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Biological response to biomechanical signals:Orthodontic mechanics to control tooth movement
Biological Response to Biomechanical Signals: Orthodontic Mechamcs to Control Tooth Movement Steven J. Lindauer and A. Denis Britto The orthodontic appliance is the clinician's primary tool for initiating and sustaining the biological processes that control tooth movement. This is achieved by applying a force system that displaces a tooth, or segment of teeth, causing stresses in the periodontal ligament which produce physical, chemical, and electrical signals that are sent to the surrounding cells and tissues. Both the quantity and quality of tooth displacement can be altered by varying the magnitude and direction of the moments and forces applied mechanically to the teeth. Understanding the relationship between appliance activation and the resultant stresses produced at the level of the periodontal ligament is the first step toward understanding the biological mechanisms that allow clinicians to move teeth predictably. (Semin Orthod 2000;6:145-154.) Copyright © 2000 by W.B. Saunders Company
he cascade of biological events that induce orthodontic tooth m o v e m e n t is initiated by mechanical stresses in the periodontium. ~,2 Forces are transferred to the teeth by the clinician using appliances designed to displace teeth a prescribed amount in a desired direction. This sends signals to the cells to remodel tissues in a way that allows teeth to move. ~ To interpret the biological responses to activation of any orthodontic appliance, each interface in the process must be thoroughly understood. Although information available about the biological processes invoked during tooth movement has dramatically increased through research efforts directed at elucidating them, the mechanical appliance itself currently remains the clinician's prima D, tool for controlling the direction and magnitude of the orthodontic response. 4,5 An active orthodontic appliance produces a force system at the crown of a tooth, or segment of teeth, that has been targeted for movement. This force system consists of a combination of m o m e n t s
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From the Department of Orthodontics, School of Dentistry, Virginia Commonwealth University, Richmond, VA. Supported in part by the Medical CoUegeof Vir~nia Orthodontic Education & Research Foundation. Address correspondence to Steven J. Lindauer, DMD, MDSc, Department of Orthodontics, School of Dentist*y, Vir~nia Commonwealth University, Richmond, VA 23298-0566. Copy,4ght © 2000 by W.B. Saunders Company 1073-8746/00/0603-0003510. 00/0 doi: 10.1053/sodo. 2000. 8081
and forces and can be controlled both quantitatively and qualitatively. The amount and m a n n e r in which a tooth is actually displaced within the periodontium as a result of applying this force system, depends on the magnitude and directional characteristics of the force system as well as on the location of the point of force application relative to the tooth as a whole. 6 Orthodontic appliances rarely, if ever, produce unifoim stresses in the periodontal ligament. 4 Only a single force or its equivalent, acting directly through the center of resistance of a tooth, could produce uniform m o v e m e n t resulting in pure translation of the tooth in the direction of the force. Even if uniform displacement occurred by applying the proper mechanical force system to cause translation, irregularities in the periodontal ligament itself make it unlikely that these stresses would be transmitted identically to the responsive cells and tissues (Fig 1). Therefore, from any orthodontic appliance activation, a gradation of stress is expected to result in the periodontal ligament along the tooth surface which would cause a gradation of biological response to that stress. This complicates the interpretation of biological informarion gathered in experiments designed to measure responses to specific stresses, which are often assumed to be homogenous. Further complicating the process of interpreting and predicting the cellular response to orthodontic force systems is the three-dimensional as-
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point of attachment, and the ratio of moment to force applied.
Controlling the Quantity of Tooth Movement
Figure 1. Irregularities in the shape of the periodontal ligament make it unlikely that uniform stresses result in a uniform cellular response. pect of the tooth itself. Teeth and the directions of forces applied can be depicted simply in 2 dimensions on a flat page as producing pressure on 1 side of a root and tension on the other. Even with a perfectly conical root and uniform periodontal ligament, pressure transitions smoothly to tension at some point in between, theoretically p r o d u c i n g a stress tangent to the bone surface at the point of transition. Interpretations are more difficult for multirooted teeth or teeth with roots that are more irregularly shaped in cross-section. An orthodontic appliance transfers mechanical stresses through the tooth to the periodontium where they are translated into physical, chemical, and electrical signals to cells that activate tissue remodeling to allow tooth movement. 3 The orthodontist is able to control the quantity and quality of the force system applied to the teeth, whereas the speed and way in which teeth move is ultimately determined by the biological response. The quantity of force application can be adjusted by altering the magnitude of activation, and the quality of the force system depends on the direction,
In a purely mechanical system, acceleration is proportional to force (F = ma). Applying a greater force makes an object move faster. In a biological system such as is found in orthodontic treatment, more force may not necessarily equate to more or faster tooth movement. The force applied to a tooth causes initial movement within the periodontal ligament space by displacing tissues and fluid, bending bone, and extruding the tooth to some degree. This rapid, initial movement is often considered the first phase in a three-phase process and has been reported to be on the order of 0.3 to 0.9 mm. 7 This is followed by a lag phase which usually lasts about 14 to 25 days and is thought to be due to hyalinization. 8 The subsequent bone resorption and formation that occurs, the postlag phase which is often considered true tooth movement, is believed to be limited by the resorption process which depends on the metabolism, recruitment, and differentiation of osteoclasts. Bone remodeling may or may not occur faster when greater forces are applied. One of the difficulties found when trying to discern the dependence of rate of tooth movement on magnitude of force is the determination of how much force is actually transmitted by the tooth through the periodontal ligament to the reactive cells. Clinically, frictional components of the appliance system that are not quantifiable may decrease the net force effected at the bracket. More significantly, it is not the force applied at the bracket that is important biologically, but the force per unit root surface that is transmitted at the level of the periodontal ligament. The force per unit area decreases for a given applied force as the amount of root surface increases. Therefore, for a large, multirooted tooth, the magnitude of force transmitted to the surrounding cells is less than for a small, single-rooted tooth in which the force is concentrated over a lesser area. Precise root surface area of any given tooth or segment of teeth is an unknown quantity but can be estimated using average data. It is not practical, however, to measure the stress transmitted at the periodontal ligament clinically using current technologyY In addition, because the stress distribution along the root is almost always nonuniform, the force per unit area exerted biologically varies depending on the
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type o f t o o t h d i s p l a c e m e n t a n d , consequently, by the l o c a t i o n o f interest a l o n g t h e root. Despite the i m p r e c i s i o n a n t i c i p a t e d in trying to relate rate o f t o o t h m o v e m e n t to m a g n i t u d e o f force, various o r t h o d o n t i c t e c h n i q u e s have evolved that are b a s e d o n a s s u m e d relationships b e t w e e n t h e 2 p a r a m e t e r s . T r e a t m e n t strategies m a y b e quite different d e p e n d i n g o n the perceived d e p e n d e n c e o f rate o f t o o t h m o v e m e n t o n force m a g n i t u d e . Based o n their analysis o f experi m e n t a l observations o f t o o t h m o v e m e n t , Q u i n n a n d Yoshikawa 1° reviewed 4 h y p o t h e s e s d e s c r i b i n g the d e p e n d e n c e o f rate o f t o o t h m o v e m e n t o n m a g n i t u d e o f force (Fig 2).
