1976 -Optimizing Anterior and Canine Retraction

American Journal of ORTHODONTICS Vohme 70, Wumber 1, JULY, 1976

ORIGINAL ARTICLES

Optimizing anterior and canine retraction

Charles J. Burstone, D.D.S., MS.,* and Herbert A. Koenig, M.S.M.E., Ph.D.** Pawnington, Corm.

D epending upon the techniques employed, a number of procedures are used for incisor, canine, or en masse retraction of the anterior segment in the treatment of extraction cases. Some mechanisms employ compression or tension coil springs or latex elastics and use an arch wire for control, allowing the canine or other teeth to slide along the arch. Unfortunately, with such mechanisms high moments are produced when translation is attempted, producing a great amount of friction between bracket and arch wire which can inhibit tooth movement. These sliding mechanisms in any application other than simple tipping movements have two disadvantages: (1) The friction may stop tooth movement cn- tirely as one approaches translatory types of movement. (2) Force magnitudes cannot be easily determined since the amount of friction is relatively unknown and unpredictable.

A second approach to the retraction of anterior teeth is the use of a frictionless system based upon incorporation of a loop (a spring) into a continuous arch wire or a section of an arch wire. This article will consider the major factors that will allow the orthodontist to optimize his use of frictionless retraction springs.

There are three primary characteristics that describe a retraction spring: (1) The moment-to-force ratio (M/F) which determines the center of rotation of the tooth during its movement : (2) The force at yield (Fyield) ; this represents the greatest force that can be delivered from a retraction spring without per-

This work was supported by Grant DE03953 from the National Institute of Dental Research, National Institutes of Health, Bethesda, Md.

*Professor and Head, Department of Orthodontics, University of Connecticut School of Dental Medicine.

“*Professor, Department of Mechanical Engineering, Unioersity of Connecticut

1

2 Burstone and Koenig Am. J. Orthod. July1976

Table I. Force system from vertical loop retraction springs

Horizontal Initial Loop loop Loop angula-

Case height length diameter tion (4) Activation No. (H) (L) B B/L (D) (degrees) (-II

1 6 I 3.5 0.5 1.0 0 0.70 2 6 7 3.5 0.5 1.0 0 1.38 3 10 7 3.5 0.5 1.0 0 0.70 4 10 I 3.5 0.5 1.0 0 3.32 5 4 1 3.5 0.5 1.0 0 0.70 6 6 21 10.5 0.5 1.0 0 0.70 7 6 21 10.5 0.5 1.0 0 1.68 8 6 14 1.0 0.5 1.0 0 1.58 9 6 7 3.5 0.5 0.5 0 0.70

10 6 1 3.5 0.5 0.5 0 1.30 11 6 7 3.5 0.5 2.0 0 0.70 12 6 I 3.5 0.5 2.0 0 1.50 13 6 I 2.0 0.3 1.0 0 0.70 14 6 7 2.0 0.3 1.0 0 1.24 15 6 I 3.0 0.4 1.0 0 0.70 16 6 7 3.0 0.4 1.0 0 1.36 17 6 7 4.0 0.6 1.0 0 1.36 18 6 1 5.0 0.7 1.0 0 1.24 19 10 21 6.0 0.3 2.0 0 4.00 20 8 14 5.5 0.4 1.5 0 0.70 21 8 14 5.5 0.4 1.5 0 2.60

Note: All forces and moments are acting on the teeth.

0.016” (0.41 mm.) stainless steel wire.

E=Zl.l (106) gm./mm.z

manent deformation : (3) Force-to-deflection rate (F/A) and/or the moment-to- rotation rate (M/O). The force-to-deflection rates define the force per unit activation or the rate of decay of the force in a retraction spring. Such secondary spring characteristics as the greatest deflection without permanent deformation (Ayield) or the greatest moment that can be applied without permanent deformation ( moment,i,la) can be derived from the primary characteristics. Although the three primary characteristics define the force properties of a retraction spring, it should be noted, that, with large activations, M/F ratio and F/A rate may change somewhat as a result of the altered shape of the spring; nevertheless, these characteristics are useful in defining the mechanical properties of a retraction spring.

In a previous article it was shown how design factors could influence the force at yield and the force-deflection rate .I Reducing the cross section of the wire, increasing the height of the spring, and strategic placement of helices was shown to facilitate the delivery of optimum magnitudes of force relatively constantly over large distances. This article will mainly consider spring design as it influ- ences the moment-to-force ratio. This is the most important characteristic of a retraction spring, since it is this ratio that determines the position to which the tooth will move (that is, whether the tooth will translate or tip). A lack of un- derstanding of the biomechanics of retraction has led to development of improper

Optimizing anterior and canine retraction 3

Maximum moment on

loop (Gm.-mm.)

