On the use of vertical loops in retraction systems M.G. Faulkner, MSc, PhD, ° A.W. Lipsett, MSc, PhD, b K. EI-Rayes, MSc, = and D.L. Haberstock, DDS, MDS d Edmonton, Alberta, Canada
While the final.goal of a retraction system is efficient, effective closure of space within the dental arch, the designs and materials of the appliances used to accomplish this effect vary considerably. Several systems use arch wires for control and elastomers for force application, to allow the canine and other teeth to be guided by this wire during movement (Fig. 1, a). A somewhat different approach uses a metallic loop or spring as the only connection between tooth segments or individual teeth (Fig. 1, b). As shown in Fig. 1, c, when this type of appliance is activated and anchored at each end, it applies forces and moments (couples) to the teeth that are equal and opposite to those necessary to maintain the appliance in its deformed state (Fig. 1, d). Effective repositioning of the teeth with this technique requires that the appliance provide a specific force system (force and moment) to segments or individual teeth in the dental arch. To provide this force system, the appliance must have the following mechanical characteristics: 1. It must provide appropriate levels of force and moment-to-force (M/F) ratios to achieve the tooth displacement desired. 2. It must be able to undergo a reasonable range of activation/deactivation in which the appliance delivers relatively constant forces and moments. 3. It must be small enough to fit comfortably in the space available for intraoral treatment. The relationship between the appliance force system and the resulting tooth movement has been investigated and described by several authors. ,-3 To understand and predict the type of movement that will occur, we describe the effective resistance of the supporting tissues in terms of a center of resistance. A force applied at From the Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta. Supported by National Sciences and Engineering Research Council of Canada grants OGP7514 and OGP6296. 1Professor. bAssociate professor. CGraduate student. dprivate practice of orthodontics, Edmonton, Alberta. 8/1/21189
328
the center of resistance would cause the tooth to translate. Therefore, if a force is applied only at the normal bracket position, it will produce uncontrolled tipping, t: For this reason, moments must also be provided to control the displacement. Since the distance between the center of resistance and the normal bracket position is approximately 8 to 10 m m , the M / F ratio at the bracket should also be in the range 8 to 10 mm to produce tooth translation. A recent numerical study by Tanne, Koenig, and Burstone 5 indicated that, for a maxillary central incisor, an M / F ratio of 9.5 mm should produce root movement with tipping at the incisal edge, 8.4 mm should produce translation, and 6.5 mm should provide controlled or crown 'tipping around the root apex. Force systems that produce M / F ratios of less than 5 to 6 mm are defined as producing uncontrolled tipping. One of the most common designs of the retraction spring is the vertical loop. Typical dimensions for its shape are shown in Fig. 2, a. These loops mhy be fabricated as independent devices (segmented arch technique) or incorporated into a continuous arch wire system. Depending on the interbracket distance (IBD), the other dimensions shown can be varied to modify the loop's mechanical characteristics. The effects of several parameters, including that of the height (h), radius (R), and IBD, were evaluated experimentally and numerically by Burstone and Koenig. 6 In addition, the effect of positioning the loop in an anteroposterior direction was investigated to assist understanding of the forces and moments that are created when the retraction appliance has been reactivated during treatment. To a limited extent, the effect of introducing a helix in the loop was also considered. The results of this study clearly showed that the ordinary vertical loop, even with the modifications outlined, is limited in its ability to produce M / F ratios that approach those necessary for translation or even controlled tipping. The largest M / F ratio was less than 4 mm and was essentially constant over a very limited activation range (0 to 1.5 mm). This means that the magnitude of force and the moment-to-force ratio are extremely sensitive to small
Volume 99 Number 4
Use of vertical loops in retraction systems
~
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1.0
329
330
Fa,dkner et al.
changes in the amount of activation. Further, this design is also limited in total activation and the spring is relatively easy to deform permanently during installation; as a result, its mechanical characteristics are altered considerably. To overcome some of the shortcomings of typical vertical loop~, we considered the effects of various amounts of preactivation. The legs of the spring were "gabled" (positioned at an angle to the direction of activation) to create larger moments, since the legs of the spring must first be brought parallel to one another before being installed and activated. This procedure increases the moment, while it has little effect on the force-deflection relationship during activation. As a result, these preactiv~ted springs have a highly nonlinear M / F ratio as a function of activation, which will be illustrated in this article. The extreme sensitivity of this design to small changes in activation has also been discussed. 6"While preactivation alters the M / F ratio, it does not appreciably alter the elastic range of activations compared with that of the typical loop. In this article, the characteristics of a typical loop are reviewed. The potential of various modifications is investigated by means of a numerical procedure that has been previously used for large-deformation analysis of rods. 7 Its application to orthodontics has been described in detail elsewhere, s The overall aim of the modifications is to produce designs that have more desirable M / F ratios, as well as larger ranges of activation, than those of typical loops. As Quinn and Yoshikawa 3 have noted, this modification can yield treatment strategies that conserve posterior anchorage while efficiently moving anterior teeth. FORCE SYSTEMS FROM A TYPICAL VERTICAL LOOP
