An evaluation of the torsional moments developed in orthodontic applications. An in vitro study

An evaluation of the torsional moments developed in orthodontic applications. An in vitro study J. ~degaard, BDS, MS, Dr.Odont.," E. Meling, b and T. Meling b

Hamburg, Germany The amount of play between brackets and wire and the magnitude of the torsional moments developed have been investigated for six different rectangular wires. The study simulated the situation occurring when torque is applied to an individual central incisor. The observed play between wire and bracket ranged from about 5~ between an 0.018 • 0.025 inch in an 0.018-inch bracket slot, to approximately 20 ~ for a 0.016 • 0.022 inch in an O.018-inch bracket slot. The observed rate of change after the play had been eliminated, varied from 0.24~ to 0.37~ for the steel wires, whereas 0.017 x 0.025-inch nitinol had a rate of change of approximately 1.07~ The use of ligatures demonstrated that a torque effect could be observed even though the play had not been completely eliminated. (AMJ ORTHOO DENTOFACORTHOP 1994;105:392-400.)

T h e moments developed by various wires in torsion is a factor in orthodontic treatment that have received very little attention. This can be assumed to be due to the difficulty of delivering an exact amount of torsional moment and at the same time measure the degree of angular twist. This article will demonstrate a new method of delivering a torsional force to a bracket and at the same time measure accurately the amount of torque achieved and the importance of play between bracket and arch wire. REVIEW OF THE LITERATURE

In 1977 Steyn, ~ applying forces to the roots of the anterior teeth in an acrylic model measured the torsional stress, developed in the arch wire to which the teeth were tied. No provision was made for the play normally observed between wire and bracket, nor were the moments delivered at each bracket calculated. Later experiments applying torque to brackets have been limited to testing brackets in torsion to measure their strength. Flores 2 tested ceramic brackets in torsion until fracture by applying a torsional force equally to both bracket wings,' whereas Holt et al.,3 using a different approach, applied torsion acting only on one side. The results from these studies showed that ceramic brackets basically had sufficient strength to withstand the torsional forces recommended in orthodontics. Other investigators have measured the amount of

*VisitingProfessor,Departmentof Orthodontics, Universityof Hamburg. Germany, and in private practice in Stavanger, Norway. bStudents at University of Oslo, Norway. Copyright 9 1994 by the American Association of Orthodontists. 0889-5406/94/$3.00 + 0 8/1/39422

392

play present between bracket and wire in torsion. Dellinger 4 and Creekmore 5 reported on the play between wire and bracket in torsion, basing their results on manufacturers information. Hixon et a]., 6 using a torquemeter assembly, measured the deviation angle, and their experimental results were different from those given by Dellinger 3 and Creekmore. 4 Later Sebanc et al. 7 reported on the importance of edge bevel and the amount of play between arch wire and bracket. Experiments giving load deflection diagrams for wires tested in torsion have, to our knowledge, not been carried out and forms the basis for this study. DESIGN OF THE TESTING APPARATUS

The apparatus is constructed from 10 mm plastic plates and rods (Perspex, ICI, Darwin, England). The principle used in the present design places the wire to be twisted in the center of a plastic crossbar (F) with a diameter of 10 ram, which can be rotated. The crossbar is held in position by two ball bearings (G), and the moment created by applying a force over a pulley (L) (Figs. 1 and 2). Eleven millimeters of the central portion of the crossbar has been removed to allow a pedestal with a bracket (C) to be moved into position. To each side of the pedestal the crossbar has been flattened, and two brackets (A) placed at each end of this flat area in such a position that the central axis of a wire with dimension 0.018 • 0.022 inches coincides with the center of the crossbar (F). The two brackets (A) serve as reference for the placement of other brackets (B), permitting variation of the interbracket distance. The brackets to be tested were mounted on plastic pedestals (d = 10 mm) 100 mm long with a fast setting

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American Journal of Orthodontics and Dentofaciat Orthopedics Volume 105, No. 4

