Prediction of orthodontic tooth movement Michael R. Marcotte, D.D.S., M.S.D.*
Farmington, Conn.
O r t h o d o n t i c s has long used mechanotherapy to implement dentofacial changes, with its p r i m a r y objective being the repositioning of teeth from a malocclusion to a normal occlusion. During this procedure, one obvious aim of the orthodontist is to accomplish this repositioning with a minimal number of undesirable side effects. Often these undesirable side effects do not appear until subsequent appointments, at which time they are either accepted or re-treated. I f it is still possible to reverse the undesirable effects, the increase in the length of treatment is often considerable. I f it were possible to predict both desirable and undesirable tooth movement, many of these undesirable side effects could be managed by using either a different appliance approach or auxiliary attachments built into one's existing appliance regimen. This article will describe the use of a relatively simple procedure to predict desirable and undesirable tooth movement. Sound orthodontic appliance t h e r a p y has its basis in mechanics, a field of engineering which is finding greater application in the health sciences. This field of mechanics is based on Newton's laws of motion, and usually three areas are identified: statics, dynamics, and strength of materials. Statics deals with Newton's first law of motion, which describes equilibrium: " E v e r y body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change by the state of forces impressed on it. ''1 A n y body in equilibrium, then, either has no forces and moments acting on it or the sum of all the forces and moments acting on it is equal to zero, that is, for equilibrium of the body, ZF = 0 and ~M=O. Orthodontic appliances are used to generate and maintain forces and moments to teeth. E i t h e r alone or in combination, these forces and moments control the manner in which a tooth moves. More specifically, the moment-to-force ( M / F ) ratio controls the center of rotation of a tooth or, if rigidly connected, a segment of teeth. ~ Some common tooth movements found in the sagittal view are simple *Assistant Professor of Orthodontics, University of Connecticut School of Dental Medicine 511
512
Marcotte
A ..... I. o , . t h , , ~ .
M ay 1 9 7 6
4
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Fig. 1. Equilibrium diagram showing the application of equal and opposite couples to lower right premolars.
tipping, pure rotation, root movement, controlled tipping, translation, and extrusion-intrusion, while in the occlusal view common tooth movements are pure rotation, controlled rotation, and translation. By varying the ratio of the moment to the force, different types of tooth movement are possible 3 ; that is, different centers of rotation can be produced. Assuming that one can identify the specific discrepancy within an arch (intersegmental) or within a segment of teeth (intrasegmental), it is possible for him to identify also the direction and the center of rotation required for its correction. If, for example, a maxillary right central incisor were tipped with its root to the labial and its crown to the lingual, the required direction and center of rotation to correct this tipped condition would be a rotation of the tooth with the crown moving to the labial and the root to the lingual about a center of rotation at or close to the center of resistance of the tooth. To produce this direction and center of rotation on the tooth, a moment of a couple must be applied, tending to rotate the tooth in this "crown-labial, root-lingual" direction. A description of the force systems required to produce specific centers of rotation can be found elsewhere) Knowing the force systems required for correction of the teeth allows one to use the concept of equilibrium, since the assumption can be made that an activated orthodontic appliance in the patient's mouth is in equilibrium. Equilibrium diagrams can be made showing the forces and moments on the orthodontic appliance or on the teeth themselves, the relationship being that the forces on the appliance are equal and opposite to those on the teeth and vice versa. F o r ease of description, the equilibrium diagrams in this articl¢ will be constructed showing the forces and moments oll the teeth, but it should be remembered that the appliance is really what is in equilibrium and that the forces are equal and opposite between the teeth and the appliance. I~'or example, for two lower right premolars, both of which require a moment of a couple for a center of rotation about their long axes, an equilibrium diagram (Fig. 1) shows that if both moments of couples on the teeth are of the same magnitude and opposite sense, the two-tooth segment is in
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Fig. 2. Equilibrium diagram showing resulting buccolingual forces when couples are opposite but not equal. equilibrium with no undesirable side effects appearing. Equilibrium exists since it can be seen that the sum of the moments equals zero ( ~ M = 0 ) . When the moments of couples are opposite and unequal (Fig. 2), equilibrium still exists; that is, the sum of the moments is still equal to zero, but a clockwise moment of couple, equal in magnitude but opposite in sense to the discrepancy between the two couples, takes the form of buccolingual forces. The force system on each tooth, then, results in a center of rotation on the first premolar somewhere between the long axis and the distal marginal ridge, and oi1 the second premolar the center of rotation lies close to the distal marginal ridge. Occasionally, this is precisely the type of movement required, and unequal and opposite moments of couples are then indicated. When the required center of rotation is at the long axis of each premolar, however, the placement of unequal and opposite moments of couples results in unsatisfactory tooth movement, which must still be corrected. This application of unequal and opposite moments of couples may result from an error in spring fabrication or in spring location or from failure to note the wire-bracket relationship. When three teeth are present in a segment, equilibrium diagrams can still be constructed to show the side effects of the required force systems, since the assumption can still be made that the activated appliance is in equilibrium. For three teeth, equilibrium diagrams are constructed by combining the equilibrium diagrams of two-tooth segments. This procedure will be illustrated in the examples which follow later in this article. Tooth models have been used to show the influence of forces and moments on centers of rotation. 4 To show the predictive qualities of equilibrium theory, twoand three-tooth segments have been placed in wax, simulating the clinical situation. To make the illustrations more dramatic, only the occlusal view has been used in this article ; it should be realized, however, that the procedure can be used in all three planes of space if it is possible to estimate the direction of the forces and moments required for appliance activation. Depending on the configuration dimensions and interbracket location of the
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Table I. Flow chart to be followed in repositioning teeth 1. 2. 3. 4. 5. 6.