The vertical loop illustrated in Fig. 2, a was analyzed by consideration of a series of activations in which forces and moments were applied to the ends of the loop as shown in Fig. 1, d. The IBD was set and maintained at 14 mm in all cases, and the wire was 0.432 × 0.432 mm (0.017 × 0.017 inch) stainless steel with a Young's modulus of 174 and a tensile yield strength of 1400 GPa. The spring was activated in a symmetric manner so that it produced only axial forces and moments as illustrated in Fig. 1, d. The forces and moments necessary to maintain these activations are illustrated in Fig. 2, b and c. During the numerical calculations for each activation, the stresses in the wire were calculated to check whether or not the wire would become permanently deformed (i.e., the material would be stressed beyond its elastic limit and would not return to its initial shape when the force was released). The
Am. J. Orthod. Dentofac. Orthop. April 1991
calculations were performed only up to the final point at which the spring remained completely elastic. The results show that the standard vertical loop spring is limited to a total activation of approximately 1 mm (Fig. 2, b) before the spring can be expected to yield. This yield occurs at the top of the loop. The force levels at this activation are relatively high (4.4 N) compared with the 1 to 2N suggested by Quinn and.Yoshikawa 3 as the level required to optimize canine retraction. (Note: a 102 gm force is equivalent to 1 N.) Also clearly seen is the fact that the force-deflection relationship is approximately linear, so that at 0.5 mm activation the force levels are only one-half of those at 1.0 ram. This means that small changes in alignment or small movements of the teeth will dramatically reduce the force levels, Fig. 2, c shows the moments produced, while Fig. 2, d illustrates the M / F ratio produced over the activation range. It shows the M / F ratio approximately constant but below 2 mm over the entire activation. (Note that the directions of force .and moment shown in Fig. 1, c are taken as the positive directions when applied to the teeth.) In general, it can be inferred that the typical stainless steel vertical loop has two major limitations. First, its activation range is very restridted; second, the M / F ratio produced is also well below ideal if controlled tipping or translation is desired. While the use of alternate materials and cross sections can change the level of force and moment to a limited extent, the M / F ratio remains unaltered. The changes in activation limits are also relatively small, so that the basic limitations cannot be overcome without a change in the design geometry. The following paragraphs consider modifications of this standard design with the objective of reducing these restrictions. EFFECT OF HELICES IN THE VERTICAL LOOP
The standard vertical loop described above can be altered by increasing or decreasing the height and the radius of the bends; however, these effects have been shown to be relatively minor. 6 In this study, consideration is given to the effects of adding a single apical loop, with a radius of 1.0 ram, and that of two lateral helices at the base, each with a radius of 0.5 mm added at the base (Fig. 3, a). Comparisons of the force-deflection, momentdeflection, and M / F ratios of the altered loops with the same variables for the standard loop are given in Fig. 3, b, c, and d. The addition of the single apical helix has the overall effect of reducing the levels of both the force and the moment for any given activation. There is a greater reduction in the force than in the moment, so that the M / F ratio increase is slightly greater than
Volume 99 Number 4
Use of vertical loops in retraction systems
331
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that of the standard. Perhaps more important is the fact that the potential activation of the spring before yield is increased to 1.6 m m - - a large change from 1 . 0 mm. The lateral helices have a considerably different overall effect. While they lower both the force and moment magnitudes at any activation, they have a greater reducing effect on the moment and, as a result, the M / F ratio is actually lower than that of the standard systeml The addition of the lateral helices does not lead to any additional total activation, since the spring will still yield at the apex of the loop, just as it would for the case of the simple loop. Fig. 3, b, c, and d also show the combined effect of all three helices on the mechanical behavior of the appliance. Combining the three helices further reduces the slope of the force/deflection curve and allows larger activations before the spring yields. The moment/deflection characteristic (Fig. 3, b) also produces an effect similar to that of the force/deflection. The M / F ratio is somewhat above that of the standard vertical loop; however, the values are still only slightly above 2 mm. In summary, the addition of the helices has "softened" the appliance so that the slopes of the force/deflection and moment/deflection characteristics
are reduced, compared with those of the standard system. The addition of the apical helix allows the activation to be increased approximately 60 percent (from 1.0 mm t o 1.6 mm) before the spring can be expected to be permanently deformed because of yielding of the material. The M / F ratios are only slightly affected by these alterations and are still well below those necessary to produce tooth translation. This means that, to develop higher moments, other design alternatives must be introduced. EFFECTS OF PREACTIVATION FOR VERTICAL LOOPS
Our analysis shows that the vertical loop can easily produce force levels that are high enough to promote tooth movement. However, because of the low M / F ratios, the resulting tooth displacement is uncontrolled tipping--not translation. To increase the level of moments produced, the spring can be preactivated, as shown in Fig. 4, a. This gabling means that a moment must be applied to make the ends of the spring parallel and collinear. Once the spring is brought into this position (the so-called "neutral" position, in which the horizontal force is zero), it can be activated in the usual manner. The effects of various amounts of preactivation are shown in Fig. 4, b, c, and d. The effect of preac-
332
Am. J. Orthod. Dentofac. Orthop. April 1991
Faulkner et al.