I n the first theory, a force t h r e s h o l d exists which, if e x c e e d e d , results in a fixed rate o f t o o t h m o v e m e n t (Fig 2A). Above the threshold, increasi n g the a m o u n t o f force results in n o f u r t h e r increases in the s p e e d at which teeth move. By this theory, t o o t h m o v e m e n t is an o n / o f f switch. Prop o n e n t s w o u l d a r g u e that cells can b e e i t h e r active o r not, a n d that i n c r e a s i n g force d o e s n o t m a k e cells metabolize, differentiate, o r r e c r u i t faster. If this t h e o r y were true, t h e o r t h o d o n t i s t w o u l d exp e c t to see e q u a l a m o u n t s o f t o o t h m o v e m e n t f r o m b o t h a n t e r i o r a n d p o s t e r i o r c o m p o n e n t s duri n g space closure w h e n a s e c o n d molar, first molar, a n d s e c o n d p r e m o l a r were u s e d to r e t r a c t a
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Figure 2. Four hypotheses relating the rate of tooth movement to force magnitude. Rate of tooth movement is constant and unrelated to force magnitude (A). Rate of tooth movement is proportional to force magnitude (B). Rate of tooth movement peaks at an optimal force level (C). Rate of tooth movement is proportional to force magnitude at low force levels and is constant at high force levels (D). (Adapted and reprinted with permission from Quinn and Yoshikawa. 1°)
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single canine. Unless the stress transmitted at the level o f the p e r i o d o n t i u m o f the posterior teeth was below the threshold at which tooth m o v e m e n t occurs (because o f the greater distribution o f force along the greater root surface area present), equal amounts of posterior versus anterior tooth movem e n t would be expected. In this case, anchorage could be maintained if a minimal a m o u n t o f force were used to retract anterior teeth because the posterior force would be distributed over multiple roots and would be too small to activate cellular r e m o d e l i n g processes. Interarch mechanics would be effective at titrating space closure only if it were the sole means o f force application used or if it were used to cancel intra-arch forces applied by a n o t h e r mechanism. Supporting this theory are studies showing n o increases in rate o f tooth movem e n t when greater forces are used. 11 T h e second theory is that rate of tooth movem e n t increases linearly with force magnitude. Increasing a m o u n t s o f force would cause teeth to move faster (Fig 2B). In this theory, greater force would n e e d to result in greater metabolic rates, recruitment, or differentiation o f cells involved in tissue remodeling. Acid phosphatase levels, indicative of b o n e resorptive activity, have b e e n shown to be higher w h e n forces are increased, supporting this theolT?2 U n d e r this hypothesis, a large, multirooted tooth would move slower than a smaller, single-rooted tooth during space closure because, although the force o n each tooth would be the same, the force per unit root area would be greater on the smaller tooth. A n c h o r a g e could be enh a n c e d by j o i n i n g large segments o f teeth into stable units that would distribute applied forces over a greater root surface area, or by using extraoral appliances or interarch mechanics to decrease effective forces o n a n c h o r units. Interarch elastics could also be used to a u g m e n t forces on units of teeth in which more movement was desired. Supporting this theopy are studies showing that teeth move faster when larger loads are appliedJ ~a4 Third, it is theorized that rate of tooth movem e n t increases with force up to a point, after which the rate decreases or ceases as force levels continue to increase (Fig 2C). A m a x i m u m a m o u n t of tooth m o v e m e n t would occur at some optimal force level. P r o p o n e n t s would argue that above the optimal force, greater forces prevent the recruitment or differentiation o f cells or that the high pressures cause tissue hyalinization, slowing tooth m o v e m e n t a n d affecting cell-tissue interactions. Supportive of this, alkaline phosphastase ac-
tivity studies suggest that b o n e formation may be related to force magnitude, peaking in an optimal range and displaying lower activities in response to both inadequate a n d excessive forces. 12 Anchorage u n d e r this hypothesis could be controlled as in the previous theory at low force levels or by increasing forces b e y o n d the optimum, thereby at-resting tooth m o v e m e n t in smaller a n c h o r units while allowing larger units to move at a maximal rate. This theory was originally p r o p o s e d by Smith and Storey ~5 and is also supported by a m o r e recent tooth m o v e m e n t study. 16 T h e last proposal is that the rate o f tooth movem e n t increases linearly with force up to a point at which increasing magnitudes o f force result in no further increases in rate (Fig 2D). A m a x i m u m a m o u n t of tooth m o v e m e n t would occur when stresses exceeded some a m o u n t per unit root area. This would be compatible with some biological limit to any increases in cell metabolism or to the n u m b e r of cells available to engage in the remodeling process. T o o t h m o v e m e n t and anchorage control would be similar to the theory o f increasing response proportional to force at low force levels a n d would be similar to the first theory with a constant rate o f tooth m o v e m e n t as a function o f force at high forces. To maximize anchorage, it would be best to have the effective forces working to p r o d u c e m a x i m u m tooth m o v e m e n t on the tooth or teeth to be m o v e d while submaximal forces were acting o n the a n c h o r unit. Some studies have shown increasing rates o f tooth m o v e m e n t with increased force up to a point where the response is constant with further force increases. 1r,18 Although Q u i n n a n d Yoshikawa 1° concluded that the last m o d e l was the one most supported by experimental a n d clinical data, it remains unclear exactly which, if any, o f the 4 theories correctly describes the relationship between rate o f tooth m o v e m e n t and applied force. Many o f the experiments cited failed to control the quality o f tooth movement, and measurements o f tooth displacem e n t were m a d e at the crowns rather than at the centers of resistance. Evaluations o f tooth movem e n t were m a d e over relatively short periods o f time, perhaps before the start of the postlag phase. What remains clear, however, is that even with relatively equal forces, rates o f tooth m o v e m e n t vary substantially a m o n g a n d even within individuals. n,14,19 Although this may be due to differences in the quality of the force systems applied, it is also likely that there is localized variability in the cellular response to otherwise shnilar applications of force.
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Figure 3. Pure tipping. A pure couple is applied at the bracket (A). Pure rotation around the center of resistance results (B). Stresses in the periodontal ligament are greatest at the apex and alveolar crest in opposite directions. There is no stress at the center of resistance (C). The center of rotation is at the center of resistance (D).
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Controlling the Quality of Tooth Movement Edgewise o r t h o d o n t i c appliances p r o v i d e clinicians with the p o t e n t i a l to apply d e s i r e d force systems in all 3 dimensions. 2° This allows for precise c o n t r o l o f the m o m e n t s a n d forces that will b e t r a n s m i t t e d t h r o u g h the t o o t h as stresses in the p e r i o d o n t a l ligament, initiating biological signals that will result in cellular activation, tissue r e m o d eling, a n d t o o t h m o v e m e n t . By c o n t r o l l i n g the c e n t e r o f rotation, the o r t h o d o n t i s t can c h a n g e the quality o f t o o t h m o v e m e n t f r o m crown t i p p i n g to p u r e translation to r o o t positioning. This is achieved by v a t t i n g the ratio o f m o m e n t s a n d forces a p p l i e d to the teeth at the o r t h o d o n t i c brackets.z1-24
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! ii:i!i : A l t h o u g h t h e terms center of rotation a n d momentto-force ratio describe m e c h a n i c a l concepts, they have real biological applicability. By effecting the p r o p e r m o m e n t - t o - f o r c e ratio a n d attaining the d e s i r e d c e n t e r o f rotation, the t o o t h is d i s p l a c e d in the m a n n e r n e e d e d to transmit the correct direction o f stresses at the level o f the p e r i o d o n t a l ligam e n t to stimulate the k i n d o f biological response r e q u i r e d . C h a r a c t e r i z i n g t o o t h m o v e m e n t as a c o m b i n a t i o n o f t i p p i n g a n d translation is a useful way o f m a t h e m a t i c a l l y calculating the n e t effects o f m e c h a n i c a l force systems in orthodontics. However, d e s c r i b i n g t o o t h m o v e m e n t a r o u n d a "center o f rotation" is a biologically accurate way o f illustrating the stresses t r a n s m i t t e d by the teeth to surr o u n d i n g tissues after a p p l i a n c e activation. 4,2]
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( Q Figure 4. Tipping. A force is applied at the bracket (A). A force and a tendency for the crown to tip in the direction of the force result at the center of resistance (B). Stresses in the periodontal ligament are greatest at the alveolar crest in the direction of the force. Stresses at the level of the center of resistance are in the direction of the force and are opposite in direction at the apex (C). The center of rotation is apical to the center of resistance and the center of resistance moves in the direction of the force (D).