943 1,860

393 1,860 1,860

116 1,860 1,860

995 1,860

864 1,860 1,050 1,860

961 1,860 1,860 1,860 1,860

499 1.860

(2.) (G:.,

-245.0 245.0 -485.0 485.0

-65.1 65.1 -311.0 311.0 -684.1 684.1 -160.0 160.0 -383.0 383.0 -415.0 415.0 -251.0 251.0 -469.2 469.2 -240.0 240.0 -516.7 516.7 -290.0 290.0 -514.0 514.0 -252.0 252.0 -488.0 488.0 -481.0 481.0 -517.0 511.0 -280.4 280.4

-91.8 91.8 -342.0 342.0

4 MP

(Gm.-mm.) (Gm.-mm.) MA/F* MdFp

536.0 -536.0 -2.19 -2.19 1060.0 - 1060.0 -2.19 -2.19

266.0 -266.0 -4.05 -4.05 1260.0 - 1260.0 -4.05 -4.05

896.0 -896.0 -1.31 -1.31 222.0 -222.0 -1.39 -1.39 531.0 -531.0 -1.39 -1.39 699.0 -699.0 -1.69 -1.69 520.0 -520.0 -2.07 -2.01 910.0 -912.0 -2.07 -2.07 580.0 -580.0 -2.42 -2.42

1248.6 - 1248.6 -2.42 -2.42 -46.5 - 1030.0 0.16 -3.55 -82.4 - 1825.0 0.16 -3.55 334.0 -111.0 -1.33 -2.85 646.0 - 1338.0 -1.33 -2.85

1388.0 -642.0 -2.85 - 1.32 1842.0 96.4 -3.56 0.19

-131.1 - 1436.0 0.41 -5.12 88.6 -339.0 -0.91 -3.69

330.0 - 1264.0 -0.91 -3.69

designs, and hence it is clinically important to consider the force systems delivered from a retraction spring.

The data in this article result from a combination of experimental and mathc- matical modeling techniques. A spring tester was designed which was capable of measuring both moments (couples) and forces delivered from retraction springs (Fig. 1). The computerized mathematical model which has previously been described is based on the theory of a small deflection beam.?

The vertical loop

One of the most commonly used mechanisms for retraction is the vertical loop. Because of its simplicity of design, let us consider first the force charaeter- istics. A vertical loop, 6 mm. high, is fabricated of 0.16 inch stainless steel wire with a yield strength of 400,000 psi, which is centered between the canine and second premolar brackets on the lower left side (Fig. 2). This retraction spring (loop) will be used as a reference and will be described as a standard loop in the remainder of the article. Its complete geometry is given as Cases 1 and 2 in Table I. The activation force system is shown in Fig. 2, 8, the deactivation force system is shown in Fig. 2, B. The orthodontic sign convention previously published is used (moments or forces tending to move crowns mesial are positive) .3

4 Burstone and Koenig Am. J. Orthod. July 1976

Fig. 1. Apparatus used to measure force system of retraction spring at the bracket. For

every millimeter of activation, force and moment are given simultaneously.

Optimizing anterior and canine retraction 5

FORCES ON THE LOOP FORCES ON THE TOOTH

IF, 1 =IFp I= 485 gm

IMAI =p%l= 1060 gm-mm

Fig. 2. Activation (A) and deactivation (6) force systems using a standard vertical loop retraction mechanism.

Two equal and opposite forces of 485 Gm. are required to activate the standard vertical loop up to the point of yield; in other words, this spring could potentially deliver 485 Gm. of force. If only forces were applied to the loop, the horizontal arms would rotate and not be parallel to the bracket. In order to maintain bracket engagement, equal and opposite moments must be placed on the vertical loop at the canine and premolar brackets. The moment at yield on the wire at each bracket is 1,060 Gm.-mm. This is significantly less than the moment generated at the critical section at the apex of the loop, which is 1,860 (&n-mm. The dcflcction at yield is 1.4 mm. Any activation over 1.4 mm. would produce permanent deformation. Since common activations of a vertical loop are approximately 1 mm., it can be seen that the force values for a 0.016 inch round loop are very large. This is sometimes not realized by the clinician, since some earlier research measured loop forces by force gauges alone, ignoring the restraint by moments at each of the two brackets. An edgewise vertical loop of 0.021 by 0.025 inch wire will deliver even larger forces. The force at yield is 2,207 Gm. ; the deflection at vield, 1.1 mm. ; thus, 1 mm. activation would delivci * 2,099 Gm. Forces of this magnitude are clearly excessive, regardless of the center of rotation required for a canine or anterior tooth retraction.