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tivation on the force/deflection relationship is minimal, while the effect on the moment/deflection is considerable. As the amount of preactivation is increased, the magnitude of the moment (Fig. 4, c) at the neutral position (zero activation) is also increased. When axial activation is now superimposed, all these moment/deflection curves are approximately parallel and increase as the amount of preactivation increases. This effect is also reflected in the plots of M / F ratio (Fig. 4, d). Because of the introduction of a large moment from the preactivation, the M / F ratio relationship has been significantly increased, especially in the very low activation range, because the force approaches zero while the moment approaches a fixed value. While the M / F ratios are above those for the standard spring, at activations approaching the elastic limit of the spring, the M / F ratio is still not very much greater than 2.5 to 3.5 mm. It is interesting to note that preactivation actually allows the spring to be activated over a slightly greater range. Fig. 4, b and c shows that the activation range has been increased from 1.0 to 1.2 m m for I0 ° preactivation because when the spring is brought to the neutral position, the critical section (at the apex of the loop) is actually compressed slightly. As the spring is acti-
vated, this compression is first released; later, it changes to tension as the activation proceeds. It is important to note that for preactivations approximately above 12.5 °, the spring would yield before reaching the neutral position. Therefore further design modifications are necessary if greater preactivations are to be used. The vastly different behavior for these preactivated springs shows how very sensitive the vertical loop is to small manufacturing or installation errors. The angulation in the 5.0 ° preactivated springs could be easily equivalent to errors in fabrication of the loop or in inadvertent yielding during installation of the appliance. When the preactivated loop is installed and fully activated, the M / F raiio is still less than 3.5 ram; however, by the time tlae activation falls to 0.2 mm, the M / F ratio could rise to 5 m m or greater. At times, this effect may be advantageous, because it could reduce tipping during the initial period of tooth movement. In summary, compared with the standard simple vertical loop, the preactivated loops have virtually the same force/deflection characteristic but with a shifted moment/deflection relationship. Also, the total possible activation has been slightly increased. The M / F characteristic is no longer constant but is considerably greater at low activations (0.0 to 0.4 mm). The spring
Volume 99 Number 4
Use of vertical loops in retraction systems
333
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is, therefore, very sensitive to small errors in manufacture or installation and is thus difficult to use in practice. EFFECTS
OF PREACTIVATION
AND HELICES
When the effects of preactivation are superimposed on those of the apical and lateral helices previously considered (Fig. 5, a), the results are analogous to those discussed for preactivation of the standard loop. Fig. 5, b, c, and d shows that the force/deflection relationship is again uninfluenced by the preactivation, but the moment/deflection--and therefore the M / F r a t i o s - are drastically altered. The original standard vertical loop, with no helices or preactivation, is included to show the magnitude of change that results from these design alterations. The maximum activation has been increased from 1.0 mm to almost 2.3 mm (in the case of l0 ° preactivation), and the maximum preactivation has been increased to about 25 °. When 10 ° of preactivation is included with the helices, the maximum moments are increased two to three times, and the M / F ratio is at least twice that of the standard loop for any given activation. These figures also show that there is an optimum amount of prcaqtivation if maximum ac-
tivation is the desired result. For this particular spring geometry in stainless steel, the value is approximately 10°; the optimum preactivation is largely dependent on the wire size, the modulus of elasticity, and the elastic limit of the particular material. Overall, this spring can still produce as high a level of force, but with considerably greater moments and a much larger range for activations. This spring design and its advantages were described by Burstone, Baldwin, and Lawless. 9 In comparison to the standard vertical loop (also shown in 5, b, c, and d), the preactivated loop produces several significant changes in the mechanical characteristics. First, the force/deflection relationship is "softened" so that much larger activations are possible without permanent deformation of the appliance. This means that, as tooth movement occurs, the level of force (for example, greater than l mm deactivation) remains at a more constant level than it would if the standard loop were used. Second, the preactivation allows application of moments that are two to three times the original magnitude to assist in creating a more desirable type of movement (i.e., higher M / F ratios). Unfortunately, the resulting moments are still not large enough to produce translation.
334
A m . J. Ordzod. Dentofac. Orthop.
F a u l k n e r et al.
April 1991 ~
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To produce the higher M / F ratios, it is necessary to further increase the activation range while increasing the moments generated. To a limited extent this change can be accomplished by increasing the size of the apical and lateral helices. Several combinations of sizes of apical and lateral helices have been considered; the maximum changes in geometry are shown in Fig. 6, a. Fig. 6, b, c, and d shows the effect of increasing the radius of each of the loops, as well as the combined effect of increasing both the apical and lateral ones for the 15° preactivated loop from Fig. 5, a. It is evident that increasing the radius of the apical loop created a more dramatic effect than increasing the lateral loops. The larger apical loop lowers force values with only marginally reduced moment values (curve B) and leads to markedly higher M / F ratios. The combined effect of increasing the size of the lateral loops as well is shown by comparing curves B, D, and E, where the M / F is marginally improved but the total activation has been increased to 3. l mm.