ii:iljll iiiiiiljli iiiiiiiiiiii !i i ii iiiii!! ! I i!iiiI¸III Pure Tipping W h e n only a couple, 2 equal a n d o p p o s i t e forces n o t acting t h r o u g h the s a m e point, is a p p l i e d to a body, p u r e r o t a t i o n a r o u n d t h e c e n t e r o f gravity o r mass occurs. Because teeth are r e s t r a i n e d by the p e r i o d o n t a l l i g a m e n t a n d t h e alveolar b o n e , rotation in r e s p o n s e to the a p p l i c a t i o n o f a p u r e coup l e occurs a r o u n d the c e n t e r o f resistance r a t h e r t h a n the c e n t e r o f mass. m D e p e n d i n g o n the p l a n e in which t h e c o u p l e is acting, this r o t a t i o n has b e e n called "rotation" (first o r d e r ) , "tipping" (seco n d o r d e r ) , o r "torque" (third o r d e r ) in o r t h o d o n tics. In any d i m e n s i o n , w h e n a p u r e c o u p l e is a p p l i e d to a tooth, the c e n t e r o f r o t a t i o n is at the c e n t e r o f resistance. T h a t is, in the a b s e n c e o f a n e t force to translate the tooth, the c e n t e r o f resistance d o e s n o t move. A p p l i c a t i o n o f a p u r e c o u p l e to effect t i p p i n g o r t o r q u e is e x p e c t e d to cause a u n i f o r m l y increasing a m o u n t o f stress to the p e r i o d o n t a l l i g a m e n t at
increasing distances f r o m the c e n t e r o f resistance (Fig 3). M a x i m u m stresses apically are at the r o o t a p e x mad coronally at the h e i g h t o f the alveolar ridge. T h e r e is n o stress t r a n s m i t t e d to the perio d o n t a l l i g a m e n t at the level o f the c e n t e r o f resistance2 Pressure is the m e c h a n i c a l signal sent to the cells at the alveolar crest toward which the crown is tipping, whereas t e n s i o n is the signal a l o n g the same surface o f the t o o t h n e a r the apex. Cellular r e s p o n s e results in n e t r e s o r p t i o n o f b o n e at the alveolar crest o n the side o f pressure, a n d b o n e d e p o s i t i o n occurs o n the same side o f the t o o t h apically.
Tipping W h e n a single force is a p p l i e d to a t o o t h at the level o f the bracket, t i p p i n g occurs which moves the crown in the d i r e c t i o n o f the force a n d the r o o t a p e x in the o p p o s i t e direction. This is similar to p u r e tipping, e x c e p t the t o o t h itself moves; the
Orthodontic Mechanics
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Figure 5. Pure translation. A force and a couple equal and opposite to the m o m e n t created by the force are applied at the bracket (A). The equivalent of a single force results at the center of resistance (B). Stresses in the periodontal ligament are relatively equal along the root surface in the direction of the force (C). The center of rotation is at infinity and the tooth moves bodily in the direction of the force (D).
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i iilill ii!iiiiii! Iiiii!i i c e n t e r o f resistance o f the t o o t h is d i s p l a c e d in the d i r e c t i o n o f the force (Fig 4). T h e c e n t e r o f rotation o f the tooth, the p o i n t at which n o stresses are t r a n s m i t t e d t h r o u g h the p e r i o d o n t a l l i g a m e n t to s u r r o u n d i n g cells r e s p o n s i b l e for tissue r e m o d e l ing, is apical to the c e n t e r o f resistance. F r o m the c e n t e r o f r o t a t i o n coronally, a n i n c r e a s i n g gradie n t o f pressure in the d i r e c t i o n o f the force causes n e t r e s o r p t i o n o f b o n e o n that side o f the tooth. Apical to the c e n t e r o f rotation, n e t d e p o s i t i o n o f b o n e occurs o n that side in r e s p o n s e to tension delivered to the p e r i o d o n t a l l i g a m e n t t h r o u g h the t o o t h f r o m the a p p l i a n c e activation. As tissue rem o d e l i n g occurs, the result is a m o v e m e n t o f the t o o t h (the c e n t e r o f resistance), in the d i r e c t i o n o f the force, with m o v e m e n t o f the crown f u r t h e r in the same direction, a n d m o v e m e n t o f t h e a p e x in the o p p o s i t e direction.
Pure Translation W h e n a clinician wants to b o d i l y move a t o o t h without tipping, (for e x a m p l e , d u r i n g space closure), a c o m b i n a t i o n o f force a n d m o m e n t m u s t b e a p p l i e d at the bracket. This can b e accomp l i s h e d using any n u m b e r o f m e c h a n i c a l devices d e s i g n e d to c o u n t e r a c t the t e n d e n c y for a t o o t h to tip w h e n a single force is acting at the crown. A force a l o n e w o u l d cause t h e c e n t e r o f resistance to move as d e s i r e d but, as m e n t i o n e d , w o u l d displace the a p e x in the o p p o s i t e direction. T o avoid this, the o r t h o d o n t i s t m u s t apply a c o u p l e to the t o o t h to c o u n t e r a c t the t e n d e n c y to tip. T o achieve translatory d i s p l a c e m e n t , the m a g n i t u d e o f this c o u p l e m u s t b e precisely equal to the a m o u n t o f the m o m e n t c r e a t e d by t h e force which will tip the crown in the d i r e c t i o n o f the force. T h e m o m e n t c r e a t e d
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Figure 6. Crown movement. A force and a couple are applied at the bracket. The couple is smaller in magnitude than the moment created by the applied force (A). A force and a small tendency for the crown to tip in the same direction result at the center of resistance (B). Stresses in the periodontal ligament are smallest near the apex and greatest at the alveolar crest in the direction of the force (C). The center of rotation is at the apex and the crown tips in the direction of the force (D).
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by the force t i p p i n g the crown in the d i r e c t i o n o f the force is equal to the m a g n i t u d e o f the force m u l t i p l i e d by the p e r p e n d i c u l a r distance f r o m the force to the c e n t e r o f resistance o f the tooth. In o t h e r words, the t e n d e n c y to tip o r " m o m e n t " increases as the force is a p p l i e d f u r t h e r f r o m the c e n t e r o f resistance. If the force c o u l d b e a p p l i e d at the c e n t e r o f resistance, the t e n d e n c y to tip w o u l d b e zero a n d the t o o t h w o u l d translate in the d i r e c t i o n o f the force. In this case, the c e n t e r o f r o t a t i o n o f the t o o t h w o u l d b e at infinity b e c a u s e the t o o t h d o e s n o t "rotate" o r tip at all. If the b r a c k e t is 10 m m f r o m the c e n t e r o f resistance, a force a p p l i e d at the b r a c k e t causes the t o o t h to tip b e c a u s e o f a m o m e n t that is 10 m m times the m a g n i t u d e o f the force. T o c o u n t e r a c t this t e n d e n c y to tip, a c o u p l e in the o p p o s i t e dir e c t i o n with a m o m e n t 10 m m times the m a g n i t u d e o f the force w o u l d n e e d to b e a p p l i e d in a d d i t i o n to the force. This w o u l d b e an a p p l i e d
m o m e n t - t o - f o r c e ratio o f 10:1, resulting in disp l a c e m e n t o f the t o o t h as if the force a l o n e h a d b e e n p l a c e d t h r o u g h the c e n t e r o f resistance. It is n o t possible to d e t e r m i n e clinically the exact distance o f the b r a c k e t f r o m the c e n t e r o f resistance o f each individual tooth, b u t it is e s t i m a t e d that the c e n t e r o f resistance o f a typical maxillary incisor is a p p r o x i m a t e l y 10 m m f r o m the bracket. T h e r e fore, a m o m e n t - t o - f o r c e ratio o f a b o u t 10:1 is pres u m e d to be necessm y to cause p u r e translation o f a t o o t h orthodontically. W h e n the c o u p l e a p p l i e d a l o n g with a three is precisely the r i g h t m a g n i t u d e to c o u n t e r a c t the m o m e n t c r e a t e d by the force, the t o o t h is disp l a c e d bodily in the d i r e c t i o n o f the force as if a single force h a d b e e n a p p l i e d t h r o u g h the c e n t e r o f resistance (Fig 5). This causes a u n i f o r m stress to b e t r a n s m i t t e d by the tooth to the p e r i o d o n t a l l i g a m e n t with pressure in the d i r e c t i o n o f the force a n d tension o n the o t h e r side. I f the p e r i o d o n t a l
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f Figure 7. Root movement. A force and a couple are applied at the bracket. The couple is greater in magnitude than the moment created by the applied force (A). A force and a small tendency for the crown to tip in the opposite direction result at the center of resistance (B). Stresses in the periodontal ligament are smallest at the alveolar crest and greatest at the apex in the direction of the force (C). The center of rotation is at the crown and the root moves in the direction of the force (D).