I

M E

,W-"~

F

~

I

F

i

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I I I

Fig. 1. A, Brackets used for mounting holding brackets. B, Holding brackets. C, Pedestal with bracket to be loaded. D, Piece of plastic uniting the two parts of the crossbar, l=, Pedestal holder. F, Crossbar. G, Ball bearings. H, Mirror. I, Laser lamp. d, Laser lamp holder. K, Weight to counterbalance the uniting part D. L, Pulley. M, Basket with lead granulate to negate moment of basket N and internal friction. N, Basket to be loaded with weights for the application of moments. O, Piece of colored plastic with 1 mm aperture to reduce the size of the laser light spot. Fig. 1 has the piece D omitted to clarify the view.

methyl methacrylate (Biocryl resin, Scheu Dental, Iserlohn, Germany). The pedestal holder (E) allows movement in the horizontal direction, as well as rotation. The brackets used in this initial experiment were wide (4.46 --- 0.003 mm) standard edgewise (0 ~ torque and angulation) with 0.018-inch slot on mesh (Ormco Corp., Glendora, Calif.). To ensure that the two holding brackets (B) were parallel mounted, a piece of 0.019 • 0.025-inch stainless steel wire (Unitek/3M, Monrovia, Calif.) was carefully ground until a tight fit existed between the wire and the brackets. Two brackets were placed 14 mm apart on this piece of wire, and then mounted equidistant from the center of the pedestal

C, using the two previously mounted brackets (A) for correct orientation. This distance was selected as it represents an average distance between a lateral and central incisor, with a second central incisor placed between them. With a mounting device identical to the pedestal holder and transfer technique, the brackets to be tested were mounted to the pedestals in such a way that the bracket C was positioned equidistant from the brackets B with its slot oriented in the same planes as the slots of brackets B. To measure the angle of twist, a surface reflecting mirror (H) was mounted on the axis with the reflecting

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American Journal of Orthodontics and Dentofacial Orthopedics April 1994

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Fig. 2. Laser lamp holder made transparent and piece of colored plastic "0" with 1 mm aperture is omitted for clarification purposes. (See Fig. 1 for description of lamp.)

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fixed to the inner surface o f the arc. According to the laws o f reflection, the angle of incidence is equal to the angle of reflection. If the mirror is twisted the angle 13, both the angle o f incidence and angle o f reflection will be changed by 13, and the change recorded will be 2 13. With a distance from the central axis to the millimeter paper o f 1432 mm, 1~ o f change will correspond to 50 mm. The size o f the projected laser light spot was 1 mm. With a little experience, it was possible to read the scale to the nearest 0.05 ram. This corresponds to V~oo~ The m a x i m u m angle o f twist that can be recorded is 35 ~. The complete construction is shown in Fig. 3.

I I I ,I ,I

Fig. 3. Position of testing apparatus inside measuring scale. The distance from the center of the axis to the millimeter paper on the inside of the arc is 1432 ram.

surface in the center o f the axis. A laser lamp (I) was mounted so that the laser beam was projected onto the mirror centrally and reflected onto a measuring scale. The measuring scale was constructed from plywood with an inner radius o f 1432 mm. Millimeter paper was

MATERIAL AND METHODS Five wide twin siamese brackets with 0.018-inch slots, without torque and angulation, were used as Zest brackets. The wires tested are given in Table I. Five pieces of each wire type were tested in the as received condition. Moments were created by adding fixed weights to a basket (N), three different types of weights being used. The mean radius of the pulley was calculated to be I7.66 mm. The resulting moments created by the three type of weights when added to the basket were 0.28 Nmm, 0.85 Nmm and 1.25 Nmm, respectively. To negate the weight of the basket and the internal friction in the apparatus, a counter moment was created by hanging a small plastic container with lead granulate (M) acting in the opposite direction. This was adjusted so that the axis barely started rotating from any starting position, when no wire was in position.