Problem identification Required direction and center of rotation Necessary force system Equilibrium state Appliance selection Preactivation bends
loops in a wire, accurate identification of all the forces and moments delivered to even two teeth is difficult, if not impossible. With more teeth connected by a wire with loops, the exact forces and moments delivered by the appliance are, in large part, unknown and thus are termed statically indeterminate. 5 The equilibrium diagrams arc presented to act as gross predictors of tooth movement under statically determinate conditions, that is, one in which the forces and moments are known or can be measured. A flow chart will be presented and used in the following examples to increase the accuracy in the prediction of tooth movement (Table I). Two-tooth segment
Knowing the required force system for a specific mode of tooth movement is basic to an understanding of equilibrium, whether it is a single tooth or a segment of teeth interconnected so that it may be considered a single multirooted tooth. The generation of this required force system requires, however, that another tooth or segment of teeth be employed as the "anchorage unit." Thus, two "teeth" form the basic unit for an understanding of the forces used in orthodonticsJ To illustrate this basic two-tooth system, two teeth are placed in wax (Fig. 3) to mimic the basic clinical condition. The conventions used in this article will be along the lines of those proposed by Burstone and Koenig2 A moment tending to move the distal portion of a tooth to the facial or the mesial portion to the lingual is positive. In the order of the flow chart (Table I), it can be seen that the problem in Fig. 3, A is that the lower right second premolar is rotated positively; that is, the distal aspect of the tooth is to the buccal. Its required direction and center of rotation are a negative rotation about its mesial marginal ridge. The required force system, then, for this direction and center of rotation is a negative (lingual) force plus a negative moment (mesial to the buccal, distal to the lingual). When this force system is applied to the second premolar, the equilibrium diagram (Fig. 3, B) shows that the first molar will experience a positive (buccal) force, tending to tip it to the buccal. It can be seen that the system is in equilibrium since the sum of the forces is zero and the sum of (1) the moment of a force (negative) and (2) the moment of a couple (positive) is also zero. The equilibrium diagram becomes a useful estimate of the forces on the teeth since some variations may be present, depending on the loop configurations employed. This force and moment are delivered to the second premolar with a horizontal loop preactivated so that the correct directions of the force and the moment are produced by engaging the wire in the molar bracket (Fig. 3, C and D).
V o l u m e 69 Number 5
Prediction of orthodontic tooth movement
I
A ~
....
515
D
Fig. 3. A two-tooth segment showing a required negative force and moment in equilibrium with a positive force.
When the wax-filled cylinder is exposed to a heat source, it can be noticed that the desired correction is accomplished; that is, the tooth rotated in a negative direction about a center somewhere close to the mesial marginal ridge (Fig. 3, E). An undesirable side effect is, of course, the positive tipping of the first molar, which can be managed either by allowing it to occur if indicated or by increasing the anchorage unit if it is contraindicated. To further illustrate how desirable and undesirable tooth movement can be predicted, a two-tooth segment is now set in wax with both lower right first and second premolars rotated positively, that is, "mesial-in, distal-out" (Fig. 4, A). The desired direction and center of rotation for each premolar is a negative rota-
516
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Fig. 4, A two-tooth segment showing two required negative moments in equilibrium with a positive moment expressed as buccolingual forces.