In Fig. 3 a canine is shown with a force directed through its center of resistance. This force, by definition, will translate the canine. The clinician prefers, however, to place the force at the bracket for convenience and comfort, and hence a force system equivalent to the one through the center of resistance must be used

at the bracket. The equivalent force system at the bracket is 200 Gm. and 2,000 Gm.-mm. The moment-to-force ratio at the bra&t is, thcreforc, 10 to 1. Momcnt- to-force ratios for translation are variable, depending upon root length and bracket placement, but usually must be greater than 8 for an incisor or canine to translate. It has already been noticed that in artivating a vertical loop, nnt

6 Burstone nnd Koenig Am. J. Orthod. JuZu 1976

Fig. 3. A force acting at the center of resistance (CR) of a tooth will translate the tooth. A couple [moment) and a force (white arrows) acting at the bracket can produce the same effect. Note that the magnitude of the moment is equal to the force times the distance from bracket to center of resistance.

only forces but also moments are produced. The deactivation moments are in a direction to move roots distally on the canine (+) and mesially on the premolar (-) . A standard vertical loop gives greater control over displacing an apex of the canine mesially than a single force applied to a canine by a wire or an elastic; however, the moment-to-force ratio is so low (2.2) that it is not capable of producing translation or even controlled tipping with the center of rotation at the apex.

In Table I, moment-to-force ratios are given for a number of identical con- figurations with varying activations. It is significant that the moment-to-force ratio is constant regardless of the number of millimeters of activation for any given configuration. If a standard vertical loop of the design and cross section described above is activated 0.1 mm., 1 mm. or greater up to the point of yield, the moment-to-force ratio is constant at 2.2. In some retraction springs of a more sophisticated design which will be discussed later, the moment-to-force ratio may not be constant with large activations since the shape of the spring mill change somewhat during activation. The basic concept, however, of the moment-to-force ratio being constant throughout the range of activation of any given retraction spring is most useful, even though slight variation will occur with some springs that have large activations.

The force-deflection rate of the 6 mm. 0.016 inch vertical loop is 351 Gm./mm. If an identical spring were fabricated of 0.021 by 0.025 inch wire, the force deflection rate would be 2,099 Gm./mm. It can be appreciated that the vertical loop has a very large force-deflection rate and even in small cross sections will have a very rapid rate of decay of the force during tooth movement.

The standard loop has some serious limitations for retraction. The moment-to- force ratio is far too low to control the tooth as it moves distally. Mu& higher ratios are needed to tip the tooth around the point at the apex or to translate it. The load-deflection rates are very high, which prevent the clinician from using

--

L

Fig. 4. Activation force system. The dotted line shows the actual shape of the loop-activated Amm. Labeled parameters are defined in nomenclature section.

optimal force magnitudes since calibration is difficult and the decay of forces is high.

Since the moment-to-force ratio is perhaps the most important characteristics of a retraction spring, let us now consider some of the design factors that could influence and, in particular, raise the moment-to-force ratio so that the clinician can gain greater control over tooth movement during c.anine and anterior tooth retraction.

Vertical loop design

In looking at how the design of a vertical loop or its dimensions can influence spring characteristics, particularly the moment-to-force ratio at the bracket, it is our intent to consider the variables of the vertical height, horizontal length, and loop diameter of the spring. The symbols for these variables are given in Fig. 4. Table I presents a number of cases in which vertical loop dimensions and their activations are varied. The moment at the anterior part of the spring (Ma), and the moment at the posterior part of the spring (Mr) , the horizontal forces (Fa and FP) , the moment-to-force ratios at both A and P are given. The 6 mm. vertical loop described in the previous section (Cases 1 and 2) referred to as the standard vertical loop will be used for purposes of comparison with loops of different dimensions and of different designs.

One of the most important factors in influencing the moment-to-force ratio is the vertical. height of the loop (H). A 4 mm. loop has a moment-to-force ratio

8 Burstone and Koewig Am J. Orthod. JuW 1976

2.4 REFERENCE DATA FOR

/

A

2.2 STANDARD VERTICAL LOOP

(k =I)

r.et 1.6 -

1.4 -

k 1.2 -

1.0 -

0.8 -

0.6 -

0.4 -

0.2 -

M, =I060 gm-mm M, =-IO60 pm-mm [M/F], , [M/F] F,, 5 =485 gm

P

01 I I I I I I 0 2 4 6 8 IO I2

LOOP HEIGHT, mm

Fig. 5. To obtain the real magnitude of each parameter, multiply k by the magnitude of the standard vertical loop.

of 1.3 ; the standard 6 mm. loop, 2.2 ; and a 10 mm. loop, 4.0, keeping all other dimensions constant (Cases 5, 2, and 4). Thus, the higher the loop, the greater the moment-to-force ratio and the better control the orthodontist would have over the root apex in preventing it from displacement forward during retraction. Even with a 10 mm. high loop, the moment-to-force ratio is much too low to produce translation, since the usual clinical situation will require moment-to- force ratios of at least 8.