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l i g a m e n t transmits this stress u n i f o r m l y to the responsive cells, the result is n e t b o n e resorption o n the side of pressure a n d n e t deposition o n the tension side, allowing the tooth to move in a translatory m a n n e r without tipping.
Crown M o v e m e n t If the moment-to-force ratio applied to a tooth is less t h a n 10:1, the result will n o t be p u r e translation of the tooth in the direction of the force. Instead, because the m o m e n t created by the applied force is n o t completely n e g a t e d by the applied couple, the result is greater m o v e m e n t of the crown t h a n the root i n the direction of the force. For a central incisor, a moment-to-force ratio of approximately 7:1 or 8:1 will cause the tooth to move a r o u n d a p o i n t n e a r the apex of the tooth, m With the c e n t e r of rotation at the apex, the magn i t u d e of the stress transmitted by the tooth to the p e r i o d o n t a l l i g a m e n t is smallest n e a r the apex a n d
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greatest at the crest of the alveolar b o n e (Fig 6). T h e result o n the side of pressure is n e t resorption of b o n e which is greater n e a r the crown t h a n at the apex. This causes the crown of the tooth to tip in the direction of the force a n d the center of resistance to move i n the direction of the force b u t to a lesser degree t h a n the crown. I n this case, the apex of the tooth moves very little or n o t at all.
Root Movement To achieve root m o v e m e n t with the center of rotation at the crown of a tooth, the applied moment-to-force ratio m u s t be greater t h a n 10:1. T h e couple applied will m o r e t h a n c o u n t e r a c t the mom e n t created by the force. As the center of resistance of the tooth moves in the direction of the force, the applied couple tips the crown in the opposite direction. A moment-to-force ratio of 12:1 or 13:1 is believed to be sufficient to effect a c e n t e r of rotation at the crown/~ I n o t h e r words,
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the crown of the tooth moves very little while the center of resistance moves in the direction of the force. The apex of the root also moves in the direction of the force but to an even greater degree. An increasing gradient of stress is transmitted by the tooth to the periodontal ligament which is greatest at the apex and least at the alveolar crest (Fig 7). Net resorption of bone occurs along the entire root surface in the direction of the force but more resorption is effected at the apex than at the alveolar crest. This results in m o v e m e n t of the root in the direction of the force while the crown remains relatively stationary. Conclusion Research into the biological reaction of h u m a n tissues to orthodontic forces has achieved much in the way o f identifying reactive cells, recruitment mechanisms, and specific signaling processes. Fundamental clinical principles, however, such as the relationship between the magnitude of applied force and the rate of b o n e resorption and deposition that effect tooth movement, have not been completely elucidated. C o m p o u n d i n g difficulties at efforts to discern these principles is misunderstanding of the mechanical components of orthodontic force application. Complications arising from an inability to measure friction at the appliance-bracket interface and to calculate precisely the distribution of stress at the level of the periodontal ligament, further impede efforts to understand biological relationships. Improving the clinician's control of orthodontic tooth m o v e m e n t is the ability to predictably vary the quality of the force system applied. This can be achieved through an understanding of the way in which varying the moment-to-force ratio applied to the teeth affects the center of rotation during tooth movement. Controlling the center of rotation is currently the clinician's most predictable way of managing the biological response to mechanical force systems applied during orthodontic treatment. References 1. Roberts WE, Goodwin WC, H e i n e r SR. Cellular response to orthodontic force. Dent Clin N A m 1981;25:3-17. 2. Yoshikawa DK. Biomechanical principles of tooth movement. Dent Clin N A m 1981;25:19-26. 3. Davidovitch Z. T o o t h m o v e m e n t . Crit Rev Oral Biol Med 1991;2:411-450. 4. Burstone CJ. Application of b i o e n g i n e e r i n g to clinical orthodontics. In: Graber TM, Swain BF (eds): O r t h o d o n tics. C u r r e n t Principles a n d Techniques. St Louis, MO: CV Mosby, 1985, pp 193-227.
5. Isaacson RJ, L i n d a u e r SJ, Conley P. T h e interface between the mechanical signal a n d the biological response in orthodontics. In: Davidovitch Z, Norton LA (eds): Biological Mechanisms of T o o t h M o v e m e n t a n d Craniofacial Adaptation. Boston, MA: Harvard Society for the A d v a n c e m e n t of Orthodontics, 1996, pp 1-5. 6. Middleton J, J o n e s ML, Wilson AN. Three-dimensional analysis of orthodontic tooth m o v e m e n t . J Biomed Eng 1990;12:319-327. 7. Burstone CJ. Biomechanics of tooth movement. In: Kraus BS, Riedel RA (eds): Vistas in Orthodontics. Philadelphia, PA: Lea & Febiger, 1962, pp 197-213. 8. Reitan 14. Some factors d e t e r m i n i n g the evaluation of forces in orthodontics. A m J O r t h o d 1957;43:32-45. 9. Lee BW. The force reqtfirements for tooth movement part III: The pressure hypothesis. Aust O r t h o d J 1996;14:93-97. 10. Q u i n n RS, Yoshikawa DK. A reassessment of force magn i t u d e in orthodontics. A m J O r t h o d 1985;88:252-260. 11. Owman-Moll P, Kurol J, L u n d g r e n D. Effects of a doubled orthodontic force m a g n i t u d e on tooth m o v e m e n t a n d root resorptions. An inter-individual study in adolescents. E u r J O r t h o d 1996;18:141-150. 12. Keeling SD, King GJ, McCoy EA, et al. S e r u m a n d alveolar b o n e p h o s p h a t a s e c h a n g e s reflect b o n e turnover d u r i n g orthodontic tooth m o v e m e n t . A m J O r t h o d Dentofac O r t h o p 1993;103:320-326. 13. A n d r e a s e n G, J o h n s o n P. Experimental findings on tooth m o v e m e n t u n d e r two conditions o f applied force. Angle O r t h o d 1967;37:9-12. 14. Owman-Moll P, Kurol J, L u n d g r e n D. T h e effects of a four-fold increased orthodontic force m a g n i t u d e on tooth m o v e m e n t a n d root resorptions. An intra-individual study in adolescents. E u r J O r t h o d 1996;18:287-294. 15. Smith R, Storey E. T h e i m p o r t a n c e of force in o r t h o d o n tics. A u s t J Dent 1952;56:291-304. 16. Lee BW. T h e force r e q u i r e m e n t s for tooth m o v e m e n t part I: Tipping a n d bodily m o v e m e n t . Aust O r t h o d J 1995;13:238-248. 17. Boester CH, J o h n s t o n LE. A clinical investigation of the concepts of differential a n d optimal force in canine retraction. Angle O r t h o d 1974;44:113-119. 18. King GJ, Keeling SD, McCoy AA, et al. Measuring dental drift a n d orthodontic tooth m o v e m e n t in response to various initial forces in adult rats. A m J O r t h o d Dentofac O r t h o p 1991;99:456465. 19. L u n d g r e n D, Owman-Moll P, KurolJ. Early tooth movem e n t pattern after application of a controlled continuous orthodontic force. A h u m a n experimental model. A m J O r t h o d Dentofac O r t h o p 1996;110:28%294. 20. Isaacson RJ, Lindauer SJ, Conley P. Responses of 3-D arch wires to vertical v bends. Semin Orthod 1995;1:5%63. 21. Smith RJ, Burstone CJ. Mechanics of tooth m o v e m e n t . A m J O r t h o d 1984;85:294-307. 22. Kusy RP, Tulloch JFC. Analysis of m o m e n t / f o r c e ratios in the m e c h a n i c s of tooth m o v e m e n t . A m J O r t h o d Dentofac O r t h o p 1986;90:127-131. 23. Andersen KL, Mortensen t-IT, Pedersen EH, et al. Determination of stress levels and profiles in the periodontal ligament by m e a n s of an improved threeqtimensional finite element model for various types of orthodontic and natural force systems. J Biomed Eng 1991;13:293-303. 24. Isaacson RJ, L i n d a u e r SJ, Davidovitch M. T h e g r o u n d rules for arch wire design: Constraints and opportunities. Semin O r t h o d 1995;1:3-11.