tion about its long axis. A moment of a couple tends to rotate a tooth about its center of resistance or, ill this occlusal view, its long axis. A negative moment of a couple must be applied to each tooth for these centers of rotation, and the equilibrium diagram (Fig. 4, B) shows that, with both couples being negative, their sum is a large net negative moment. F o r equilibrium of the two-tooth segment, a positive moment of a couple of the same magnitude appears. Two equal and opposite forces (a couple) in the form of a negative (lingual) force on the first premolar and a positive (buccal) force on the second premolar produce this positive moment of a couple which,
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for equilibrium, is an unavoidable by-product. Thus, in addition to the desired moments of couples, positive-negative forces are produced which are usually undesirable. To demonstrate this system of two negative moments of couples acting, a " T " loop was preactivated so that each arm crossed the center of each bracket (Fig'. 4, C). With the " T " loop fully activated (Fig. 4, D) and with exposure to the heat source, the predicted movements can be observed (Fig. 4, E ) . It can be seen that the negative rotations have improved the alignment between both teeth but, as predicted, the resulting positive-negative forces result in a different segmental alignment which is usually undesirable. I f it is found to be undesirable, and hence contraindicated, an alternative appliance system must be used to prevent its occurrence. The inclusion of more teeth into each segment can be comfortably managed if each segment, when rigidly connected, is thought of as a large, multirooted tooth. The six teeth in Fig. 5 are still basically a two-tooth system, that is, two large teeth, since both buccal segments of teeth are aligned normally intrasegmentally and connected passively with a rigid wire (0.021 by 0.025 inch). The existing problem is that each segment of teeth is rotated positively; that is, the premolars are lingual and the second molars are buccal. The required direction and center of rotation for each segment of teeth is a negative rotation about the long axis of each "large tooth" or approximately at the long axis of the first molar. As was mentioned previously, for any tooth or group of connected teeth to rotate about its long axis (in an ocelusal view), moments of couples are required. I f the required moments are equal and opposite, the equilibrium diagram of the "twotooth segments" (Fig. 5, B) shows that no undesirable side effects will occur. The appliance that can be used to generate these equal and opposite moments of couples is an 0.036 inch lingual arch. To ensure that these equal and opposite moments of couples exist, each tab of the lingual arch is preactivated to cross the lingual arch sheath at the same angles as on the opposite side. This may be confirmed by inserting the tab of one side into its sheath and adjusting it so that the opposite tab lies at an appropriate point on the opposite molar (Fig. 5, C and D). The procedure is then repeated on the opposite tab until both right and left lingual arch tabs have the same relationship to their respective sheaths. The tabs are then inserted for activation of the appliance (Fig. 5, E), and when the wax cylinder is exposed to the heat source, the desired tooth movement can be observed (Fig. 5, F ) . B y reducing a problem of six malpositioned teeth to two "large teeth" in malposition, the clinician is able to move teeth more expeditiously and with a minimum number of side effects. Intersegmental discrepancies, unfortunately, are not always symmetrical as in the last example, but the same procedure can be used to predict tooth movement in the presence of asymmetries. In Fig. 6, A the left bucca] segment of teeth is rotated positively in such a way that the premolar is to the lingual and the second molar is to the buceal, while the right buccal segment of teeth is positioned normally. A correction of this asymmetric intersegmental discrepancy could be made if the left buecal segment of teeth were rotated negatively around the long axis of the first molar. A negative moment of a couple is required to produce this direction and center of rotation. The equilibrium diagram of the "two-tooth seg-
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A~,..1. O~'thod. May 197(;
Fig. 5. "Two-tooth" segment comprising six teeth.
m e n t " (Fig. 6, B) shows that for this unilateral negative moment of a couple to exist on the left buceal segment, an equal and opposite moment of a couple, in the f o r m of anterior-posterior forces, also exists. These anterior-posterior forces produce a moment of a couple equal in magnitude and opposite in direction to the required moment of a couple on the left buccal segment. The sum of both moments is now equal to zero. Thus, on the left buccal segment, one can expect to observe the desired negative rotation as well as some mesial movement due to the positive force ; on the right, one expects some distal movement due to the negative force. Wires measuring 0.021 by 0.025 inch fit passively in the brackets of each segment of teeth, while an 0.036 inch ling u m arch can be preactivated to deliver the negative moment of a couple on the left buecal segment. These preaetivation bends are made so that when the right lingual arch tab is inserted into the right molar sheath, the left lingual arch tab
voz,~.,e 69 Number 5
Prediction of orthodontic tooth movement
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R
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Fig. 6. Asymmetric "two-tooth" segment. lies directly over its sheath on the left first molar (Fig. 6, C). When the left lingual arch tab is inserted into its sheath, the right tab should lie distal to its sheath (Fig. 6, D). F o r illustrative purposes, two marks are placed on the reference wires to indicate the initial position of the mesial surfaces of both premolars in Fig. 6, D. When the appliance is fully inserted and the wax-filled container exposed to the heat source, it can be seen that the desired negative segmental rotation occurs on the left along with the predicted positive (mesial) movement, while the right segment underwent slight negative (distal) movement. These anterior-posterior movements of the buccal segments may or may not be undesirable, depending on the nature of the intermaxillary relationships.
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Fig. 7. Three-tooth segment with the first molar being normal.