SUMMARY OF PARAMETRIC
CHANGES
To summarize the effects of the various design parameters considered above, Fig. 7, a, b, and c com-
pares the force/activation, moment/activation, and M / F ratios for several selected cases. In addition, to illustrate the effects of changes in material and crosssectional dimensions, examples with a reduced crosssectional depth and a lower modulus material are also included. As we discussed above, the standard simple vertical loop is relatively stiff, has a limited range of activation (0 to 1.0 mm for the 0.432 mm 2 wire), and produces less than a 2.0 mm M / F ratio (Fig. 2). The addition of apical and lateral loops "softens" the spring and increases its range of activations (Fig. 3). If, in addition to adding loops, one preactivates the ends, the maximum moments rise by a factor of 2 to 3 (Fig. 5). These changes in mechanical behavior are shown in curves A and B of Fig. 7, which compares the standard vertical loop (curve A) with one that contains both apical and lateral loops (curve B) and has been preactivated 15° (S denotes small radii). Not only has the modified spring's range of activation been doubled, but the maximum moments have been increased from approximately 6 N mm to 15 N mm. The M / F ratio characteristic has also been drastically changed. For example, at 1.0 mm activation (maximum for the standard loop)
Volume 99 Number 4
Use of vertical loops in retraction systems
335
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Activation (ram) C
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the M / F ratio has been increased to 5.0 mm from somewhat below 2.0 ram. As the activation decreases, the modified loop produces even higher values. The spring's behavior can be further enhanced by increasing the radii of all the loops. While Fig. 6 showed these effects for a range o f radius combinations, curve C in Fig. 7 shows the larger-radius loops (1.5 mm for the apical and 1.0 mm for the lateral ones) producing a larger activation range (0 to 3.0 mm), with M / F ratio even higher than for the small-radius loops. These modifications have produced a spring that now has three times the activation range, with the same level of force and an M / F ratio that varies from 3.6 mm to well above 10 mm within the activation range. To further soften this appliance, the depth of the cross section can be reduced. To illustrate the effect o f this alteration, a stainless steel wire with a cross section of 0.406 mm x 0.432 mm was analyzed and shown as curve C' in Fig. 7. While this reduced cross section produced softening of the spring, the M / F ratio remained almost the same, except that it extends over a larger range o f activations (up to 3.4 ram). To obtain higher M / F ratios than those described, it is necessary to increase the amount of preactivation. Curve D shows the effect of 30 ° of preactivation for
the 0.406 × 0.432-mm wire. While M / F ratio has been increased, it has been increased at the expense of reducing the range of possible activations. Too much preactivati0n has caused yielding of the spring at a relatively low activation. Higher amounts of preactiration are possible if a material with lower modulus of elasticity and relatively high yield strength is substituted for the stainless steel. 6 For example, the use of 13titanium (titanium molybdenum alloy, TMA) with a modulus of approximately 2/5 steel allows the use of larger preactivation angles. The mechanical characteristics of 0.017-inch ~ TMA are shown in curve E for a 45 ° preactivation. The r a n g e o f activations is now up to 6 mm with M / F ratios markedly above those of the 15° preactivated stainless steel wire. Reducing the cross section (curve E') again allows a further increase in activation, with the resulting extension of the M / F ratio curve. The alterations, which have been illustrated, have produced a retraction spring with force and moment characteristics similar to those of preactivated T springs, which have been described by Burstone. ~° While the additional alterations we have considered illustrate the possible increase in moments and M / F ratio when larger amounts of preactivation are feasible, these are only examples.
336
Am. J. Orthod. Dentofac. Orthop. April 1991
Faulkner et al.
CONCLUDING REMARKS The limited utility of a standard vertical loop can be drastically increased through the judicious use of more wire, which allows larger total activation and considerable preactivation of the appliance. Adding helices to the apex and lateral sides of the loop is an effective technique that increases the amount of wire being bent. If the proper gabling (preactivation) is coupled to the design with the helices, it also produces considerably higher moments and M / F ratios compared with those of the standard loop. These redesigned loops then produce force/deflection; moment/deflection, and M / F characteristics that make them less sensitive to minor manufacturing and placement errors than the standard vertical lodp. The most common material used in these appliances is stainless steel; however, other materials can also have a very significant effect on the final mechanical characteristics. The ideal material is one with a large ratio of yield stress to the elastic modulus. In this regard, T M A is superior to stainless steel for the appliance designs considered in this study. REFERENCES 1. Burstone CJ. Application of bioengineeringto clinical orthodontics. In: Graber Tgl, Swain BF, eds. Current orthodontic
2. 3. 4.
5.
6. 7.
8.
9. 10.
concepts and techniques. Vol. 1. Philadelphia:WB Saunders. 1975:chap. 5. Smith RJ, Burstone CJ. Mechanics of tooth movement. AM J ORTtlOD 1984;85:294-307. Quinn RS, Yoshikawa D. A reassessmentof force magnitudein orthodontics. AM J OR~IOD 1985;88:252-60. FaulknerMG, FuchshuberP, ttaberstock D, MioduchowskiA. A parametric study of the force moment systems produced by T-loop retractionsprings. J Biomech 1989;22:637-47. TanneK, Koenig HA, BurstoneCJ. Momentto f~rce ratios and the center of rotation. AM J OR'HIODDENTOFACORTHOP1988; 94:426-3I. Burstone CJ, Koenig HA. Optimizing anterior and canine retraction. AM J ORTIIOD1976;70:1-16. FaulknerMG, StredulinskyDC. Nonlinearbending of inextensible thin rods under distributed and concentrated loads. Trans Can Soc Mech Eng 1977;4:77-82. Lipsett AW, FaulknergiG, EI-RayesK. Large deformationanalysis of orthodontic springs. ASME J Biomech Eng 1990;I12: 29-37: BurstoneCJ, Baldwin JJ, Lawless DT. The applicationof continuous forces to orthodontics. Angle Orthod 1961;31:1-14. Burstone CJ. The segmented arch approach to space closure. AM J OR'I'HOD1982;82:361-78.