he cascade of biological events that induce orthodontic tooth m o v e m e n t is initiated by mechanical stresses in the periodontium. ~,2 Forces are transferred to the teeth by the clinician using appliances designed to displace teeth a prescribed amount in a desired direction. This sends signals to the cells to remodel tissues in a way that allows teeth to move. ~ To interpret the biological responses to activation of any orthodontic appliance, each interface in the process must be thoroughly understood. Although information available about the biological processes invoked during tooth movement has dramatically increased through research efforts directed at elucidating them, the mechanical appliance itself currently remains the clinician's prima D, tool for controlling the direction and magnitude of the orthodontic response. 4,5 An active orthodontic appliance produces a force system at the crown of a tooth, or segment of teeth, that has been targeted for movement. This force system consists of a combination of m o m e n t s
T
From the Department of Orthodontics, School of Dentistry, Virginia Commonwealth University, Richmond, VA. Supported in part by the Medical CoUegeof Vir~nia Orthodontic Education & Research Foundation. Address correspondence to Steven J. Lindauer, DMD, MDSc, Department of Orthodontics, School of Dentist*y, Vir~nia Commonwealth University, Richmond, VA 23298-0566. Copy,4ght © 2000 by W.B. Saunders Company 1073-8746/00/0603-0003510. 00/0 doi: 10.1053/sodo. 2000. 8081
and forces and can be controlled both quantitatively and qualitatively. The amount and m a n n e r in which a tooth is actually displaced within the periodontium as a result of applying this force system, depends on the magnitude and directional characteristics of the force system as well as on the location of the point of force application relative to the tooth as a whole. 6 Orthodontic appliances rarely, if ever, produce unifoim stresses in the periodontal ligament. 4 Only a single force or its equivalent, acting directly through the center of resistance of a tooth, could produce uniform m o v e m e n t resulting in pure translation of the tooth in the direction of the force. Even if uniform displacement occurred by applying the proper mechanical force system to cause translation, irregularities in the periodontal ligament itself make it unlikely that these stresses would be transmitted identically to the responsive cells and tissues (Fig 1). Therefore, from any orthodontic appliance activation, a gradation of stress is expected to result in the periodontal ligament along the tooth surface which would cause a gradation of biological response to that stress. This complicates the interpretation of biological informarion gathered in experiments designed to measure responses to specific stresses, which are often assumed to be homogenous. Further complicating the process of interpreting and predicting the cellular response to orthodontic force systems is the three-dimensional as-
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point of attachment, and the ratio of moment to force applied.
Controlling the Quantity of Tooth Movement
Figure 1. Irregularities in the shape of the periodontal ligament make it unlikely that uniform stresses result in a uniform cellular response. pect of the tooth itself. Teeth and the directions of forces applied can be depicted simply in 2 dimensions on a flat page as producing pressure on 1 side of a root and tension on the other. Even with a perfectly conical root and uniform periodontal ligament, pressure transitions smoothly to tension at some point in between, theoretically p r o d u c i n g a stress tangent to the bone surface at the point of transition. Interpretations are more difficult for multirooted teeth or teeth with roots that are more irregularly shaped in cross-section. An orthodontic appliance transfers mechanical stresses through the tooth to the periodontium where they are translated into physical, chemical, and electrical signals to cells that activate tissue remodeling to allow tooth movement. 3 The orthodontist is able to control the quantity and quality of the force system applied to the teeth, whereas the speed and way in which teeth move is ultimately determined by the biological response. The quantity of force application can be adjusted by altering the magnitude of activation, and the quality of the force system depends on the direction,
In a purely mechanical system, acceleration is proportional to force (F = ma). Applying a greater force makes an object move faster. In a biological system such as is found in orthodontic treatment, more force may not necessarily equate to more or faster tooth movement. The force applied to a tooth causes initial movement within the periodontal ligament space by displacing tissues and fluid, bending bone, and extruding the tooth to some degree. This rapid, initial movement is often considered the first phase in a three-phase process and has been reported to be on the order of 0.3 to 0.9 mm. 7 This is followed by a lag phase which usually lasts about 14 to 25 days and is thought to be due to hyalinization. 8 The subsequent bone resorption and formation that occurs, the postlag phase which is often considered true tooth movement, is believed to be limited by the resorption process which depends on the metabolism, recruitment, and differentiation of osteoclasts. Bone remodeling may or may not occur faster when greater forces are applied. One of the difficulties found when trying to discern the dependence of rate of tooth movement on magnitude of force is the determination of how much force is actually transmitted by the tooth through the periodontal ligament to the reactive cells. Clinically, frictional components of the appliance system that are not quantifiable may decrease the net force effected at the bracket. More significantly, it is not the force applied at the bracket that is important biologically, but the force per unit root surface that is transmitted at the level of the periodontal ligament. The force per unit area decreases for a given applied force as the amount of root surface increases. Therefore, for a large, multirooted tooth, the magnitude of force transmitted to the surrounding cells is less than for a small, single-rooted tooth in which the force is concentrated over a lesser area. Precise root surface area of any given tooth or segment of teeth is an unknown quantity but can be estimated using average data. It is not practical, however, to measure the stress transmitted at the periodontal ligament clinically using current technologyY In addition, because the stress distribution along the root is almost always nonuniform, the force per unit area exerted biologically varies depending on the
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type o f t o o t h d i s p l a c e m e n t a n d , consequently, by the l o c a t i o n o f interest a l o n g t h e root. Despite the i m p r e c i s i o n a n t i c i p a t e d in trying to relate rate o f t o o t h m o v e m e n t to m a g n i t u d e o f force, various o r t h o d o n t i c t e c h n i q u e s have evolved that are b a s e d o n a s s u m e d relationships b e t w e e n t h e 2 p a r a m e t e r s . T r e a t m e n t strategies m a y b e quite different d e p e n d i n g o n the perceived d e p e n d e n c e o f rate o f t o o t h m o v e m e n t o n force m a g n i t u d e . Based o n their analysis o f experi m e n t a l observations o f t o o t h m o v e m e n t , Q u i n n a n d Yoshikawa 1° reviewed 4 h y p o t h e s e s d e s c r i b i n g the d e p e n d e n c e o f rate o f t o o t h m o v e m e n t o n m a g n i t u d e o f force (Fig 2).