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Prediction of orthodontic tooth movement R
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Fig. 8. Intrasegmental corrections accomplished with intersegmental anchorage. Three-tooth segment
Dental discrepancies cannot always be reduced to a "two-tooth" segment as in the previous examples. Segments containing more teeth can be treated following the same order in the flow chart (Table I). In the construction of three-tooth equilibrium diagrams, equilibrium diagrams of two two-tooth segments are combined to give the equilibrium condition for the three-tooth segment. For example, the intrasegmental discrepancies in the lower right quadrant of teeth (Fig. 7, A) are (1) the second premolar is rotated positively or "mesial-in" and (2) the second molar is rotated negatively o~" "distal-in." To correct these intrasegmental discrepancies, a negative rotation of the second premolar about its distal marginal ridge and a positive rotation of the second molar about its mesial marginal ridge are required. The direction and center of rotation on the second premolar are
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produced with a positive force and a negative nloment, while on the second molar the required direction and center of rotation are produced with a positive force and a positive moment. The equilibrium diagrams (Fig. 7, B) showing these required force systems on the teeth also indicate that the first molar will experience a negative force when both two-tooth equilibrium diagrams are combined. An 0.018 by 0.025 inch wire with vertical loops is first fabricated passively (Fig. 7, C) and then preactivated to produce the desired force systems (Fig. 7 D). With the appliance fully engaged (Fig. 7, E) and the wax cylinder exposed to the heat source, the desired tooth movement is seen to have occurred, while the undesirable lingual movement of the first molar is minimal. The entire procedure can be summarized by using an example in which intrasegmental corrections are accomplished with intersegmental anchorage to minimize the predicted undesirable side effects. In Fig. 8, A both segments of teeth appear bilaterally symmetrical, with the second premolars rotated positively (mesial-in) and the second molars rotated negatively (mesial-out). Required directions and centers of rotation for the correction of the second premolars are negative rotations about the distal marginal ridges of the second premolars and, for the second molars, a positive rotation about the distal marginal ridges. To produce these directions and centers of rotation, both right and left second premolars must receive a negative moment and a positive force. Both second molar teeth, on the other hand, must receive a positive moment and a negative force. For the forces and moments required on the teeth, an equilibrium diagram for both right and left sides is shown in Fig. 8, B. Equilibrium diagrams of the two two-tooth segments for each side are combined, and it can be seen that, for equilibrium to exist in each segment, the first molars tend to rotate positively, or "distal-out, mesial-in." These undesirable first molar rotations can be prevented by connecting the right first molar to the left first molar, and an equilibrium condition will exist intersegmentally since the moments are equal and opposite to each other. The required forces and moments are delivered to the teeth with 0.018 by 0.025 inch wire with loops and preactivated as shown in Fig. 8, C and D. A 0.036 inch lingual arch fabricated passively will provide this intermolar connection, which makes one undesirable side effect equal and opposite to the other. With the buecal wires activated and the lingual arch in place (Fig. 8, E), the corrections can be seen to occur when the wax cylinder is exposed to the heat source (Fig. 8, F). ]t is noticed that the right buccal wire was not preactivated with as much moment preactivation as on the left (Fig. 8, D). This omission appears as a rotation in the correct direction along with a slight positive (buceal) displacement of the right premolar in Fig. 8, F. Although the preceding examples have been only in the ocelusal view, similar results can be observed in the other two views. This procedure may also be used for the treatment of the entire arch if it is realized that each arch, once each segment of teeth has been ideally positioned and rigidly connected, consists of but three large, multirooted teeth.
Volu.~e 6a Nun~ber 5
P r e d i c t i o n of orthodo)#ic tooth m o v e m e n t
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Summary
O r t h o d o n t i c s is r a p i d l y a d v a n c i n g f r o m the s t a g e of f o r t u i t o u s success to one of p l a n n e d success. W h e n a p p l i a n c e d e s i g n is b a s e d on s i m p l e c o n c e p t s of e q u i l i b r i u m , p r e d i c t i o n of d e s i r a b l e a n d u n d e s i r a b l e tooth m o v e m e n t becomes possible. T h e flow c h a r t t h a t h a s been p r e s e n t e d allows t h e c l i n i c i a n to s y s t e m a t i c a l l y t r e a t d e n t a l d i s c r e p a n c i e s m o r e effectively in p o t e n t i a l l y less t r e a t m e n t time, since t h e n u m b e r of d e s i r a b l e a n d u n d e s i r a b l e side effects can be k n o w n in adv a n c e a n d m a n a g e d a c c o r d i n g l y . S i m p l e l a w s of e q u i l i b r i u m , b a s e d on a knowle d g e of t h e f o r c e s y s t e m s r e q u i r e d f o r specific t o o t h m o v e m e n t , also p e r m i t t h e d e s i g n of p r o p e r p r e a c t i v a t i o n b e n d s f o r a n a p p l i a n c e . T h e p r o c e d u r e m a y be e n l a r g e d to i n c l u d e t r e a t m e n t of each a r c h since, once the t e e t h a r e i d e a l l y posit i o n e d a n d r i g i d l y h e l d w i t h i n each segment, each a r c h can be c o n s i d e r e d to consist of t h r e e " l a r g e , m u l t i r o o t e d t e e t h . " REFERENCES 1. Goff, R. H., and Hardenbergh, D. E.: Introduction to statics, New York, 1967, Holt, Rinehart & Winston, p. 4. 2. Burstone, C. J.: Biomechanics of the orthodontic appliance. In Graber, T. M.: Current orthodontic concepts and techniques, Philadelphia, 1969, W. B. Saunders Company, p. 161. 3. Burstone, C. J. : Biomechanics of tooth movement. In Kraus, B. S., and Reidel, R. A. : Vistas in orthodontics, Philadelphia, 1962, Lea & Febiger, pp. 200, 202. 4. Haack, D. C., and Weinstein, S.: Geometry and mechanics as related to tooth movement studied by means of a two-dimensional model, J. Am. Dent. A. 66: 157-164, 1963. 5. Koenig, H. A., and Burstone, C. J.: Analysis of generalized curved beams for orthodontic applications, J. Biomechanics 7: 429, 1974. 6. Burstone, C. J., and Koenig, H. A.: Force systems from an ideal arch, A~t. J. ORTHOD. 65: 270-289, 1974.
Until we have succeeded in definitely proving what the real results of our manipulations are, much energy will be expended in the periodical reintroduction of methods of treatment, which a past generation of operators has used and discarded without reporting that, as to permanence, they were unsuccessful. (Lundstrom, Axel F.: Concerning the Effects of Orthodontic Treatment, The international Journal of Orthodontia, Oral Surgery and Radiography, predecessor of the American Journal of Orthodontics, February, 1928, p. 14O.)