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I, Wires tested. Wire a-d were supplied in straight lengths, whereas e and f as arch blank. Each test piece came from a separate length or a separate arch blank. For the latter only the distal portion was used

Table

Wire Manufacturer a. b. c. d. e. f.

Unitek/3M Unitek/3M Unitek/3M Rocky Mountain Ormco Unitek/3M

I

Material Resilient SS Resilient SS Resilient SS Yellow elgiloy TMA Nitinol

MEASUREMENT OF THE METHOD ERROR To test for accuracy stainless steel wires with dimensions 0.016 x 0.022 inch, 0.017 • 0.022 inch and 0.018 x 0.025 inch, five pieces of each were tested. The test piece was fixed in position to the brackets B and C with plastic modules, care being taken not to stretch the modules unduly. It was decided to use plastic modules, as it is difficult to place a stainless steel ligature exerting equal pressure, and this pressure will have a restraining effect on the wire. Each piece of wire was tested against five brackets. The load placed on the wire was 16.25 Nmm. The reason for using a relatively low load was to prevent overloading the brackets and the wires. After a piece of wire had been tested with the first bracket C, the procedure was repeated with a new bracket C, each time with a new ligature for a total of 75 bracket/wire/test piece combinations. After a period of I week the tests were repeated with the same wires and the same brackets. The method error that was based on 75 double measurements was calculated with the Dahlberg formula: i,---.-

method error = d / 2~ The difference distribution was tested for systematic errors with a paired t test. The importance of error variance was tested by estimating the coefficient of reliability (Houstong): 1 - S,~/S,2 9 (S,2 = error variance, Si = total variance in the material). The experiment was carried out with the same procedure as used when testing for method error, but for each wire five separate pieces all from different lengths were used against five brackets for 25 wire/bracket combinations. Wire "a" was tested three times. First without any elastic ligature holding the wire in position, then elastic ligatures in brackets B, and finally with elastic ligature in brackets B and C. The rest of the wires were only tested with elastic ligatures holding the wire in position. After the test of wire "a" had demonstrated that no distortion of the wire or bracket had taken place, the heavier wires were tested, but the maximum load was kept at 16.25 Nmm to make sure that the test brackets would not be distorted. The two titanium wires tested were only loaded to I 1 Nmm. At this load the maximum amount of rotation for the most elastic wire was around 30~. There was no sign of distortion of the wires up to these

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Dimension(inches) 0.016 0.017 0.018 0.016 0.017 0.017

x x x • x x

0.022 0.022 0.025 0.022 0.025 0.025

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Identification

Batch no. IB79B Lot #293

loads with one exception. One piece of wire "d" showed an angular change from 11~ to 30~ at a load of 5 Nmm. This test piece was replaced with a new sample from a new length.

RESULTS The method error was found to be = 0.087 ~ There was no significant variation between the three pieces of wires tested. The paired t test revealed no systematic error. The coefficient of reliability indicated that for all the wires tested, the method error accounted for 1.86% of the observed variation in the worst case. After each test run, the test piece was measured for its cross-sectional dimensions, with an electronic micrometer (Mitutoyo Corp., Japan) capable of reading with an accuracy of V~oco mm. The results of these measurements are given in Table II. For each wire type, the mean deflection at the various loads with respect to the brackets and the test pieces were calculated, as well as a total mean based on 25 measurements at the same load. These last values were used to plot a curve representing the relationship between load and rotational deflection. The resulting curves are found in Figs. 4 and 5. The curves for wires a through d showed a linear relationship for loads between 7.5 and 16.25 N m m and for wires e and fbetween loads of 6 and 11 Nmm. A straight line was fitted to this portion of the curves with the least square method, and the results are given in Table III. The table gives the values for the slope of this straight line, as well as the intercept with the x-axis. The value for the intercept of the curve itself with the x-axis is also given. The degree of torque present at 7.5 N m m for wires a through d and at 6 N m m for wires e and f are also given. To illustrate the relative importance of bracket and wire variation, the mean standard deviation of the angular twist was calculated with respect to the brackets and the wires, with the values obtained at a load of 16.25 N m m for wires a through d, and 11 N m m for wires e and f. The results are given in Table IV. Analysis of variance was carried out with the data

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Fig. 4. Curves demonstrating relationship between applied moment and angular twist for wires a through d. For each curve a straight line has been fitted to the linear portion of the various curves using the least square method, to get an expression for the slope of the line and an intercept with the x-axis.