I n the first theory, a force t h r e s h o l d exists which, if e x c e e d e d , results in a fixed rate o f t o o t h m o v e m e n t (Fig 2A). Above the threshold, increasi n g the a m o u n t o f force results in n o f u r t h e r increases in the s p e e d at which teeth move. By this theory, t o o t h m o v e m e n t is an o n / o f f switch. Prop o n e n t s w o u l d a r g u e that cells can b e e i t h e r active o r not, a n d that i n c r e a s i n g force d o e s n o t m a k e cells metabolize, differentiate, o r r e c r u i t faster. If this t h e o r y were true, t h e o r t h o d o n t i s t w o u l d exp e c t to see e q u a l a m o u n t s o f t o o t h m o v e m e n t f r o m b o t h a n t e r i o r a n d p o s t e r i o r c o m p o n e n t s duri n g space closure w h e n a s e c o n d molar, first molar, a n d s e c o n d p r e m o l a r were u s e d to r e t r a c t a
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Figure 2. Four hypotheses relating the rate of tooth movement to force magnitude. Rate of tooth movement is constant and unrelated to force magnitude (A). Rate of tooth movement is proportional to force magnitude (B). Rate of tooth movement peaks at an optimal force level (C). Rate of tooth movement is proportional to force magnitude at low force levels and is constant at high force levels (D). (Adapted and reprinted with permission from Quinn and Yoshikawa. 1°)
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single canine. Unless the stress transmitted at the level o f the p e r i o d o n t i u m o f the posterior teeth was below the threshold at which tooth m o v e m e n t occurs (because o f the greater distribution o f force along the greater root surface area present), equal amounts of posterior versus anterior tooth movem e n t would be expected. In this case, anchorage could be maintained if a minimal a m o u n t o f force were used to retract anterior teeth because the posterior force would be distributed over multiple roots and would be too small to activate cellular r e m o d e l i n g processes. Interarch mechanics would be effective at titrating space closure only if it were the sole means o f force application used or if it were used to cancel intra-arch forces applied by a n o t h e r mechanism. Supporting this theory are studies showing n o increases in rate o f tooth movem e n t when greater forces are used. 11 T h e second theory is that rate of tooth movem e n t increases linearly with force magnitude. Increasing a m o u n t s o f force would cause teeth to move faster (Fig 2B). In this theory, greater force would n e e d to result in greater metabolic rates, recruitment, or differentiation o f cells involved in tissue remodeling. Acid phosphatase levels, indicative of b o n e resorptive activity, have b e e n shown to be higher w h e n forces are increased, supporting this theolT?2 U n d e r this hypothesis, a large, multirooted tooth would move slower than a smaller, single-rooted tooth during space closure because, although the force o n each tooth would be the same, the force per unit root area would be greater on the smaller tooth. A n c h o r a g e could be enh a n c e d by j o i n i n g large segments o f teeth into stable units that would distribute applied forces over a greater root surface area, or by using extraoral appliances or interarch mechanics to decrease effective forces o n a n c h o r units. Interarch elastics could also be used to a u g m e n t forces on units of teeth in which more movement was desired. Supporting this theopy are studies showing that teeth move faster when larger loads are appliedJ ~a4 Third, it is theorized that rate of tooth movem e n t increases with force up to a point, after which the rate decreases or ceases as force levels continue to increase (Fig 2C). A m a x i m u m a m o u n t of tooth m o v e m e n t would occur at some optimal force level. P r o p o n e n t s would argue that above the optimal force, greater forces prevent the recruitment or differentiation o f cells or that the high pressures cause tissue hyalinization, slowing tooth m o v e m e n t a n d affecting cell-tissue interactions. Supportive of this, alkaline phosphastase ac-
tivity studies suggest that b o n e formation may be related to force magnitude, peaking in an optimal range and displaying lower activities in response to both inadequate a n d excessive forces. 12 Anchorage u n d e r this hypothesis could be controlled as in the previous theory at low force levels or by increasing forces b e y o n d the optimum, thereby at-resting tooth m o v e m e n t in smaller a n c h o r units while allowing larger units to move at a maximal rate. This theory was originally p r o p o s e d by Smith and Storey ~5 and is also supported by a m o r e recent tooth m o v e m e n t study. 16 T h e last proposal is that the rate o f tooth movem e n t increases linearly with force up to a point at which increasing magnitudes o f force result in no further increases in rate (Fig 2D). A m a x i m u m a m o u n t of tooth m o v e m e n t would occur when stresses exceeded some a m o u n t per unit root area. This would be compatible with some biological limit to any increases in cell metabolism or to the n u m b e r of cells available to engage in the remodeling process. T o o t h m o v e m e n t and anchorage control would be similar to the theory o f increasing response proportional to force at low force levels a n d would be similar to the first theory with a constant rate o f tooth m o v e m e n t as a function o f force at high forces. To maximize anchorage, it would be best to have the effective forces working to p r o d u c e m a x i m u m tooth m o v e m e n t on the tooth or teeth to be m o v e d while submaximal forces were acting o n the a n c h o r unit. Some studies have shown increasing rates o f tooth m o v e m e n t with increased force up to a point where the response is constant with further force increases. 1r,18 Although Q u i n n a n d Yoshikawa 1° concluded that the last m o d e l was the one most supported by experimental a n d clinical data, it remains unclear exactly which, if any, o f the 4 theories correctly describes the relationship between rate o f tooth m o v e m e n t and applied force. Many o f the experiments cited failed to control the quality o f tooth movement, and measurements o f tooth displacem e n t were m a d e at the crowns rather than at the centers of resistance. Evaluations o f tooth movem e n t were m a d e over relatively short periods o f time, perhaps before the start of the postlag phase. What remains clear, however, is that even with relatively equal forces, rates o f tooth m o v e m e n t vary substantially a m o n g a n d even within individuals. n,14,19 Although this may be due to differences in the quality of the force systems applied, it is also likely that there is localized variability in the cellular response to otherwise shnilar applications of force.
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Figure 3. Pure tipping. A pure couple is applied at the bracket (A). Pure rotation around the center of resistance results (B). Stresses in the periodontal ligament are greatest at the apex and alveolar crest in opposite directions. There is no stress at the center of resistance (C). The center of rotation is at the center of resistance (D).
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Controlling the Quality of Tooth Movement Edgewise o r t h o d o n t i c appliances p r o v i d e clinicians with the p o t e n t i a l to apply d e s i r e d force systems in all 3 dimensions. 2° This allows for precise c o n t r o l o f the m o m e n t s a n d forces that will b e t r a n s m i t t e d t h r o u g h the t o o t h as stresses in the p e r i o d o n t a l ligament, initiating biological signals that will result in cellular activation, tissue r e m o d eling, a n d t o o t h m o v e m e n t . By c o n t r o l l i n g the c e n t e r o f rotation, the o r t h o d o n t i s t can c h a n g e the quality o f t o o t h m o v e m e n t f r o m crown t i p p i n g to p u r e translation to r o o t positioning. This is achieved by v a t t i n g the ratio o f m o m e n t s a n d forces a p p l i e d to the teeth at the o r t h o d o n t i c brackets.z1-24
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! ii:i!i : A l t h o u g h t h e terms center of rotation a n d momentto-force ratio describe m e c h a n i c a l concepts, they have real biological applicability. By effecting the p r o p e r m o m e n t - t o - f o r c e ratio a n d attaining the d e s i r e d c e n t e r o f rotation, the t o o t h is d i s p l a c e d in the m a n n e r n e e d e d to transmit the correct direction o f stresses at the level o f the p e r i o d o n t a l ligam e n t to stimulate the k i n d o f biological response r e q u i r e d . C h a r a c t e r i z i n g t o o t h m o v e m e n t as a c o m b i n a t i o n o f t i p p i n g a n d translation is a useful way o f m a t h e m a t i c a l l y calculating the n e t effects o f m e c h a n i c a l force systems in orthodontics. However, d e s c r i b i n g t o o t h m o v e m e n t a r o u n d a "center o f rotation" is a biologically accurate way o f illustrating the stresses t r a n s m i t t e d by the teeth to surr o u n d i n g tissues after a p p l i a n c e activation. 4,2]
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B
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( Q Figure 4. Tipping. A force is applied at the bracket (A). A force and a tendency for the crown to tip in the direction of the force result at the center of resistance (B). Stresses in the periodontal ligament are greatest at the alveolar crest in the direction of the force. Stresses at the level of the center of resistance are in the direction of the force and are opposite in direction at the apex (C). The center of rotation is apical to the center of resistance and the center of resistance moves in the direction of the force (D).