Reprint requests to:
Dr. M.G. Faulkner Department of Mechanical Engineering Universityof Alberta 4-9 MechanicalEngineeringBuilding Edmonton T6G 2G8 Alberta, Canada
While the final.goal of a retraction system is efficient, effective closure of space within the dental arch, the designs and materials of the appliances used to accomplish this effect vary considerably. Several systems use arch wires for control and elastomers for force application, to allow the canine and other teeth to be guided by this wire during movement (Fig. 1, a). A somewhat different approach uses a metallic loop or spring as the only connection between tooth segments or individual teeth (Fig. 1, b). As shown in Fig. 1, c, when this type of appliance is activated and anchored at each end, it applies forces and moments (couples) to the teeth that are equal and opposite to those necessary to maintain the appliance in its deformed state (Fig. 1, d). Effective repositioning of the teeth with this technique requires that the appliance provide a specific force system (force and moment) to segments or individual teeth in the dental arch. To provide this force system, the appliance must have the following mechanical characteristics: 1. It must provide appropriate levels of force and moment-to-force (M/F) ratios to achieve the tooth displacement desired. 2. It must be able to undergo a reasonable range of activation/deactivation in which the appliance delivers relatively constant forces and moments. 3. It must be small enough to fit comfortably in the space available for intraoral treatment. The relationship between the appliance force system and the resulting tooth movement has been investigated and described by several authors. ,-3 To understand and predict the type of movement that will occur, we describe the effective resistance of the supporting tissues in terms of a center of resistance. A force applied at From the Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta. Supported by National Sciences and Engineering Research Council of Canada grants OGP7514 and OGP6296. 1Professor. bAssociate professor. CGraduate student. dprivate practice of orthodontics, Edmonton, Alberta. 8/1/21189
328
the center of resistance would cause the tooth to translate. Therefore, if a force is applied only at the normal bracket position, it will produce uncontrolled tipping, t: For this reason, moments must also be provided to control the displacement. Since the distance between the center of resistance and the normal bracket position is approximately 8 to 10 m m , the M / F ratio at the bracket should also be in the range 8 to 10 mm to produce tooth translation. A recent numerical study by Tanne, Koenig, and Burstone 5 indicated that, for a maxillary central incisor, an M / F ratio of 9.5 mm should produce root movement with tipping at the incisal edge, 8.4 mm should produce translation, and 6.5 mm should provide controlled or crown 'tipping around the root apex. Force systems that produce M / F ratios of less than 5 to 6 mm are defined as producing uncontrolled tipping. One of the most common designs of the retraction spring is the vertical loop. Typical dimensions for its shape are shown in Fig. 2, a. These loops mhy be fabricated as independent devices (segmented arch technique) or incorporated into a continuous arch wire system. Depending on the interbracket distance (IBD), the other dimensions shown can be varied to modify the loop's mechanical characteristics. The effects of several parameters, including that of the height (h), radius (R), and IBD, were evaluated experimentally and numerically by Burstone and Koenig. 6 In addition, the effect of positioning the loop in an anteroposterior direction was investigated to assist understanding of the forces and moments that are created when the retraction appliance has been reactivated during treatment. To a limited extent, the effect of introducing a helix in the loop was also considered. The results of this study clearly showed that the ordinary vertical loop, even with the modifications outlined, is limited in its ability to produce M / F ratios that approach those necessary for translation or even controlled tipping. The largest M / F ratio was less than 4 mm and was essentially constant over a very limited activation range (0 to 1.5 mm). This means that the magnitude of force and the moment-to-force ratio are extremely sensitive to small
Volume 99 Number 4
Use of vertical loops in retraction systems
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1.0
329
330
Fa,dkner et al.
changes in the amount of activation. Further, this design is also limited in total activation and the spring is relatively easy to deform permanently during installation; as a result, its mechanical characteristics are altered considerably. To overcome some of the shortcomings of typical vertical loop~, we considered the effects of various amounts of preactivation. The legs of the spring were "gabled" (positioned at an angle to the direction of activation) to create larger moments, since the legs of the spring must first be brought parallel to one another before being installed and activated. This procedure increases the moment, while it has little effect on the force-deflection relationship during activation. As a result, these preactiv~ted springs have a highly nonlinear M / F ratio as a function of activation, which will be illustrated in this article. The extreme sensitivity of this design to small changes in activation has also been discussed. 6"While preactivation alters the M / F ratio, it does not appreciably alter the elastic range of activations compared with that of the typical loop. In this article, the characteristics of a typical loop are reviewed. The potential of various modifications is investigated by means of a numerical procedure that has been previously used for large-deformation analysis of rods. 7 Its application to orthodontics has been described in detail elsewhere, s The overall aim of the modifications is to produce designs that have more desirable M / F ratios, as well as larger ranges of activation, than those of typical loops. As Quinn and Yoshikawa 3 have noted, this modification can yield treatment strategies that conserve posterior anchorage while efficiently moving anterior teeth. FORCE SYSTEMS FROM A TYPICAL VERTICAL LOOP