Table II. The table gives the theoretical and measured mean values for height and width for the various wires in addition to the difference between theoretical and measured mean value. All values are given in 1/100 m m (0.13001 inch = 0.25 1/100 mm) Measured values

Theoretical vahtes Wire

Thickness

I

I

Width

Thickness

n

a b c d e f

40.6 439 45.7 40.6 43.2 43.2

55.9 559 63.5 55.9 63.5 63.5

40.66 42.94 44.86 40.66 43.38 43.04

-4- 0.23 -'- 0.39 • 0.17 --- 0.47 • 0.20 • 0.45

at maximum load to get an estimate o f the relative importance of bracket and wire size with respect to the observed torque. The results o f these analyses are given in Table V. To our knowledge, no formulas are available that can be used to calculate accurately the modulus o f rigidity from the present data. The modulus of rigidity is, however, inversely proportional to the angular change per newton millimeter (Nmm) "b" and the polar moment of inertia Ip. To get an impression o f the relative rigidity o f the four materials tested, a comparison was made using the expression b * Ip, where b is the slope o f the linear portion o f the curve and lp is the polar moment of inertia. For stainless steel, a mean from the three wire dimensions was used. The results are given in Table VI. To check for bracket distortion after the conclusion o f the experiment attempts were made to fit a piece of unground 0.019 • 0.022-inch wire from the same

I

Difference Width

56.68 55.56 63.36 55.34 62.30 62.62

_ • • • • •

0.25 0.83 0.49 0.42 0.42 0.29

Thickness

0.06 - 0.26 -0.84 0.06 0.18 -0.16

I

Width

0.78 - 0.34 -0.14 -0.56 - 1.20 -0.88

length that was used for mounting. In no case was it possible to fit the wire into the slot. It can be concluded that the experiment lead to very little, if any distortion o f the brackets 9 DISCUSSION

The present experimental design allows for a multitude o f experiments, such as comparison between bracket types, bracket material, a n d wire material 9 This article is limited to a comparison between a number of wire materials to evaluate their ability to deliver torque, as well as an analysis o f the amount o f play present in torque applications. The result of the error analysis indicated a method error of 0.087 ~. Compared with previous investigations where the torque has been measured to the nearest 0.25o, 7 this error is very small. Even though it was possible to measure to the nearest 0.01 ~ with the laser light, variation in the internal friction and insufficient

American Journal of Orthodonticsand DentofacialOrthopedics Volume 105,No. 4 . . . . . "a" 016" x 022" SS W,1JG. . 9 a .016" x .022" SS WO,tlO ~ . . . . . . "a" .016" x .Q22." SS WO,'LIG "8.0"

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T O R Q U E IN DEGREES Fig. 5. Curves demonstrating relationship between applied moment and angular twist for wires e and f. For each curve a straight line has been fitted to the linear portion of the various curves using the least square method, to get an expression for the slope of the line and an intercept with the x-axis.