Farmington, Conn.
O r t h o d o n t i c s has long used mechanotherapy to implement dentofacial changes, with its p r i m a r y objective being the repositioning of teeth from a malocclusion to a normal occlusion. During this procedure, one obvious aim of the orthodontist is to accomplish this repositioning with a minimal number of undesirable side effects. Often these undesirable side effects do not appear until subsequent appointments, at which time they are either accepted or re-treated. I f it is still possible to reverse the undesirable effects, the increase in the length of treatment is often considerable. I f it were possible to predict both desirable and undesirable tooth movement, many of these undesirable side effects could be managed by using either a different appliance approach or auxiliary attachments built into one's existing appliance regimen. This article will describe the use of a relatively simple procedure to predict desirable and undesirable tooth movement. Sound orthodontic appliance t h e r a p y has its basis in mechanics, a field of engineering which is finding greater application in the health sciences. This field of mechanics is based on Newton's laws of motion, and usually three areas are identified: statics, dynamics, and strength of materials. Statics deals with Newton's first law of motion, which describes equilibrium: " E v e r y body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change by the state of forces impressed on it. ''1 A n y body in equilibrium, then, either has no forces and moments acting on it or the sum of all the forces and moments acting on it is equal to zero, that is, for equilibrium of the body, ZF = 0 and ~M=O. Orthodontic appliances are used to generate and maintain forces and moments to teeth. E i t h e r alone or in combination, these forces and moments control the manner in which a tooth moves. More specifically, the moment-to-force ( M / F ) ratio controls the center of rotation of a tooth or, if rigidly connected, a segment of teeth. ~ Some common tooth movements found in the sagittal view are simple *Assistant Professor of Orthodontics, University of Connecticut School of Dental Medicine 511
512
Marcotte
A ..... I. o , . t h , , ~ .
M ay 1 9 7 6
4
Q
Fig. 1. Equilibrium diagram showing the application of equal and opposite couples to lower right premolars.
tipping, pure rotation, root movement, controlled tipping, translation, and extrusion-intrusion, while in the occlusal view common tooth movements are pure rotation, controlled rotation, and translation. By varying the ratio of the moment to the force, different types of tooth movement are possible 3 ; that is, different centers of rotation can be produced. Assuming that one can identify the specific discrepancy within an arch (intersegmental) or within a segment of teeth (intrasegmental), it is possible for him to identify also the direction and the center of rotation required for its correction. If, for example, a maxillary right central incisor were tipped with its root to the labial and its crown to the lingual, the required direction and center of rotation to correct this tipped condition would be a rotation of the tooth with the crown moving to the labial and the root to the lingual about a center of rotation at or close to the center of resistance of the tooth. To produce this direction and center of rotation on the tooth, a moment of a couple must be applied, tending to rotate the tooth in this "crown-labial, root-lingual" direction. A description of the force systems required to produce specific centers of rotation can be found elsewhere) Knowing the force systems required for correction of the teeth allows one to use the concept of equilibrium, since the assumption can be made that an activated orthodontic appliance in the patient's mouth is in equilibrium. Equilibrium diagrams can be made showing the forces and moments on the orthodontic appliance or on the teeth themselves, the relationship being that the forces on the appliance are equal and opposite to those on the teeth and vice versa. F o r ease of description, the equilibrium diagrams in this articl¢ will be constructed showing the forces and moments oll the teeth, but it should be remembered that the appliance is really what is in equilibrium and that the forces are equal and opposite between the teeth and the appliance. I~'or example, for two lower right premolars, both of which require a moment of a couple for a center of rotation about their long axes, an equilibrium diagram (Fig. 1) shows that if both moments of couples on the teeth are of the same magnitude and opposite sense, the two-tooth segment is in
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P r e d i c t i o n of orthodo~tic tooth ~wveme~d
S13
Fig. 2. Equilibrium diagram showing resulting buccolingual forces when couples are opposite but not equal. equilibrium with no undesirable side effects appearing. Equilibrium exists since it can be seen that the sum of the moments equals zero ( ~ M = 0 ) . When the moments of couples are opposite and unequal (Fig. 2), equilibrium still exists; that is, the sum of the moments is still equal to zero, but a clockwise moment of couple, equal in magnitude but opposite in sense to the discrepancy between the two couples, takes the form of buccolingual forces. The force system on each tooth, then, results in a center of rotation on the first premolar somewhere between the long axis and the distal marginal ridge, and oi1 the second premolar the center of rotation lies close to the distal marginal ridge. Occasionally, this is precisely the type of movement required, and unequal and opposite moments of couples are then indicated. When the required center of rotation is at the long axis of each premolar, however, the placement of unequal and opposite moments of couples results in unsatisfactory tooth movement, which must still be corrected. This application of unequal and opposite moments of couples may result from an error in spring fabrication or in spring location or from failure to note the wire-bracket relationship. When three teeth are present in a segment, equilibrium diagrams can still be constructed to show the side effects of the required force systems, since the assumption can still be made that the activated appliance is in equilibrium. For three teeth, equilibrium diagrams are constructed by combining the equilibrium diagrams of two-tooth segments. This procedure will be illustrated in the examples which follow later in this article. Tooth models have been used to show the influence of forces and moments on centers of rotation. 4 To show the predictive qualities of equilibrium theory, twoand three-tooth segments have been placed in wax, simulating the clinical situation. To make the illustrations more dramatic, only the occlusal view has been used in this article ; it should be realized, however, that the procedure can be used in all three planes of space if it is possible to estimate the direction of the forces and moments required for appliance activation. Depending on the configuration dimensions and interbracket location of the
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Table I. Flow chart to be followed in repositioning teeth 1. 2. 3. 4. 5. 6.