ii:iljll iiiiiiljli iiiiiiiiiiii !i i ii iiiii!! ! I i!iiiI¸III Pure Tipping W h e n only a couple, 2 equal a n d o p p o s i t e forces n o t acting t h r o u g h the s a m e point, is a p p l i e d to a body, p u r e r o t a t i o n a r o u n d t h e c e n t e r o f gravity o r mass occurs. Because teeth are r e s t r a i n e d by the p e r i o d o n t a l l i g a m e n t a n d t h e alveolar b o n e , rotation in r e s p o n s e to the a p p l i c a t i o n o f a p u r e coup l e occurs a r o u n d the c e n t e r o f resistance r a t h e r t h a n the c e n t e r o f mass. m D e p e n d i n g o n the p l a n e in which t h e c o u p l e is acting, this r o t a t i o n has b e e n called "rotation" (first o r d e r ) , "tipping" (seco n d o r d e r ) , o r "torque" (third o r d e r ) in o r t h o d o n tics. In any d i m e n s i o n , w h e n a p u r e c o u p l e is a p p l i e d to a tooth, the c e n t e r o f r o t a t i o n is at the c e n t e r o f resistance. T h a t is, in the a b s e n c e o f a n e t force to translate the tooth, the c e n t e r o f resistance d o e s n o t move. A p p l i c a t i o n o f a p u r e c o u p l e to effect t i p p i n g o r t o r q u e is e x p e c t e d to cause a u n i f o r m l y increasing a m o u n t o f stress to the p e r i o d o n t a l l i g a m e n t at
increasing distances f r o m the c e n t e r o f resistance (Fig 3). M a x i m u m stresses apically are at the r o o t a p e x mad coronally at the h e i g h t o f the alveolar ridge. T h e r e is n o stress t r a n s m i t t e d to the perio d o n t a l l i g a m e n t at the level o f the c e n t e r o f resistance2 Pressure is the m e c h a n i c a l signal sent to the cells at the alveolar crest toward which the crown is tipping, whereas t e n s i o n is the signal a l o n g the same surface o f the t o o t h n e a r the apex. Cellular r e s p o n s e results in n e t r e s o r p t i o n o f b o n e at the alveolar crest o n the side o f pressure, a n d b o n e d e p o s i t i o n occurs o n the same side o f the t o o t h apically.
Tipping W h e n a single force is a p p l i e d to a t o o t h at the level o f the bracket, t i p p i n g occurs which moves the crown in the d i r e c t i o n o f the force a n d the r o o t a p e x in the o p p o s i t e direction. This is similar to p u r e tipping, e x c e p t the t o o t h itself moves; the
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Figure 5. Pure translation. A force and a couple equal and opposite to the m o m e n t created by the force are applied at the bracket (A). The equivalent of a single force results at the center of resistance (B). Stresses in the periodontal ligament are relatively equal along the root surface in the direction of the force (C). The center of rotation is at infinity and the tooth moves bodily in the direction of the force (D).
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i iilill ii!iiiiii! Iiiii!i i c e n t e r o f resistance o f the t o o t h is d i s p l a c e d in the d i r e c t i o n o f the force (Fig 4). T h e c e n t e r o f rotation o f the tooth, the p o i n t at which n o stresses are t r a n s m i t t e d t h r o u g h the p e r i o d o n t a l l i g a m e n t to s u r r o u n d i n g cells r e s p o n s i b l e for tissue r e m o d e l ing, is apical to the c e n t e r o f resistance. F r o m the c e n t e r o f r o t a t i o n coronally, a n i n c r e a s i n g gradie n t o f pressure in the d i r e c t i o n o f the force causes n e t r e s o r p t i o n o f b o n e o n that side o f the tooth. Apical to the c e n t e r o f rotation, n e t d e p o s i t i o n o f b o n e occurs o n that side in r e s p o n s e to tension delivered to the p e r i o d o n t a l l i g a m e n t t h r o u g h the t o o t h f r o m the a p p l i a n c e activation. As tissue rem o d e l i n g occurs, the result is a m o v e m e n t o f the t o o t h (the c e n t e r o f resistance), in the d i r e c t i o n o f the force, with m o v e m e n t o f the crown f u r t h e r in the same direction, a n d m o v e m e n t o f t h e a p e x in the o p p o s i t e direction.
Pure Translation W h e n a clinician wants to b o d i l y move a t o o t h without tipping, (for e x a m p l e , d u r i n g space closure), a c o m b i n a t i o n o f force a n d m o m e n t m u s t b e a p p l i e d at the bracket. This can b e accomp l i s h e d using any n u m b e r o f m e c h a n i c a l devices d e s i g n e d to c o u n t e r a c t the t e n d e n c y for a t o o t h to tip w h e n a single force is acting at the crown. A force a l o n e w o u l d cause t h e c e n t e r o f resistance to move as d e s i r e d but, as m e n t i o n e d , w o u l d displace the a p e x in the o p p o s i t e direction. T o avoid this, the o r t h o d o n t i s t m u s t apply a c o u p l e to the t o o t h to c o u n t e r a c t the t e n d e n c y to tip. T o achieve translatory d i s p l a c e m e n t , the m a g n i t u d e o f this c o u p l e m u s t b e precisely equal to the a m o u n t o f the m o m e n t c r e a t e d by t h e force which will tip the crown in the d i r e c t i o n o f the force. T h e m o m e n t c r e a t e d
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Figure 6. Crown movement. A force and a couple are applied at the bracket. The couple is smaller in magnitude than the moment created by the applied force (A). A force and a small tendency for the crown to tip in the same direction result at the center of resistance (B). Stresses in the periodontal ligament are smallest near the apex and greatest at the alveolar crest in the direction of the force (C). The center of rotation is at the apex and the crown tips in the direction of the force (D).
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by the force t i p p i n g the crown in the d i r e c t i o n o f the force is equal to the m a g n i t u d e o f the force m u l t i p l i e d by the p e r p e n d i c u l a r distance f r o m the force to the c e n t e r o f resistance o f the tooth. In o t h e r words, the t e n d e n c y to tip o r " m o m e n t " increases as the force is a p p l i e d f u r t h e r f r o m the c e n t e r o f resistance. If the force c o u l d b e a p p l i e d at the c e n t e r o f resistance, the t e n d e n c y to tip w o u l d b e zero a n d the t o o t h w o u l d translate in the d i r e c t i o n o f the force. In this case, the c e n t e r o f r o t a t i o n o f the t o o t h w o u l d b e at infinity b e c a u s e the t o o t h d o e s n o t "rotate" o r tip at all. If the b r a c k e t is 10 m m f r o m the c e n t e r o f resistance, a force a p p l i e d at the b r a c k e t causes the t o o t h to tip b e c a u s e o f a m o m e n t that is 10 m m times the m a g n i t u d e o f the force. T o c o u n t e r a c t this t e n d e n c y to tip, a c o u p l e in the o p p o s i t e dir e c t i o n with a m o m e n t 10 m m times the m a g n i t u d e o f the force w o u l d n e e d to b e a p p l i e d in a d d i t i o n to the force. This w o u l d b e an a p p l i e d
m o m e n t - t o - f o r c e ratio o f 10:1, resulting in disp l a c e m e n t o f the t o o t h as if the force a l o n e h a d b e e n p l a c e d t h r o u g h the c e n t e r o f resistance. It is n o t possible to d e t e r m i n e clinically the exact distance o f the b r a c k e t f r o m the c e n t e r o f resistance o f each individual tooth, b u t it is e s t i m a t e d that the c e n t e r o f resistance o f a typical maxillary incisor is a p p r o x i m a t e l y 10 m m f r o m the bracket. T h e r e fore, a m o m e n t - t o - f o r c e ratio o f a b o u t 10:1 is pres u m e d to be necessm y to cause p u r e translation o f a t o o t h orthodontically. W h e n the c o u p l e a p p l i e d a l o n g with a three is precisely the r i g h t m a g n i t u d e to c o u n t e r a c t the m o m e n t c r e a t e d by the force, the t o o t h is disp l a c e d bodily in the d i r e c t i o n o f the force as if a single force h a d b e e n a p p l i e d t h r o u g h the c e n t e r o f resistance (Fig 5). This causes a u n i f o r m stress to b e t r a n s m i t t e d by the tooth to the p e r i o d o n t a l l i g a m e n t with pressure in the d i r e c t i o n o f the force a n d tension o n the o t h e r side. I f the p e r i o d o n t a l
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l i g a m e n t transmits this stress u n i f o r m l y to the responsive cells, the result is n e t b o n e resorption o n the side of pressure a n d n e t deposition o n the tension side, allowing the tooth to move in a translatory m a n n e r without tipping.