The vertical loop illustrated in Fig. 2, a was analyzed by consideration of a series of activations in which forces and moments were applied to the ends of the loop as shown in Fig. 1, d. The IBD was set and maintained at 14 mm in all cases, and the wire was 0.432 × 0.432 mm (0.017 × 0.017 inch) stainless steel with a Young's modulus of 174 and a tensile yield strength of 1400 GPa. The spring was activated in a symmetric manner so that it produced only axial forces and moments as illustrated in Fig. 1, d. The forces and moments necessary to maintain these activations are illustrated in Fig. 2, b and c. During the numerical calculations for each activation, the stresses in the wire were calculated to check whether or not the wire would become permanently deformed (i.e., the material would be stressed beyond its elastic limit and would not return to its initial shape when the force was released). The
Am. J. Orthod. Dentofac. Orthop. April 1991
calculations were performed only up to the final point at which the spring remained completely elastic. The results show that the standard vertical loop spring is limited to a total activation of approximately 1 mm (Fig. 2, b) before the spring can be expected to yield. This yield occurs at the top of the loop. The force levels at this activation are relatively high (4.4 N) compared with the 1 to 2N suggested by Quinn and.Yoshikawa 3 as the level required to optimize canine retraction. (Note: a 102 gm force is equivalent to 1 N.) Also clearly seen is the fact that the force-deflection relationship is approximately linear, so that at 0.5 mm activation the force levels are only one-half of those at 1.0 ram. This means that small changes in alignment or small movements of the teeth will dramatically reduce the force levels, Fig. 2, c shows the moments produced, while Fig. 2, d illustrates the M / F ratio produced over the activation range. It shows the M / F ratio approximately constant but below 2 mm over the entire activation. (Note that the directions of force .and moment shown in Fig. 1, c are taken as the positive directions when applied to the teeth.) In general, it can be inferred that the typical stainless steel vertical loop has two major limitations. First, its activation range is very restridted; second, the M / F ratio produced is also well below ideal if controlled tipping or translation is desired. While the use of alternate materials and cross sections can change the level of force and moment to a limited extent, the M / F ratio remains unaltered. The changes in activation limits are also relatively small, so that the basic limitations cannot be overcome without a change in the design geometry. The following paragraphs consider modifications of this standard design with the objective of reducing these restrictions. EFFECT OF HELICES IN THE VERTICAL LOOP
The standard vertical loop described above can be altered by increasing or decreasing the height and the radius of the bends; however, these effects have been shown to be relatively minor. 6 In this study, consideration is given to the effects of adding a single apical loop, with a radius of 1.0 ram, and that of two lateral helices at the base, each with a radius of 0.5 mm added at the base (Fig. 3, a). Comparisons of the force-deflection, momentdeflection, and M / F ratios of the altered loops with the same variables for the standard loop are given in Fig. 3, b, c, and d. The addition of the single apical helix has the overall effect of reducing the levels of both the force and the moment for any given activation. There is a greater reduction in the force than in the moment, so that the M / F ratio increase is slightly greater than
Volume 99 Number 4
Use of vertical loops in retraction systems
331
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that of the standard. Perhaps more important is the fact that the potential activation of the spring before yield is increased to 1.6 m m - - a large change from 1 . 0 mm. The lateral helices have a considerably different overall effect. While they lower both the force and moment magnitudes at any activation, they have a greater reducing effect on the moment and, as a result, the M / F ratio is actually lower than that of the standard systeml The addition of the lateral helices does not lead to any additional total activation, since the spring will still yield at the apex of the loop, just as it would for the case of the simple loop. Fig. 3, b, c, and d also show the combined effect of all three helices on the mechanical behavior of the appliance. Combining the three helices further reduces the slope of the force/deflection curve and allows larger activations before the spring yields. The moment/deflection characteristic (Fig. 3, b) also produces an effect similar to that of the force/deflection. The M / F ratio is somewhat above that of the standard vertical loop; however, the values are still only slightly above 2 mm. In summary, the addition of the helices has "softened" the appliance so that the slopes of the force/deflection and moment/deflection characteristics
are reduced, compared with those of the standard system. The addition of the apical helix allows the activation to be increased approximately 60 percent (from 1.0 mm t o 1.6 mm) before the spring can be expected to be permanently deformed because of yielding of the material. The M / F ratios are only slightly affected by these alterations and are still well below those necessary to produce tooth translation. This means that, to develop higher moments, other design alternatives must be introduced. EFFECTS OF PREACTIVATION FOR VERTICAL LOOPS
Our analysis shows that the vertical loop can easily produce force levels that are high enough to promote tooth movement. However, because of the low M / F ratios, the resulting tooth displacement is uncontrolled tipping--not translation. To increase the level of moments produced, the spring can be preactivated, as shown in Fig. 4, a. This gabling means that a moment must be applied to make the ends of the spring parallel and collinear. Once the spring is brought into this position (the so-called "neutral" position, in which the horizontal force is zero), it can be activated in the usual manner. The effects of various amounts of preactivation are shown in Fig. 4, b, c, and d. The effect of preac-
332
Am. J. Orthod. Dentofac. Orthop. April 1991
Faulkner et al.