T a b l e III. The change in the degree of torsion per newton millimeter for loads between 7.5 and 16.25 Nmm for wires a through d, and between 6 and 11 N m m for wires e and f

Wire rype

a a a b c d e f

WIL WO/LC WO/L 9

Change hz torque per newton millimeter (degrees)

Intercept linear stresslstrain

0.364 0.373 0.369 0.325 0.241 0.383 0.575 1.071

(degrees)

Intercept main curve (degrees)

Degree of torque at 7.5 resp. 6 nm (degrees)

20.4 20.3 20.3 14.4 5.3 18.1 11.8 15.7

7.8 11.2 18.5 7.6 2.6 7.7 7.2 9.0

23.2. 23.1 23.1 16.9 7.1 20.5 15.3 22.1

curve

aWIL, wire "a" with ligatures in both brackets B and C. aWOILC, wire "a" without ligature in bracket C. a W O I L wire "'a'"without any ligatures.

accuracy in the construction o f the apparatus must be assumed responsible for the resulting method error. The curves obtained show an initial portion where the degree o f rotation changes rapidly for a small change in the applied moments with the exception o f the curve for wire " c , " w h i c h changes its direction relatively q u i c k l y . Then the degree o f rotation decreases, and the curves reach a linear relationship for moments around 7.5 N m m for wires a through d and around 6 N m m for wires e and f. The question then arises, what is the actual amount

o f play between brackets and wires. The experiment was set up to imitate a clinical situation, and elastic ligatures were used to hold the wires in position. One experiment was performed without any ligatures at all, and one with only ligatures in brackets B. The results from the test without ligatures demonstrated that the intercept between the curve and the x-axis had a larger value when no l i g a t u r e s w e r e used, but that the effect o f no ligatures disappears very quickly, being eliminatcd by a moment of 3.75 N m m . As a result the ligatures have a restraining effect that will lead to a small

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Table IV. The mean degree of rotation wire a through d for loads of 16.25 Nmm and for wires e and f at 11 Nmm. The main standard deviation is based on the 25 wire/bracket measurements at this load. The standard deviation for the bracket and the wire is a mean value based on an average of five standard deviations, each in turn based on five measurements. They should only be looked on as a guide to the relative variation between brackets and wires

Wire a a a b c d e f

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Oegreeo O s,

W/L WO/LC WO/L .

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26.3 26.4 26.3 19.8 9.2 24.4 18.2 27.6

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SO rocket

0.74 0.79 0.76 1.05 0.57 0.96 0.69 2.13

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S , ir s

0.46 0.51 0.50 0.36 0.46 0.57 0.47 0.82

0.74 0.79 0.78 1.05 0.58 0.99 0.71 2.18

SD, Standard deviation.

Table V. The resulting F-values base on the data obtained at a load of 16.25 N m m for wires a through d, and at a load of 11 N m m for wires e and f Wire

brackets

o,ue

brackets

a W/L a WO/LC a WO/L b c d e f

43.2 36.6 34.3 135.5 272.5 50.8 5.6 171.8

<0.001 <0.001 <0.001 <0.00l <0.001 <0.001 <0.001 <0.001

delivery of torque even though the play between wire and bracket has not been eliminated. The amount of torque delivered is low, and its clinical effect doubtful. A more true expression for the actual amount of play is better given by the intercept between the continuation of the straight line fitted to the curve and the x-axis, rather than the intercept between the curve and the xaxis at the start of the experiment. The actual play in each bracket will then be half this value. With this in mind, the actual amount of play observed in this experiment is similar to the results reported by Sebanc et ~l. 7They observed a deviation angle between a 0.016 x 0.022-inch wire in an 0.018-inch slot ranging from 10.3 ~ to 14.1 ~ for the various bracket/wire combinations. This compares well with the mean value for the intercepts for wire "a" and "d" of 20.4 ~ and 18.2 ~ respectively. The actual play in each bracket is, as explained before, half this value. Still, this is the play we must expect in the clinic if we want to torque one tooth in relation to the neighboring teeth. Both the play in the bracket receiving the torque and in the brackets delivering the torque must be negated.