Comparing springs at yield, higher vertical loops deliver larger moments and small forces ; in other words, considering the force alone, shorter vertical loops can potentially give greater magnitudes of force if activated to the point of permanent deformation.

A 4 mm. vertical loop can be activated 0.7 mm. without permanent deformation, and a 10 mm. loop can be activated at 3.3 mm. Thus, the vertical height of the loop is important in increasing the amount of activation possible. The actual values of force and moments in situations where the vertical height is altered and other design variables remain constant are given in Table I, Cases 1 to 5.

It can be seen that the relationship between vertical height and force, moment, or deflection values is not linear. Fig. 5 plots M/F, MSleld, Fyieid, and Ayield for various loop lengths. The numbers on t,he ordinate represent a constant (K) . A constant of 1 refers to the standard vertical loop (Case 2). In order to obtain the actual force, moment, or deflection magnitudes for any given vertical loop length, it is necessary to multiply the constant for that loop by the actual values for the standard vertical loop. Referring to Table I and Fig. 5, it should be ap-

Optimizing &e&or and canine retractio?b 9

REFERENCE DATA FOR STANDARD VERTICAL LOOP

1.6 -

1.4 -

1.2 -

I.0 -

k 0.8 -

0.6 -

0.4 -

0.2 -

(k =I) M, =I060 gm-mm M, =-IO60 gm-mm

F,, F, = 485 gm Ayield = 1.38 mm

0”““““““” 0 4 8 12 16 20 24 28

INTERBRACKET DISTANCE, mm


parent to the reader how the vertical height of a loop can significantly influence spring properties.

The horizontal length of a vertical loop (L) in most clinical situations is determined by the interbracket distance and, hence, the position of the teeth. There are ways, however, of altering the horizontal length if one were to combine relatively rigid wires with the lighter ones used for the loop (segmentation ) Since horizontal length can be considered a variable, it was studied in Cases 2, 6, 7, and 8, where the vertical loop height was kept constant and horizontal length varied from 7 to 21 mm. The effect of varying the horizontal loop length is shown in Fig. 6. Keeping all other variables constant as one increases the horizontal length of the loop, the moment-to-force ratio decreases; both the moments and forces at yield decrease and the deflection at yield increases. The horizontal length of the loop can be a factor in spring properties, but its effect on the mommt-to- force ratio is not as great as a change in the height of the loop.

As the diameter (D) of the vertical loop increases, there will also be an increase in the moment-to-force ratio, in the moment and force magnitudes at yield, and in the amount of deflections at yield. The relationship between the vertical loop force characteristics and loop diameter are shown in Fig. 7. Cases 2, 9, 10, 11, and 12 demonstrate the effect on the force system of changing the diamet,cr of the loop.

The T loop

Data on the vertical loop retraction spring suggest that additional wire placed apically at the loop would have the effect of raising t,he moment-to-force ratio

10 Bnrsto~le cmd Koeltig


(k =I) M, =I060 gm-mm

MP =-IO60 gm-mm F,, Fp =485 gm Agi.ld = 1.38 mm 1.4 I-

0.8 k

0.6

01 ” ” ” ’ “‘I ” 0 0.4 0.8 1.2 1.6 2.0 2.4

Loop DIAMETER, mm


while simultaneously reducing the load-deflection rate. With this in mind, an experiment was carried out in which added wire was placed gingivally, forming a T loop. As can be seen in Fig. 8, as one increases the gingival-horizontal length (G ) of the loop, the moment-to-force ratio increases. It will be remembered that a standard vertical loop 6 mm. in height had a moment-to-force ratio of 2.2. When the additional horizontal wire is added, the moment-to-force ratio increases but never quite approaches 6, which is the vertical height of the T loop. With a horizontal length (G) of 50 mm., the moment-to-force ratio is 5.3.

There are two important conclusions that can be drawn from this experiment. First, the moment-to-force ratio can never be higher than the vertical length of the loop that is used. Thus, if a higher moment-to-force ratio is required, it is necessary to use a greater vertical height. Second, additional horizontal wire placed gingivallp will raise the moment-to-force ratio for any given loop length. At the same time, a very dramatic lowering of the load-deflection rate can be obtained. In Fig. 9, moment-to-force ratio and load-deflection rate are plotted for a series of 8 mm. tall T loops. Note that the effect of adding horizontal wire lengths gingivally is much greater upon the load-deflection rate than upon the moment-to-force ratio and that the moment-to-force ratio approaches 8 but never quite reaches it.