Problem identification Required direction and center of rotation Necessary force system Equilibrium state Appliance selection Preactivation bends
loops in a wire, accurate identification of all the forces and moments delivered to even two teeth is difficult, if not impossible. With more teeth connected by a wire with loops, the exact forces and moments delivered by the appliance are, in large part, unknown and thus are termed statically indeterminate. 5 The equilibrium diagrams arc presented to act as gross predictors of tooth movement under statically determinate conditions, that is, one in which the forces and moments are known or can be measured. A flow chart will be presented and used in the following examples to increase the accuracy in the prediction of tooth movement (Table I). Two-tooth segment
Knowing the required force system for a specific mode of tooth movement is basic to an understanding of equilibrium, whether it is a single tooth or a segment of teeth interconnected so that it may be considered a single multirooted tooth. The generation of this required force system requires, however, that another tooth or segment of teeth be employed as the "anchorage unit." Thus, two "teeth" form the basic unit for an understanding of the forces used in orthodonticsJ To illustrate this basic two-tooth system, two teeth are placed in wax (Fig. 3) to mimic the basic clinical condition. The conventions used in this article will be along the lines of those proposed by Burstone and Koenig2 A moment tending to move the distal portion of a tooth to the facial or the mesial portion to the lingual is positive. In the order of the flow chart (Table I), it can be seen that the problem in Fig. 3, A is that the lower right second premolar is rotated positively; that is, the distal aspect of the tooth is to the buccal. Its required direction and center of rotation are a negative rotation about its mesial marginal ridge. The required force system, then, for this direction and center of rotation is a negative (lingual) force plus a negative moment (mesial to the buccal, distal to the lingual). When this force system is applied to the second premolar, the equilibrium diagram (Fig. 3, B) shows that the first molar will experience a positive (buccal) force, tending to tip it to the buccal. It can be seen that the system is in equilibrium since the sum of the forces is zero and the sum of (1) the moment of a force (negative) and (2) the moment of a couple (positive) is also zero. The equilibrium diagram becomes a useful estimate of the forces on the teeth since some variations may be present, depending on the loop configurations employed. This force and moment are delivered to the second premolar with a horizontal loop preactivated so that the correct directions of the force and the moment are produced by engaging the wire in the molar bracket (Fig. 3, C and D).
V o l u m e 69 Number 5
Prediction of orthodontic tooth movement
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A ~
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D
Fig. 3. A two-tooth segment showing a required negative force and moment in equilibrium with a positive force.
When the wax-filled cylinder is exposed to a heat source, it can be noticed that the desired correction is accomplished; that is, the tooth rotated in a negative direction about a center somewhere close to the mesial marginal ridge (Fig. 3, E). An undesirable side effect is, of course, the positive tipping of the first molar, which can be managed either by allowing it to occur if indicated or by increasing the anchorage unit if it is contraindicated. To further illustrate how desirable and undesirable tooth movement can be predicted, a two-tooth segment is now set in wax with both lower right first and second premolars rotated positively, that is, "mesial-in, distal-out" (Fig. 4, A). The desired direction and center of rotation for each premolar is a negative rota-
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Fig. 4, A two-tooth segment showing two required negative moments in equilibrium with a positive moment expressed as buccolingual forces.