Crown M o v e m e n t If the moment-to-force ratio applied to a tooth is less t h a n 10:1, the result will n o t be p u r e translation of the tooth in the direction of the force. Instead, because the m o m e n t created by the applied force is n o t completely n e g a t e d by the applied couple, the result is greater m o v e m e n t of the crown t h a n the root i n the direction of the force. For a central incisor, a moment-to-force ratio of approximately 7:1 or 8:1 will cause the tooth to move a r o u n d a p o i n t n e a r the apex of the tooth, m With the c e n t e r of rotation at the apex, the magn i t u d e of the stress transmitted by the tooth to the p e r i o d o n t a l l i g a m e n t is smallest n e a r the apex a n d
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greatest at the crest of the alveolar b o n e (Fig 6). T h e result o n the side of pressure is n e t resorption of b o n e which is greater n e a r the crown t h a n at the apex. This causes the crown of the tooth to tip in the direction of the force a n d the center of resistance to move i n the direction of the force b u t to a lesser degree t h a n the crown. I n this case, the apex of the tooth moves very little or n o t at all.
Root Movement To achieve root m o v e m e n t with the center of rotation at the crown of a tooth, the applied moment-to-force ratio m u s t be greater t h a n 10:1. T h e couple applied will m o r e t h a n c o u n t e r a c t the mom e n t created by the force. As the center of resistance of the tooth moves in the direction of the force, the applied couple tips the crown in the opposite direction. A moment-to-force ratio of 12:1 or 13:1 is believed to be sufficient to effect a c e n t e r of rotation at the crown/~ I n o t h e r words,
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the crown of the tooth moves very little while the center of resistance moves in the direction of the force. The apex of the root also moves in the direction of the force but to an even greater degree. An increasing gradient of stress is transmitted by the tooth to the periodontal ligament which is greatest at the apex and least at the alveolar crest (Fig 7). Net resorption of bone occurs along the entire root surface in the direction of the force but more resorption is effected at the apex than at the alveolar crest. This results in m o v e m e n t of the root in the direction of the force while the crown remains relatively stationary. Conclusion Research into the biological reaction of h u m a n tissues to orthodontic forces has achieved much in the way o f identifying reactive cells, recruitment mechanisms, and specific signaling processes. Fundamental clinical principles, however, such as the relationship between the magnitude of applied force and the rate of b o n e resorption and deposition that effect tooth movement, have not been completely elucidated. C o m p o u n d i n g difficulties at efforts to discern these principles is misunderstanding of the mechanical components of orthodontic force application. Complications arising from an inability to measure friction at the appliance-bracket interface and to calculate precisely the distribution of stress at the level of the periodontal ligament, further impede efforts to understand biological relationships. Improving the clinician's control of orthodontic tooth m o v e m e n t is the ability to predictably vary the quality of the force system applied. This can be achieved through an understanding of the way in which varying the moment-to-force ratio applied to the teeth affects the center of rotation during tooth movement. Controlling the center of rotation is currently the clinician's most predictable way of managing the biological response to mechanical force systems applied during orthodontic treatment. References 1. Roberts WE, Goodwin WC, H e i n e r SR. Cellular response to orthodontic force. Dent Clin N A m 1981;25:3-17. 2. Yoshikawa DK. Biomechanical principles of tooth movement. Dent Clin N A m 1981;25:19-26. 3. Davidovitch Z. T o o t h m o v e m e n t . Crit Rev Oral Biol Med 1991;2:411-450. 4. Burstone CJ. Application of b i o e n g i n e e r i n g to clinical orthodontics. In: Graber TM, Swain BF (eds): O r t h o d o n tics. C u r r e n t Principles a n d Techniques. St Louis, MO: CV Mosby, 1985, pp 193-227.
5. Isaacson RJ, L i n d a u e r SJ, Conley P. T h e interface between the mechanical signal a n d the biological response in orthodontics. In: Davidovitch Z, Norton LA (eds): Biological Mechanisms of T o o t h M o v e m e n t a n d Craniofacial Adaptation. Boston, MA: Harvard Society for the A d v a n c e m e n t of Orthodontics, 1996, pp 1-5. 6. Middleton J, J o n e s ML, Wilson AN. Three-dimensional analysis of orthodontic tooth m o v e m e n t . J Biomed Eng 1990;12:319-327. 7. Burstone CJ. Biomechanics of tooth movement. In: Kraus BS, Riedel RA (eds): Vistas in Orthodontics. Philadelphia, PA: Lea & Febiger, 1962, pp 197-213. 8. Reitan 14. Some factors d e t e r m i n i n g the evaluation of forces in orthodontics. A m J O r t h o d 1957;43:32-45. 9. Lee BW. The force reqtfirements for tooth movement part III: The pressure hypothesis. Aust O r t h o d J 1996;14:93-97. 10. Q u i n n RS, Yoshikawa DK. A reassessment of force magn i t u d e in orthodontics. A m J O r t h o d 1985;88:252-260. 11. Owman-Moll P, Kurol J, L u n d g r e n D. Effects of a doubled orthodontic force m a g n i t u d e on tooth m o v e m e n t a n d root resorptions. An inter-individual study in adolescents. E u r J O r t h o d 1996;18:141-150. 12. Keeling SD, King GJ, McCoy EA, et al. S e r u m a n d alveolar b o n e p h o s p h a t a s e c h a n g e s reflect b o n e turnover d u r i n g orthodontic tooth m o v e m e n t . A m J O r t h o d Dentofac O r t h o p 1993;103:320-326. 13. A n d r e a s e n G, J o h n s o n P. Experimental findings on tooth m o v e m e n t u n d e r two conditions o f applied force. Angle O r t h o d 1967;37:9-12. 14. Owman-Moll P, Kurol J, L u n d g r e n D. T h e effects of a four-fold increased orthodontic force m a g n i t u d e on tooth m o v e m e n t a n d root resorptions. An intra-individual study in adolescents. E u r J O r t h o d 1996;18:287-294. 15. Smith R, Storey E. T h e i m p o r t a n c e of force in o r t h o d o n tics. A u s t J Dent 1952;56:291-304. 16. Lee BW. T h e force r e q u i r e m e n t s for tooth m o v e m e n t part I: Tipping a n d bodily m o v e m e n t . Aust O r t h o d J 1995;13:238-248. 17. Boester CH, J o h n s t o n LE. A clinical investigation of the concepts of differential a n d optimal force in canine retraction. Angle O r t h o d 1974;44:113-119. 18. King GJ, Keeling SD, McCoy AA, et al. Measuring dental drift a n d orthodontic tooth m o v e m e n t in response to various initial forces in adult rats. A m J O r t h o d Dentofac O r t h o p 1991;99:456465. 19. L u n d g r e n D, Owman-Moll P, KurolJ. Early tooth movem e n t pattern after application of a controlled continuous orthodontic force. A h u m a n experimental model. A m J O r t h o d Dentofac O r t h o p 1996;110:28%294. 20. Isaacson RJ, Lindauer SJ, Conley P. Responses of 3-D arch wires to vertical v bends. Semin Orthod 1995;1:5%63. 21. Smith RJ, Burstone CJ. Mechanics of tooth m o v e m e n t . A m J O r t h o d 1984;85:294-307. 22. Kusy RP, Tulloch JFC. Analysis of m o m e n t / f o r c e ratios in the m e c h a n i c s of tooth m o v e m e n t . A m J O r t h o d Dentofac O r t h o p 1986;90:127-131. 23. Andersen KL, Mortensen t-IT, Pedersen EH, et al. Determination of stress levels and profiles in the periodontal ligament by m e a n s of an improved threeqtimensional finite element model for various types of orthodontic and natural force systems. J Biomed Eng 1991;13:293-303. 24. Isaacson RJ, L i n d a u e r SJ, Davidovitch M. T h e g r o u n d rules for arch wire design: Constraints and opportunities. Semin O r t h o d 1995;1:3-11.