In our discussion up to this point, only one variable has been considered at a time. Using the constants given on the graphs (Figs. 5, 6, and 7), it is possible to determine the spring characteristics of any vertical loop. To accomplish this, we multiply each constant for a given characteristic together and then multiply this derived constant by the values given for a standard vertical loop. An example

0’ ---- EXPERIMENTAL RESULTS z-

THEORETICAL RESULTS

‘0 I I I I I I I I I

5 IO 15 20 25 30 35 40 45 - G, mm

Fig. 8. The moment-to-force ratio increases as the gingival horizontal (G) increases on a ‘T” loop, The ratio approaches but will never be larger than the loop height (H).

6.5

40

6.0

35

5.5 EE

30: lL 2 5.0 d

il

Fig. 9. As the gingival-horizontal length increases, the M/F ratio levels out; however,

the F/a rate continues to decrease.

G, mm

12 Burstone and Koewig Am. J. Orthod. July 1976

Table II. Prediction of spring constants (k) varying parameters D, B, H, L

D = 1.5. B = 5.5, H = 8.0. L = 14.0 Actual Dev. (per cent)

kA= (1.04)(0.96)(1.62)(1.14)= 1.844 1.879 1.86 k~p~=(1.09)(1.44)(1.12)(0.66)= 1.16 1.19 2.5 kFR = (1.03) (I .02) (0.78) (0.85) = 0.697 0.705 1.2 kMA = (1.09)(0.4)(1.12)(0.66)=0.322 0.311 3.6 k,w~p=(1.04)(1.44)(1.41)(0.77)= 1.63 1.685 3.5 kmmla =(1.04) (0.4) (1.41) (0.77) = 0.45 0.441 2.4

is given in Table II for a vertical loop which is different from the standard loop in dimensions D, L, H, and B. It should be noted that the estimated constants determined by this method are close to the actual.

Centering the loop

During the clinical use of a vertical loop for retraction, there may be considerable variation in the placement of the loop in an anteroposterior direction. For canine retraction, it is common to,, place the loop closer to the canine to allow for a longer range of activation without, having to remake a retraction section. The horizontal placement of the loop is’most critical if one considers its effect upon the developing force system. Cases 2, 13, 14, 15, 16, 17, and 18 in Table I demonstrate this significant effect. The column labeled “Distance B” gives the distance between the center of the loop and distal portion of the wire. “L” is the total horizontal length of the loop, measured from its mesial and distal extrem- ities, and the ratio B/L describes the amount of eccentricity of the loop placement. The higher the ratio, the closer the vertical loop is to the canine bracket. Let us describe the situation in which the B/L ratio is 0.3 and a 6 mm. tall vertical loop is activated to yield (Case 14). The force delivered is 514 Gm., which is considerably higher than the 485 Gm. produced with a symmetrically placed vertical loop of the same dimensions. Even more significant is the fact that the moment at A is opposite in direction from the moment normally delivered and, although it is small, has the effect of moving the crown distally and the root mesially (a negative moment). A very large moment (1,825 Gm.-mm.) is produced on the posterior segment which tends to move crowns back and roots forward. Since the moments acting on the retraction spring do not sum to zero, vertical forces are produced at each bracket. The magnitude of the vertical force is considerable in this instance, approximately 249 Gm., tending to intrude the canine, and 249 Gm., tending to extrude the premolar. If the loop is placed further mesially so that the B/L is 0.7, the reverse would be true. The larger moment would be found at the canine region, and smaller moment in the posterior region. The vertical forces acting on the teeth would also be reversed, with the canine being extruded by a vertical force. The actual forces and moments produced by altering the B/L ratio are given in Table I, Cases 13 to’l8. It should be remembered that the forces and moments that are given are those that act on the arch wire and that the sign must be reversed to determine the force system on teeth.

Fig. 10 plots loop position given as the B/L ratio against a constant where 1

1.8 -

1.6 -

1.4 -

1.2 -

1.0 -

k 0.8 -

0.6 -

0.4 -

0.2 -

0

-0.2 r


M, =I060 gm-mm

F,, = 485 gm A~WC = 1.38 mm [M/F]*, [MIFlp = 2.19

I 0.14 0.281 0.42 0.58 lo.70

ECCENTRICITY OF LOOP PLACEMENT, 8/L


Table III. Effect of helices placed in a vertical loop (0.010 by 0.020 inch wire)

No. ofturns in the helix of 8 mm. vertical loop

0 1 5

10

M/F F/A (Gm.lmm.)