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tivation on the force/deflection relationship is minimal, while the effect on the moment/deflection is considerable. As the amount of preactivation is increased, the magnitude of the moment (Fig. 4, c) at the neutral position (zero activation) is also increased. When axial activation is now superimposed, all these moment/deflection curves are approximately parallel and increase as the amount of preactivation increases. This effect is also reflected in the plots of M / F ratio (Fig. 4, d). Because of the introduction of a large moment from the preactivation, the M / F ratio relationship has been significantly increased, especially in the very low activation range, because the force approaches zero while the moment approaches a fixed value. While the M / F ratios are above those for the standard spring, at activations approaching the elastic limit of the spring, the M / F ratio is still not very much greater than 2.5 to 3.5 mm. It is interesting to note that preactivation actually allows the spring to be activated over a slightly greater range. Fig. 4, b and c shows that the activation range has been increased from 1.0 to 1.2 m m for I0 ° preactivation because when the spring is brought to the neutral position, the critical section (at the apex of the loop) is actually compressed slightly. As the spring is acti-
vated, this compression is first released; later, it changes to tension as the activation proceeds. It is important to note that for preactivations approximately above 12.5 °, the spring would yield before reaching the neutral position. Therefore further design modifications are necessary if greater preactivations are to be used. The vastly different behavior for these preactivated springs shows how very sensitive the vertical loop is to small manufacturing or installation errors. The angulation in the 5.0 ° preactivated springs could be easily equivalent to errors in fabrication of the loop or in inadvertent yielding during installation of the appliance. When the preactivated loop is installed and fully activated, the M / F raiio is still less than 3.5 ram; however, by the time tlae activation falls to 0.2 mm, the M / F ratio could rise to 5 m m or greater. At times, this effect may be advantageous, because it could reduce tipping during the initial period of tooth movement. In summary, compared with the standard simple vertical loop, the preactivated loops have virtually the same force/deflection characteristic but with a shifted moment/deflection relationship. Also, the total possible activation has been slightly increased. The M / F characteristic is no longer constant but is considerably greater at low activations (0.0 to 0.4 mm). The spring
Volume 99 Number 4
Use of vertical loops in retraction systems
333
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is, therefore, very sensitive to small errors in manufacture or installation and is thus difficult to use in practice. EFFECTS
OF PREACTIVATION
AND HELICES
When the effects of preactivation are superimposed on those of the apical and lateral helices previously considered (Fig. 5, a), the results are analogous to those discussed for preactivation of the standard loop. Fig. 5, b, c, and d shows that the force/deflection relationship is again uninfluenced by the preactivation, but the moment/deflection--and therefore the M / F r a t i o s - are drastically altered. The original standard vertical loop, with no helices or preactivation, is included to show the magnitude of change that results from these design alterations. The maximum activation has been increased from 1.0 mm to almost 2.3 mm (in the case of l0 ° preactivation), and the maximum preactivation has been increased to about 25 °. When 10 ° of preactivation is included with the helices, the maximum moments are increased two to three times, and the M / F ratio is at least twice that of the standard loop for any given activation. These figures also show that there is an optimum amount of prcaqtivation if maximum ac-
tivation is the desired result. For this particular spring geometry in stainless steel, the value is approximately 10°; the optimum preactivation is largely dependent on the wire size, the modulus of elasticity, and the elastic limit of the particular material. Overall, this spring can still produce as high a level of force, but with considerably greater moments and a much larger range for activations. This spring design and its advantages were described by Burstone, Baldwin, and Lawless. 9 In comparison to the standard vertical loop (also shown in 5, b, c, and d), the preactivated loop produces several significant changes in the mechanical characteristics. First, the force/deflection relationship is "softened" so that much larger activations are possible without permanent deformation of the appliance. This means that, as tooth movement occurs, the level of force (for example, greater than l mm deactivation) remains at a more constant level than it would if the standard loop were used. Second, the preactivation allows application of moments that are two to three times the original magnitude to assist in creating a more desirable type of movement (i.e., higher M / F ratios). Unfortunately, the resulting moments are still not large enough to produce translation.
334
A m . J. Ordzod. Dentofac. Orthop.
F a u l k n e r et al.
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Fig. 6. Effect of loop radii on the forces, moments, and M / F ratio for preactivated vertical loops with helices.
To produce the higher M / F ratios, it is necessary to further increase the activation range while increasing the moments generated. To a limited extent this change can be accomplished by increasing the size of the apical and lateral helices. Several combinations of sizes of apical and lateral helices have been considered; the maximum changes in geometry are shown in Fig. 6, a. Fig. 6, b, c, and d shows the effect of increasing the radius of each of the loops, as well as the combined effect of increasing both the apical and lateral ones for the 15° preactivated loop from Fig. 5, a. It is evident that increasing the radius of the apical loop created a more dramatic effect than increasing the lateral loops. The larger apical loop lowers force values with only marginally reduced moment values (curve B) and leads to markedly higher M / F ratios. The combined effect of increasing the size of the lateral loops as well is shown by comparing curves B, D, and E, where the M / F is marginally improved but the total activation has been increased to 3. l mm.