tion about its long axis. A moment of a couple tends to rotate a tooth about its center of resistance or, ill this occlusal view, its long axis. A negative moment of a couple must be applied to each tooth for these centers of rotation, and the equilibrium diagram (Fig. 4, B) shows that, with both couples being negative, their sum is a large net negative moment. F o r equilibrium of the two-tooth segment, a positive moment of a couple of the same magnitude appears. Two equal and opposite forces (a couple) in the form of a negative (lingual) force on the first premolar and a positive (buccal) force on the second premolar produce this positive moment of a couple which,
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for equilibrium, is an unavoidable by-product. Thus, in addition to the desired moments of couples, positive-negative forces are produced which are usually undesirable. To demonstrate this system of two negative moments of couples acting, a " T " loop was preactivated so that each arm crossed the center of each bracket (Fig'. 4, C). With the " T " loop fully activated (Fig. 4, D) and with exposure to the heat source, the predicted movements can be observed (Fig. 4, E ) . It can be seen that the negative rotations have improved the alignment between both teeth but, as predicted, the resulting positive-negative forces result in a different segmental alignment which is usually undesirable. I f it is found to be undesirable, and hence contraindicated, an alternative appliance system must be used to prevent its occurrence. The inclusion of more teeth into each segment can be comfortably managed if each segment, when rigidly connected, is thought of as a large, multirooted tooth. The six teeth in Fig. 5 are still basically a two-tooth system, that is, two large teeth, since both buccal segments of teeth are aligned normally intrasegmentally and connected passively with a rigid wire (0.021 by 0.025 inch). The existing problem is that each segment of teeth is rotated positively; that is, the premolars are lingual and the second molars are buccal. The required direction and center of rotation for each segment of teeth is a negative rotation about the long axis of each "large tooth" or approximately at the long axis of the first molar. As was mentioned previously, for any tooth or group of connected teeth to rotate about its long axis (in an ocelusal view), moments of couples are required. I f the required moments are equal and opposite, the equilibrium diagram of the "twotooth segments" (Fig. 5, B) shows that no undesirable side effects will occur. The appliance that can be used to generate these equal and opposite moments of couples is an 0.036 inch lingual arch. To ensure that these equal and opposite moments of couples exist, each tab of the lingual arch is preactivated to cross the lingual arch sheath at the same angles as on the opposite side. This may be confirmed by inserting the tab of one side into its sheath and adjusting it so that the opposite tab lies at an appropriate point on the opposite molar (Fig. 5, C and D). The procedure is then repeated on the opposite tab until both right and left lingual arch tabs have the same relationship to their respective sheaths. The tabs are then inserted for activation of the appliance (Fig. 5, E), and when the wax cylinder is exposed to the heat source, the desired tooth movement can be observed (Fig. 5, F ) . B y reducing a problem of six malpositioned teeth to two "large teeth" in malposition, the clinician is able to move teeth more expeditiously and with a minimum number of side effects. Intersegmental discrepancies, unfortunately, are not always symmetrical as in the last example, but the same procedure can be used to predict tooth movement in the presence of asymmetries. In Fig. 6, A the left bucca] segment of teeth is rotated positively in such a way that the premolar is to the lingual and the second molar is to the buceal, while the right buccal segment of teeth is positioned normally. A correction of this asymmetric intersegmental discrepancy could be made if the left buecal segment of teeth were rotated negatively around the long axis of the first molar. A negative moment of a couple is required to produce this direction and center of rotation. The equilibrium diagram of the "two-tooth seg-
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A~,..1. O~'thod. May 197(;
Fig. 5. "Two-tooth" segment comprising six teeth.
m e n t " (Fig. 6, B) shows that for this unilateral negative moment of a couple to exist on the left buceal segment, an equal and opposite moment of a couple, in the f o r m of anterior-posterior forces, also exists. These anterior-posterior forces produce a moment of a couple equal in magnitude and opposite in direction to the required moment of a couple on the left buccal segment. The sum of both moments is now equal to zero. Thus, on the left buccal segment, one can expect to observe the desired negative rotation as well as some mesial movement due to the positive force ; on the right, one expects some distal movement due to the negative force. Wires measuring 0.021 by 0.025 inch fit passively in the brackets of each segment of teeth, while an 0.036 inch ling u m arch can be preactivated to deliver the negative moment of a couple on the left buecal segment. These preaetivation bends are made so that when the right lingual arch tab is inserted into the right molar sheath, the left lingual arch tab
voz,~.,e 69 Number 5
Prediction of orthodontic tooth movement
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R
519
I
Fig. 6. Asymmetric "two-tooth" segment. lies directly over its sheath on the left first molar (Fig. 6, C). When the left lingual arch tab is inserted into its sheath, the right tab should lie distal to its sheath (Fig. 6, D). F o r illustrative purposes, two marks are placed on the reference wires to indicate the initial position of the mesial surfaces of both premolars in Fig. 6, D. When the appliance is fully inserted and the wax-filled container exposed to the heat source, it can be seen that the desired negative segmental rotation occurs on the left along with the predicted positive (mesial) movement, while the right segment underwent slight negative (distal) movement. These anterior-posterior movements of the buccal segments may or may not be undesirable, depending on the nature of the intermaxillary relationships.
520
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Fig. 7. Three-tooth segment with the first molar being normal.