3.3 75.0 3.9 61.5 4.2 46.0 3.9 35.5

represents the values for the standard 6 mm. height vertical loop. A quick glanc(a at the figure clearly shows that, with a vertical loop placed symmetrically as in the standard spring, the moment-to-force ratios at A and P are identical. As ow places the loop off center, the ratios will be different ; if the eccentricity is great’, 11~~ sign ma,v change. Furthermore, with greater eccentricity at one of the brackets. a larger moment-to-force ratio could be produced than would occur in a symmt4- rically placed loop. Practically, this means that during canine retraction, with the vertica.1 loop placed closer to the canine, a higher ratio would hc present OII the canine which could better control the root apex. Unfortunately, the vert,ic*al force produced would be undesirable since it would tend to extrude the caninc.

As the vertical loop is moved off center, the magnitude of the horizontal forc*es increases and the magnitude of one of the moments becomes considerablp larger than the moment capable of being produced 1)~ a symmetrical loop. I’inally, thli deflection at yield will dccreasc slibhtly as eccentricity is placed in the Gnfigura- tion.

Fig. 11 plots vertical forces against loop position. It is evident that~ a vc*r?;

14 Burstoue and Komig Am. J. Orthod. JUZ$/l9TF

300 t

200 -

w IOO- ii e

-I o- I 1

s 0.14 0.28 F - ECCENTRICITY OF E >-lOO-

- 300 t

Fig. 11. As a vertical loop is placed off center, vertical forces in addition to horizontal forces are produced. As the loop approaches the canine, vertical eruptive forces increase.

significant extrusive or intrusive force can be produced at either end of a vertical loop if it is placed off center. In summary, it should be re-emphasized how critical the placement of a vertical loop is in an anteroposterior direction. Very small asymmetries can alter the moments at the canine or the premolar bracket with a concomitant development of unexpected vertical forces as well as the anticipated horizontal ones.

The role of the helices

The placement of a helix in a loop of a given configuration primarily in- fluences the load-deflection rate and not the moment-to-force ratio. Table III gives some experimental data on moment-to-force ratios and load-deflection rate for 8 mm. vertical loops with varying numbers of turns in an apical helix. There are no significant differences in the moment-to-force ratios. However, the additional wire lowers the load-deflection rate dramatically. Notice that the load-deflection rate of a plain vertical loop fabricated from 0.010 by 0.020 inch stainless steel wire is 75 Gm./mm. With ten turns at the helix, the load-deflection rate is 35.5 Gm./mm. Our data on T loops with and without helices further document that helices affect primarily the load-deflection rate and not the M/F ratio.

Increasing the moment-to-force ratio by angulation

By trial and error, the clinician has learned that the placement of a gable bend (which will be referred to as angulation) in the horizontal legs of a vertical

L = 8.384 mm. (INTERSRACKET DISTANCE)

I

700

600

500

5400 l&Y 300 I

II

0

-100 1

E -1600 - E & -1400 -

; -1200 -

-1000 -

-800 -

-600 -

-400 -

Fig. 12. Small differences in the horizontal length (L,) of an angulated vertical loop produce large modifications in the M/F ratio. A difference of 0.3 mm. will produce varying tooth movement from tipping to root movement. Gable bends in a vertical loop are therefore difficult to calibrate.

loop could increase the moment-to-force ratio. A large enough increase in the moment-to-force ratio theoretically could translate a canine distally. There are some inherent problems, however, in using the principle of angulation for con- trolling canine or anterior retraction.

Let us consider the situation in which the interbracket distance between a second premolar and the canine is 8.4 mm. and a standard 6 mm. high vertical loop is to be used. In order to increase the moment-to-force ratio, a 10 degree angulation is placed at the junction of the vertical and horizontal legs, both mesially and distally (total angulation, 20 degrees). By \-arying the horizontal length of t)hc spring, different magnitudes of the moment-to-force ratio arc produc.ed. ln Fig. 12 the moment-to-force ratio is plotted against the horizontal length of the spring (1~~). If the horizontal distance is the same as the interbracket distance. 8.4 mm., the clinician might intuitively think that no force would be producrtl, only equal and opposite moments on two adjacent brackets. Actually, this activation produces a relatively low moment-to-force ratio (4.3) which would tend to tip the tooth with the apex displacing in the opposite direction. If a moment-io- force ratio of 10 was required to produce translation, a horizontal length slightly less than 8.9 mm. would be required. Exactly at 8.9 mm. of horizontal length, only equal and opposite moments are produrcd with no force. ~211 a~timtiotI

16 Burstone and Koenig Am J. Orthod. July 1976

- DEACTIVATED

---- ACTIVATED \

u

L./

b) WITH ANGULATION

~SCRATCH MARK

SCRATCH MARKS

A a4m P T o) BEFORE ANGULATON

Fig. 13. A vertical loop angulated 40 degrees is placed in the brackets by applying only a a couple (moment). Note that the loop shortens. Scratch marks placed at the proximals of the brackets are 1 mm. short of each bracket after activation. If the orthodontist is un- aware of this change, too much force or permanent deformation of the loop will occur.

of a spring, where no force is produced, only moments, is referred to as the neutral activation.