SUMMARY OF PARAMETRIC
CHANGES
To summarize the effects of the various design parameters considered above, Fig. 7, a, b, and c com-
pares the force/activation, moment/activation, and M / F ratios for several selected cases. In addition, to illustrate the effects of changes in material and crosssectional dimensions, examples with a reduced crosssectional depth and a lower modulus material are also included. As we discussed above, the standard simple vertical loop is relatively stiff, has a limited range of activation (0 to 1.0 mm for the 0.432 mm 2 wire), and produces less than a 2.0 mm M / F ratio (Fig. 2). The addition of apical and lateral loops "softens" the spring and increases its range of activations (Fig. 3). If, in addition to adding loops, one preactivates the ends, the maximum moments rise by a factor of 2 to 3 (Fig. 5). These changes in mechanical behavior are shown in curves A and B of Fig. 7, which compares the standard vertical loop (curve A) with one that contains both apical and lateral loops (curve B) and has been preactivated 15° (S denotes small radii). Not only has the modified spring's range of activation been doubled, but the maximum moments have been increased from approximately 6 N mm to 15 N mm. The M / F ratio characteristic has also been drastically changed. For example, at 1.0 mm activation (maximum for the standard loop)
Volume 99 Number 4
Use of vertical loops in retraction systems
335
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the M / F ratio has been increased to 5.0 mm from somewhat below 2.0 ram. As the activation decreases, the modified loop produces even higher values. The spring's behavior can be further enhanced by increasing the radii of all the loops. While Fig. 6 showed these effects for a range o f radius combinations, curve C in Fig. 7 shows the larger-radius loops (1.5 mm for the apical and 1.0 mm for the lateral ones) producing a larger activation range (0 to 3.0 mm), with M / F ratio even higher than for the small-radius loops. These modifications have produced a spring that now has three times the activation range, with the same level of force and an M / F ratio that varies from 3.6 mm to well above 10 mm within the activation range. To further soften this appliance, the depth of the cross section can be reduced. To illustrate the effect o f this alteration, a stainless steel wire with a cross section of 0.406 mm x 0.432 mm was analyzed and shown as curve C' in Fig. 7. While this reduced cross section produced softening of the spring, the M / F ratio remained almost the same, except that it extends over a larger range o f activations (up to 3.4 ram). To obtain higher M / F ratios than those described, it is necessary to increase the amount of preactivation. Curve D shows the effect of 30 ° of preactivation for
the 0.406 × 0.432-mm wire. While M / F ratio has been increased, it has been increased at the expense of reducing the range of possible activations. Too much preactivati0n has caused yielding of the spring at a relatively low activation. Higher amounts of preactiration are possible if a material with lower modulus of elasticity and relatively high yield strength is substituted for the stainless steel. 6 For example, the use of 13titanium (titanium molybdenum alloy, TMA) with a modulus of approximately 2/5 steel allows the use of larger preactivation angles. The mechanical characteristics of 0.017-inch ~ TMA are shown in curve E for a 45 ° preactivation. The r a n g e o f activations is now up to 6 mm with M / F ratios markedly above those of the 15° preactivated stainless steel wire. Reducing the cross section (curve E') again allows a further increase in activation, with the resulting extension of the M / F ratio curve. The alterations, which have been illustrated, have produced a retraction spring with force and moment characteristics similar to those of preactivated T springs, which have been described by Burstone. ~° While the additional alterations we have considered illustrate the possible increase in moments and M / F ratio when larger amounts of preactivation are feasible, these are only examples.
336
Am. J. Orthod. Dentofac. Orthop. April 1991
Faulkner et al.
CONCLUDING REMARKS The limited utility of a standard vertical loop can be drastically increased through the judicious use of more wire, which allows larger total activation and considerable preactivation of the appliance. Adding helices to the apex and lateral sides of the loop is an effective technique that increases the amount of wire being bent. If the proper gabling (preactivation) is coupled to the design with the helices, it also produces considerably higher moments and M / F ratios compared with those of the standard loop. These redesigned loops then produce force/deflection; moment/deflection, and M / F characteristics that make them less sensitive to minor manufacturing and placement errors than the standard vertical lodp. The most common material used in these appliances is stainless steel; however, other materials can also have a very significant effect on the final mechanical characteristics. The ideal material is one with a large ratio of yield stress to the elastic modulus. In this regard, T M A is superior to stainless steel for the appliance designs considered in this study. REFERENCES 1. Burstone CJ. Application of bioengineeringto clinical orthodontics. In: Graber Tgl, Swain BF, eds. Current orthodontic
2. 3. 4.
5.
6. 7.
8.
9. 10.
concepts and techniques. Vol. 1. Philadelphia:WB Saunders. 1975:chap. 5. Smith RJ, Burstone CJ. Mechanics of tooth movement. AM J ORTtlOD 1984;85:294-307. Quinn RS, Yoshikawa D. A reassessmentof force magnitudein orthodontics. AM J OR~IOD 1985;88:252-60. FaulknerMG, FuchshuberP, ttaberstock D, MioduchowskiA. A parametric study of the force moment systems produced by T-loop retractionsprings. J Biomech 1989;22:637-47. TanneK, Koenig HA, BurstoneCJ. Momentto f~rce ratios and the center of rotation. AM J OR'HIODDENTOFACORTHOP1988; 94:426-3I. Burstone CJ, Koenig HA. Optimizing anterior and canine retraction. AM J ORTIIOD1976;70:1-16. FaulknerMG, StredulinskyDC. Nonlinearbending of inextensible thin rods under distributed and concentrated loads. Trans Can Soc Mech Eng 1977;4:77-82. Lipsett AW, FaulknergiG, EI-RayesK. Large deformationanalysis of orthodontic springs. ASME J Biomech Eng 1990;I12: 29-37: BurstoneCJ, Baldwin JJ, Lawless DT. The applicationof continuous forces to orthodontics. Angle Orthod 1961;31:1-14. Burstone CJ. The segmented arch approach to space closure. AM J OR'I'HOD1982;82:361-78.
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Dr. M.G. Faulkner Department of Mechanical Engineering Universityof Alberta 4-9 MechanicalEngineeringBuilding Edmonton T6G 2G8 Alberta, Canada