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wire

wire

113.7 90.3 78.7 1.064.1 440.7 464.4 177.8 370.3

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.00l

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DF 4.16 4.16 4.16 4.16 4.16 4.16 4.16 4.16

The results from the wire measurements are similar to those reported by Sebanc et al. 7 and within the nominal sizes presented by the manufacturers. No micro: scope was available to measure the bracket slot, but the transfer and mounting wire had an original dimension of 0.019 • 0.025 inch. Before it could be used, it had to be ground to fit the brackets tightly. Control measurements of the wire before grinding gave a mean thickness of 0.483 ram. According to most manufacturers a 0.018-inch bracket should have a theoretical width of 0.457 mm + 0.018 mm if0.0007 inch is used as maximum tolerance. The ground wire was found to have a thickness of 0.474 mm. This places the brackets used just around the tolerance limits. Increasing cross-sectional dimensions (wires a through c) demonstrated as expected a reduction in the amount of play, and the slope of the linear portion of the curves indicated increasing stiffness (Table II). According to mechanical theory, the change in deflection per newton millimeter should be inversely proportional to the polar moment of inertia for the different wires. A calculation that used the mean measured dimensions for wire "a,b,c" revealed this to be very closely adhered

American Journal of Orthodontics and Dentofacial Orthopedics Volume 105, No. 4

to. If the result for wire "c" is taken as a basis, the change in angular twist per newton millimeter for wire "a" should be 0.365 ~ and for wire "b" 0.332 ~ Considering the nature of the experiment and the observed method error, these results demonstrate very well the relationship between deflection and polar moment of inertia. Two set of wires had the same dimension (a and d) and (e and f). A comparison between their intercepts demonstrate that the amount of play is different. It is reasonable to assume that this is due to variation in bevel edge such as reported by Sebanc et al. 7 They observed a deviation angle of about 8 ~ for wire "e" with the same dimensions, but did not test wire "f." From these results, it can be assumed that wire "e" has more rounded comers. Similarly, the variation in the results for "a" and "d" must be attributed to the same cause. The analysis of variance demonstrated that the variation in the observed torque at a load of 16.25 Nmm for wires a through d, and at loads of 11 Nmm for wires e and f, was caused by both variation in the bracket and the wire, both being significant forp < 0.001. The analysis of variance indicated that the variation between the wires was the more important factor. Bracket 1 consistently gave larger values for angular twist, but this difference was impossible to detect through the attempt of fitting a wire into the slot. Another possible explanation for this result could be the width of the bracket, but bracket 1 was one of the wider brackets, and from this point of view one would expect the opposite result. The rrlean standard deviations between the brackets for the observed degree of torque at maximum load, was smaller than the mean standard deviation between the test pieces for each wire dimension tested. This indicates that variation in the wire is the more important factor when variation in the observed torque is analyzed. The most reasonable explanation is that it is due to variation in the degree of bevel and material properties, because the observed variation in the wire dimensions were small. The results raise another important point, what is an acceptable magnitude of moment for torque applications. It has been suggested that 75 gm of net force is a proper force for distal driving canines. Assuming a center of rotation about 12 mm from the point of force application, this will give a moment with respect to the center of rotation 900 gm mm or 9 Nmm. For torque applications it has been stated that the moment must be higher. Wainwright 9 concluded from his studies on Macaca speciosa monkeys that 2000 g m / m m was a physiologic force for torquing human central incisors. Nikolai ~~ is of the opinion that an average maxillary incisor segment requires from 3000 to 3500 g m / m m

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Meling, and Meling 399

Table VI. Relative rigidity of the four materials

tested based on the slope of the linear portion of the curve and the polar moment of inertia, using the mean of the observed values for the various wires. Stainless steel is used as reference Stainless steel/chrome-cobalt Stainless steel/Nitinol Stainless steel/TMA TMA/Nitinol