It is apparent from the graph that if one changes the horizontal length of a vertical loop, keeping the interbracket distance and angulation constant, moment-to-force ratios will go through a range from a little over 2 through infinity. Thus, an angulated vertical loop has the capability of producing any type of tooth displacement required (crown tipping, translation, or root movement). Keeping the angulation constant, the clinician could vary the horizontal length of the loop (L1) to produce the desired moment-to-force ratio. Unfortunately, as shown by the graph, small differences in the horizontal length of the loop produce large changes in the moment-to-force ratio. Erring 0.3 mm. would make the difference between tipping at the apex and root movement of an incisor.

The critical nature of the horizontal length of the loop has a number of inherent disadvantages in applying angulation to a vertical loop: (1) Since the operator cannot practically determine horizontal lengths within 0.3 mm., it is unlikely or purely accidental that he would have the moment-to-force ratio that he desires. This is particularly true in tooth movements as translation. (2) As the tooth moves, the moment-to-force ratio will radically change for a small amount of tooth displacement; hence, as a canine is retracted, a constant center of rotation would not be found. This means that the canine would “wiggle” back and forth until it reaches its final position.

0 t e IO LL 2

DEACTIVATED

DIMENSIONS OF THE LOOP

,- , , / > “‘“‘Eg”~*~~~” 0 I 2 3 4 5 6 7 8 L:7mm

A, mm ‘#J = ANGULATION * 65” NO. OF TURNS IN THE HELIX = I

Fig. 14. Lowering the load-deflection rate by placing helices in a “T” loop reduces the change in the M/F ratio for every millimeter of activation (A]. In comparison to a standard vertical loop, the center of rotation of the tooth is more constant and clinical determina- tion of proper activations for any given center of rotation is less critical.

Previous studies have attempted to demonstrate the effect of various retraction springs on tooth displacement and the centers of rotation. One should br careful about generalizing on what a retraction spring would do unless the geometry of the spring is very accurately clctermined. Small changes in dimrn- sion or angulation can produce very large changes in the moment-to-force ratio and, hence, in the may a tooth will move.“. 5

The absolute values of the forces and moments produced by varying the horizontal activation of a loop with a 10 degree angulation arc given in Fig. 12. At, the neutral activation, the horizontal loop length is 8.9 mm. and an initial moment of 480 Cm. mm. is delivered. The smallest horizontal length that can be placed between two brackets is 7.3 mm., since any smaller distance would pause the spring to be stressed beyond its yield strength.

It is important for the clinician to recognize that angulation changes tht> neutra,l activation of a spring and that the traditional ways of determining activation may err considerably. For example, let us suppose that during canine retraction an interbracket distance of 8.4 mm. was present. A standard vertical loop is constructed (Fig. 13, 11) that fits passively between the two brackets and a scratch mark is placed at the distal aspect of the canine bracket (A) and the mesial aspect of the premolar bracket (P) If such a loop is tied in place, with the scratch marks at the proximals of the brackets, no force would bc osertccl. Kow Ict us place a 40 degree angulation in each horizontal arm of the loop ( F’ig. 13, R) . The scratch marks are now at A,P,. If two equal and opposite couples arc

18 Bwrsto~re trnd Kocwig Am. J. Orthod. July1976

placed at tither end of the loop to insert it in the brackets, the loop will tend to cross. At this neutral activation where no force will bc exerted, the scratch marks (A,, PZ) on the horizontal legs of the loop will now be a total of 2.0 mm. short of the brackets. If the loop is tied in so that the scratch marks line up with the brackets, 2.0 mm. of activation would exist. Since a neutral activation with a 10 tlcgrec angulation produces 1,860 (:m-mm. in the spring, it is not possible to have any significant horizontal activation before permanent deformation would

occur. Thus, if an attempt were made to place this loop into the brackets so that the scratch marks are aligned with the proximal of the brackets, permanent deformation would occur. It should bc noted that angulation tcntls t,o increase the amount, of force protlucetl by a r&action spring beyond what the clinician might expect using the common clinical tcchniqucs of activation.