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Prediction of orthodontic tooth movement R
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Fig. 8. Intrasegmental corrections accomplished with intersegmental anchorage. Three-tooth segment
Dental discrepancies cannot always be reduced to a "two-tooth" segment as in the previous examples. Segments containing more teeth can be treated following the same order in the flow chart (Table I). In the construction of three-tooth equilibrium diagrams, equilibrium diagrams of two two-tooth segments are combined to give the equilibrium condition for the three-tooth segment. For example, the intrasegmental discrepancies in the lower right quadrant of teeth (Fig. 7, A) are (1) the second premolar is rotated positively or "mesial-in" and (2) the second molar is rotated negatively o~" "distal-in." To correct these intrasegmental discrepancies, a negative rotation of the second premolar about its distal marginal ridge and a positive rotation of the second molar about its mesial marginal ridge are required. The direction and center of rotation on the second premolar are
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May 1976
produced with a positive force and a negative nloment, while on the second molar the required direction and center of rotation are produced with a positive force and a positive moment. The equilibrium diagrams (Fig. 7, B) showing these required force systems on the teeth also indicate that the first molar will experience a negative force when both two-tooth equilibrium diagrams are combined. An 0.018 by 0.025 inch wire with vertical loops is first fabricated passively (Fig. 7, C) and then preactivated to produce the desired force systems (Fig. 7 D). With the appliance fully engaged (Fig. 7, E) and the wax cylinder exposed to the heat source, the desired tooth movement is seen to have occurred, while the undesirable lingual movement of the first molar is minimal. The entire procedure can be summarized by using an example in which intrasegmental corrections are accomplished with intersegmental anchorage to minimize the predicted undesirable side effects. In Fig. 8, A both segments of teeth appear bilaterally symmetrical, with the second premolars rotated positively (mesial-in) and the second molars rotated negatively (mesial-out). Required directions and centers of rotation for the correction of the second premolars are negative rotations about the distal marginal ridges of the second premolars and, for the second molars, a positive rotation about the distal marginal ridges. To produce these directions and centers of rotation, both right and left second premolars must receive a negative moment and a positive force. Both second molar teeth, on the other hand, must receive a positive moment and a negative force. For the forces and moments required on the teeth, an equilibrium diagram for both right and left sides is shown in Fig. 8, B. Equilibrium diagrams of the two two-tooth segments for each side are combined, and it can be seen that, for equilibrium to exist in each segment, the first molars tend to rotate positively, or "distal-out, mesial-in." These undesirable first molar rotations can be prevented by connecting the right first molar to the left first molar, and an equilibrium condition will exist intersegmentally since the moments are equal and opposite to each other. The required forces and moments are delivered to the teeth with 0.018 by 0.025 inch wire with loops and preactivated as shown in Fig. 8, C and D. A 0.036 inch lingual arch fabricated passively will provide this intermolar connection, which makes one undesirable side effect equal and opposite to the other. With the buecal wires activated and the lingual arch in place (Fig. 8, E), the corrections can be seen to occur when the wax cylinder is exposed to the heat source (Fig. 8, F). ]t is noticed that the right buccal wire was not preactivated with as much moment preactivation as on the left (Fig. 8, D). This omission appears as a rotation in the correct direction along with a slight positive (buceal) displacement of the right premolar in Fig. 8, F. Although the preceding examples have been only in the ocelusal view, similar results can be observed in the other two views. This procedure may also be used for the treatment of the entire arch if it is realized that each arch, once each segment of teeth has been ideally positioned and rigidly connected, consists of but three large, multirooted teeth.
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P r e d i c t i o n of orthodo)#ic tooth m o v e m e n t
523
Summary
O r t h o d o n t i c s is r a p i d l y a d v a n c i n g f r o m the s t a g e of f o r t u i t o u s success to one of p l a n n e d success. W h e n a p p l i a n c e d e s i g n is b a s e d on s i m p l e c o n c e p t s of e q u i l i b r i u m , p r e d i c t i o n of d e s i r a b l e a n d u n d e s i r a b l e tooth m o v e m e n t becomes possible. T h e flow c h a r t t h a t h a s been p r e s e n t e d allows t h e c l i n i c i a n to s y s t e m a t i c a l l y t r e a t d e n t a l d i s c r e p a n c i e s m o r e effectively in p o t e n t i a l l y less t r e a t m e n t time, since t h e n u m b e r of d e s i r a b l e a n d u n d e s i r a b l e side effects can be k n o w n in adv a n c e a n d m a n a g e d a c c o r d i n g l y . S i m p l e l a w s of e q u i l i b r i u m , b a s e d on a knowle d g e of t h e f o r c e s y s t e m s r e q u i r e d f o r specific t o o t h m o v e m e n t , also p e r m i t t h e d e s i g n of p r o p e r p r e a c t i v a t i o n b e n d s f o r a n a p p l i a n c e . T h e p r o c e d u r e m a y be e n l a r g e d to i n c l u d e t r e a t m e n t of each a r c h since, once the t e e t h a r e i d e a l l y posit i o n e d a n d r i g i d l y h e l d w i t h i n each segment, each a r c h can be c o n s i d e r e d to consist of t h r e e " l a r g e , m u l t i r o o t e d t e e t h . " REFERENCES 1. Goff, R. H., and Hardenbergh, D. E.: Introduction to statics, New York, 1967, Holt, Rinehart & Winston, p. 4. 2. Burstone, C. J.: Biomechanics of the orthodontic appliance. In Graber, T. M.: Current orthodontic concepts and techniques, Philadelphia, 1969, W. B. Saunders Company, p. 161. 3. Burstone, C. J. : Biomechanics of tooth movement. In Kraus, B. S., and Reidel, R. A. : Vistas in orthodontics, Philadelphia, 1962, Lea & Febiger, pp. 200, 202. 4. Haack, D. C., and Weinstein, S.: Geometry and mechanics as related to tooth movement studied by means of a two-dimensional model, J. Am. Dent. A. 66: 157-164, 1963. 5. Koenig, H. A., and Burstone, C. J.: Analysis of generalized curved beams for orthodontic applications, J. Biomechanics 7: 429, 1974. 6. Burstone, C. J., and Koenig, H. A.: Force systems from an ideal arch, A~t. J. ORTHOD. 65: 270-289, 1974.
Until we have succeeded in definitely proving what the real results of our manipulations are, much energy will be expended in the periodical reintroduction of methods of treatment, which a past generation of operators has used and discarded without reporting that, as to permanence, they were unsuccessful. (Lundstrom, Axel F.: Concerning the Effects of Orthodontic Treatment, The international Journal of Orthodontia, Oral Surgery and Radiography, predecessor of the American Journal of Orthodontics, February, 1928, p. 14O.)