1.02 3.99 2.15 1.85

of torque. The implication is that this amount of torque will be developed by the wires tested at a higher degree of angular twist. It is fair to assume that the linear portion of the obtained curves can be extended a certain amount before a permanent set will occur. It is then obvious that to deliver a precise moment will be very difficult as it requires a very accurate degree of torque. This problem is demonstrated by the slope of the linear portions of the various curves. With the exception of wire "f," all the other tested wires showed a change in the degree of torque between 0.24 ~ to 0.60 ~ per newton millimeter, which is a very rapid change from an orthodontic point of view. Furthermore, if a desirable moment in torque applications is considered to be 20 Nmm for an individual central incisor, this moment will be very quickly dissipated through a small change in the tooth axis. Considering the 0.017 • 0.022-inch wire, the rate of change was 1 Nmm per 0.325 ~ If we assume little or no torque effect is left at a moment of 6 Nmm, the change from 20 Nmm to 6 Nmm will take place over 4.6 ~ To work with such close tolerances will be difficult. The tested nickel titanium wire demonstrated the greatest change in torque per newton millimeter, and should thus be the wire of choice. The fact that it is not possible to bend this wire makes its use for individual torque applications difficult. The only realistic approach seems to be the use of a bracket that has a degree of torque different from the neighboring teeth, appropriate for the tooth movement desired. This also seems to be the method of choice in those cases where overtreatment is desired from the point of view of stability. With respect to rigidity, the results indicate that steel is four times stiffer than Nitinol in torsion (Table VI), whereas it is a little less than twice as stiff as TMA. Steel and chrome-cobalt have about the same rigidity. These results are comparable to those found when measuring the modulus of elasticity in test machines. CONCLUSIONS AND CLINICAL IMPLICATIONS

The results have demonstrated that the amount of play between bracket and wire in torsion for individual

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Odegaard, Meling, attd Meling

tooth movement is considerably larger than the amount expected. It has also been shown that the initial portion o f the load/deflection curves are relatively flat for the smaller dimensions before a linear relationship between moment and deflection is achieved, indicating a restraining effect caused by the ligature. The resulting curves using wire "a" without ligature illustrates this point. The linear portions o f the curves show that the change in effective rotational moment will change rapidly for small changes in the tooth axial inclination, suggesting that reactivation o f the wires should take place at frequent intervals. For individual tooth torque, a more efficient method can be the use of highly elastic wires in combination with brackets with variable torque. REFERENCES 1. Steyn CL. Measurement of edgewise torque force in vitro. AM J ORTLtOD 1977;71:565-73.

2. Flores DA. The fracture strength of ceramic brackets: a comparative study. [Master's thesis]. Loma Linda, California: Loma Linda University, 1988. 3. Holt MH, Nanda RS, Duncanson MG. Fracture resistance of ceramic brackets during arch wire in torsion. AM J ORTltOD DENTOFACORTItOP1991;99:287-93.

American Journal of Orthodontics and Dentofaciat Orthopedics April 1994

4. Dellinger EL. A scientific assessment of the straight-wire appliance. A~f J ORTHOD1978;73:290-9. 5. Creekmore TD. On torque. J Clin Orthod 1979;13:305-10. 6. Hixon ME, Brandtley WA, Pincsak JJ, Conover JP. Changes in bracket slot tolerance following recycling of direct-bond metallic orthodontic appliances. AM J ORTttOD1982;81:447-54. 7. Sebanc J, Brandtley WA, Pincsak JJ, Conover JP. Variabilityof effective root torque as a function of edge bevel on orthodontic arch wires. A~.IJ ORTHOD1984;86:43-51. 8. Dahlberg G. Statistical methods for medical and biol~ical students. London: George Allen and Unwin, Ltd., 1940. 9. Houston WJB. The analysis of errors in orthodontic measurements. AM J ORTttOD 1983;83:382-90. 10. WainwrightWM. Faciolingual tooth movement: its influence on the root and cortical plate. AM J OR'roOD 1973;64:278-302. I 1. Nikolai RJ. Bioengineering analysis of orthodontic mechanics. Philadelphia: Lea & Febiger, 1985:299-305.

Reprint requests to: Prof. dr. Jan Odegaard Abteilung fiir Kieferorthop.~die Universit~itskrankenhausEppendorf Mortinistrasse 52 2000 Hamburg 20 Germany

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