If one of the objectives of treatment is to assure that the change in M/F rates is minimal for crery millimeter of retraction, the angulation principle can be used, provided that the F/a rate is low. Data for an S mm. 0.010 by 0.020 inch T loop with a helix at each corner is shown in Fig. 14, in which 65 degrees of angulation has been placed in both horizontal arms. Note that the M/F ratio changes only a small amount over the 7 mm. range of activation. A low-rate retraction spring is, therefore, superior in its ability to give a, more constant center of rotation of the tooth and in allowing the clinician greater margin of error during retraction.

Summary

Vertical loops or modified vertical loops are basically frictionless springs which are used for canine and anterior tooth retraction. The design and selec- tion of a proper loop or retraction spring should be based on a number of scientific criteria. Foremost among these would be a sufficiently high moment-to- force ratio so that root apices are not displaced mesially or anteriorly. A retraction spring with zero angulation of its horizontal-occlusal arms delivers a moment when activated to produce a force. The ratio of this moment and force is constant throughout the elastic range of activation of the spring. The higher the moment- to-force ratio, the greater is the clinician’s control over the apices of the anterior teeth. An analysis of design factors demonstrates that the higher the loop occlnso- gingivally, the shorter its horizontal length occlusally, and the greater the gingival horizontal length as in a T loop; these are significant factors in increasing the moment-to-force ratio. The placement of helices is a useful design considera- tion but the main effect is in reducing the load-deflection rate. By keeping these design factors in mind, the clinician can build into his retraction springs, without the placement of any gable bend, the largest possible moment-to-force ratio so as to optimize his tooth movement.

Although it may be possible to design retraction springs to deliver an adequate moment-to-force ratio for controlled tipping around the apex of an incisor or a canine, translatory movements are not possible, considering the intraoral limitations on spring height. This can be overcome by the placement of gable bends or angulation in a vertical loop or retraction spring. Unfortunately, with the typi- cally used high-load-deflection-rate vertical loops, activation to achieve the desired

Optimi.zi?~g f/7( terior crnd ctrwi?le rr!trnctio?l 19

moment-to-force ratio is too critical, exacting, and changeable with small displaced movements of the tooth. This can be partly overcome by utilizing designs that have not only the highest possible moment-to-force ratio during pure horizontal act,ivation of their arms but low-load deflection rates as well. Bccausc of the low load-deflection rate, moment-to-force ratios arc relatively more consta~!i if a gable bend (angulation) is placed.

The science of spring design as applied to the problems of canine and anterior tooth retraction in this article allows the clinician to optimize the design of his retraction springs. More important, with properly designed springs, it allows him to estimate with relative accuracy the force systems producetl and to avoid undesirable side effects which might not have been apparent from super- ficial observation.

Nomenclahwe

A = Anterior B = Distance between the center of the loop and the distal position of the wire

B/L = Eccentricity of the loop placement D = Diameter of the vertical loop E = Young’s modulus F = Horizontal force

F.< = Horizontal force at the anterior side of the loop Fr = Horizontal force at the posterior side of the loop

F/, = Force to deflection rate G = Gingival-horizontal length

H = Vertical height of the loop L = Horizontal length of a vertical loop

L, = Horizontal length of an angulated vertical loop M = Moment (force times distance)

M, = Moment at the anterior side of the loop Mr = Moment at the posterior aide of the loop

M/F = Moment-to-force ratio M/G = Moment-to-rotation rate

M y*e,a = Greatest moment that can be delivered at the bracket without permanent deformation at the loop

M n,ax = Maximum moment within the loop (at the critical cross section) P = Posterior A = Deflection of the loop

A yield = Greatest deflection without permanent deformation of the loop 8 = Angle of rotation ‘p = Angulation of the horizontal arm of the loop

REFERENCES

1. Burstone, C. J., Baldwin, J. J., and Lawless, D. T.: The application of continuous forces to orthodontics, Angle Orthod. 31: l-14, 1961.

2. Koenig, H. A., and Burstone, C. J. : Analysis of generalized curved beams for orthodontic applications, J. Biomech. 7: 429-435, 1974.

3. Burstone, C. J., and Koenig, H. A.: Force systems from an ideal arch, AM. J. ORTHOD. 65: 270-289, 1974.

4. Yang, T. Y., and Baldwin, J. J.: Analysis of space closing springs in orthodontics, J. Biomech. 7: 21-28, 1974.

5. Baeten, L. R.: Canine retraction: A photoelastic study, AM. J. ORTHOD. 67: 11-23, 1975.

Documents

1976 -Optimizing Anterior and Canine Retraction