termed root movement. Contrast these two M/F ratios with an M/F ratio of 10/l, which will result in bodily movement. Obviously, small changes in this ratio have major ef- fects on clinical tooth movement.

SUMMARY

Undesired or inefficient tooth movement during orthodontic treatment results from individual variations in biologic response and the improper use of forces. Application of the rules of biomechanics allows one of these sources of variation to be reduced or eliminated. The ability to measure and control couple-to-force ratios at the bracket is a key to predictable and con- trolled tooth movement.

We thank David Gipe for preparation of the illustrations and Barbara Bass for typing several drafts of the manuscript.

REFERENCES 1.

2.

3.

4.

5.

Burstone CJ, Pryputniewicz RJ: Holographic determination of centers of rotation produced by orthodontic forces. AM J OR- THOD 77: 396-409, 1980. Burstone CJ, Pryputniewicz RJ, Weeks R: Centers of resistance of the human mandibular molars (abstract). J Dent Res 60: 515, 1981.

Pryputniewicz RJ, Burstone CJ: The effect of time and force magnitude on orthodontic tooth movement. J Dent Res 58: 1754-1764, 1979.

Mulligan TF: Common sense mechanics. 2. Forces and mo- ments. J Clin Orthod 13: 676-683, 1979. Hocevar RA: Understanding, planning, and managing tooth movement: orthodontic force system theory. AM J ORTHOD 80: 457.477, 1981.

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6. Thurow RC: Edgewise orthodontics, ed. 4, St. Louis, 1982, The C.V. Mosby Company.

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Reprint requests to: Dr. Richard J. Smith Department of Orthodontics University of Maryland Dental School 666 W. Baltimore St. Baltimore, MD 21201